© 2017 Kirk Shellko All rights reserved.
Science cannot be defined in only one way; it is a kind of inquiry that never finds complete description. Thinking and verifying one's thought has always been a part of the human experience; we have always observed, found patterns, argued and systematized, but science puts these things into a specific order and demands a certain kind of proof for its assertions. It is thus a natural ability to order that comprises science over the ages, and it is not foreign to any people, though it emerges gradually over long periods. Saying that there was no science here or there – then or later – is misleading, but because neither our particular ordering, nor method, was not present in the ancient world, we must say that science did not exist in antiquity. Still, it existed in pieces. The root of the word is scio, a Latin verb meaning “to know.” Modern science is an attempt at knowledge of a given subject through reason and verification, relying on probabilities. It attempts to reason about the likelihood of a given event or process. In knowing the probabilities of things, science allows humans to control their environment and manipulate specific processes in that endeavor. Science has many geneses: political, social, economic, geographic, ethnographic and intellectual, but as Bernard Pullman points out science cannot flourish without free expression, a willingness to consider different theories, openness toward debate and tolerance from public authorities.[i] It is a body of theoretical knowledge that employs technology in the solution of problems and the current belief about the nature of reality. It is a rigorous and precise attempt to attain and understand an assumed objective reality, and as such it filters out speculation, conjecture and imagined beliefs in favor of empirical verification of hypotheses. Its reach extends to almost every human endeavor: physics, medicine, politics, economics et alia. Science has made humans master of nearly all of the earth and because it did not exist in its complete form in antiquity, it is necessary to provide a tentative definition, one that will allow us to see in some measure and how ancient inquiry survives in contemporary scientific analysis. This text focuses specifically on physics, and so our definition must center around physical change, a central unifying principle and a system that explains how that principle functions in the universe. Science is a manner of understanding the behavior of nature, using deductive reasoning in the form of logic or mathematics to establish hypotheses. It empirically verifies these hypotheses through a process of experimentation, which involves understanding the interactions of things in material reality – other belief systems notwithstanding. Here is a tentative, general definition of science that is useful for understanding the continuity between ancient and modern methods for understanding. G.E.R. Lloyd claimed [ii] that science is “certain investigations that are continuous with what is normally included in our term science [which were] initiated in ancient Greece”, yet while valuable and accurate, Lloyd’s perspective leads to R.K. French’s suggestion [iii] that “to see science in antiquity we need to have a definition of science so broad as to be meaningless.” Our tentative definition is needed in order to begin discussion and to refrain from meaninglessness. It is not complete, but it allows a discussion about what may be called "ancient science.”
On the one hand, science is an intellectual inquiry into a particular subject-matter and on the other, it is the empirical or experiential verification of that continued inquiry. One part cannot be science without the other. Physics comes to mind when one uses the word “science”, perhaps because we have inherited Aristotle’s prejudice that knowledge of physics is knowledge of everything.[iv] Our aim is to discuss science on a fundamental level, so a discussion focusing on the thought process of science will reveal some measure of how ancient inquiry was a continuous investigation without the definition being meaningless. The thinking in science and its manner of verification in its most fundamental form are then our topics, and in order to understand the epistemology of science, we must understand some significant part of logic. Along the way, we will be able to discuss some of the empirical dimensions of ancient investigation.
Deduction and Induction
Logic is the engine of reason and science is only as strong as its reasoning. Reasoning involves the drawing of inferences. The connections that exist between accepted facts or postulates provide a new assertion. These new assertions proceed from what one does know to what one did not know: inferences. One calls such connections logic and the stronger the connection the more potent the reasoning; the more potent the reasoning the truer the inference. There are two significant kinds of logic that are part of a rigorous system of thought used to investigate nature. These are deduction and induction.[v] The ancients understood these terms differently than we do, naturally, and it is valuable to point out the differences between the ancient and the modern conceptions of logic in order to bring into sharper relief how ancient thought founds modern science. In the ancient view, deduction is a thought process that proceeds from universals to particulars. One begins with a universal conceptual thought, perhaps an abstraction, and proceeds to comprehend a subject on the basis of characteristics that define the whole of a category. What is consistent, what is cohesive and what follows thus from generalities are primary concerns of the ancient notion of deduction. Ancient induction proceeds from the particular to the universal. One begins with an observation or a sense that some of a category possess specific characteristics and proceeds to understand universals through perhaps comparison. A syllogism is a set of assertions about the world that together allow for a “bearing in” or inference from statements that are accepted as true to a previously unrealized assertion.
If the connections are strong and real, the inference is said to be true. If the connection is faulty or non-existent, then it is said to be false. Because it is true that all cats are quadrupeds and all lions are cats, it is true that all lions are quadrupeds. Here is a possible connection between thoughts, entities, or happenstances that possesses an internal consistency. It speaks of universals in the sense that it expresses the whole of a category of things. All members of the set of lions are quadrupeds if they fit into the class of cats, which in turn is a subset of quadrupeds. There is inherent in this simple syllogism a mathematical dimension in the sense that it makes an assertion about sets of things. Ancients did make distinctions between what they perceived as universals in particular manifestations of things – what we may call abstraction – and the particular things themselves. The particular things exist in the realm of becoming; they come into being and proceed to grow to fruition and decrease, then perish. They have no permanence in the sense that they change. In seeking fixed and absolute aspects of reality, ancients generally sought permanent categories of things perceived in the world. The expression of a connection between particulars is a different form of assertion.
There is no conclusion here because the connection made between particular statements is not strong enough to generate a strong inference (more below). These assertions talk of only a portion of a class of things: at least one member of a category. In the first sentence at least one member of the class of humans is a member of the class of safe drivers, and the same with owners of sports cars and safe drivers in the second sentence. So, when making assertions of this kind, one makes statements about only parts of whole categories of things. In particular and universal statements such as these, there are subject terms and predicate terms, indicated by the “S” and the “P” in the examples below.
These are the subject and the predicate of the given statement. There are also middle terms, indicated by the “M” in the examples below.
The middle term is the category that is common to both the other terms when a logical connection can be made. So, in example one “cats” is the middle term. It has something in common with both “lions” and “quadrupeds.” This commonality is what allows one to draw an inference. Two statements of categories together coupled with a conclusion comprise a syllogism. A syllogism draws an inference when it makes a connection, and that connection thus allows a conclusion. “All lions are quadrupeds” is the inference made through the connection and thus it is the conclusion. In Aristotelian, syllogistic logic there are four types of assertions whose elements possibly have a connection. An “A” statement asserts that one whole subject set is a subset of the predicate set:
All S are P.
An “E” statement asserts that one whole subject set is excluded from a predicate set.
No S are P.
An “I” statement asserts that one part (at least one) of a subject set is a subset of the predicate set.
Some S are P.
An “O” statement asserts that one part (at least one) of a subject set is excluded from the predicate set.
Some S are not P.
The first two sentences are universal; they assert something about all of at least one set of things. The third and fourth sentences are particular; they assert something about at least one of a set of things. The majority of ancient thinkers looked for first principles or fundamental first elements that would allow them to explain change and continuity in processes encountered in physical happenstances. They believed they needed to find a permanent aspect of reality that was the foundation for impermanence. They thus sought what was fixed and lasting and attempted to base their arguments on permanence even as some ancient thinkers embraced change as a fixed element. In other words, they sought universals. A and E statements were the kinds of assertions, then, that they wished to employ in their reasoning.
Induction as understood by ancients operates from the other direction. One proceeds to make the same kinds of connections, but from particulars to universals. I and O statements are inductive statements, asserting something about at least one member of a set of things. The connections made between parts of sets by their nature cannot be as reliable as those made about all of the members of a certain set. An assertion about all of one set and all of another set encompasses all possibilities, and so is obviously a much more reliable connection. Again, the above example.
There is certainty in the statement “All lions are quadrupeds” because the sentences above it assert something about all of a given category. If we alter these statements and make them particular, they have a very different meaning.
There is no universal, and therefore reliable, connection between these statements because they make assertions about only some of a given category; we do not even know how many of a given category possess the characteristics asserted. If some cats are quadrupeds and some lions are cats, then the lions that are quadrupeds may not be the same cats, all of them may be or only some. We have no way of drawing a firm inference about the “some” in either sentence. Ancient thinkers generally sought the certainty that comes from the strong connection of a universal. In fact, many ancients perceived particulars as unreliable because they thought of them as unstable. One can see how dissatisfying it would have been for ancient thinkers to discover rules about logical connections only to find that logic cannot find the very universals it was meant to discover. Centuries of investigation passed before scientists disposed of the attempt to find universals in favor of probabilities.
Deduction and Induction in Modern Science
The modern view of logic and its limitations is different. When an argument has a good form and the connections it makes seem real and verifiable in terms of their relation to one another, we call the argument valid. So, that we have made a firm connection between lions and quadrupeds in the above example is that argument's validity. There is a conceptual connection between the different categories. When, however, an argument possesses not merely good form and connection but also has as its premises verifiable, true elements of the world (like facts or accepted conditions of existence), then one calls it sound. So, that lions who are cats exist in the world and that cats are actually quadrupeds along with the connections between them makes the argument sound. As one may imagine, an argument can be valid and unsound. But an argument cannot be sound and not valid because the connections as well as true statements need to exist for an argument to be sound. So, the following argument is valid.
The statements do not need to be true, but the connections between the stated categories need to exist. If the form of the argument is good and if the connection is present, then the argument is valid. Additionally, one must accept the conclusion. If it has good form it is valid, even if it is nonsensical. Our initial example argument is sound.
There are such things as cats in the world and cats are actually quadrupeds. Also, there are such things as lions and lions are actually cats. Because the categories make a strong connection, the argument is valid; because the statements are true and because the argument is valid, the argument is sound.
Deduction in the modern sense means that if we accept the premises of the argument, then the conclusion will not only naturally follow, but we must accept the conclusion. The certainty of a very strong connection is deductive, given its assumptions. Mathematics is a good example of deductive reasoning. It presents clear and certain conclusions from commonly accepted constants, or the most certain calculations possible. Modern deductive reasoning is not necessarily in harmony with its subject. In other words, the internal consistency of a deductive argument may compel someone to accept the conclusion, but the assertions used in the argument may have little to do with reality. If one creates a perfectly cohesive mathematical thesis, it may be valid, but have little to do with how things interact in material reality. Scientists verify their arguments through an empirical means, which is, of course, the experiment.
Induction for moderns is the discovery of connections, but they are considerably weaker than those made by deduction, and modern induction is not concerned with particulars in the same way that ancient induction concerns itself with particulars. A modern inductive argument asserts something about the probability of some conclusion to be true. Obviously, there are different degrees of probability. So, an inductive argument in the modern sense is one that will make a kind of prediction about an event. Given the facts that we possess about the earth and its relation to the sun we can predict with a great amount of certainty that the sun will rise in the east and set in the west tomorrow. So, it is an argument that asserts a specific occurrence will take place with a great deal of probability, given certain circumstances, but not that the occurrence must take place. There is a predictability to the argument, but not absolute certainty. Thus, the probability. One sees this kind of argument when one reads the prediction for the weather. There is a certain percentage chance that rain will fall, for instance. The percentage is the probability of rain, induced by the circumstances that are taking place at a given time. So, if there are dark clouds in the sky and one hears thunder, one can make a relatively certain induction that rain will fall. Inductive arguments guide modern scientific investigation because the scientific search for universals has ended, or it has at least paused. We will return to the particulars of logic and Aristotelian logic rules later, but now it is necessary that we understand what part of this manner of thinking we are investigating. Other kinds of logic exist besides the syllogism. Propositional logic is similar to syllogistic logic, but it is a form of logic in which the fundamental components are whole statements or assertions. Propositional logic is the combining of two or more sentences, or propositions, in order to find more complicated assertions of truth. Propositional logic operates in the same manner as syllogistic logic, but with more assertions. In fact, all logic operates on the same level in one sense. All of logic attempts to make connections based on the truth and falsity of certain statements, the categories that they represent and the connections or relationship between them. Symbolic logic is a development of propositional logic, but with different sigla. It replaces purely linguistic statements with symbolic ones and is comprised of a given set of proven connections that lead to more complex connections. Predicate logic makes use of the same symbols as symbolic logic, but accounts for the complete sets and partial sets that one sees in syllogistic logic. Mathematics is only a short step away from these kinds of logic in that it is in part a quantification of similar assertions. We will cover only a few of these kinds of logic, but we want to keep in mind that logic is not monolithic. There is a great similarity between kinds of logic, but they have different forms. The most fundamental aspect of logic is that it makes connections between things. Every kind of logic performs that function.
Induction as a Part of Scientific Method
We are examining the engine of science, its logical apparatus, in order to learn about ancient contributions made to the current scientific method. We are interested in the history of science, but merely stating at what conclusion a particular thinker arrived is hardly satisfying when we want to comprehend science as a way of determining truth and a way of discovering “laws” of the natural world. The awareness and manipulation of natural processes have allowed humans to possess great control over their environment and given us the modern age, as well as the virtual age. Humans have even journeyed outside the planet because of the thought that drives science. We want to keep the distinction between deduction and induction in mind as we examine systems of thought that have contributed to an overall scientific method employed today. We will come to some understanding of how much and in what way each element of thinking survives in current scientific method.
How, then, precisely does the thought of science operate and how does it demonstrate what it asserts is actually true about the world are fundamental questions for us. One part of science looks for the causes of things, so we need to have some way of understanding causes.[vi] There are three definitions of cause that apply for us. There are necessary conditions, sufficient conditions and sufficient and necessary conditions. Necessary conditions are conditions needed for something to happen, but in themselves are not enough to make something occur. Water is necessary to produce life, but it is hardly sufficient. If it were, we would have been able to create new forms of life at will, even in the ancient world. Water is a necessary cause for life. A sufficient condition is a condition, or cause, that itself is enough to make something happen. The chopping off of someone's head is a sufficient cause of death, but other methods prove efficacious. Shooting, stabbing or strangling a person will result in their death if done in a specific way. A necessary and sufficient condition is a condition that will in itself produce the desired effect, and nothing more than it is necessary. The acceleration of a body is caused by the action of some force upon it.[vii] Nothing more or less is needed for acceleration to come about.
If X is a sufficient condition for Y, then if X occurs then Y occurs. Beheading is our example. If someone beheads another person, that person will die. Additionally, if X is a necessary condition for Y, then if Y occurs X occurs, but not necessarily the other way around. So, if I am driving my car, my car's engine must be running, but simply because my car's engine is running does not mean that I am driving it. Similarly, the absence of X is a sufficient condition for the absence of Y, if and only if Y is a sufficient condition for X. So, if there is no acceleration of my car, then there must be no force, its engine, acting upon it.
In a similar fashion, the absence of X is a necessary condition for the absence of Y if and only if Y is a necessary condition for X. If my car's engine is not running, then it must be that it has not been started, because starting a car is needed for its engine to be running. Lastly, X is not a sufficient condition for Y if X occurs without Y. If it so happens that I started my car and it is not moving, then starting my car is not a sufficient condition for it to move. And, X is not a necessary condition for Y if Y occurs without X. It is necessary, but not sufficient. If I started my car without wearing my hat, then wearing my hat is not needed for starting my car.
We talk about these conditions because they will be necessary for understanding how the reasoning process works in evaluating causes of things. Remember that for our purposes there are two realms in which science works. The one is more theoretical, and it involves making abstract arguments about things in the world. We use statements, propositions, axioms, mathematics and the like to make these arguments. They may apply to material reality and they may not. The other is more empirical, and it involves verifying those valid but perhaps not sound arguments upon circumstances in the world. When we experiment upon something, we find out if our seemingly logical standpoint really speaks to some objective truth about how things operate. So, now we need to understand what methods there are for verifying our reasoning. We will cover five methods for demonstrating causal connections.[viii] These are basic ways our reasoning may be involved in making scientific inferences.
The Direct Method of Agreement demonstrates the relationship between an effect and a necessary condition. One looks for a single element that is present in many occurrences where the effect sought is also present. The one element that is present when the effect is present is thought to be the cause. An example:
The following table shows how the conclusion came about. The occurrences are the individuals who ate at the same restaurant. The possible necessary conditions indicate the possible foods eaten. An asterisk shows the presence of something and the dash absence. The condition that is present in all of the occurrences is the condition taken to be the cause of the malady.[ix]
Because the hot sauce (B) is the only common condition in all occurrences, it is taken to be the necessary cause. The idea is to eliminate as many of the necessary conditions as the information permits. When only one condition remains, one has the possible cause. Note that this argument is inductive. It uses evidence to examine what may be present when the effect is present in order to determine a connection and that connection is probable to some degree. Some other cause may have been overlooked; the malady may have been transmitted through a combination of things. It may have been possible that some other condiment caused the malady. In other words, the argument does not state that all those not using the hot sauce did not fall ill. Here is what it says:
X is not a necessary condition for Y if X is absent when Y is present.
The Inverse Method of Agreement demonstrates the difference between an effect and a sufficient condition. The idea is roughly the same, but one tries to identify an element that is absent from a number of occurrences in which the effect is also absent. The absent element is taken to be the cause. An example:
The argument asserts that the absence of a cell phone contributed to the diminished performance of the students. Here is a representation of the argument:
The idea is similar to the Direct Method of Agreement, except we look for an element that is absent. B represents their cell phones. Among all the other possible causes the only one that is absent when good performances are absent is a cell phone. One attempts to eliminate as many possible causes as one can. The rule used here is one for sufficient conditions: X is not a sufficient condition for Y if X is present when Y is absent. This argument is inductive as well, since not all of the important conditions have been determined. Also, two or more elements can be acting in conjunction in order to cause the phenomenon. One can only say that we have narrowed the possible causes and that the one absent possible cause may be the actual cause. The argument also pertains only to the persons who have participated, not to all possible students. Students in general may act in a very different way. One can see the probability involved in this argument. It is informative, but one can obtain only a degree of probability because one does not possess all possible information. If more occurrences are added, more possible causes may be considered. Additional conditions may significantly impact the conclusions. Giving cell phones to students may improve their performance, but that is not the only element that may improve performance. Perhaps more rigorous study or less consumption of alcohol may improve performance as well.
The Double Method of Agreement is the combination of the direct method of agreement and the inverse method of agreement. This method identifies necessary and sufficient conditions. This method is often used by researchers to determine the effectiveness of drugs on humans or animals. An example:
Here is a table representing the Double Method of Agreement:
This method measures the sufficient and necessary causes, as one can see from the four asterisks on top and the four dashes on the bottom. The asterisks represent the occurrence of the cure and the dashes represent the absence of the cure. F is the element that is present when the cure is present and absent when the cure is absent. The first four asterisks eliminate A, B, C, D, and E as necessary conditions because they are absent when the cure is present. The last four dashes eliminate A, B, C, D, and E as sufficient causes because they are present when the cure is absent. The argument is still probable, because there may be things overlooked, but there seems to be a high degree of probability that the cure is F because it is the only factor that is present when the cure is present and absent when the cure is absent. A combination of things with F may be a possible cure, but because we have argued for necessary and sufficient causes the likelihood that a combination of things with F cured the patients is significantly diminished.
The Method of Difference finds one single, present element in an occurrence that is absent in another, similar occurrence where the phenomenon is absent. This element is how the two occurrences differ and it is supposed that this element is the cause. Sometimes called the laboratory method, the Method of Difference is used by researchers in very controlled conditions. This method is obviously one that interests us a great deal. One can see parts of the other methods in it, and it is what many scientists use in order to verify arguments. Their arguments, or their hypotheses, may have good connections, and so they may be valid, but the verification testifies that the argument applies to a given situation. An example:
The only differentiating factor between the mice was the drug. The idea is to eliminate certain conditions as possible causes, just as in the other methods.
One sees that in occurrence 1 A, B, C, D and E are present while the phenomenon is present. In occurrence 2 only A, B, C, D are present and the phenomenon is absent. A sufficient condition is identified. The method of difference differs from the inverse method of agreement because the conclusion is less general. In other words, here “the conclusion applies directly only to the specific occurrence in which the phenomenon is present, whereas in the inverse method of agreement it applies to all the occurrences listed.”[xii] Notice that the method of difference makes use of particular instances and establishes a kind of probability, though one based on the specific instance used in the argument. One must keep in mind the likelihood that two occurrences will be precisely the same. The need to verify the repeat-ability of the conditions that the hypothesis concerns is the reason experiments like this are repeatedly performed and verified by different laboratories. The differences can be slight or great and the results possibly will be quite different. A thesis that may be experimented upon and rejected is falsifiable, meaning any given thesis can be verified or rejected by these given inductive methods. A new hypothesis replaces the old. The effect of any given difference is unknown, so much can rely on only a few – seemingly insignificant – changes. Also, one cannot take into account each and every element that may go into an experiment like this one. One would need all possible elements of a given cause and effect in order to make an absolute determination, and that is obviously impossible. The degree to which one can exhaust all the possibilities is related to the degree to which one can be certain. One can use this method with the absence of a given element and obtain fruitful results. An example:
The only real difference here is that one counts the absence of vitamin D as a kind of presence. In other words, the fact that there is no vitamin D is what caused the rickets and so one can say that vitamin D prevented the rickets. The same kind of probabilities and critiques can be said about the method of difference used with the absence of something as when the method is used with the presence of something.
The Joint Method of Agreement and Difference combines the direct method of agreement with the method of difference. This method identifies elements that are both necessary and sufficient, since the direct method identifies the necessary element while the method of difference identifies the sufficient element. An example:
The joint method of agreement is more general because it pertains to all the occurrences in the experiment. Occurrence 2 eliminates A, B and C as elements because they are absent when the phenomenon is present. This is the direct method of agreement. Occurrence 3 eliminates A, B and C as sufficient causes because they are present when the phenomenon is absent. The conclusion concerns the particular case of George and so pertains only to George's specific circumstances. One may claim then only that this experiment cured George, but as long as there exist characteristics similar to George but found in other men, the results will be the same. The more similar the characteristics that others have to George, the more likely will the antibiotic cure them as well. So, there is only a probability that the antibiotic will work.
Finally, the Method of Concomitant Variation matches elements' variations in one element with variations with another. This method is used when it is not possible that an element be completely present or absent. Increases of one element are matched with increases in another element and decreases in one element are matched with decreases in another element. Also increases can be matched to decreases and decreases matched to increases.
Therefore, B is causally connected to b. This graphic signifies any of the above: increases matched to increases, decreases matched to decreases, increases matched to decreases or decreases matched to increases. This method establishes the probability that some one element has a causal relation to another element. It is not clear what that relationship is precisely because the complete causes and effects are not present as with the other methods, but one can establish some causal connection, no matter what the probability is. Because there are not more rigorous comparisons and because the connection between elements is unknown, the argument here is probable. And, it is difficult to determine the level of probability as well.
We have then what amounts to several significant pillars of scientific inquiry: inductive and deductive reasoning (both modern and ancient versions), definitions of types of causes and methods of verification of the causes, a comprehension of the necessity of falsifiable hypotheses. Science uses more than these things, depending on what a particular science investigates, but we have a good, working idea of what constitutes reasoning and verification. There are observations of nature that lead researchers to speculate on what is the “law” of nature that operates in a given occurrence of something. These are empirical aspects – in the form of recognized patterns – that assist in knowing what will happen; they are perceptions that come from experience and our senses. We use these pieces of information in order to then make arguments and apply them to the world in the form of controlled experimentation. We form hypotheses with our reasoning and test the hypothesis, using one or more of the above methods for verifying cause and effect perhaps. Researchers suppose that there is a certain probability X will happen when Y happens. They then form notions about how nature will act and these notions are inductive in the modern sense; there are certain probabilities that Y will occur when X happens. Thus, researchers suspect that they know what will happen when certain conditions are met, or perhaps they reason in the form of a language-based argument or a mathematical calculation. Naturally, they need the strongest arguments possible in order to make the best possible inference about material reality. They thus need some foundation for their arguments that no-one disputes, or at least that everyone is able to accept.
As ancient thought progressed, mathematicians and philosophers found what we now call constants. These are accepted aspects of material reality or mathematical calculations give thinkers starting points from which they can draw inferences about the patterns seen in nature and how they may be predicted. These constants are things like the Pythagorean theorem, the calculation of π, sine, cosine and tangent, as well as the calculation of the area of a circle. Constants allow thinkers to have an anchor such that systematic reasoning becomes more potent. The development of argumentation in science relies upon these accepted truths. We will see how ancient thinkers first speculated about the nature of material reality and then questioned the answers that other thinkers gave. Generally, ancient thinkers engaged in reasoned arguments and not in empirical verification. There are, of course, empirical elements to every kind of reasoning, but ancient thinkers in general thought that reasoning itself was more potent and more true than mere observation. There was a slow but regular progress in the development of what we call science, but it was not concerted in its efforts the way science is today. Progress of ancient “science” was intermittent. We will review a number of thinkers in order to discern what pieces of science they employed, but we are only reviewing interpretations of some possible arguments of each, since believing the definitive argument to be found is to cease thinking. We do not have enough evidence to argue definitively, but we want to discuss method and findings. Thus, we imagine a bit and look at some of the evidence, keeping in mind that ancient thought was not scientific and that ancient thought has its own merits. It is not inferior to science but rather ancient thought occupies another dimension of thinking. We will use Aristotelian syllogisms to examine how ancient thinkers thought.
Beginning in the sixth century B.C. the ancient Greeks attempted to answer the questions scientists still seek to answer. They looked for fundamental first principles of things, and it was something material they initially thought was the first “element” of all things. These Greeks thought that the universe is inherently ordered and discernible with the mind. They thought the parts of the universe fit together almost like a machine and ancient Greeks thus thought that they were able to dissect and explain the workings of the universe. Yet, sixth-century Greeks were not the first to seek explanations for physical phenomenon. The poets explained the universe in terms of Gods who controlled natural forces, and other earlier civilizations attempted explanations of their own. These Gods the poets thought were appeaseable through sacrifice and thus humans could not only comprehend the universe but they could control it and predict what would occur, if the Gods could be asked to act in certain ways once given the proper sacrifice. So, we begin with two poets and explain the universe in their terms, but we ought to know what we seek in any given thinker because each one presents us with a different account of what is fundamental about the order of the universe. What we look for in a thinker are many things: inductive or deductive reasoning, first principles, constants, an hypothesis, mathematical reasoning, language-based logical reasoning, pattern-finding. So, ancient Greeks sought patterns; patterns are repeated instances of things; repeated instances of things are categories; categories are compared; comparisons of the categories show how the patterns overlap and the overlapping of the categories is the reasoning of the thinker. This kind of reasoning is scientific in that it finds patterns and determines how the patterns relate to one another. Here is a significant portion of science and each thinker will possess some piece of science, like an explanation as to what element is the most fundamental. Thales, by example, seems to have thought that material reality was composed of water in various forms. We will review and then analyze some aspect of thought for each thinker. We do not have space or time to investigate every aspect of every thinker, but we will attempt to draw some essential insight from each. Because these ancient Greeks were physicists, we will look to understand how their notion of material reality operated. We look for how their arguments worked and what, if any, empirical aspect of their thought existed. We will then get some idea of what ancient “science” was, though there really was no such thing in the complete sense.
© 2017 Kirk Shellko All rights reserved
HOMER AND HESIOD
© 2017 Kirk Shellko All rights reserved
Life and Works
When the dark ages of the ancient world began, circa 1200 B.C., Greek-speaking people moved into what would become a center of western culture. Their skills, laws and customs survived in the form of an oral tradition: folktales, fables, legends, myths. Basic needs that arise from migration and the necessity for cultural cohesion fueled the linguistic and artistic creativity of the Greeks, and as a result Greek culture spread out among different locales. Small city-states (poleis) later sprouted and each community produced a particular interpretation of the central Greek culture. The creativity needed in order to survive fueled discussions and innovations in thought and culture. Explanations for how the universe arose and what it essentially is grew in myth and story-telling, representatives of which are the ancient Greek bards Homer and Hesiod who wrote about the origin of the universe, actions of gods and men and the relationship humans have to the physical happenstance they encounter, among other things. Scholars know very little about the floruit of the Greek bard Homer, who wrote his epics – Iliad and Odyssey – in the late eighth century B.C. Many places claimed to be his home, and we know little else than that his name means “hostage” and his poetry is beautiful, but perhaps he came from Ionia. Likely, he was the greatest of a long line of distinguished poets who progressively synthesized earlier works into twenty-four book epics; he was in the least an extraordinary weaver of already-written tales. Hesiod was a competitor of Homer who settled in Boeotia and wrote perhaps in the early seventh century, attempting to explain the universe in terms of origins. His topic specifically was the birth of gods. His two major works are Works and Days and Theogony.
Homer conceived of the order of the universe through common sense and imagination. He wrote poems that reinforced for the Greeks not only habits and customs but also conceptions of the universe as a whole. There exist physical forces more powerful than humans, which Homer and poets like him thought of as divinities.[i] What one sees is what exists and what one does not see is speculated over or imagined and in most cass given anthropomorphic form[ii]. No thoroughly systematized logical analysis and critique is present in the Homeric epics, Iliad and Odyssey.[iii] These gods were forces, blind and potent, acting upon the human family without feeling or concern and if they were human-formed, then they must be doing so because of human flaws. Yet, the gods do not need morality. They are forces of nature well beyond any human need, unless they please to have human sentiment. They represent a kind of order in that they have domains, but should they choose to cease their present course they may do so with impunity. The gods in one sense are the attempt by Greeks to impose order on a universe lacking apparent order and an attempt to understand what is there; Homer describes what he sees. The sky is bowl-like in shape and it sits atop the flat earth.[iv] Sky thus covers earth and the section between earth and sky is composed of air near the ground and aither near the dome of sky itself. The earth, Homer says, stretches below the surface and has roots in the lowest part of the underworld, Tartarus – the chasm below the earth:
Having taken him I will throw him into cloudy Tartarus, where exists the deepest pit under Earth, iron gates and bronze threshold are there, just so far below Hades as Sky is from land.( Homer Il. 8, 13).[v]
Some conceptions of the underworld claimed that there exists a symmetry between the earth and sky and the underworld below, but the symmetry was not perfect. Xonophanes's conception of the underworld made the chasm infinite:
This here the upper boundary of earth is seen beside our feet in contact with air, contrarily the bottom boundary goes on without limit. (Xenophanes fr. 28 (=183)).
Oceanus, a vast river, flows along the edge of the flat disk of Earth. As a source of water it became the origin of all water bodies:
...lord Acheloios does not vie with him equally, nor the great strength of deep-flowing Oceanus, out of which all rivers and each sea and all founts and deep wells rum. (Homer Il. 21, 194).
Thus earth appears to be a circular disk and travel to the outer edges of the dome leads to water. Homer is geographer as well as story-teller:
I go in order to see the ends of much-nourishing earth, both Oceanus source of gods and mother Tethus.... (Homer Il. 14, 200).
The idea that water surrounded the flat disk of Earth was common and Oceanus becomes for Homer the source of all things:
Another of the always-living gods I (Hypnos) would send to sleep easily, even the flow of river Oceanus, who as source for all makes [all].... (Homer Il. 14, 244).
Even later thinkers who criticized Homer believed that there is some value to discussing Oceanus' central place. Plato and Aristotle:
...Homer, who having said 'Oceanus source of gods and mother Tethys' said all things are born of streams and motion. (Plato Theatetus 152e).
...there are some who...made Oceanus and Tethys the parents of coming-to-be, and water the oath of the gods, which by the poets themselves is called Styx; for the oldest thing is most valued, and the most valued thing is an oath. (Aristotle Met. 1.983b27-33).[vi]
Night in Homer is a force of the universe even more powerful than the king of gods:
And he would have thrown me from the aither into the sea, unless Night, tamer of gods and men, had not saved me. To her I came fleeing and an exceedingly angry [Zeus] ceased. He dreaded that he would do things displeasing to swift Night. (Homer Il. 14, 258).
The gods are usually anthropomorphic in their form and motivation and so they look and act like most powerful aristocrats, spoiled and capricious as well as arbitrary. More importantly they are natural forces. These are the forces of activity and reaction personified in Homer's Iliad and Odyssey and they may be taken as natural forces and abilities found in the universe, encapsulated in anthropomorphic metaphor. Again, Homer's Iliad:
The father of the gods and men fiercely thundered from above, while from beneath Poseidon shook boundless earth, and lofty heads of hills. The feet and heads of many-springed Ida quaked, as well the city of the Trojans and the ships of Achaeans. Beneath them Hades, king of those below, grew afraid; he sprang fear-stricken from his throne and cried aloud in terror lest earth-shaker Poseidon should tear open the ground above, and his aweful, dank house become visible to mortals and immortals–things all gods abhor. Such a blare arose at the coming together of the gods in discord. Phoibus Apollo stood opposed to lord Poseidon, holding winged arrows, Athena opposed the furious warlike god; golden-arrowed Artemis, noise-sounding arrow-pourer, sister of far-shooter Apollo, stood to face Juno; stout Hermes bringer of good luck stood against Leto, while the deep-eddying river, whom gods call Xanthus but men Scamander, opposed Hephaistus. (Homer Il. 20.54-74).
Again, the gods personify not only natural forces, but the skills and actions of human beings. Athena is the good ideas of thinkers and politicians; Zeus is the unspoken agreement between aristocrats to give good hospitality, or Xenia. Apollo is the art of healing, the skill of bow-making and all of the skills needed to shoot arrows as well as augury and in later antiquity light. Hephaistus is the skill of carpentry, engineering and all the skills needed to be a blacksmith. The gods are everywhere and are responsible for everything that transpires. The gods' whims must be met with obeisance lest they strike down humans with anger or disfavor. Thus, ancient Greeks found themselves in a subordinate position that made them vulnerable, and the attempt to understand natural forces in terms of human experience demonstrates an intent to organize the universe in a comprehensible and manageable way. One sacrificed to a god when one met with success, in order to thank the god or gods. One also sacrificed before a battle or when one married, or on some other important occasion, in order to gain the gods' favor and make the event successful. The domain of a god determined to what deity one must sacrifice. A thief, for example, might have sacrificed to Hermes the God of thieves before attempting to steal.
Gods and men in Homer interact with one another in various ways. Daniel Turkeltaub[vii] describes five ways that humans encounter the gods in epiphanies: 1) mortal perception of a god after the god's departure, 2) a disguised god reveals their identity, 3) a mortal recognizes the voice of a god, 3) a mortal sees a god, 5) recognition of a god is taken for granted and not elucidated. These moments of recognition seem to be universal in Homer's Iliad and represent the moment that an activity or happenstance is perceived as coming from a divinity. In all cases they are parts of happenstance, or nature, and so humans recognize the divine in everything that is a natural occurrence, but in different ways. The gods are then explanations for natural events, but they are not complete, nor consistent explanations. The gods's actions are not pure; they are seemingly contradictory.[viii]
Causation and Cosmology
Additionally, telling aspects of the interrelations of gods and men in Homer are cause and effect. Gods are internal as well as external agents: Athena is a good thought and Zeus is thunder and lightning. Humans act and have responsibility for what they do, yet the gods regularly interfere. The gods' actions and human action are intermixed until it is not certain if material happenstance or historical events are the result of physical processes, human volition or divine will. Yet, there is an explanation for what transpires. In the Iliad Homer makes Zeus send a dream to Agamemnon in order to make him win, but not in the way Agamemnon believes. The Greeks will win only after suffering and struggle:
This plan seemed best to Zeus's heart to send a baneful dream to Atreidian Agamemnon. 'Go, baneful Dream, upon the quick ships of the Achaians. Having arrived in the hut of Atreidian Agamemnon announce everything very precisely as I prescribe. Command the long-haired Achaians to arm with cuirass and all speed. For now he might take the wide-walled city of Trojans. (Homer, Iliad 2.5-13).
Agamemnon's actions, then, will prove effective, but only by losing many men in the immediate and regaining the aid of Achilles who had argued with him just prior to Agamemnon receiving the dream from Zeus. It is necessary that Agamemnon fight and lose in order to produce an ultimate goal – final victory – that lies hidden, and the agency of the final result is far from direct. History, divinity and physics are bound together. In Homer's Odyssey Ino saves Odysseus from drowning, giving him advice and encouragement when Poseidon, angered at Odysseus, discovers that he is on a raft and about to find land:
He [Poseidon] will not destroy you, though exceedingly desiring it. Do this immediately. You seem to me not to lack understanding. Shed these clothes and leave your raft to be born on the winds. Swimming with your hands, strive for a return to land, that of the Phaiacians where it is fate for you to escape. Stretch this divine headband on your breast. No fear to suffer or to perish. (Homer Odyssey 5.341-7).
One may interpret Ino's advice as the thoughts of a drowning man as he panics, or perhaps a stress-induced vision. The thoughts appear to be his, but they still come from the gods. The agency of mental activity and its conclusions are mixed. Athena gives Telemachus advice about when travel is safe and how to act:
I am such a kind as a friend to your father who will ready for you a quick ship and I myself will follow. But going home consort with the suitors, fit provisions and store all in vessels, wine in amphoras and barley and good-marrowed nourishment of men in thick skins. And going through the town, I will gather comrades willing. There are many ships in sea-girt Ithaka. (Homer Odyssey 2.286-293).
Here she appears as a man who directs Telemachus, so she is an external agent. She acts not only as a father-figure by prodding him when needed, but she gathers men for him. Athena also prevents Odysseus' boyhood nurse, Eurykleia, from revealing his presence when he returns home disguised to a house filled with enemies. His wife Penelope would be in danger, if she knew her husband had returned. Eurykleia's cry is an utterance that is at its sounding not yet appropriate to hear. She washes his feet and recognizes him by a scar on his knee:
All at once joy and pain gripped her mind. Both eyes filled with tears and her sturdy voice was held back. Having taken hold of Odysseus' chin she said “Truly you are Odysseus, dear child. At first I did not know you before examining all my master.” She looked with her eyes at Penelope, wishing to make known her husband had come home. But Penelope was unable to see her directly or understand, for Athena had turned her mind away. (Homer Odyssey 19.471-9).
Athena turns away the attention of Penelope and as elsewhere the causes of things that transpire in the universe are not merely the physical interactions of objects or even the interactions of objects, other objects and humans, but rather they are a combination of those things and gods who suffer from human pettiness and caprice. Direct cause and effect are not a part of how actions and their consequences manifest when Gods involve their judgment, and the Gods' judgment is necessary for anything to happen. Homer and ancient Greeks like him explained the vicissitudes of fortune by inserting divinities as explanations for random events and for the unexpected in general, as in Athena turning Penelope's mind from Eurykleia's cry. Homer and other poets do not set out to explain cause and effect explicitly, but they offer what may be called a semi-rational literary explanation of cause and effect. Reasoning in Homer and Hesiod is analogous. That is, the actions and reactions of gods in the epics can be taken as metaphors in epic allegory. Men are not only similar to leaves, but advice that comes from a family friend and mentor is seemingly god-inspired or is a god come down to speak. That is, the gods breathe life into the activities of humans as well as natural forces.
The Homeric epics are rich with explanation of cause and effect when one takes them to be metaphorical explanations. Each hearing or reading may take the interpreter to a different and perhaps better comprehension of what transpires. In this manner the Gods as metaphors give multiple explanations for happenstance by reason of analogy. All literature accomplishes a hermeneutic in this manner, yet the literature of the ancient Greeks was the center of their culture as well as the center of their explanation of the universe. Civilization had collapsed and the ancient Greek stories carried their culture with them as they migrated and remained once they settled. In brief, the Homeric and Hesiodic epics provide metaphorical explanations that in turn provide an arena of causal interpretation. The universe is explained, but the Greeks had to wait for thinkers like Aristotle to provide them with a more pure analytic system, one composed of less literature and metaphor and more logic and mathematics. Understanding and participation in these forces in the form of sacrifice to capricious gods and direct interaction with divine natural forces gave Greeks some measure of control over their destiny – another aspect of science – yet there remains one more important aspect of literature that bears the basic human ability to create scientific systems. It is a fundamental function of a literature that bears and supports the laws, habits and customs of a people.
A very important aspect of ancient literature and especially epic is mimesis. A member of the ancient community is expected to read or hear about the characters of an epic in order to know the laws, habits and customs of the culture. Plato points out the habit formation of imitation. Mimeseis “if continued in the future from youth, establish themselves as habits and nature with respect to body, voice and thought.” (Plato, Republic, 3.395d1-4).
Humans do not merely mimic behavior; mimesis is the development of character. They imitate themselves, gods and other animals as well. Imitation requires the cognizance of a recognizable pattern in action. One imitates what one has seen repeated in the actions of another human, like a parent. Sometimes, humans imitate other creatures, but the pattern of imitation is part of human psychology in that we learn to act as others, human or non-human, do. We perceive an activity that is useful or one that will provide something we desire. Then, we copy that pattern of behavior. It is, for Plato and Aristotle at least, basic learning behavior, and so it must have been for many other ancient Greeks. In his Poetics Aristotle claims that humans are the most imitative creatures, learning initially by imitation.[ix] Mimesis for Aristotle continues throughout life and is responsible for pleasure in witnessing artistic representations of things that otherwise are painful. The goal of art, like epic, for Aristotle is the discovery of a universal[x] and humans naturally make art in the way that nature makes natural things, using forms to make categories of things.[xi] Perhaps Aristotle is incorrect about learning first by imitation, as research has thrown doubt onto that long-held assumption,[xii] but imitation seems to be a function of the human mind related to language acquisition and resulting from different mechanisms[xiii], a possible innate ability vital to the development of theory of mind and empathy in humans.[xiv]
Mere mimicry was not the only aspect of imitation the ancients meant when they used the term mimesis; repetition of acts of nature[xv] is important for our purposes. One related psychological theory is that of Carl Jung's archetypes. Anthony Stevens[xvi] explains that archetypes are virtual images. These are psychological categories that have an aggregate character, which are images of anything in general: child, mother, wife etc. They are not the particulars encountered in nature, but those categories that make the apprehension of nature possible. They come into solidity when humans experience them. These are the psychological categories that a literary culture immerses itself into and employs in order to understand, or to think. When one reads or hears of Zeus, one thinks of regal power, likewise wise counsel for Athena. These literary characters serve to awaken general patterns in human beings. In other words, humans have for Jung already-set patterns of thought that are actualized when they process stories. They have a pattern-seeking ability as part of their psyche.[xvii] Jung's archetypes are another way of articulating imitation, and there is no doubt that humans imitate one another as well as other creatures; such actions are an integral part of learning. Again, one perceives a pattern of behavior somewhere and whether oriented toward the goal of the action or merely the repetition of the action one imitates the pattern. Such internal patterning is directed outward when humans perceive events that are repeated in nature and when they create art. Aristotle speaks of mimesis as art imitating nature:
The poet (as maker) being an imitator just like the painter or other image-maker, it is necessary to imitate some one number of three existing things, either what kind of things as they were or are, or what sort are said to be, or seem to be, or to have been, or as what kind it is necessary to be. (Aristotle, Poetics 1460b8-12)
Nature has a pattern whose organization and structure art imitates, even if art imitates human behavior. We recognize repeated occurrences in nature, like gravity and photosynthesis. So, humans must be able to discern patterns in their own behavior, yet they also recognize patterns in animal behavior and even in objects as well as in change. They look for patterns in nature in order to understand what she does. When they see an action repeated, human beings look for similarities between occurrences and when they find enough similarities, they think to have found a “law” of nature, and this “law” is the repeated occurrence or pattern we have located. Humans are able not only to predict what will happen when X and Y occur, but they will be able to understand things as they become; they may even reproduce specific happenstance. It is the human quality of imitation that compels them to seek patterns in nature, and those similarities not only become natural “laws”, but they are the beginning of logical inquiry. Logic, in fact, is the making of connections through the recognition of similarities, or patterns.[xviii] What was imagined and woven into a narrative as an activity was what ancient Greeks thought proper to do, though Homer and Hesiod may not have intended that humans imitate the gods they depicted. The humility of Odysseus before an angry god; the obedience of Telemachus to Athena; the proper sacrifice to a god in thanks for victory all are imitable activities. When one wants to be like Odysseus and when one acts as he does, the same pattern-finding human apparatus is in effect. Since science is in part the discovery of patterns in nature and the ascertainment of natural law through the recognition of patterns in repetition, part of scientific method is latent in the literature of the ancient Greeks as it is in all literature. And the ancient Greek culture was nothing if it was not literary. While all literature is imitative of nature, a culture like the ancient Greek culture had an inherent tendency toward seeking after patterns because literature was a primary element of education, and Greek culture was in one sense decentralized because there was no overarching authority governing all the city-states (poleis). Each city interpreted their own version of the gods and vied for influence. There existed thus a competition about what was true and what not about god. The central aspect of the culture–that the gods are the same–remained the same while Greeks sought different ways to understand the divine. Their cultural dynamic, while the same as other literary cultures, was that of the particular and the universal in tension. This tension allowed for many interpretations of physical happenstance and the divine. Thus, different theories of nature. And Aristotle verifies for us the direction that mimesis in literature took within the ancient Greek culture: imitating how things were, are and how they will be. The seeking after of patterns then must have become second nature to ancient Greeks via their literature and music. If seeking after patterns in nature is a part of science as well as an integral aspect of nature, then inherent in Homeric epic was a significant part of the scientific endeavor. Other cultures employed literary mimesis in different ways, but the Greeks became thinkers, then philosophers and scientists. The decentraled-centralization of the culture accented the creative aspect of the poets, and some, as we will see, sought radically new ways of understanding gods, mostly through nature.
In Hesiod's Theogony, gods are conceived similarly to how Homer depicts them. They are abilities and forces of nature. Hesiod begins his genealogy of gods with more abstract deities that lead gradually to fully anthropomorphic metaphors of natural forces:
…the very first did Chaos come to be, and then broad-breasted Gaia [earth], always an unfailing setting-place of all immortal things...and cloudy Tartaros in the inmost corner of wide-wandered earth, and Eros, most beautiful among immortal gods, looser of limbs, who subdues mind and thoughtful counsel of all gods and all men in their breasts. Out of Chaos, Erebos and black Night came into being; and out of Night, again, came Aither and Day, whom she begat and bore, having mingled in love with Erebos. And Earth first brought forth star-like Ouranus [sky], equal to herself, so that he would cover her in every way, so that she would be an unfailing setting-place for blessed gods always. Then she brought forth high Mountains, charming beds of divine Nymphs.... She also bore unharvested sea, seething with its swell, Pontos, without delightful love; and having lain with Ouranus she bore deep-eddying Oceanus, and Koios and Krios and Hyperion and Iapetos.... (Hesiod Theogony 116-134).
Hesiod is perhaps more scientific than Homer in that there is a greater abstraction in his work, perhaps more of an attempt at systematic explanation. Kirk and raven [xix] talk in some detail about Chaos as the first force of nature arising in Hesiod. They take as credible Francis Cornford's interpretation of Chaos as a separation of the first deities to appear in the universe. There the account of the universe is semi-rational and what Kirk and Raven seem to mean is that there is some logical consistency in the order of the birth of the natural forces: Chaos first, then Gaia, Tartaros, Eros etc. Chaos is the initial separation of the divine forces, which seems to be the act of existence being separated from other acts in the order of things. There are for our purposes two ways that one may interpret this separation. The first is the actual separation itself. Chaos is the gap between the earth and Tarataros and the earth is a fertile agent, producing the gods. She is the mother of the gods. Chaos, if interpreted as the continuing separation or force of separation of the universe, then, is part of the generation of living organisms and deities. Chaos is difference itself, without which no individual thing may come to be other than any other individual thing. Chaos not only plays a role in the initial separation that created the mother of all things, but it continues to break the universe itself into different beings, a part of birth and certainly a portion of death. It continues to operate as a driving force. The second and more anthropomorphic interpretation is that Chaos is the breaking of things into different beings and the ability of ancient Greeks to reason and separate one thing from another as well as the ability to examine and differentiate the parts of things. Chaos and differentiation then become the very process of marking out boundaries of objects and living creatures in such a way as to understand and perhaps participate in their being. So, ancient Greeks separated and then combined things in order to comprehend anything that they encounter. Inherent in the metaphor of primordial separation is the scientific mindset itself. Reason is, among other things, that which discovers pieces of objects, breaks them and then sews them together again in logos, reason or word, in order to comprehend their parts and workings, and to participate in becoming. One may object that Hesiod mentions Chaos seldomly in Theogony, that Chaos does not continue to act in nature, but first, this force of nature is a psychological category anthropomorphized into the makeup of the universe. It is human ability to separate and distinguish in order to comprehend and manipulate. One may think of it as a psychological state imposed upon the physical universe. Chaos is also a stated act of separation, and separation in the form of violence and strife exists throughout Theogony, though Chaos is the initial stated act of violence and separation and not the continued acts of violence. Suffering and strife are long a part of the implications Hesiod incorporates into the relationships between Gods and men, the initial being the generational conflict between Ouranus and his children:
However so much that came from Gaia and Ouranus, the most terrible of children, from the beginning were detested by their own parent; and when first any of them came into being he hid them all and did not allow them into the light, in the hollow of Gaia; and Ouranus delighted in the ugly deed. And she, colossal Gaia, groaned within, being crowded; and she pondered a cunning, ugly deceit...having concealed him [Chronos] she placed him into an ambush, placed in his hands a saw-toothed sickle, and advised him of the whole trick. Bringing on Night great Ouranus came, and over Gaia, desiring love, he extended himself, and spread all over her; and the boy [Chronos] from his ambush stretched out with his left hand, and with his right he grasped the gigantic sickle, long saw-toothed, and furiously cut off the genitals of his dear father, and flung them back to be carried away.... (Hesiod Theogony, 154).
Ouranus abhors his children; Gaia abhors Ouranus; Chronos and the titans make war upon their father; Chronos atempts to control his children, begetting in them hatred for him. Zeus and his allies then make war on Chronos and the titans. When the last war between the Olympian Gods and their parents the Titans – the titanomachia – ends, strife continues in the form of rivalry and strife mixed with desire. In destruction as well as generation the activity of separation is implied. Reason has broken into differing elements the analyzable aspects of the universe and Hesiod's inherent ability to break objects and concepts into pieces–itself a species of violence–enables him to organize these forces into his narrative. There is a psychological need to understand that necessitates the violence of analysis. Inherent in the sadistic events of Theogony is the forcefulness of the scientific impulse to learn.
Homer and Hesiod did set out to explain the universe, but they do not make use of arguments like ones hypothesized and then verified. There is no attempt at mathematical analysis of the forces of the universe, nor is there an explicit deductive or an inductive argument. There are, however, explanations for material reality: gods for Homer, and gods, separation and unification for Hesiod. These are mythological explanations that allowed ancient Greeks at least to believe they were able to alter or even direct in some manner the very forces of the universe that control fate and physical happenstance. Most important is that Homer and Hesiod make use of mimesis, which is the term used by ancients to discuss the same pattern-seeking that modern psychologists pursue. It begins with mere repetition of some pattern seen in nature and then evolves into a term used to discuss metaphysical issues in Plato and scientific ones in Aristotle. Whatever is part of the human psyche is also a part of some comprehension of regular activities that serve as not only symbols but patterns of behavior. When they remove the self from that pattern-seeking, humans look after patterns in the universe alone. When they seek patterns in the universe, they look to particular manifestations that lead to an expectation that the same particular manifestation will happen again, and thus look for the universal in the particular. Drawing universal from particular is one part of the dynamic of the ancient notion of logic: induction is particular to universal and deduction is universal to particular. The patterns come from everywhere in human culture as well as the universe, but strongly in literature. The Greeks had a decentralized religious system that allowed them to create the spiritual pattern in different ways; they sought the nature of god because they felt compelled to seek after something of god; they had no centralized notion dictated to each locality. Thus, local aristocrats with the leisure and education needed to create systems of thought about nature–which they perceived as god–created different systems that possessed the same subject. Then, Greeks began to argue with one another, using this inherent logic that comprehended the universe by means of the relationship between the universal and the particular. Here is the birth of science, yes, but it is a birth from an already-present manner of comprehension. Ancient Greeks made use of this way of comprehension without necessarily recognizing its rules and what they called mimesis , whatever that really is, is part of that function of the mind. Humans employ their tendency to imitate one another in the discovery of repeated patterns in nature itself. These patterns are compared and what their relationship tells the scientist is a new piece of knowledge about them, an inference as we have already seen. These inferences lead to more comparisons and speculation. More comparison and speculation lead to an ability to manipulate or control or even predict nature's outcomes. Such a dynamic is science, and science is fundamentally comprised of the natural state of human inquiry that compels one human to copy another. This ability is focused outward onto the physical makeup and material happenstance of the universe. There is a logic that humans possess already, one which is only later articulated but not created by Aristotle. A use of syllogistic logic, which systematizes the relationship between the universal and the particular, is justified in analyzing the different thinkers that follow. Lastly, the nature of analysis in the western tradition is forceful. Logos is our reason, an intellective function that breaks its subject into parts in order to understand it. Dissecting of animals is an empirical example and examining the different parts of an element – like the proton, the neutron or the nucleus of an atom – is another. These absolutely necessary aspects of human endeavor are discernible in Homer and Hesiod, though in implied, embryonic form. Science could not exist except for those things which are a natural part of every culture but which were intimately a part of ancient Greek expression, politics, culture and thinking. It is no wonder the ancient Greeks began the analysis of material reality.
© 2017 Kirk Shellko All rights reserved
HOMER AND HESIOD BIBLIOGRAPHY:
© 2017 Kirk Shellko All rights reserved
© 2018 Kirk Shellko All rights reserved
Life and Works
Heraclitus, the weeping philosopher, was an aristocrat who lived in Ephesus perhaps circa 540-480 B.C., but little else is known about his life. According to one legend he inherited the title of king of the Ionians, and he rejected it. Known for his misanthropy and riddling manner of writing,[i] he wrote a book on the universe, politics and theology and dedicated it in the temple of Artemis. Some ancients believed that he wrote in an obscure manner in order that only educated aristocrats would be able to access his meaning,[ii] or to make his sayings similar to oracular utterances. One can see clearly why such an assemblage of declarations would be dedicated in an ancient temple; the gods gave advice as well as augury and many valuables were placed in temples. His book seems to have taken the form of a collection of aphorisms, in ancient Greek gnomai, which may have been gathered together by his followers or arranged by Heraclitus himself. It was divided into three parts – cosmology, politics and theology – though these topics are intertwined in Heraclitus. His seemingly theological [iii] musings on nature and its relationship to human beings survives in fragments taken from ancient quotes.
Arguably a monist, Heraclitus claimed that there is an inherent structure to things, which must be understood through logos, a term ordinarily signifying “reason” or “word.”
It is a wise thing for those hearing not me but the logos to agree that all things are one. ( Fr. 50, Hippolytus Ref. IX, 9 I).
That logos exists forever men always come to be uncomprehending, both prior to hearing and having heard it. That all things have come about according to this logos they are like those with no experience, attempting to understand such words and deeds as I relate while I distinguish each thing according to its nature and declare how it exists; but as much as escapes the notice of some men what they do while awake just so much they forget while sleeping. (DK 1B, Fr. 1, Sextus adv. math. VII, 132).
Logos for Heraclitus signifies a specific kind of composition, not mere logic or even reasoning, but logic is a significant part of it. Logos seems to be for Heraclitus a process-structure that can be found in the universe, but it is also the order found in the reasoning mind.[iv] Human beings are able to access the order of the universe through reason. In other words, the universe is ordered in some way and reason allows us to apprehend the structure that is at the same time a process. What that order is Heraclitus partially explains, admitting that things are revealed to human beings even as they are concealed. Reason orders by assessing perception after the senses take in sights, sounds and understanding:
However-so-much things of which there is sight, hearing and learning I hold in high regard. (Fr. 55, Hippolytus Ref. IX 9, 5).
Yet, Heraclitus seems to have thought that the eyes and ears are deceptive; he understood that perception is necessary in order to assess the inherent order of things.
The eyes and the ears of human beings are unable to interpret correctly what the senses tell them and so they are foreign to knowledge, which is directed by logos. According to Heraclitus, not many human beings use reason appropriately because instead of attempting to find the commonalities of the universe they employ logos in a private manner:
It is necessary to follow the general; while there exists a general logos the many live as if having a private thought. (Fr. 2, Sextus adv. math. VII, 133).
One is able to discern immediately a sense of what moderns may call objectivity in the notion that what is general in the universe is most important, while what is private – or perhaps subjective – Heraclitus disdained.[v] Again, logos is that which finds the inherent order of things and so reason finds universals, but perception deceives. Because perception deceives it needs to be ordered by the mind, or logos.
Causation and Cosmology
Such an inherent order was not straightforward, but a riddle; some scholars claim it is a unity of opposites:
One understands through perception only some incomplete part of material reality. Once one uses reason, one sees an interaction between or unity of opposites in common happenstance, which seems to be partly what Heraclitus means when he asks men to comprehend the general and not the private. Again, private things may here be linked to subjective interpretation, or to opinion. Heraclitus himself employs literary devices and hides a significant part of his meaning, but he seems to have been aware of the need to be more objective or rational in making assessments. Kirk and Raven [vi] point out four important ways that Heraclitus demonstrates the supposed “unity of opposites.” First, similar things produce opposing effects on differing classes of objects, as in the sea's affect on men and fish. Second, differing facets of the same thing elicit opposing descriptions, as in the differing directions being the same road.[vii] Third, things are recognizable as what they are only by means of comparison to their opposites, as in weariness making rest good. Fourth, some opposites are connected because they come after or are followed by each other, as in old age following youth; there is a unity in their opposition (see below). These are aspects of material reality that to Heraclitus seemed to be concrete, physical things and processes, again as in the water's affect on men and fish and weariness making rest more prominent. One is struck immediately by the relative relationship of the opposites. That is, the supposed continuum of opposites seems to be the same as the relative relationship of one opposing thing or happenstance to another. The south road and the north road being the same are not merely perspectives. They are absolutes that function in relation to one another, producing an apparent contradiction that encompasses the same thing. The dynamic is relative because of the relationship between things, but Heraclitus describes it as an interaction between or a unity of opposites rather than a relative relation. Here he gives us a reference point and seems to comprehend and employ some element of relativism in his description of one thing being its own contradiction. The reference point is needed in order to determine if the road is the south road or the north road, and while these opposites seem to be absolute there is no way to determine the absolute answer to whether the road is north or south. There is no real, fundamental and eternal answer, other than the road north is the south road and vice versa. That is Heraclitus's point. Much of the unity of opposites in his thinking seems to come from an insight into relativity, and so some part of relativity theory is inherent in his thought, but note also that he has observed first what seems to be a relative relationship. He then employs his reason, logos, in order to determine what is the deeper or one may say higher truth: the relationship of opposites that comprise a single happenstance. Within his way of comprehending things is a scientific framework: observation and later reasoning in order to bring out the inherent truth to the observations. Neither thought nor perception themselves suffice, and perception is taken to be decidedly inferior to thought combined with perception. Heraclitus may have taken much more concrete examples and used them to show the structure of their dynamic, rather than an abstraction made solid. If he did as much, it is possible that his thought possesses an empirical dimension worthy of mention. Yet, there are many kinds of opposites Heraclitus describes; a common way of understanding his insights comes from unity and plurality, a common topic among ancient thinkers:
A taking-together is whole and not-whole, a being-brought-together and separated, accorded-discorded. Out of all things is one and from one are all things. (Fr. 10, [Aristotle] de mundo 5, 396b20).
We do not know enough about Heraclitus's writing to make a definitive judgment of his meaning, and his literary manner along with his enigmatic expression inspire very various interpretations, but such a hermeneutic may have been Heraclitus' intention, since he asserts that things reveal and conceal themselves at once. It is important to remember that for Heraclitus the implicit structure beneath all rest and activity is that of opposition or perhaps tension. All things interact in an opposing way, even things that seem to have no opposition or even conflict at all, like the lyre:
They do not apprehend how a thing differing with itself agrees with itself; a stretched-back harmony [viii] exists, like the bow and the lyre. (Fr. 51, Hippolytus Ref. IC, 9, I).
The tension of the strings makes the lyre possible as a lyre. No music emerges, if the strings and tension are not present. Such a structure of conflict is needed for the instrument to be itself:
In this manner for Heraclitus conflict and strife compose the fundamental arena in which all things transpire:
War is the father of all things and of all things king, and on the one hand he put forth the gods and on the other human beings; some he made slaves and others free. (Fr. 53, Hippolytus Ref. IX, 9, 4).
What lies hidden is where the tension exists:
All things are for Heraclitus in constant change. His famous river image [ix] gives the metaphor:
Perhaps he means that all things change constantly, as in even seemingly unchanging things are changing in imperceptible ways:
Somewhere Heraclitus says that all things make room for another and nothing remains still, and likening things existing to a rush of a river he says that twice into the same river you would not enter. (Plato Cratylus 402A).
What is interesting for us is the constancy of the interchange of opposites and change in general. The constant wrangling of the universe Heraclitus expresses in more than one metaphor, but his primary element of the universe seems to be fire.
All things are an exchange for fire and fire for all things – just as useful things for gold and gold for useful things. (Fr. 90, Plutarch de E 8 p. 388E).
Fire is commonly thought to be Heraclitus' first principle, but in his work fire is also at times a metaphor for change of various kinds; nonetheless it is also a physical first element. So, fire is both a first principle and a metaphor for Heraclitus. Anaximander thought that there was vengeance between forces in the universe and some scholars believe that Heraclitus's sense of a constant give and take is a development of that thought, yet both thinkers seem to have thought change to be a part of a constant in the sense that there is an exchange of dominance without loss of activity or inherent ability. Thus, all powers of the universe are exchanging their ability to perform specific tasks, somewhat similar to our notion of the preservation of energy. Heraclitus' idea that the universe always has been and always will be, coupled with his assertions that opposites are at variance with one another and that they exchange forces, recall the first two laws of thermodynamics:
This order of the universe, the same for quite everything, neither some god nor some man made, but it always was and is and will be fire ever-living, flame-fixed in measures and squelched in measures. (Fr. 30 Clem. Strom. IV 105 [II 396, 10] [Plutarch d. anim. 5 p. 1014 A]).
The first law of thermodynamics is the law of conservation of energy. It states that all energy of an individual system is constant. Energy is not destroyed, but transformed from one kind of energy to another. Change in the internal energy of a system equals the heat supplied to that system minus the work done by that system on its surroundings. It is noteworthy that the first law of thermodynamics centers around heat and heat exchange. Heraclitus' notion of a closed system of constant change through fire, the entire universe, is similar in that the system always will be the same and yet constantly alters through the exchange of its fundamental process or element, fire. It itself is a closed system, and the change that physical objects undergo is that of heat, associated with fire, which can be interpreted as energy. The second law of thermodynamics states that all the entropy of an isolated system never diminishes because isolated systems alter into a thermodynamic equilibrium, the greatest entropy. Here is an equality similar to that of Heraclitus' notion. Fire for him is both a metaphor and an element; it is a constant change that alters constantly. In his universe the equilibrium is that of change itself, like the river metaphor. It is the running of the river that makes it constant. So, the very act of change is the remaining the same, a kind of equilibrium already balanced by the tension needed in order for something to be what it is. [xi]
It is important to note a fundamental criticism of Heraclitus before we proceed to analyze a part of his argument.[xii] In logic there is a conception known as the principle of non-contradiction. It states that something cannot be true and not true in the same manner at the same time. [xiii] One immediately sees how scholars and thinkers may believe that Heraclitus is unaware of this logical principle, and perhaps he lacked understanding of it. Still, what he asserts is some ontology of the universe that seems to go beyond logical categorization. His unity of opposites may be a semi-rational way of delving into an inner working of the universe that itself is important to comprehend and yet beyond the conventions of human logic. Logic does not answer all questions, as the universe does not lend itself to mere human reason. It follows that thinking beyond reason benefits those attempting to comprehend the universe. In this sense, Heraclitus is more of a metaphysician than a scientist, but his status does not force him into the shape of a mere empirical observer. It seems that some part of his reasoning is as follows.
One can see a piece of his reasoning here in that perceptions, or private thoughts, are common thoughts at the same time. There is an objective aspect of an observation that may be used to comprehend a universal law. Science does not, of course, claim that scientific law is universal, but only probable. Yet, the notion that an observation may prove to be a reliable general expectation is inherent in the notion that the subjective, private thought is also common. Reason for Heraclitus is able to make the distinction, and logos takes these private thoughts and makes them common. In other words, laws of nature come from seemingly subjective observations. Heraclitus argues thus for fit or unfit witnesses to truth:
The above argument is a warning that the common apprehensions can become unreliable, if logos is not properly applied. So, we must be careful what we do with our subjective assessments. If we find that they apply in at least a semi-regular manner, we may be justified in saying that they apply in most cases, or at least that they can be relied upon as probable expressions of truth. What Heraclitus seems to be saying is that observations made ought to be rigorously structured through reasoning. And here is one aspect of scientific reasoning. Much more paradoxical and revealing about Heraclitus' view of the structure of things is his argument that shows how his opposites operate:
One need not change the first two premises in order to come to another conclusion: All paths up are paths up. Still, here is the other configuration.
These two arguments seem to be a redundancy until one recognizes that the paths up and down, when these arguments are taken together, are the very same path that differs from itself. Their being the very same path is a spurious contradiction; it is a contradiction that does not contradict, if one may be inspired by Heraclitus's manner of expression. This contradiction is the implicit kind of order about which Heraclitus talks; the paths up are the paths up and the paths down are the paths down. One cannot say definitively which is the actual path, up or down, because they are the same, and that they are also radically different is one aspect of opposites that is part of a relative comprehension of what they are. If one begins with the reference point of paths down, then all paths down are paths up, and if one begins with the reference point of paths up, then all paths up are paths down. Recall the comment about the sea being unhealthy to men in some sense and healthy for fish. The aspects that make the sea both healthy and unhealthy are the same, and the very thing that elicits opposing description is the thing that has a different affect on differing things. What Heraclitus notices about these relationships is not the differences alone, but the similarities that lead to and sometimes themselves are the differences. Again, there is for him an absolute in each aspect of these things that leads to a shifting relationship between the different aspects that are the same. The path up becomes the path down and the unhealthy sea becomes the healthy one, yet not by changing into anything else but rather by changing into itself. Inherent in them is a constancy that is regular; it is also the result of a never-ending tension, which can be thought of as energy. It seems, then, that Heraclitus incorporated some aspects of what we call relativity into his insight about the apparent contradictions that compose things. His arguments are not those of conventional logic and they seem to be redundancies, but they reveal something about the nature of objects and processes; they are observations that have been reviewed and turned about by reason in order to find a more fundamental, general meaning. In this sense Heraclitus' thought is inductive; he proceeds from observations to general conclusions. He assumes that specific aspects of his observations are always true, and that is his most prominent flaw, yet he seems to be aware that the senses fool those who seek general truths and he attempts to eliminate perceptual illusion through reason, or logos. His arguments have an enigmatic soundness to them in that they have good form and they talk about things that actually exist in the world, yet they seem redundant or even trivial until one sees the depth to them. Heraclitus was no mathematician, and so no formula was used in order to discover insights about material reality. The significance of math in the ancient world really begins with the Pythagoreans. Heraclitus may have believed that fire was actually the material first element and principle of the universe, but it is advisable to take his fire as “fire.” Fire for Heraclitus was a metaphor as well as an element. And here, of course, we may find some legitimate criticism of his cosmology. Naturally, fire is not an element at all, but a chemical process. Yet, if one takes fire as a metaphor, then Heraclitus' view applies still to our 21st century world. It is in part relativism. Heraclitus pointed out the strangeness and paradox of relative relationships between things. He claims that the universe is concealed and it seems as if he says the concealing is at the same time a revealing. His writing conceals and reveals in the same way, it seems, like his universe. And a part of his thought is consistent with commonly accepted scientific thought.
© 2018 Kirk Shellko All rights reserved.
© 2018 Kirk Shellko. All rights reserved.
Life and Works
Little is known about the life of Pythagoras. Born on the island of Samos, he emigrated to Magna Graecia, a Greek colony in southern Italy.[i] A student of the Babylonians, Egyptians, Indians or Zoroastrians, he became a statesman of Croton[ii] and developed a large following. He has been labeled mystic, guru, shaman, philosopher, cosmologist and scientist. Perhaps all of these labels are not quite true. His floruit occurred around 532 B.C. and so he was born perhaps around 572 B.C. He may have met Thales as an old man and he may have spent time studying in Egypt. Plato[iii] and Aristotle[iv] mention him indirectly because his actual contributions were in doubt, as they are today:
The learning of many things does not teach one intelligence; if it did it would have taught Hesiod and Pythagoras, and again Xenophanes and Hecataeus. (Diogenes Laertius IX, I) Pythagoras...engaged in learning by inquiry more than all other men and, having made a selection of these writings, claimed a wisdom of his own, badly wrought and superficial. (Diogenes Laertius VIII, 6)
Pythagoras founded a religious order that advocated vegetarianism and espoused metempsychosis, a cosmic system of reincarnation. Again, Diogenes Laertius:
What [Xenophanes] says about Pythagoras goes like this: 'Once they say that while passing a puppy that was being mistreated he had compassion and said: “Stop, do not hit it; for it is the soul of a friend that I recognized when I heard it crying.” (Diogenes Laertius VIII, 36). [Pythagoras] says the soul is immortal; next that it changes into other kinds of creatures; that in certain cycles things that have occurred again come to be, and that nothing is absolutely new; and finally, that there is need to believe all ensouled things are of the same race. Pythagoras appears to have been the first to import these beliefs to Greece. (Porphyrius, Vita Pythagorae 19) Diogenes Laertius (VIII, 4-5) also says that Pythagoras recalled four of his incarnations.
The Pythagorean community forbade divulging of secrets[v] because Pythagoras and his followers wrote nothing and communicated orally. Thus little was written by the order initially. His followers for one reason or another developed a precedent of attributing insights to their master even when new ideas arose after his death and so at least some mathematical insights attributed to Pythagoras were not actually his. After his death his school separated into two sects. The one adhered to the mystical metempsychosis, the other – being mathematicians – concentrated on applying number to reality.[vi] No reliable source of Pythagoras' own engagement in scientific inquiry exists, and so we talk of “Pythagoreans” because the ones most likely responsible for the mathematical insights attributed to Pythagoras are his followers.[vii] Still, Pythagoras himself perhaps deserves credit for discovering that principal musical intervals can be expressed in numeric ratios between the first four integers.[viii]
Most probably, Pythagoras discovered the ratio of musical intervals by using a monochord, a single-stringed ancient device that measured sound frequencies. He may have examined its sounds and learned what musicians knew about the scales;[ix] thus an inductive and empirical aspect of his discovery some claim is an experiment.[x] The ratios he measured are the octave = 2:1, the fifth = 3:2, and the fourth = 4:3. The octave is an interval between one pitch and another in half or double its frequency. Thus, the ratio of 2:1. A fifth and fourth are like intervals of different spacing. One may explain much of Pythagoras' own inspiration through his thought about the musical scale. There are low and high pitches. The low pitches are vibrations that are slow and the high pitches are those that are fast. On the musical scale the different vibrations create different pitches. When one travels up or down the scale one-half or twice as much, one finds a pitch that is similar to a pitch positioned that distance. Now, we measure these pitches by hertz. So, a pitch of 50 hertz will be an octave of a pitch of 100 hertz. The same is true of a pitch at 200 hertz. The ratio of the scale determines the distance from one pitch to another. The ratio for a fifth is, as above, 3:2, and the ratio for a fourth is 4:3. The musical scale may be thought of as an unlimited series because of an unlimited number of vibrations while the points on the scale and their ratio is a measurement determined by number and therefore bounded, limit that is. That is to say, number makes a boundary that produces a specific pitch. Number demarcates and therefore makes things, and Pythagoras theorized that all of the objects of the universe actually are the proportion of some ratio upon what may be called the unlimited.[xi] Most of the Pythagorean insights that arise from his initial thought are applications of some kind of mathematical measurement to all objects in the universe.[xii] For example, the Pythagoreans believed that the number ten was the nature of number:
The nature of number is ten. All Greeks...count up to ten, and having come to ten return again to the unit. And again, Pythagoras says, the power of the number ten lies in the number four, the tetrad. This is the cause: if one starts at the unit and adds successively the numbers up to four, one will come to the number ten; and if one overshoots the tetrad, one will exceed ten too. Thus, if one takes the unit, adds two and three and to these adds four, one will make up the number ten. The result is that number as a unit exists in the number ten, but potentially in the number four. (Aetius I, 3, 8)
They conceived the number one as a whole and after the whole unit of ten is reached the counting begins again, added to the number of ten-units that have already been counted. Because the ten-unit is the renewal of the counting, that amount of numbers is fundamental to the measurement of mathematics; is it again the unit. The number four is the number of numbers that comprise the unit of ten and it is also the number that contains one, two and three (in a seemingly literary sense) and thus produces the new unit, a unit which is the foundation of all numbers. In this sense it seems that the Pythagoreans thought of four as being potentially ten. 1 is not truly a number because it is not yet plural; 2 is the beginning of even numbers, and 3 the beginning of odd. 5 has the distinction of being the first product of the first principles.[xiii] The Tetractys of the Decad (essentially the number 10) is a visual representation of this conception of number.
One can see not only the addition of four numbers that comprise ten in interesting ways,[xiv] but also the initial insight into the musical scale fits conveniently into the scheme of this conception of number, the octave and the fifth as well as the fourth being set within the ten-unit. It is no wonder that Pythagoras conceived of material reality as itself composed of number and number composed of music and thus the universe was conceived as musical.
Causation and Cosmology
Something about number was not only a part of each thing; it was constant. Number had determined something essential about the musical scale. It was not merely a tool for measurement, but its ratio was the musical scale. Not only was number always able to be applied with certainty; it was divine in its constant certainty.[xv] Number becomes vital in the comprehension of material reality because one only needs to apply the proper number or set of numbers to an object in order to know fundamentally what it is. Justice had a specific number, as did a line, a plane and a solid.
Since of these principles numbers are by nature the first, and in numbers [Pythagoreans] seemed to see many things most similar to existing things and things coming to be, more than in fire and earth and water so that this sort of manifestation of numbers is justice and another sort mind and reason, and another being opportunity–and each of the others similarly manifest so to speak.... (Aristotle Met. A5, 985b23).
Aristotle claims that the Pythagoreans confused abstract number with objects. In other words, numbers themselves were supposed to have spatial extension. While such a misapplication led the Pythagoreans into many imaginative corners, certain calculations applied as measurement of spatial extension operated with seeming flawlessness. One can see how the musical scale and the Decad impressed, but the Pythagorean theorem, still used today, is one representative example of the potency for lasting influence and truth in Pythagoras' perspective on number: the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the sides enclosing the right angle (A²+B²=C²).[xvi] The formula is relatively easy. Given a right triangle, the sum of the squares of the two sides opposite the hypotenuse (the side opposite to the right angle) is equal to the square of the hypotenuse.
So, here if a = 3 and b = 4, then c = 5. The unit of measure makes no difference and this formula arrives at the required number no matter the size of the triangle, as long as one of the angles is a right angle. Most important for our purposes, it is accurate whether there is an actual triangle present or not. It seems the ability of number to perform universal tasks like the musical ratios and the “Pythagorean” theorem is what convinced Pythagoras and his followers that they had a means to access a fundamental aspect of all things. One needed only to discover the number or ratio that was the shape, mass, depth et cet. of things as well as the number that was the objects themselves in order to find its essence, fundamental and applicable on any occasion. Number was bounded and it operated with the boundless. No other principle or element could be applied so universally. Aristotle sums up their conception:
...the Pythagoreans have said...that there are two principles, but added this much, which is their specific thing, that they thought the bounded and the limitless and the one were not certain other natures, such a thing as fire or earth or some other sort of thing, but that the limitless itself and the one itself were the very existence of which things they are predicated. This is why number was the very existence of all things. (Aristotle Met. A5, 987a13)
It is the universal application that we must remember because if and when the Pythagoreans arrived at proofs, they were discovering theorems that were deductive. In other words, the logic they were using compelled them to arrive at specific conclusions. We have seen that deduction is precisely that: if one accepts the premises (in this case the increments and their measurements), then the conclusion naturally follows. Numbers were, of course, responsible for the potential reproduction of their findings and thus a constant, fundamental principle. Qualities of numbers were for Pythagoreans characteristics of the material universe. Oddness and evenness were the limit (light or fire) and the unlimited (air, void, or space),[xvii] as we have seen in the determination of the musical scales. Their insight into the musical scale lead them to believe they were able to find similar applications of number, or the numbers themselves, in whatever they analyzed. Limit was oddness, unity, rest, goodness and the unlimited was evenness, multiplicity, motion, bad. The whole of their dualism relied upon this mathematical underpinning to opposites.[xviii]
The left figure shows the gnomon, or shapes comprised of points. As we said above, the Pythagoreans thought of numbers as actual points. In other words, they seemed to believe that numbers have magnitude in themselves. In the left figure the square is comprised of odd-numbered points stacked around one another. It remains a square no matter what odd number is added to it, and a clear line divides it into halves. The Pythagoreans believed that this number represents limit and thus a figure that possesses a beginning, middle and end. The addition of an odd number of points is necessary to compose a new square, but always will there be a new square, given the addition of a new set of odd numbers. Only the size changes. In figure on the right, the rectangle, even numbers are added in order to compose a figure, but the ratio, that number found in the musical scale, changes with each addition of an even number. Pythagoreans took this altering of ratio as a sign of indefiniteness; also there is no clear middle part of this construction. Thus, they associated evenness with the unlimited, which lacked necessary parameters of a more balanced constant form, like the square:
[The Pythagoreans] meant by the limitless even number, 'because everything even',...'is divisible into equal things, and the thing that is divisible equally is limitless with respect to division in two; for halving into equals continues (without limit), but the addition of the odd makes a limit; for it prevents the division of it into equal things'. In this way the commentators attach the limitless to the even in respect of division into equal things, and it is clear that they take limitless division in terms not of numbers but of magnitudes.... (Simplicius Physics 455, 20)[xix]
A similar way of applying numbers and their quantity to the musical scale and the Pythagorean theorem also applied to everything.[xx] One needed only to find what that application is. The difficulty in this kind of thinking, of course, is that number is assumed to have already some kind of spatial magnitude or that number generates spatial magnitudes. After that assumption the formulas are constructed. Aristotle critiques their position:
...that bodies are composed of numbers, and that this number is mathematical, is impossible. For to speak of magnitudes as indivisible is not true; as much as there may be magnitudes that exist in this manner, units at any rate have no magnitude; and how is it possible that a magnitude come from indivisibles? But certainly arithmetical number, at any rate, is unitary, and they [Pythagoreans] say number is real things; their propositions they apply to bodies as if they consisted of those numbers. (Aristotle Met. M8, 1083b8)[xxi]
Some scholars assert[xxii] that Pythagoreans believed number to be a kind of atom and that Greeks were slow to think of things as having no spatial extension, but atomistic principle of number and spatial extension may not be precisely what Pythagoreans intended. Number may have possessed a more metaphysical characteristic. Numbers may have been necessary to produce many kinds of substantiality from seemingly insubstantial abstraction. Fire and earth have a different principle than justice or soul, yet Pythagoreans seem to have believed that number was responsible for the genesis of each.
Whether the Pythagoreans attribute coming-to-be to [eternal things] or not there is no need to doubt; for clearly they say that once the one had been constructed, either out of planes or of surface or of seed or of [other] things they are at a loss to say, immediately the nearest part of the limitless began to be drawn in and limited by the limit. (Aristotle Met. N3, 1091a12)
It may be that they thought of number as being capable of generating extension as well as justice, thought and the like:
For since the dyad is the first distance (for the unit first made distance into the dyad, and in this way to the triad and the successive numbers), if we mark out the line, the Pythagoreans say, it is not necessary to say that it is quantity in one direction, but a line is the first distance made. (Alexander Met. 512, 37)
Yet if one believes Aristotle, a sometimes notorious critic, they never articulated precisely how that genesis took place. Void existed for the Pythagoreans and it performed a necessary function, coming from the unlimited:
The Pythagoreans...held that void exists and that [a kind of] breath...and void enter from the limitless into the sky itself as if inhaling and that the void marks out the natures of things, the void being a kind of separating and distinguishing factor of successive things. This is the first thing among the numbers; for the void demarcates their nature. (Aristotle Physics D6, 213b22)[xxiii]
It seems to have been a negative aspect of things that distinguished parameters. Naturally, void would come from an indefinite aspect of reality, and the unlimited was that aspect for the Pythagoreans. Though they may have conceived of number as possessing more than one substantiality, we are at least justified in claiming that the Pythagoreans thought of number as the beginning of spatial extension, as long as the point itself has extension:
For 1 is the point, 2 the line, 3 the triangle and 4 the pyramid. All these are the first things, the principles of individual things of the same kind...and the same things [come to be] in generation too; for the first principle in magnitude is the point, the second the line, the third surface and the fourth the solid. (Speusippus ap. Theologumena Arithmeticae p. 84, 10 de Falco)[xxiv]
The Pythagoreans also held a conception of the universe based upon the initial insight of the musical intervals that the entire universe made sound:
It is clear from these things that the assertion that a harmony came to be from the stars carrying themselves about, i.e. that the sounds they make are consonant...is not true. It seems to some [Pythagoreans] that a sound comes to be from bodies of such a size being carried, since locally things having neither equal bulk [to the celestial beings] nor being carried by such a speed do [actually produce] sound; Also, when the sun and the moon...and the mass and great number of stars are moving with so rapid a motion, it is not possible that a sound of such magnitude not come to be. Positing these things and that their speeds, as measured by their distances, have the ratios of musical consonants, they assert that the [musical] sound of the stars being carried around in a circle is a harmony. (Aristotle de caelo B9, 290b12)
According to Aristotle the Pythagoreans adhered to their theories rather than observation:
Most say that the earth lies at the center of the universe,...but the...Pythagoreans say otherwise. At the center...is fire, and the earth, one of the stars being carried in a circle around the middle, makes night and day. Still another earth opposed to this one they construct, which they call by the name counter-earth, not seeking reasoning and causes pertaining to appearances but forcing the appearances upon some reasoning and opinions of their own. But to many others it would be agreed that there is no need to assign to the earth the middle place, fitting their belief not on appearances but rather on arguments. They think that the most honored place befits the most honored thing: and fire is more honored than earth and limit more honored than intermediate, and the outermost point and the middle are limits. The result of reasoning from these things is that they do not think it [the earth] lies in the middle of the sphere, but rather fire [does]. (Aristotle de caelo B13, 293a18)
The very universe, then, is not only configured differently than common perception allowed, but it produces vibrations by its very essence and is therefore musical. In this manner the Pythagoreans took Anaximander's three wheels (sun, moon and stars) that composed the universe and gave them ratios;[xxv] the earth was shaped like a sphere because of the solid figures it is most beautiful. The Pythagorean cosmology begins when the numbers are generated as the unlimited is taken in (breathed in) by the limit.[xxvi] Notice that in a proto-scientific manner the Pythagoreans make use of number as a means to measure and understand the universe. They do not use an hypothesis and then test it, but they seem to have believed that number not only measured the universe but was that measure. The initial insight of the musical scale applied in a fundamental way to points, lines, planes, plane figures, solids and then other topics like the cosmos. Remember that for them number also applied to abstractions like justice, the soul, god and love. The very universe itself is musical, and perhaps one reason Pythagoreans adhered to this assertion is that such a perspective is somewhat romantic and literary. There is a charm to comprehending the whole of the universe in terms of music.
It remains to evaluate some small part of Pythagorean argumentation and see how it applies precisely. We take only the beginning insight of the musical scale and the Pythagorean theorem as our points of discussion. Number seems to have been applied as a tool in the measurement of the musical scale and the Pythagorean theorem. Yet, number is applied to everything possible because the unlimited and limit seemed to be a fundamental part of measurement and calculation. One mistake was that the Pythagoreans took measurement as reality. In other words, even in antiquity Aristotle and others took numbers as abstractions; we have seen Aristotle say as much. The Pythagoreans wished to apply them to reality as the objects they analyzed. Number provides a foundation for the determination of parameters certainly in shapes and abstract solids, but there is no clear argument that number actually is existing solids or abstract notions like justice, and application of number to certain things simply does not work. Thus, Aristotle's complaint. There was, however, a deductive aspect to Pythagorean thinking that must have been seductive. The application of number to right triangles had produced a calculation that unerringly, seemingly, produced the correct assessment no matter the size or presence of a right triangle. Numbers may have seemed to be at once eternal and substantial in every sense because they were capable of measuring despite size or even manifestation.
This argument is something like the attitude Pythagoreans seem to have taken. Number as fundamentally all things is assumed once the insight into right triangles and its implications have been observed. One has only to determine how. Imaginative use of number application abounded, and sometimes their efforts came to fruition, as in the Pythagorean theorem. Other times their efforts were merely playful, as in four points actually being a solid. Yet, one cannot apply number to all things simply because its application to a right triangle or a musical scale seems to have revealed something hidden in them. The revelation of what is a right triangle is not the revelation of any given thing or object. Nevertheless, the use of numbers in the musical scale as well as the Pythagorean theorem is perfectly legitimate. One cannot adequately represent its calculation in a syllogism, but the reasoning makes seemingly impervious connections. In each and every case of a right triangle where C is the hypotenuse and where A and B are the opposing lines that comprise the right angle, the formula A² + B² = C² is true. One must think in terms of deduction compared to induction. Math is inherently deductive. 8 x 4 always equals 32, at least in traditional mathematics. There is no other conclusion to draw as there are no other conclusions to draw when other operations are in order. Addition, subtraction, multiplication, division all give one constant answer. This simple truth makes mathematics absolutely certain in its own realm. One may then apply mathematical reasoning to things in the universe, but as we have seen the application is partial or limited and an element of observation and thus empiricism is necessary. Again, we cannot form a mathematical calculation of justice or of love. The basic outlines of solids are amenable to mathematical calculation, but numbers are not in objects. Objects are in objects. Pythagoreans in their apparent excitement attempted to apply imaginatively a very potent deductive system to things that do not lend themselves completely to such a system. In other words, math as they employed it seems to be limited to measurement of material reality and Pythagoreans wished to transform math into an ontology, or some system that is the study of being. Here is one of the primary divergences of philosophy and science. Science (here physics) is not an ontology; it is a measurement of an object, or a thing, as it exists materially. Pythagoreans thought that what is meta-material about reality and what is material were the same. Therein lies their mistake. Perhaps they were aware of the difficulties of their position, but given that we have only some scattered references from ancient scholars and a not always reliable Aristotle to tell us, we simply cannot make a better determination. The lasting power of Pythagoreanism is that it gave rise to measurement by number, not equation of number with being.
One most important thought about the Pythagoreans: the Pythagorean theorem is a constant. We will see that as scientific progress continued the use of mathematics in revealing unseen aspects of the universe grew. Yet, mathematical systems are systems of logic. Systems of logic need solid foundations upon which they make assertions about reality. Constants in such systems are essential in creating solid connections and reasoning in a strong way. The old analogy of a building applies. A building with a solid foundation stays up. The constants are the solid foundation, or the reference points used to make calculations about how objects react to certain conditions and what they are. There are many constants in scientific reasoning: sine, cosine, tangent, Pythagorean theorem, E = Mc² are only some examples. These constants are aspects of material reality that are so regular as to be reliable to insert into hypotheses. And science has used them ever since humans realized that these constants do not change. In fact, we have even seen that constants like the Pythagorean theorem do not even need to apply to existing things. The formula itself will be true no matter what the state of the universe is. That is how reliable the formula of the Pythagoreans is, and the Pythagoreans began the process of deductive reasoning along with Parmenides and Zeno.[xxvii] Modern science makes use of these constants in order to form theoretical, mathematical calculations that are then tested. This is the difference between theoretical and applied mathematics. One can see easily how lasting and incredible such an insight as the Pythagorean theorem has been and continues to be. It has not only endured thousands of years of scrutiny. It also has found its way into the modern world of hyper-mathematical experimentation. Though Pythagoras did not come up with all the insights attributed to him, his one insight endures. It seems, in fact, to be eternal.
© 2018 Kirk Shellko. All rights reserved.
THE PYTHAGOREANS BIBLIOGRAPHY:
© 2018 Kirk Shellko. All rights reserved.
Life and Works
Parmenides of Elea's floruit occurred circa 475 B.C., which puts his birth around 515. Diogenes Laertius [i]says he was a student of Xenophanes. Speusippus [ii] claims he established laws for Elea, a Greek colony in southern Greece. Strabo [iii] claims that he was a Pythagorean. He wrote in dactylic hexameter a work of poetry called On Nature. [iv] Plutarch claims that Parmenides “left nothing important unsaid.”[v] Its two parts were Way of Truth and Way of Seeming consist of approximately 160 lines of perhaps 800. These lines are preserved in Sextus' and Simplicius' commentary and are believed by scholars to compose Parmenides' primary philosophical argument. Much debate surrounds the interpretation of the Way of Truth and the Way of Seeming, but one primary aim of current Parmenidean scholarship is to reconcile the two paths, a not insignificant task. The proem [vi] tells of a man who rides on a chariot to the path of the goddess where maidens convince her to open mighty doors. The goddess welcomes him and tells him that it is right he learn everything, as well as truth. She claims the opinions of mortals have no truth to them.[vii] The most important aspect of the proem for our purposes is a phrase used to describe reason: well-rounded truth. Some scholars claim that one may enter any part of Parmenides' thought and it will take them through to the whole:
It is a common origin from where I will begin; for there again I will come once more. (Fr. 5, Proclus in Parm. I, 708, 16 Cousin).
Parmenides' may not truly have first principles; his point may be that they do not exist, but that does not mean we are to refrain from thinking about origins and a foundation of all that exists. His way of truth establishes a primary Parmenidean assertion: if something is, it has being and possesses no non-being at all:
If you come I will tell you...the only paths that exist for thinking: the one that is is and that it is not possible to not be is the way of persuasion. For it follows Truth. The other that it is not and that necessarily it is not to be, I tell you is pathless and filled with ignorance; for neither would you know the thing not being (for it is not possible) nor would you speak it. (Fr. 2 Proclus in Tim I, 345, 18 Diehl). For the same thing exists for thinking and for being. (Fr. 3)
Thus, all that is is. Another way of putting this insight is that being implies being and never non-being. Some scholars[viii] distinguish between the predicative and existential being in Parmenides. Predicative being is that something is a certain quality or state while existential being is that it is present as an existing thing, whatever that is. Parmenides seems to have existed at a time when these two were not yet differentiated, and yet he is on the path himself towards making the distinction. Parmenides appears to be saying that the only topics one is able to talk about are the things that exist, as qualities or as things. Yet, the qualities or things as individual entities are not their being; that they exist is their being and there is no ontological particularity that makes the being of something a particular kind of existing thing. One is able to think only of things that are, and the non-existent cannot be apprehended by the mind, and so Parmenides is concerned about knowledge, which is a point of contention among scholars. It does seem that Parmenides makes no distinction between the existence that one finds in a particular thing as opposed to that thing's mere existence; again that something is present is the being of existence or existential being; that something exists as a particular thing or a particular kind of thing or process is predicate being. Herein lies much interpretive disagreement over what Parmenides seems to express as his first principles. One must keep in mind that Parmenides is the first genuine metaphysician because he talks about first principle as a non-spatial and non-temporal foundation. And perhaps Parmenides meant the discussion to continue. He expressed his thought in a poem after all, and poetry is interpreted variously by its very nature. Additionally, one must consider both paths of his poem. The Way of Truth is an exposition of being, the Way of Seeming one of appearance and opposites. In time, science and philosophy give up on answering the question of what being is in favor of a categorization of particulars,[ix] but Parmenides seems to talk of both.[x]
Parmenides' poem talks of being as if it is uniform to all particulars, and he concentrates on that aspect of mere existence that is uniform and vastly general rather than the particulars that are of it. Existence is and cannot not be:
It is necessary that speaking and being abide in being; it is possible to be but it is not possible for not-being to be. I call on you to reflect upon these things. I restrain you from this first path of inquiry and from the path on which knowing-nothing mortals wander, two-headed; a lack of means in their breast guides their wandering mind; they are born deaf and blind at the same time, astounded thoughtless masses, for whom being and not-being are believed to be the same and not the same thing, and for whom the way of things is backward-turning. (Fr. 6 Simplicius Phys. 117, 4).
In fact, his confusion of predicate and existential being gives rise to his monism. There is a continuous being that pervades everything. That is existence, an eternal being which is an eternal now, and this eternal being is all particular and therefore predicate being as well as a continued, present being of existence. Yet, Parmenides makes the distinction between the two when he talks about the different paths. Because existence is, it cannot have not been and because it cannot not-be it will not cease to exist:
One word of the path remains, that it is; on this path are full-many signs, that being is un-generated and non-destructible, for it is whole-limbed and unmoving and without end. It did not exist in the past nor will it be, since now it is all the same, one, continuous; for what generation will you seek of it? In what way and how increased? Not from not-being will I permit you to say nor think; for not said nor thought is it that it is not. What need would stir it to grow later or before, having begun from nothing? It is necessary that it altogether is or not. Nor at some time will strong belief say something comes to be from non-being beside it (what is). On this account Justice does not slacken its bonds to permit it to become or pass away, but she holds it fast. The judgment concerning these things is in this: it is or it is not. It has been decided , just as necessity does, to leave the one path nameless and thoughtless (for it is not a true path), and the other path to judge that it is and to decide that it is true. How would being then be destroyed? How would it come to be? For if it came to be, it is-not nor if ever it is about to be. Then coming-to-be is extinguished and destruction unknown. (Fr. 8 Simplicius Phys. 145, I).
One interpretation sees being in temporal terms: it is the ever-present present and not the past or future because only the present exists. Parmenides is talking about what is the is of what is. This is is everywhere and nowhere less or more, and so nowhere is is different:
Nor is it divisible since it is all the same; nor is it in some way more, nor some way less, the thing that denies its holding-together, but all is filled-in with is. Its all is continuous; for being draws itself into is. (Fr. 8, 1. 22, Simplicius Phys. 145, 23).
Seemingly repetitive, Parmenides' reasoning attempts to cover all aspects of the being of existence. It is motionless:
But motionless in the limits of great bonds it is non-begun and ceaseless, since coming-to-be and destruction have been driven off; true belief repelled them. The same in the same thing it lies, remaining in itself, and in this way it stays in its own place – here. Strong necessity holds it in the bonds of limit that holds it on both sides, since it is not right that being be unlimited; for it is not lacking. The non-being-is would lack all. (Fr. 8, 1. 26, Simplicius Phys. 145, 27).
And thought is not merely intertwined with being; thought is being like any other particular is being:
It is the same thing both to think and the thing for the sake of which is the thought. For you will not find thinking without the is in which it is spoken. For there is nothing nor will there be anything along the outside of being, because Fate bound the whole to be un-moving; it will possess all the names, howsoever much mortals established, holding them to be true: to come to be and to perish, to be and not and change of place and exchange of bright color. (Fr. 8, 1. 34, Simplicius Phys. 146, 7)
Being is then a first principle of sorts for Parmenides, but it is a non-spatial and non-temporal one and it is not therefore discernible in the way earlier thinkers believed. One can see that Parmenides in some senses blends being and seeming together, yet he does talk of the structure of the world.
Causation and Cosmology
The above passage (Fr. 8,1.26, Simpicius Phys. 145) is, like almost all of Parmenides' work, variously interpreted and argued. Its end result seems to be that change and alteration along with movement are illusions in which humans are engaged. Some scholars [xi] claim Parmenides rejected Pythagorean dualism of the limit and unlimited only to accept the limit as intelligible being. He is compelled to talk about what being is, but he must do so with sensibles, or in other words with perception and specific terms. He must therefore travel his own path of seeming, though he knows it is fraught with difficulties:
Parmenides having transitioned from things thought to things perceived, he proceeds from truth to opinion, as he says, when he states “I stop trustworthy discourse and truthful thought; learn mortal opinions of this listening to the deceiving order of my words”, he himself makes the origins of coming-to-be the primary opposition, which he calls light and darkness or fire and earth or density and rarefaction or the same and difference, saying after the above lines “for they established a naming of two known forms, of which there is no need to name only one – in which they have gone astray – they separated the bodily opposites and established signs separate from one another, to one aither's fire-flame, gentle and light, in all directions the same as itself, but not the same as the other; and that in itself is the opposite, dark-ignorant night, dense of body and weighty. I tell you the whole likely ordering, that no thought of mortals will surpass you.” (Fr. 8, 1. 50 and Fr. 8,1. 53, Simplicius Phys. 30, 14)
Parmenides rejects the typical dualistic positions of many ancient thinkers: limit and unlimited, change and stasis among other things. At the same time he believed he must take the path of sensibles, and so opposites, in order to elucidate truth. This position reveals an important part of his thought. Parmenides believed in an aspect of reality that lies outside of our perception. He is not attempting to be objective, but he rejects mere opinion and thus a form of subjectivity in favor of finding what is intellective in the universe. Here is an attempt to reach beyond the everyday and common sense into an aspect of reality that is permanent. Such permanence seems for him to lie in another realm accessible only through deduction. In other words, there is a strong mathematical element to Parmenides' thinking. He seems to be a precursor for theoretical mathematics in that he requires reason to create a system of thought where premises lead inevitably to conclusions. His poem is partially axiomatic in that it employs conclusions from earlier arguments as premises in later arguments. What seems like repetition scholars interpret as thorough systematic reasoning, which was perhaps the aspect of Parmenides' thought that so impressed ancient Greek thinkers.[xii] Parmenides, though he does not employ mathematics in his poem, remains faithful to its spirit. His intellective being is rendered by deduction. He does take as his pair of opposites light and darkness, but again one must remember that Parmenides merely writes in the field of opposites in order to express something about how being manifests. He does not believe in opposites in the manner that others believed in them. They are manifestations of being that operate with and for the senses. They become and produce the appearance that things come to be and pass away; being on the one hand and opposites on the other are intertwined. Becoming is the illusion of perception. In one sense then the senses are distrustful and in another manner the senses are being, simply because they are, and Parmenides makes a distinction between predicate being and the being of existence in that predicate being is not perhaps the truest being. The being of existence is true being; the opposites that he employs, light and darkness, are likely metaphors for the revelation of a more fundamental being, manifestation and concealment of the being of existence. Predicate being is perhaps then the way of falsity that misleads while at once it has in itself truth, or being. Theophrastus gives a somewhat incomplete interpretation of Parmenides' view:
Parmenides say[s sensation] is of like by like.... Parmenides did not demarcate it completely but [said] only that, there being two elements, knowledge exists by means of their excess. If the hot or the cold take over, another thought comes to be, but a better and more pure [thought comes about] on account of the hot, not but what even that is in need of a balance; for he says as each man has a mixture in his wandering limbs so does thought come to mankind; for that very thing which thinks is the nature of the limbs for men, it is the same thing for each and all men; for the more is thought. For he says that perception and thought are the same thing; [and he says] memory and forgetfulness come to be from these things on account of their mixture; but if they are made equal by mixture, whether there will be thought or not, and what condition [would exist], he did not establish. But that he makes perception due to an opposite is clear in those passages where he says that a corpse does not perceive light and heat and sound on account of the departure of fire, but [a corpse] does perceive cold and silence and the opposites. [He says also] that all of existence has some knowledge. (Theophrastus de sensu I ff. (DK 28A 46)).
Parmenides' cosmology is composed of those opposites:
Parmenides said there were rings wound round one other, one from the rare, the other from the dense; and that there were others mixed of light and darkness between these. That thing that surrounds them all like a wall is, he says, by nature solid on which is again a fire-ring. The mid-most of all is a solid around which again is a fire-ring. The mid-most of the mixed rings is the origin and cause of movement and of coming-to-be, and he calls it the key-holding goddess who steers all, Justice and Necessity. He says the air is separated off from the earth, vaporized because of the earth's stronger compression; the sun is an out-breathing of fire, and so is the stellar-circle of milk (the Milky Way). He says the moon is mixed of both air and fire and that aither lies at the edges, surrounding all; then there is the fiery thing that we call the sky under which are the things around the earth. (Aetius II, 7, I)
Still, this order remains a world that seems a certain way. It is not as it seems, and the Parmenidean deduction is the only path that will lead one to what actually is, a more fundamental being of existence that underlies what mortals perceive as being, but which is also predicate being on some level. These positions are confusing in that Parmenides denies that opposites are fundamental principles of reality, yet he engages with them. One may interpret the range of opposites as an appearance that is a manifestation of what is fundamentally underlying existence. In that way his system is consistent. The being that is and predicate being make up his cosmology. They are a sphere:
But since a farthest limit exists, it is bound everywhere like the mass of a well-rounded sphere, in every way from the middle equal-balanced. For neither greater nor smaller is there need that it be in some way or other. For neither does not-being exist, which would stop it from coming into similarity, nor is it possible that is be more here and less there than is, since all is intact; for everywhere equal to itself, in equal parts it rests in its limits. (Fr. 8, 1. 42, Simplicius Phys. 146, 15).
This passage is variously interpreted because it seems to talk about the being of existence, true being, as well as predicate being. For some scholars it is metaphorical and for others it ought to be taken literally.[xiii] We recognize that there is an empirical element to Parmenides' thought, but that empirical element he perceives as the way of falsity. It is a necessary falsity because it is the realm in which we exist and it is what makes perception, but he seeks the reality beneath perceptions. The sphere seems to be both in some way, but it is important to note that seeking an underlying reality is consistent with all of science. One thinks of the difference between Newtonian and quantum mechanics as just one example. These systems are the same, but they operate in very different ways.
What we here present is one interpretation of Parmenides' arguments. He writes literature and logic itself is interpretive, so there are myriad interpretations of what Parmenides claims; perhaps that is one of his aims.[xiv] Parmenides' manner of reasoning is for us a kind of closed circuit, resembling axiomatic reasoning that reveals aspects of reality not immediately perceived. Our simple way of articulating his argument does not entirely suffice and so we begin with an explanation of the principle of sufficient reason, which in Anaximander's thought claims if one cannot deduce that something ought to come about at one time rather than another, then there is no reason so argue that it comes to be at any one time at all. There is inherent in Parmenides' thought the premise that either being is or being is not. These are the categories of being and they are coupled with the law of the excluded middle: either B or not B. Everything follows from these assumptions.[xv] Once Parmenides is allowed to make this assertion, he follows its necessary consequences involving space, time,[xvi] unity and other aspects of nature.
On the one hand, one can see that it is necessary to take parts of one syllogism to create another.[xvii] Parmenides' thought is bound up within itself in the sense that one needs to understand one element in order to break into his overall argument; notice that the conclusions of each syllogism are themselves used in other syllogisms. When the premise that being does not come from non-being is accepted, all the rest follows as if in a strong edifice. Here Parmenides creates an axiomatic system that arises from a very simple, and devastating, premise. Once accomplished, entering into Parmenides' thought allows one to understand his whole system. Many scholars see the organization of Parmenides' thought as deductive and in a sense mathematical. We agree to the extent that his somewhat repetitive system coheres from one premise to another, but he is not the only nor the first thinker of the ancient world who crafted a coherent, consistent manner of reasoning. What Parmenides did was to see an aspect of material reality that countered ordinary experience through a reasoning that appears repetitive but is actually in part axiomatic. We will see how Euclid builds a system of geometry based upon earlier and simpler reasoning. Such systems are iron-clad in themselves, and so once one accepts the premises they must accept the conclusions. Such potent reasoning coupled with imagination is integral to scientific endeavor. Here is a major accomplishment the likes of which is a part of mathematical investigation of matter and reality.
Parmenides' argument is that nothing really comes to be or perishes, which seems untrue to ordinary perception. Thus the path of falsity. Yet, if one follows his reasoning and uses some imagination, one is able to see that quite another aspect of matter and being is a part of what we call existence. His first element seems to be being, and not simply being but a specific kind of existence: the being of existence, or existential being, is his first principle. It lies beyond coming to be and passing away, and access to it comes through logos, or reasoning, and not through experience ordinary or otherwise. His logos reveals something unexpected to common experience like Heraclitus' logos reveals something hidden and contradictory to common experience. Reasoning that appears to contradict common experience or reasoning that reveals something hidden to common experience is a fundamental part of scientific endeavor. Theoretical mathematics engages in that kind of reaching into another part of material reality that remains unseen, bizarre even. Parmenides reached not with mathematics but with reason and he is in this sense at least an impressive figure in ancient inquiry and insight. One can also compare his sense of one being and monism with Heraclitus' sense of no loss of energy. Both Parmenides and Heraclitus seem to have noticed that in the interactions of objects there is some manner of constancy, like a closed system. We can think of this constancy as the preservation of energy, as we have seen with Heraclitus. Both thinkers appear to be monists[xviii] as well. Both contributed handsomely to the revelation of some aspect of reality previously unknown. All they did really was reason, and reasoning lead them to previously unacknowledged, imperceptible reality. Thus, imagination coupled with deduction was a potent combination. It remains as powerful today, of course.
[i] Diogenes Laertius, IX, 21-3.
[ii] Speus. fr. 3 Taran ap. D.L. 9.23; cf. Plu. Col. 1126A.
[iii] Strabo 6, p. 252 Cas. (DK 28A12). For a detailed exposition on precisely how Parmenides can derive his first principles, in the form of being and opposites, from Pythagoreanism see Francis Cornford (1922 and 1923). The idea that Parmenides was initially a Pythagorean is an attractive one that seems to be going out of style. We include it here because it has merit, though room does not permit a defense of the position.
[iv] On Nature was likely not the title of the work, but it is how scholars have referred to it.
[v] (adv. Col. 1114B). Plutarch refers to the many scientific topics in the Way of Seeming.
[vi] Fr. I, Sextus adv. math. VII, III and Simplicius de caelo 557, 25.
[vii] These initial lines in Matthew Cosgrove's words “connote an 'illuminated observer'”, 30. They “depict a journey within the world of appearance....” 30.
[viii] Kirk and Raven, The Presocratic Philosophers, 270-2.
[ix] We will see that Aristotle is the first authentic analytic philosopher who categorized not only statements in logic, but also things in the universe. He begins a process that continues in science to this day. The question of being is revisited in the twentieth century by philosophers on the continent. There are now myriad interpretations of being now discussed (predicate, existential, relational, veridical, as a copula of definitional identity are only a few). Patricia Curd takes Parmenides' being as “an informative identity claim, an assertion that, when true, reveals the nature of a thing, saying just what something is,” just as “what we know in knowing what-is is the real or genuine character of a thing.” For Mourelatos The “x” in the Parmenidean “x is F” “ranges over any and all ordinary physical objects, whereas the 'F' ranges over the physis or aleitheia ('nature', 'reality' or 'true identity') of the ordinary object at issue.” 123. S. Wheeler makes a claim for predicate being in that “if x lacks anything to be F, it lacks everything, that is, it is not F.” John Palmer claims that being and not-being for Parmenides are modalities “as ways of being or ways an entity might be rather than as logical patterns” and so thought and being a closely related. Ronald Hoy asserts that time is not real for Parmenides “because temporal becoming requires one to affirm that what is ostensibly real both is and is not.” 379. So, for Hoy the contradiction of being is the crux of the matter. Charles Kahn takes the veridical position that Parmenides is concerned with knowing how being is true more than he is concerned with the being in another sense. This position changes the conversation fundamentally. Kahn's view has the virtue of understanding the simplicity from which complexity arises in Parmenides' thought. S. Tugwell discusses the ambiguous nature of the verb “to be” in ancient Greek. Frank Lewis perhaps over-reads and thus his view becomes anachronistic, but its philosophical content remains of interest. G.E.L. Owen claims that Parmenides asserts the non-being of what is not because of a misplacement of a modal qualifier. Significantly, McKirahan elucidates how the realm of seeming and that of truth operate together in that “mortals believe, on the basis of what their senses report, that things come to be and perish. They come to be at one time and perish at a later time. Before coming to be, a thing is not (it does not exist); after it perishes, it is not (it no longer exists), and in between, it is. So mortals suppose that it both is and is not, just as they suppose that the fig (as it changes color through time) is the same and not the same. Neither case involves contradiction.”, 158. Parmenides' divine logic is responsible for drawing out the contradictions. In the least, as Stephen Makin points out, we are confronted with a thinker whose conclusions confound common sense on the topic of being. For more on the distinctions between predicate being and the being of existence (as well as the epistemological state of being), see Schlagel, Bredlow, Curd, Mourelatos, Wheeler, Palmer, Hoy, Kahn, Owen, McKirahan, Warren (who gives a good overview of the entire poem), Roochnik,
[x] There are scholars who claim that Parmenides does not write of nature or first principles if only because of his monism. Margaret Scharle is representative of this group when she claims that “to investigate whether being is one and motionless is not to investigate concerning nature. For just as the geometer has nothing more to say to one who denies the principles of science...so a man investigating principles cannot argue with one who denies their existence. For there is no longer a principle if there is one thing only, and one in this way. For a principle is of some thing or things.” 330. Other scholars look to reconciling the Way of Seeming with the Way of Truth. John Sisko claims that “the goddess suggests that light and night, far from being substances, are actually different phase-states of some one underlying substance.” 46. We take this view, at least in part. Like Sisko we take it that being is a cosmic arche, or fundament and Parmenides is in some sense a monist; being is in Sisko's terms a “substrate”. Light and night are forms of this fundament but have no independence from it. Reality is the being of what-is. A.H. Coxon thought that the realm of opposites is that of relativity, which has no real existence (one thinks immediately of Heraclitus). Mitchell Miller rightly claims that the “goddess does not object to the dualism as such; rather she objects to the failure of mortals to go beyond it and name a higher, unifying form” 19, namely being. The opposites may be illusory in that they do not bring one to the ultimate reality, but they are perhaps more than mere illusion in that being is somehow responsible for them; in what remains of the poem Parmenides does not explain how. For more, see Sisko, Coxon, Scharle.
[xi] Kirk and Raven, 270.
[xii] Ancient thinkers contemporary with Parmenides also used reasoning that employed earlier conclusions, but not to the extent perhaps that Parmenides did, and he let his premises lead him to whatever conclusions they did, which meant that common sense was reduced to rubble. We are unable to come to a firm conclusion about the matter given what survives, but Parmenides seems to have created a thorough system, moreso than others. Deductive in their own manner and possessing a vigorous system of their own, the Pythagoreans used math in a more imaginative manner, making points into lines and specific amounts of points into solids and specific shapes. Pythagoreans were more geometric in their enterprise.
[xiii] Parmenides is credited for realizing the circuit of the moon and the sun with respect to the earth, a scientific breakthrough. For more, see Mourelatos (2031) and Bertman.
[xiv] For example, McKirahan says of 8.34 that “several renderings...are possible; in each case the different translations reflect two different ways of interpreting the passage. The phrase esti noein can mean (a) “thinking is” (Owen, Sedley); and also (b) “is to be thought” (Mourelatos), “is there to be thought” (KRS), “is for thinking” (Curd), “is to be thought of.”, 203. We have here only one small phrase and the above interpretations direct the argument in quite different ways.
[xv] Lewis claims that Parmenides alters what he claims and that alteration is no longer contradictory and so not an instance of the law of the excluded middle. Wedin's Governing Deduction provides answers to scholars who find Parmenides' thought to be fallacious. Presenting a thorough introduction to the logical niceties of Parmenides' work, Schlagel points out that Parmenides is the first to draw his conclusions not from experience or sensation so much as from a more pure reason, or speculation in language analysis. He sees the emergence of pure space as a necessary outcome of Parmenides' thought, saying that “the logic of the arguments is not unlike Einsteins's who also concluded that the universe has a spherical shape and this is both finite and infinite: finite in the sense that the dimensions are not infinite, but infinite in the sense that the universe is continuous so that one could never come to its limits, there being nothing beyond it to limit it.”, 110. McKirahan takes a healthy perspective when he claims “the interpreter's job is not to aim for formal validity, but to attempt a reconstruction of Parmenides' train of thought, showing how he might have supposed that the conclusion followed from premises he gives.”, 193. Parmenides may have deliberately left out premises because he believed he had already related them. He did not possess the method of later mathematicians, where the argument is formally proven with explicit premises and clear relationships between them. For more on Parmenidean argumentation, see Schlagel, Kahn, Lewis, Wedin, McKirahan,
[xvi] For a contrary view, see Tarán.
[xvii] Parmenides' reasoning is complicated and sophisticated. Lack of space keeps us from evaluating his way of thinking more thoroughly. His reasoning is much more complicated and thorough than it is here represented.
[xviii] There are different kinds of Parmenidean monism articulated by different scholars: material, numerical and predicational. One kind of material composing reality; one thing underlying all things and oneness of each thing in its particularity is permitted to be only one thing. Aristotle claims that Parmenides must mean that “is...means exactly what is and precisely what one is.” (Phys. 186A33-34). The different types of monism sometimes coincide, but it seems necessary to determine the kind of monism in order to understand how the opposites of Parmenides' universe relate to his being. I. Crystal argues that Parmenides asserts a monistic thesis that “entails the strict identification of the epistemic subject and object” and in fact there is great debate on whether Parmenides is talking about being or about thinking-being or merely thinking. Demetris Nicolaides points out that according to modern quantum theory there is instantaneous effects on the universe no matter what distance between things. In other words, because something is measured or because it is somehow affected makes the whole universe change in some manner. Nicolaides demonstrates how modern theory supports a certain kind of Parmenidean monism, stating “the view of Being as an indivisible whole is supported by Einstein's theory of general relativity: for everything that there is, space, time, matter, and energy are no longer independent of each other (that is, they are not absolute), as was the case with Newtonian physics, but are ultimately interwoven, affecting one another constantly.”, 175 and “One of the most fascinating consequences of quantum theory is the phenomenon of quantum entanglement. According to it, there are no perfectly isolated particles (or systems). The notion of an individual particle disconnected from the rest of the universe is inaccurate. Rather, all particles in the universe are part of a unified whole. They are in constant and instant interaction, affecting and determining the behavior of each other regardless of how far apart they are.”, 176. For more on Parmenides' monism, see Graham, Coxon, Curd, Crystal, Mourelatos and Wedin. For more on thinking and being, see Crystal, Curd and Long. For a brief but stimulating overview of some modern implications of quantum theory on the notion of monism, see Nicolaides.
LEUCIPPUS AND DEMOCRITUS
© 2018 Kirk Shellko. All rights reserved.
Life and Works
We know very little about Leucippus, but he did live in the fifth century BC and he may have lived in Elea, Abdera, or Miletus. He may also have been a student of Zeno of Elea. Some scholars suggest that Leucippus may not even have existed. Democritus was born about 460 BC and lived in Abdera or Miletus, though he visited Athens, apparently enjoying his lack of fame there. He wrote a work on euthemia, or “good-spiritedness,” which gave him the moniker of the Laughing Philosopher in the ancient world. The Roman author Lucian also satirized him. Thus Heraclitus was the crying philosopher while Democritus was the laughing one .[i] Democritus allegedly saw life as a mere drift of atoms while Heraclitus saw it as subjective, filled with stupidity, and doomed to perish in change. It is not certain when Democritus died, but he may have lived to advanced old age. His large body of work has regrettably survived only in fragments––accounts given by other thinkers critical of his work that are not altogether reliable—and a group of ethical aphorisms. It is said that Plato wanted all of Democritus’s works burned, though this account[ii] is not reliable. A polymath student of Leucippus, Democritus built a system of thought around the teachings of his mentor. His interests included mathematics, astronomy, meteorology, and even economics as well as other subjects. Two primary works on ancient atomism from Leucippus and Democritus have been found: Great World System and Little World System. Sometimes Great World System is attributed to Leucippus, but no one is certain who wrote it.
Scholars have tended to bundle Democritus and Leucippus together as one, but some differences between them are known.[iii] The ancient Greeks were generally animists with respect to physical matter, and Democritus and Leucippus were no different. Only in modern times has the notion of pure materialism (an exclusive focus on matter rather than metaphysics or religion) emerged.[iv] In fact, modern atomists like John Dalton (best known for proposing the modern theory of atomism) owe a great debt to Leucippus, Democritus, Epicurus, and Lucretius, who thought of atoms as tiny pieces of matter. The ancient theory of the atom as we have it, first proposed by Leucippus, was most probably developed and refined by Democritus with later development and some alterations by Epicurus[v] and the Roman Lucretius. The ancient atomic theory covered here is primarily Democritus's effort, though we cannot say definitively where Leucippus leaves off and Democritus begins. Thus several quotes below refer to Leucippus, but they may just as well be those of Democritus.
Ancient atomists had differing attitudes about the senses, but Democritus seems to have held the most scientific position on them. He believed that the senses captured something quite different from atoms. In other words, things like color, taste, and smell were conventional, arising from within the perceiver rather than from the thing perceived. All things were made only of motions, collisions, and atoms. Some scholars interpret this stance as a sense of objectivity.[vi] While there is some truth to this assertion, the full and proper notion of objectivity was far from the grasp of ancient thinkers. Leucippus and Democritus followed the Eleatic path and believed that being and non-being are not mixed.[vii] They claimed that there were two aspects of material reality ascertainable to a certain degree with the senses: fullness and void. Fullness and void make the objects that we encounter in the world, as Aristotle confirms:
Having agreed that these things pertain to appearances, [Leucippus] also said to those who construct an argument for the one that movement would not exist without void, that void is nonbeing and that no non-being is being. For the thing existing properly is altogether full (of being). But to be this sort of thing is not to be one; rather, the great multitude of them are boundless and not visible on account of the smallness of their size. These things are born along in the void (for void exists); on the one hand, bringing themselves together, they produce a genesis; on the other, being loosened, they fall to destruction. (Aristotle, De Gen. et Corr. A8, 325a2)
Aristotle states further,
Democritus . . . calls space by these names: “the void,” “nothing,” and “the boundless”; and he calls each of the existing things “hing” [“nothing” without “not”], the “close-pressed,” and “being.” He believes these beings to be so small as to flee our senses, and he believes there are all sorts of forms and shapes and differences in magnitude of them that exist. From these things already, just as from elements, come forth compoundings, sizes perceived by the eyes and the senses. (Aristotle, On Democritus ap. Simplicium, De Caelo 295, I)
Being does not change into non-being, and so what exists cannot change because it would then alter into what is not. As a result, there is a lack of being in objects that is not actually the object itself, but rather is part of something only in that an object has non-being, or what we may call empty space, as a part of it. Atoms are literally “uncut” tiny bits of material reality––being––that compose the objects and things in the universe. Change and becoming in the universe are the breaking up of these bits and their rearrangement.[viii] There is much scholarly discussion on the precise nature of fullness, void, and the general composition of atoms in ancient atomism.[ix] Likely, Democritus meant “nothing” when he wrote “nothing.”
Causation and Cosmology
Void, according to Democritus, was that lack-into-which shape formed and bodies moved. For him it had to be nothing. Nothing else had the ability to make shape and movement possible. The shape, arrangement, and position of atoms are thus essential to the characteristics and therefore actions of objects in the universe. Once again Aristotle:
These men [Leucippus and Democritus] . . . say that the differences [between atoms] are the causes of other things. They say that these [differences] are three: shape, placement, and position. They say being differs by means of rhythm, by touch, and by turn. Of these, rhythm is shape, touch is placement, and turn is position; for A differs from N in shape, AN differs from NA in placement, and Z differs from N in position. (Aristotle, Met. A4, 985 b4)
Atoms are composed of a single substance with one nature to them. Each of the individual atoms are “one.” Leucippus seems to claim that the atom's indivisibility is due to its smallness. But for Democritus atoms fit into one another by means of shape and no matter how closely packed they are there is always some space between them. Things are composed by mere place and conjunction of atoms as well as conflict among them.[x] It is important to note that for Leucippus compounds in the modern sense––where the characteristics of the atoms themselves change–– do not truly form. The qualities of each atom retain their individuality for Leucippus and no characteristics are altered in the compounding. Democritus claims that there is a tendency of similar things to gather together. As animals gather in herds and as people gather in communities, so also do inanimate objects come together. A fragment of atomist thought survives in Sextus's commentary:
For creatures (he says) flock together with creatures similar in kind, doves with doves, cranes with cranes, and just the same with the others. And the same thing occurs with inanimate things, as can be seen with seeds put through a sieve and pebbles on the wave-breaking beach. (Fr. 164, Sextus, Adv. Math. VII, 117)
Such a gathering of similar objects is not the same as the ancient notion of like attracting like necessarily, but rather it is a tendency. It seems that Leucippus believed that all atoms are uniformly small, and so he might not have believed that shape and size were central to the properties that atoms exhibit. Democritus seems to have believed that atoms have different sizes and shapes, and those characteristics produce properties that we see in the conglomerations of atoms when we perceive objects. Apparently there was even more disagreement between the two men.[xi] Democritus claimed that atoms have mathematical rather than physical parts. Leucippus believed that all atoms were small, while Democritus believed they varied in size. For Leucippus the motion of atoms was always in all directions and they moved of their own nature.[xii] Still, the primary qualities of atoms for both men remain size and shape, and these qualities are their most significant differences. They agreed that atoms move in some manner, moving within themselves while seemingly at rest or moving as the things that they comprise.
Whatever the constitution and behavior of atoms, these uncut bits are infinite in number and are “being” in the Eleatic manner, which is to say that they do not change. Thus, they were uncut and eternal. According to the ancient atomists change and rearrangement are due to the void in the conglomerations of atoms packed together in objects. Again, Simplicius describes their position:
[Leucippus, Democritus, Epicurus] said that the beginning principles were limitless in number, which they thought to be indivisible and unaffected atoms on account of their close-pressed-ness, and they thought [them] to have no share of void; for they said that division comes about according to the void, the one in the bodies. (Simplicius, De Caelo 242)
Void and being exist equally, giving both fullness and emptiness to the universe. They interact in an equally powerful manner.[xiii] Fullness and void interact so as to form the earth and the apparent dome over it. The order of the universe is the infinite interactions of these two forces. They are responsible for creating the shape of the universe by attraction and repulsion from a whirl, which gathers together the heavy pieces of the universe and spreads out the thinner elements, “atoms,” into the void. We have some description from Diogenes Laertius:
Leucippus says that the all is infinite. . . . One section of this is full and another, different, section is void. . . . From this [combination] exist countless world orders, and into these things they dis-integrate. He says the world orders come to be in this manner: from a cutting away out of the infinite many bodies with all sorts of shapes are carried into a great void; the very things that are joined together complete one eddy, along which—striking against one another and in all sorts of ways circling—they separate apart, like to like. But on account of the multiplicity of them being no longer able to be carried around in an equal position, the fine ones separate into the outside void just like things sifted; the rest remain in conjunction and, weaving about themselves, run down together, and make a first sphere-formed conglomerate. (Diogenes Laertius, IX, 31)
The atoms that cling to one another owing to their shape and perhaps their size[xiv] conglomerate into the earth and its objects, he continues:
This is a kind of membrane that stands apart, holding around in itself all kinds of bodies; and the membrane of those bodies rotating around down along the resistance of the middle becomes thin, the bodies running together on account of the contact with the eddy. In this manner the earth came to be, the bodies remaining together after having been held in upon the middle. (Diogenes Laertius, IX, 32)
The fundamental forces that create and sustain objects are the same forces that shaped the entire universe. Still, different worlds arise through constant motion. These worlds are not only infinite, but also sometimes partial and sometimes complete, always in transition, as the ancient commentator Hippolytus writes:
Democritus speaks similarly to Leucippus about elements, full and void. . . . He said that beings themselves always become moved in the void; and the world orders are infinite and they differ in magnitude. He says in some there is neither sun nor moon, and in others they are bigger than the ones nearby us, and in yet others they are more numerous. (Hippolytus, Ref. I, 13, 2)
The motion and activity of the universe is different in Democritus's and Leucippus's conception, but the basic forces of attraction demonstrate similarities. The explanation for all of these happenstances is necessity, which is the same as the whirl.[xv] The whirl is necessary by means of determined and mechanical conglomerations and collisions of atoms by means of its necessary collisions and binding-together of atoms; these are mechanical and determinable.[xvi] The motion of atoms therefore plays a fundamental role in the formation of objects in this materialistic universe, although the composition of the world is not entirely mechanistic.[xvii] The ancient atomists emphasized that the chain of collisions along with the shape and size of atoms are responsible for everything that emerges in the universe. Nothing for Democritus and Leucippus occurs in vain because necessity or the eddy makes something out of the combinations; gravity is a primary force in the composition and dissolution of things because gravity is also in part necessity. Theirs is, however, not a teleological universe. At the same time that nothing transpires in vain, there are no fundamental essences of a deeper reality that comprise the universe. In other words, the combinations that make things are accidental; they are inadvertent actions of minuscule being, necessitated by their natures and gravity.
Weight for the ancient atomists is the tendency of objects to press down and it is the result of being that an object has weight, which means naturally that lighter objects have less being in them. Void is thus responsible for lightness, writes Theophrastus:
Democritus distinguishes heavy and light by magnitude. . . . Yet in things mixed together the lighter is the thing containing more void; the heavier is the thing containing less. (Theophrastus, De Sensu 61 [DK 68 A47])
Weight is here the tendency of an object to press down in the world by means of the eddy. Otherwise, objects may have no inherent heaviness to them and thus do not press down.[xviii] Movement among and around the atoms results from dissimilarities among them, which speaks to the gathering of similarities mentioned earlier.
[Atoms] clash and are carried in the void because of the dissimilarity and the other differences that have been said, and being carried they fall upon and are entwined round one another. . . . (Aristotle On Democritus ap Simplicium De Caelo 295). Democritus says, “Always the first bodies are moved.” (Aristotle, De Caelo, G. 2. 300b: cf. Hipp., Ref. I. 1; D.A.40: al).
Atoms collide with one another constantly and in that mass of tiny corpuscles is a “vibration,” which is the movement of the thing itself, as Bailey noted. So the very existence of atoms and void carried with them atomic motion and the first motion comes about from “necessity” as an element of the constitution of things, yet forced motion and motion from “blows” are both derived from it.[xix] In other words, the gathering of atoms in similarities and the cohesion of atoms seem to arise from the weight, size, and shape of atoms along with a tendency of these bits of material reality to flock together.[xx] There seem to be two attracting forces: one attracts like atoms to like atoms and the other binds them together. How precisely these attractions occur is not clear; modern science would call them forces, electromagnetic or otherwise. Thus arise objects in the natural world, as Simplicius states:
Carrying themselves and being carried [the atoms] fall upon and are entwined round one another in an entanglement that makes them touch one another and puts them near to one another, and that truly engenders not any sort of single nature out of them whatsoever; for it is altogether simple minded that two or more would come to be at some time one. [Democritus] claims that up to a certain point the interchanges and the exchanges of bodies cause these beings to remain together, for some of them are uneven and others are hooked, still others are hollow, while others are curved, possessing innumerable other differences. He believes them to cling to one another in turn and remain together up to such time when some more powerful necessity, coming to be from that which surrounds, thoroughly shakes and breaks them apart. (Aristotle, On Democritus ap. Simplicius, De Caelo 295, II)
Atoms seem to conglomerate into objects more by congruence and less by similarity of kind. Perhaps their combination and dissolution result from randomness,[xxi] while atoms themselves in their continued sameness preserve a kind of necessity. Still, the compounding of Democritus's and Leucippus's atoms into new objects and living things is debated.
The chemical molecule that results from the combining of atoms may have been possible in the atomists' universe, but the issue is far from clear. On the one hand, atoms may have cohered because of their shapes or perhaps their sizes, creating temporary new substances, as we have seen. On the other hand, even considering all the ancient atomist theories, it is hard to imagine how a molecule could arise if atoms must lose certain characteristics for the new substance to come to be.[xxii] It is certain that the atomists believed the atoms themselves to be unchangeable, which would prevent such chemical processes from taking place.[xxiii] The theory of Democritus and Leucippus may be interpreted as a play of forces that interact with one another. Modern science supposes something similar: that there are forces operating in the universe that comprise all objects (the strong, weak, electromagnetic, and gravitational). Understanding these forces is understanding the universe.[xxiv] Something similar is true of ancient atomists who thought of the void as a driving force in conjunction with the shape and size of atoms. No matter the particulars, ancient atomists thought in terms of powerful active pieces of the universe that drive into and compose things from their conglomerations. All of these notions arise from imagination, which Democritus and Leucippus seem to have possessed in abundance.
Democritus and Leucippus do not speculate on or categorize the precise shapes of different atoms except to say that mind and fire are sphere shaped[xxv] and that other elements are comprised of conglomerations of differing size and magnitude. As Aristotle writes,
What sort and what shape of each of the elements Leucippus and Democritus did not demarcate, but only to fire did they assign the sphere; air and water and the others they differentiated by largeness and smallness, as if their nature was a mixture of all kinds of seeds of all elements. (Aristotle, De Caelo G 4, 303a 12)
Atoms, it seems, must be differently shaped in order to conglomerate, yet in some instances—like fire and mind—they must conglomerate through the concept of like to like, which seems antithetical to the original conception of atoms as forming objects by differences of shapes. In other words, atoms need to be compatible, which implies that they have different shapes, yet they must have a similar shape in certain substances, like fire and mind, in order for them to function. Fire and mind are the most penetrative of substances because of their consistency of shape, a conception inconsistent with the idea of compatibility through different shapes. If Leucippus claimed that all atoms are the same size (small), then the idea of size producing the activity and function of atoms may have been a development of Democritus. It is difficult to determine.
One may ask whence came the first motion of atoms, but surviving material gives no definitive explanation about an initial motion, only the continued motion of the atoms themselves or the motions of conglomerations of atoms. Perhaps it was believed that motion simply always existed, but given the difficulty of determining the origin of motion, it is plain to see why an answer would not be forthcoming. Indeed, science has no sufficient answer for that question today, except perhaps to suggest that at some time there was an initial explosion, or bang, and even that is now doubted. Nevertheless, we have seen that motion in the void is suggested by the dissimilarities among atoms and by gravity. If atoms always existed and dissimilarities cause movement, then motion would be inevitable; and if gravity is somehow an aspect of a physical thing, then another sort of motion exists because of the conglomerations that comprise things.[xxvi] It seems safe to say that the ordinary movements that comprise objects arise from the above-mentioned collisions and “rebounds,” whether motion is of one kind and origin or another.
It remains for us to examine a few arguments central to the atomist theory in order to make clear some essential parts of their doctrine. Recall that these are not necessarily the precise arguments that Democritus or Leucippus formed, but they follow the same way of reasoning we saw first in Greek literature; they seek patterns and then seek the relationships between patterns discovered. The patterns are the empirical observations that a thinker perceives as regular occurrences. These observations are compared and an inference results. It may have been that Leucippus and Democritus reasoned with constants and in a much more systematic way, and so their thought would have been much more similar to that of a modern scientist,[xxvii] but we do not know. The arguments assessed here are based on some of the evidence we possess, in an attempt to understand some of the more important aspects of ancient atomic theory. The most fundamental aspects of the atomists’ argument come from their conception of being and non-being. In some places we must assume elements of the argument, and by necessity some elements must be left out.
“Things that have being” are here the atoms themselves, not the more complex conglomerations making up objects. The objects we perceive and use have void in them. Otherwise, they would not be able to change and no motion would be possible. So these things that have being must not be in the field of change. This argument assumes that void exists and that void is nonbeing.
Naturally, there is no middle ground here. Atomists seem to believe that either a complete being—fullness—exists and there is no lack to being at all, or an absolute lack exists. The absolute fullness is their acceptance of Parmenides's stance. Still, we are able to interpret this atomist argument as valid. It is not sound, but it makes some proper connections. Coming-to-be results from the movement of being into that part of material reality where absolute lack exists. There is no conception of void as an actual existing thing or a partial lack—some existing thing or space into which another existing thing can be placed—yet void “exists” no less than being. Movement comes from being, fullness, sliding into the void that is a fundamental force in the universe equal in realness to fullness.
The above two syllogisms are a part of the atomist assertion that void and absolute fullness exist. They appear to be based on common sense observation and experience in the world—Democritus cannot have analyzed the changes and consistencies of the universe on the microscopic level, obviously. Here is where he naturally engages in speculation and an observational form of induction, but Democritus also has found patterns through observation and linked them to other patterns: things with absolute fullness (being) and void-spaces (empty spaces where things may stand). His argument for the interchange of void and fullness explains being and becoming in a materialistic manner:
Changes come from the void space and the lack that void makes possible by its nature:
The changes made possible from void-space are then motions. The observations taken as universals make the argument for motion as a fundamental part of change and seeming creation. The things not subject to change are the elements of the universe that have an Eleatic aspect to them. The atomists get around the problem of the illusion of change, found in Parmenides's thought, by claiming there is an aspect of material reality that does not change and that has as part of its nature continued existence. These bits of reality are rearranged and instead of comprehending all change as an illusion, as does Parmenides, atomists claim that change happens on the predication of changeless elements that are too small to see. The atomist arguments are valid, yet the arguments themselves are based on unproven and as-yet unaccepted premises, like that there is such a thing as an absolute fullness in the field of material reality. Thus, one flaw of the above arguments is that they, like other ancient arguments we have seen, take these premises as universal and already apparent; they are accepted patterns in need of argument or proof. Atomists like Leucippus and Democritus seem to craft arguments with an empirical dimension to them, all the while accepting as part of their premises empirically unproven elements. There is for them an absolute fullness to material reality, which explains why you cannot pass your hand through a leaden ball. The leaden ball does also have in it void, which makes the ball able to be cut or broken into parts. We think immediately of the modern notion of matter being made up of mostly empty space, but of course modern empty space is not the ancient atomists' void. The ancient void is absolute nothing, while science is finding today that “empty” space is not as lacking in activity as once thought.
The conglomerations of atoms and their separation are made possible by specific aspects of atoms' size and shape, as in the following argument:
It seems that Leucippus and Democritus believed that atoms have a specific character, and that character accounts for the integrity of a given object. When a “necessity” or a movement that results in a collision, or some other material force, interacts with them, their present constitution changes. The interactions of these tiny bits of fixed material reality could be called coming-to-substance or dissolutions, though the only real beings are the atoms themselves. Complex structures made from the atoms are only beings in that they are composed of tiny beings:
Again, movement into void is primary for things in the world, cohesion and change. As can be expected, Leucippus and Democritus seem to have made little attempt to differentiate the various atoms in the material universe, but rather they seem to have imagined how only some of them may cling to one another or possibly repel one another. If we accept their premises, which are based on a great deal of speculation, then their arguments are reasonable, but if we question one of these unsubstantiated and very difficult to prove assertions, then their system falls apart quickly. Atoms are not eternal and unchanging; change does not come about through mere rearrangement of void-spaces and completely filled beings. The ancient atomists simply had no way to prove, or even to assert with any degree of certainty, what they wished to assert and prove, yet they perceived repeated manifestations of things, patterns, that they explained with speculation. They saw that some objects are hard and cannot be penetrated, except by “blows”. They reasoned that there must be holes or “void” where things may separate, which is a not terrible explanation–given their ability to observe. Their theory does assert something true about material reality. Atoms do exist, of course, but not in the way they believed. Objects are filled with mostly empty space, but there is no absolute void in them that makes space possible. The atomists discovered truths still held to be true in the modern world, but these truths were not precisely as they imagined them. The observations and the repeatable interactions of substances in the universe were explainable by means of the atomists' doctrine, but only because the ancient Greeks did not have the tools and methods to disprove them. They possessed falsifiable theories that they were unable to falsify. Again, they speculated. Their first principles of everything were ultimately provable necessary pieces of reality: uncuttable pieces of matter that were innumerable and perhaps quite varied in size and shape. Perhaps there were empirical proofs of these arguments. We possess the thoughts of the atomists only indirectly, so their arguments may have been much more nuanced, or different. They made use of earlier insights into material reality and explained how things operate by advancing a radical idea, and their idea was proven correct, in a qualified sense, several centuries later.
© 2018 Kirk Shellko. All rights reserved.
[i] See Cora E. Lutz, “Democritus and Heraclitus,” Classical Journal 49, no. 7 (1954): 309-314.
[ii] Diogenes Laertius, IX 40.
[iii] For more on the disagreements between the two men, see Cyril Bailey, The Greek Atomists and Epicurus: A Study (Oxford: Clarendon, 1928). For the dates of Democritus and a discussion of the debated existence of Leucippus, see Herman de Ley, “Democritus and Leucippus: Two Notes on Ancient Atomism,” L'Antiquité Classique 37, no. 2 (1968): 620–633.
[iv] For a greater sense of the debt of modern atomists to Democritus and Leucippus, see Joshua C. Gregory, “Dalton's Debt to Democritus,” Science Progress in the Twentieth Century (1919–1933) 14, no. 55 (1920): 479–486.
[v] This book lacks space for Epicurus, but one section of his work will give an idea of his perspective. In his Letter to Herodotus (DLX 39–40) Epicurus claims, “Everything is body and empty. For how bodies exist, perception itself bears witness to all (of them), according to which it is necessary to show what is unclear to reason by signs. If there was not a thing which we call empty and place and impalpable nature, there would be no place for bodies to exist, nor anywhere for them to move themselves just as bodies appear to be moving things.” For more on Epicurus, see Saul Fisher, Pierre Gassendi's Philosophy and Science: Atomism for Empiricists (Boston: Brill, 2005).
[vi] For some discussion on objectivity and sense perception in the atomists' system, see Demetris Nicolaides, In the Light of Science: Our Ancient Quest for Knowledge and the Measure of Modern Physics (New York: Prometheus, 2014); Bailey, The Greek Atomists and Epicuris; and S. Sambursky, The Physical World of the Greeks (New York: Routledge, 1956).
[vii] Kirk and Raven suggest that the full being of materialism in the atomist universe is a response to Parmenides and his fullness of being. G. S. Kirk and J. E. Raven, The Presocratic Philosophers (Cambridge: Cambridge University Press, 1979) 408. Alfred Lloyd claims that the atomists needed paradox: plenum (fullness of being) and void must be immanent in all objects. Plenum and void involve paradox because they must be immanent. “A Study in the Logic of the Early Greek Philosophy: Pluralism: Empedocles and Democritus,” Philosophical Review 10, no. 3 (1901): 261–270. David Konstan claims that Democritus believed that atoms are being with no nothing in them; the void is nonbeing with nothing in it. “Democritus the Physicist,” Apeiron: A Journal for Ancient Philosophy and Science 33, no. 2 (2000): 125–144. For a sense of how the notion of fullness and void limited the atomists, see J. R. Milton, “The Limitations of Ancient Atomism,” in Science and Mathematics in Ancient Greek Culture, ed. C. J. Tuplin and T. E. Hill. (New York: Oxford University Press, 2002).
[viii] For an in-depth summation of and one perspective on the debate on mathematical divisibility or indivisibility of the Democritean atom, see Richard W. Baldes, “‘Divisibility’ and ‘Division’ in Democritus,” Apeiron: A Journal for Ancient Philosophy and Science 12, no. 1 (1978): 1–12.
[ix] Werner Heisenberg writes that the void “was the carrier for geometry and kinematics, making possible the various arrangements and movements of atoms.” Physics and Philosophy: The Revolution in Modern Science (New York: Harper & Row, 1962). Demetris Nicolaides argues that Democritus's notion of void was not really nothing because it was responsible in part for movement and geometry; it made possible different arrangements and motion. In the Light of Science, 89. Bernard Pullman asserts that clustering of atoms “is promoted by the diverse shapes atoms can have—polished, rough, pointed, hooked, twisted, bent.” The Atom in the History of Human Thought (New York: Oxford, 1998), 33. Thomas Cole points out that the ordering of atoms for Democritus tended to become larger, thus making perceptible things. He claimed that atoms are a multiplicity that is one. They are infinite in number and “one” in two senses. Democritus and the Sources of Greek Anthropology (Chapel Hill, NC: Press of Western Reserve University, 1967),107. Cyril Bailey claims that “the atoms continue to perform tiny trajects, greater or less according to the texture of the compound, colliding with one another in infinitesimal periods of time, and recoiling again to another collision: every compound body, every 'thing' that we perceive by the senses is in a constant state of internal atomic vibration.” Atoms do form compounds because “into the tiny intervals of empty space in the compound new atoms will enter in their flight from outside. Occasionally their 'shape or position or order' will fit them to become entangled in their turn and increase the bulk of the compound: as long as this happens the thing grows.” The Greek Atomists and Epicuris, 88–89.
[x] See Cole, Democritus and the Sources of Greek Anthropology.
[xi] For more on the properties that atoms manifest when they are combined with other atoms, see Vijay Tankha, Ancient Greek Philosophy, 2nd ed. (Delhi, India: Pearson Education India, 2014).
[xii] See Bailey, The Greek Atomists and Epicurus, 78.
[xiii] Some scholars claim that Democritus did not necessarily take void and fullness to be equal in potency. For a discussion on a fragment relating to the problem, see W. I. Matson, “Democritus, Fragment 156,” Classical Quarterly 13, no. 1 (1963): 26–29.
[xiv] For a discussion about why atoms cohere and their shape, see Sambursky, Physical World of the Greeks. Milton correctly points out that “the atoms postulated by Democritus and Epicurus were assigned their properties on the quite different ground of analogical extrapolation from macroscopic bodies, regulated by metaphysical debate.” “The Limitations of Ancient Atomism,” 186. The ancients were compelled to make analogies and to speculate. We ought not to fault them for their position in history, but rather we ought to admire their imagination and insight.
[xv] Diog. Laertius, IX, 45.
[xvi] Kirk and Raven, The Presocratic Philosophers, 412.
[xvii] For the universe as a living thing, see Milton, “The Limitations of Ancient Atomism.” For mechanistic aspects of ancient atomism, see Raymond Godfrey, “Democritus and the Impossibility of Collision,” Philosophy 65, no. 252 (1990): 212–217; and David Kline and Carl A. Matheson, “The Logical Impossibility of Collision,” Philosophy 62, no. 242 (1987): 509–515.
[xviii] See John Burnet, Early Greek Philosophy, 4th ed. (London: A. and C. Black, 1930).
[xix] Bailey, The Greek Atomists and Epicurus, 133–136.
[xx] David Konstan talks about ancient atoms adhering to one another without amalgamating. See “Democritus the Physicist.” Taylor talks about the attraction of like to like as in the animal analogy above. C. C. W. Taylor, The Atomists: Leucippus and Democritus (Toronto: University of Toronto Press, 1999).
[xxi] See J. F. Duvernoy, L'Epicurisme et sa tradition antique (Paris: Bordas, 1990).
[xxii] Pullman writes that “the elementary corpuscles of matter are indivisible. This property is due . . . to their 'impassivity' (hardness, incompressibility). They are compact and full, without parts, of homogeneous composition, and exhibit no qualitative difference.” The Atom in the History of Human Thought, 32.
[xxiii] For more on the debate about compounds and molecules, see Pullman, The Atom in the History of Human Thought; and Benjamin Farrington, Greek Science: Its Meaning for Us (Nottingham: Spokesman, 1980).
[xxiv] For more on the similarities between ancient and modern atomism, see Nicolaides, In the Light of Science, and Giorgio de Santillana, The Origins of Scientific Thought: From Anaximander to Proclus, 600 B.C. to 300 A.D. (Chicago: University of Chicago Press, 1961).
[xxv] Democritus says that the most easily mobile of shapes is the sphere, and these kinds are both mind and fire. Aristotle De An. A2, 405a II.
[xxvi] Kirk and Raven suggest that “irregular atoms are in a state of disequilibrium in the void, and so undergo movement.” Presocratic Philosophers, 417.
[xxvii] Archimedes is said to have given Democritus a great deal of credit for first claiming that the cone is a third part of the cylinder, and the pyramid a third part of the prism, which have the same base and an equal height. Yet Archimedes claims that Democritus accomplished this feat without providing a proof. So Democritus may have learned some things from the Egyptians when he visited there, but he seems to have not reasoned deductively, that is to say, mathematically. Several of Democritus’s works may have demonstrated his mathematical ability, including On a Difference in an Angle; On Contact with the Circle or the Sphere; Geometrica; and Numbers, Irrational Lines and Solids. See Diogenes Laertius 9.47–48. We simply do not possess these works.
LEUCIPPUS AND DEMOCRITUS BIBLIOGRAPHY:
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Life and Works
Plato was an aristocratic Athenian philosopher and student of Socrates and of Cratylus who lived circa 427-347 B.C. His mammoth contribution to western culture and thought cannot be overestimated. Responsible for foundation questions about metaphysics, ontology, politics, practical wisdom and other issues, Plato demonstrates a depth of interest and knowledge rarely seen in philosophers and thinkers in general. He wrote dialogues, which largely are narrative discussions between his philosophical hero Socrates and Athenian aristocrats, many times young men. These dialogues are part of his method, which is fundamentally a continued questioning without the expectation of an absolute answer. Plato is one of the only thinkers of the ancient world whose work survives practically extant, so great was his reputation and influence. He was not what we may call a physicist, and so his main preoccupation was not the material composition of the universe.[i] He did, however, write a dialogue specifically addressing the issue of the genesis of the universe and its “likely” physical composition. It is necessary to review Plato's conception of the origin and composition of material reality in his Timaeus, but of great concern for scientific inquiry is his method of questioning, refuting and answering – his elenchus – which permeates his thinking and is found in his dialogues. In Timaeus is located his account of the universe's genesis [ii] and in Meno one finds a way of reasoning whose foundation remains an integral part of scientific inquiry.
Timaeus reveals what Plato considered to be “probably the case” of the composition of material reality. Yet, Plato disdained matter and material explanations for concepts, things and objects.[iii] He believed in a deeper aspect of reality than matter because matter comes to be, brings itself to fruition and falls away. It is thus unstable. Things that come to be are for Plato unreliable examples of what things really are. He directed his methods at understanding a metaphysical aspect of reality that infused becoming with being, which is to say that the most fundamental aspects of reality are not the things we see, touch, hear etc., but rather they are the things that exist always and have no movement; they are being. This being we cannot directly sense, but the mind connects with being through reason:
The one thing truly grasped by the mind with reason always exists in the same way, and another thing again [grasped by] belief with unreasoning perception is a matter of opinion, always coming to be and passing away but never truly existing. (Plato, Timaeus 28a)[iv]
In the majority of Plato's dialogues there are at least two interlocutors who talk to one another, engaging a topic with questions and answering, but Plato's Timaeus is largely a monologue of a single character explaining how the universe arose and what is its material and metaphysical composition. The awareness that it is a monologue is important in that Plato likely treated the topic of the material composition of the universe with a bit more than a little distaste, and so there is no dialogue [v] about physics. Timaeus explains gargantuan topics with almost no questions and insufficient description. Again, Plato's interest resided in what was for him more real, a reality accessible to the mind and a fundamental but almost hidden part of material becoming.[vi] Still, Plato discussed material processes, but with respect to his metaphysics of being and becoming. The universe for Plato is made from earth and fire because the characteristics of those two elements seem to enable sensation and thus perception:
...the seen and the touched things that come to be must be in body-form, but nothing would ever come to be seen apart from fire, nor touched without some solid, nor solid without earth. Setting out to organize from fire and earth did the god make all bodies. (Plato Timaeus 31b).
Earth and Fire make up parts of the universe, but the universe itself must be bound together, and its bind resides for Plato in proportion, a naturally mathematical relationship:
...it isn't possible to combine two things well by themselves without a third; some unifying bond is needed in the middle of both. The finest of bonds is the one that makes the things bound together and itself one as much as possible, and naturally mathematical proportion accomplishes this in the best way. For whenever of three numbers that either have mass or (mathematical) powers the middle term between any two of them is such that what the first term is to it, it is to the last, and, in reverse, what the last term is to the middle, it is to the first, then, since the middle term comes to be both first and last, and the last and the first again both turn out to be middle terms, all things in this way from necessity will happen to be the same things, and all the same things shall have come to be one in relation to one other. (Plato Timaeus 31c-32a).
We must remember that for Plato, like for Euclid later, the relationship that exists between numbers and spatial extension has a greater significance than mere measurement, which is how moderns understand that relationship. How things have cohesion is that part of objects and creatures that bonds them; it is mathematical and because spatial extension and number are intimately related – which is to say that proportion makes things one and thus unified – proportion is a genuine bond in material things. Another way to understand what is happening in Timaeus is to see that being is the part of reality, for Plato, that does not change. It is static and it is the foundation for the things that are not static, things that change. Being is un-cuttable and, as eidoi (Forms), provides a basis for things that come-to-be. A mathematical equation is a truth that will exist no matter if there is a manifestation of it or not. A circle will always have as part of its measurement π (ratio of a circle's diameter to its circumference) or πr² (area of a circle). There is something in the mathematical equation that goes beyond mere happenstance, and its foundation is being (Forms) that is the permanent aspect of the universe. So, a mathematical equation that demonstrates a bond between two numbers really is bond itself, a more permanent bond than physical things engage. Such being makes number generative in that the bond between elements makes things. Number is more real for Plato than things that become and so proportion creates a most significant bond. One example is that cubes, or a number multiplied by itself three times, are considered to be solids. So, Plato's elements themselves are comprised of shapes that themselves have proportions because this proportion is needed for the bond:
Some things have come to be bound up with it itself and are solid-formed, but one term never fits together solids, but two middle terms do. In this manner the god having set water and air between fire and earth [and] having worked out as much as possible the same proportion for each to each – so that what fire is to air, air is to water, and what air is to water, water is to earth – he bound together and organized a sky both visual and tangible. (Plato, Timaeus, 32b).
The elements themselves have a mathematical relationship to one another that is similar to the proportion just described. Additionally, material reality is for Plato atomistic. One says “atomistic” because atomists, in addition to believing in uncut and uncuttable bits of reality, believed in a void that Plato did not and Plato's atoms have separate parts. In fact, a significant part of the atomist doctrine was matter combined with void in different ways, but Plato believed that matter is composed of tiny, imperceptible bits of physical substance:
It is necessary to think of these things as so small, so that, due to their small size, each of each kind individually is not visible to us. When, however, many of them are assembled together, the mass of them are seen. (Plato, Timaeus, 56b-c).
Gaps and density make chemical processes possible:
Neither fire nor air will melt masses of earth, because naturally they [fire and air] are things of smaller nature than the composition of the gaps in the earth. They pass through the wide gaps of earth, not being constrained, leaving it continuous and unmelted. But since the parts of water are greater by nature, forcibly making an outlet for themselves, and loosening it [earth] they melt the earth. For water alone in this way loosens earth not compressed by force, but nothing but fire dissolves compressed earth, because an inside-path is left behind by nothing except fire. And fire alone disperses the most forcibly concentrated water, yet both fire and air scatter a looser state of water. Air enters the gaps, and fire breaks up the triangles. Nothing loosens air condensed by force except along [its] elements, and fire alone melts down [water] not condensed by force. (Plato, Timaeus, 60e-61a).
The logic of Plato's chemical process seems to be that of gaps between minuscule bits of matter functioning together with the size and shape of the elements. When there exist gaps through which elements can pass, there is a chemical mixture, or perhaps we may call it a chemical reaction. These tiny bits of matter interact through these gaps, yet there is no void, or complete lack of being, anywhere in the Platonic universe; only Parmenidean being exists. These bits of material, mathematical reality that Plato did accept themselves were composed of aspects of reality more fundamental than becoming. Plato accepted the four elements of Empedocles – earth, air, fire and water – but only as atomistic geometric shapes.[vii] These shapes are themselves elements too small to perceive with the senses, but in larger groupings they become visible: Fire elements took the shape of the tetrahedron. Earth elements took the shape of the cube. Air elements took the shape of the octahedron. Water elements took the shape of the icosahedron. The overall universe was shaped like a dodecahedron:
To earth let us [designate] cube because earth is the least moving and most moldable of the four kinds of bodies, and it is most necessary that this sort of thing has come about having the most steadfast faces. The face of the initially postulated triangles that belongs to the ones having equal sides is more steadfast than the one that belongs to triangles having unequal sides, and the surface composed of the two triangles, the equilateral quadrangle [the square], holds its position with greater stability than does the equilateral triangle, both in their parts and on the whole. ...and of the solid figures that are left, we designate the form most difficult to move to water, to fire the easiest [form] to move, and the middle [form] to air. The smallest body is fire's, the largest water's, and the middle one air's; the sharpest body is fire, the second sharpest is air, and the third sharpest water. (Plato, Timaeus, 55d-56a).
The atomistic and geometric shapes were used up in the construction of the universe:
The construction of the universe used up each of the four [elements], for from fire and all of water and air and earth the one who set it up put it together, having left behind no part of anything nor any power. He intended these things: first, that it be a living whole as much as possible and complete from completed parts; next, it should be one, so as nothing be left over from which another one of this sort could come to be; and further that it should be ageless and lacking in disease. He observed that heat or cold or anything else that has strong powers surrounds a composite body from outside and assails it, it destroys the body prematurely, bringing diseases and old age upon it and makes it waste away. (Plato, Timaeus, 32c-33a).
The atomistic shapes themselves were composed of isosceles and scalene right-angled triangles:
First that fire, earth, water and air are bodies is clear in some degree to all. Every form of body has depth. There is every necessity that surface surrounds depth by nature, and a linear surface is put together from triangles. All triangles are derived from two triangles, each having one right angle and two acute angles. One of these triangles [isosceles right-angled triangle] has on each side a part of a right angle that is divided by equal sides; the other [scalene right-angled triangle] has unequal parts of a right angle on each side divided by unequal sides. This we posit as the source of fire and of the other bodies, furnishing them according to the likely account of necessity. Principles of these [triangles] a god [alone] knows, and whoever among human beings is dear to him. (Plato, Timaeus, 53c-d).
In fact, the fundamental shape that gave rise to material reality was the triangle, whose different shapes composed the Platonic solids that interacted on a minute scale. The geometric atoms are the atomic universe and the triangles are the subatomic universe in the modern sense. These fundamental triangles sometimes broke one another down and sometimes built upon one another, depending on the kinds of triangles involved:
Four kinds of bodies come to be from the [right-angled] triangles we have selected; three of those come from triangles possessing unequal sides, but the fourth alone is fitted together from isosceles triangles. Certainly not all of them, breaking up from many small bodies and turning into each other, are able to become many small bodies [converting] into a small number of large bodies and the reverse. Three are able to do this. For all three come about from a single [type] of triangle; when the larger bodies have been broken up, many small [bodies] will be put together out of the same [triangles], taking on shapes appropriate for them. And again, when many small bodies are broken up into their triangles, having come to be one great mass number, they may come to fruition as one single form. (Plato, Timaeus, 54b-d).
Plato had his own subatomic theory: two kinds of right triangles were the base units of all elements. These had three legs that were never separated. One kind of triangle was the isosceles triangle formed by cutting a square in half: 1, 1, √ 2. Another was formed by cutting an equilateral triangle in half: 1, 2, √ 3. He constructed the first four solid faces with these triangles.
One cannot make a pentagon from these triangles, but other sources indicate that Plato thought a pentagon ought to be constructed like the five-sided shape above. The triangle face consisted of six triangles, the square face of four and the pentagon of thirty triangles. Because the tetrahedron possessed four sides it was composed ultimately of 24 triangles, the octahedron of eight sides 48 triangles and the icosahedron with twenty sides of 120 triangles. These triangles were of the 1, 2, √ 3 kind. The hexahedron or the cube having six sides was composed of 24 triangles of the 1, 1, √ 2 kind. The dodecahedron consisted of 360 triangles because of the complicated manner in which each face was composed. One can see how elements break down and reform, if the most fundamental, immutable particles are the triangles. These elements interact with one another and so account for changes in material reality. [viii] In this manner Plato gives an ancient account of chemistry, the elements themselves interacting with one another in fairly predictable ways.
When earth encounters fire and is broken up by its sharpness, it may drift about–either having happened to break up within fire itself, or within a mass of air or water – to a given point its parts happen upon it somewhere, they [the parts] refit themselves together and become earth. For the parts of earth will never pass into another form. But water broken into parts by fire or even by air, it is possible that the parts reunite to become one corpuscle of fire and two of air. And the pieces of air, broken up from any single part that is broken up, may become two bodies of fire. And again, whenever a bit of fire is enveloped by a sizable quantity of air or water or some earth – being moved inside the ones being carried about and while resisting is overcome and shattered – then any two bodies of fire may combine to constitute one form of air. And when air is overpowered and cut to pieces, then two and one half entire forms of air will be compacted into a single, entire form of water.(Plato, Timaeus, 56c-e).
The Platonic elements are not random shapes that Plato chose. They are “the only perfectly symmetrical arrangements of a set of (non-planar) points in space.”[ix] Plato adopted his colleague Theatetus' assessment of the five shapes. The tetrahedron was composed of four composite equilateral triangles formed from the faces above, three triangles meeting at each vertex. The fourth triangle is the base formed from the meeting of the three triangles. These triangles, the tetrahedron, form fire.
Four composite triangles meeting at one vertex make a pyramid with a square at its bottom. Two of these fused at the square base become the octahedron, which is the shape with eight sides, or air.
Five composite triangles meeting at one vertex make a twenty-sided shape when they are joined with other triangles that form sets of triangles meeting at each vertex. The shape thus produced is the icosahedron, or water.
It is not possible to construct perfectly symmetrical solids by adding more triangles. The shapes become irregular to the point of being useless to ancient Greeks. In other words, adding another triangle does not produce a finite solid, and the ancient Greeks would not have believed an infinite solid could be a fundamental part of the composition of the universe. Three composite squares fused at one vertex form a solid angle, and if three more squares are added to those three a hexahedron, or cube, arises.
No triangle that forms the side of a square can be equilateral, so as we have seen the square is composed of triangles that differ from the other elements: 1, 1 √2. The square is the shape of the element earth and because its triangles are not the same as in the other elements, earth elements cannot exchange their triangles with the other elements.
Three composite pentagons meeting at one vertex make one part of a twelve-sided figure, if the pentagons are arranged such that twelve pentagons make a solid. The resulting twelve-sided shape is the dodecahedron, or the shape of the universe.
Not insignificant is the fact that these shapes are in a sense proportionate to one another. The icosahedron and the dodecahedron are duals of one another. In other words, while the icosahedron has twenty faces and twelve vertices the dodecahedron has twelve faces and twenty vertices. Having six faces and eight vertices and then eight faces and six vertices respectively, the cube and the octahedron are duals of one another. The tetrahedron is the dual of itself. Plato's requirement that the universe has proportion seems to have been met in the relationship of these elemental shapes to each other. We will see below that geometric shapes are similar in kind to Platonic eidoi, whose more fundamental reality is both unchanging and eternal. In other words, the elements, being essentially shapes, themselves reduce to mathematical aspects. These mathematical aspects were themselves non-material, and thus the most fundamental aspect of material reality was itself immaterial. In other words, the very material atoms in which Plato believed have at their root an immaterial reality. They are being (Forms) that the mind reasons.[x]
Causation and Cosmology
The deity that acted as a catalyst for being to come to be was a demiurge,[xi] or a kind of false god. He believed himself to be the creator of the universe, but he merely made use of already-existing perfect models to compose its overall order:
[The universe] has come to be. For it is both visible and tangible and it has a body – and all things of this kind are perceptible, and perceptible things are grasped by opinion with sense perception. Again we say that for what comes to be must come to be through some cause. ...Which of the two models did the maker use when he completed [the universe]? Was it the one possessing the same things and being always as it is, or the one that has come to be? If this universe is beautiful and its craftsman good, then it is clear he looked at the eternal model. But if what is not right to say is the case, [then he used] the one that came to be. Now truly it is clear to all that [he used] the eternal model, for [our universe] is the most beautiful of all the things that have come to be, and the best of causes is its craftsman. (Plato, Timaeus, 28b-29a).
The demiurge shaped the universe into a sphere.
And he gave to it [the universe] a shape appropriate and natural to what it was. That is the appropriate shape for a living thing that is about to contain within itself all shapes there are on account of which it became sphere-formed, its center equally holding distance from its extremes in all directions, turned round in a circle. It is the most complete and most like itself of all shape, which he gave to it because he believed that likeness is infinitely more excellent than unlikeness. (Plato, Timaeus, 33b).
The universe was itself a living organism.
...out of the things naturally seen nothing without a mind on the whole will ever be more beautiful than anything having a mind on the whole, and it is impossible that mind came about from it separate from soul. On account of this reasoning [the demiurge] crafted the whole [universe] having set mind in soul and then soul in body so that [the universe] would be as naturally beautiful as possible and so that he completed the best work. In this manner then, according to a likely account, there is need to speak of this as a living universe both en-souled and with mind, and truly through the forethought of the god it came to be. (Plato, Timaeus, 30b-c).
Fundamental aspects of the universe are Forms – accessible through reason.[xii]
If mind and true opinion are two kinds [of things], then absolutely these things exist by themselves–Forms unperceived by us, [but] being objects of mind alone. But if...true opinion differs in no way from mind, then however many things we perceive through the body must be acknowledged as the most steadfast things that exist. Those two must be spoken of, on account of them having come to be separate and being dissimilar. The one of them [mind] comes about for us through instruction, and the other [true opinion] by persuasion. Mind always comes about with a true reasoning while [true opinion] comes about without reason. And the one [mind] is unmoved by persuasion, and the other [true opinion] is open to persuasion. (Plato, Timaeus, 51d-e).
They are eternal and unchanging realities that existed prior to the formation of the order of things, what the demiurge used to create the living and en-souled universe:
The god...gave priority and seniority to the soul both in its coming to be and in virtue, as the master of the body ruling over it [the body] as one ruled for its own good. He established it from these things and in this manner: Between indivisible and always changeless being and what [being-becoming] is divisible and comes to be in the bodies, he mixed a third form of being in the middle of the other two. And as far as the constitution of the Same and of the Different he established things in the same way in the middle of the indivisible [things] and of the divisible, somatic parts. And having taken the three things he mixed them together into a uniform kind, fitting together by force the nature of Different, which resisted mixing, into the Same. Having mixed these two with Being from three he had made one, [and] again he divided the whole into as many parts as was fitting, each part itself mixed from the Same, the Different and from Being. (Plato, Timaeus, 34c-35b).
The Forms, or eidoi, here are Same, Difference and Being, but one can easily imply a Platonic form into the triangles that compose the elements and thus material reality. Remember that the formulas for triangles and circles exist even though there is no manifestation of them. The same is true of the forms. Forms are metaphysical arrangements of sorts that are the things that become in the universe. One may think of them not as the spatially extended numbers of the Pythagoreans, but as fundamental, spaceless and timeless most real organizations that the becoming universe imitates. Number expresses geometric shape without an actual shaping of substance into geometric form, and in this way numeric equations are excellent metaphors for the nature of Platonic Forms.[xiii] They are similar to the equation for geometric shape. When we say that A²+B²=C², we are saying the shape without seeing it and without its manifestation as a shape. Geometric shapes, like Platonic Forms, need not become manifest in order to remain possible or even actual in a more fundamental sense of the word “actual.” In other words, no matter what becomes in the universe the Pythagorean theorem is true for certain types of triangles. No time or space or anything else needs to exist; no triangle needs to come to be in order for that calculation and a fundamental reality to be true about triangles. Similar but more fundamental is the immaterial nature of Platonic eidoi; and Same, Difference and Being are themselves eidoi, possessing a more fundamental, more real nature. So, the geometric shapes of the Platonic atoms themselves imply eidoi that are more fundamentally real than material manifestations. It is on this kind of metaphysical basis that the Platonic elements are based, and thus geometry allows Plato to make a connection between his theory of forms and his “probable account” of material, atomistic reality. These forms operate on anything and everything in the universe. Forms even create the levels of the cosmos for Plato, who believed that the revolutions of the stars and celestial bodies are eternal whereas the becoming of the realm near the earth is mortal:
Next, having sliced this whole compound in two along its length and having thrown each one middle to middle like an X [the demiurge] bent them down into one circle, binding together each half to itself end to end and to the ends of the other half at the point opposite to the one where they had been joined together. He then included them in that motion revolving in the same place in the same way, and began to make one the outer, and the other the inner circle. And he assigned to the outer movement the movement of the Same, and to the inner the movement of the Different. The one of the same he led around to the right along the side, and the different to the left along the diagonal, and he gave strength to the revolution of the Same and Similar in that he left this one alone undivided, as he divided the inner one six times, to make seven unequal circles.... (Plato, Timaeus, 36c-d).[xiv]
These circles are the celestial spheres on which the stars and other bodies rotate, creating an eternal world of the outer sphere and an inner world of mortality. Yet, if all of these aspects of the universe are immaterial at their root, then there must be something unlike a form that is the foundation for becoming. Plato realized the necessity of explaining what is it that becomes.[xv] His answer is that there is some aspect of reality, both changing and itself unstable, that allows the shapes of the triangles and the eidoi of the universe to come into being without actually being born, growing and wasting away:
[The receptacle] is perceived by the senses, begotten, always borne along, coming to be in a certain place and again perishing out of it, apprehended by opinion with perception. (Plato, Timaeus, 52b).
Since that for which an image has come to be is not at all inherent in the image, an image perpetually borne along to picture something else, on account of these things it is fitting that the image should therefore come to be in some other thing, in some way adhering to being, or else be absolutely nothing. But for that which really is the account of true precision is an aid: as long as the one is one thing and the other another, neither of them has ever come to be in the other in such a way that they at the same time will become one and the same, and also two. (Plato, Timaeus, 52c-d).
This aspect of the universe Plato called the receptacle. It was the “space” into which these models were placed and an unstable, ever-changing mass of nothing-in-itself that permitted the Forms to come to be in it, while at the same time never becoming the Forms nor being changed fundamentally by the Forms participating in it.[xvi] Some scholars think of it as matter. When one considers the ultimate reality of geometric shapes as mathematical objects, the relationship of the Platonic eidoi and the elements as material reality comes closer. The elements are mathematical and so also intermediate or similar in kind to Platonic forms, yet they come to be in the receptacle; number and mathematics then are intermediate to eidoi (forms) for Plato. The being of the universe for Plato is fundamentally immaterial and eternal and also much more real than the objects that we perceive.
Plato's conception of material reality is obviously not what scientists believe now, but there are aspects of his thought that survive in the sciences. His insistence on continued questioning forms the basis of scholarly publication and thus conversation.[xvii] The manner of determining commonalities and thus comprehending patterns in nature is a fundamental part of Plato's dialogues. Plato's dialogue form was a conversation about a given topic. His philosophical hero Socrates directs the discussion on most occasions and a given dialogue possesses a specific structure. A man, usually young and aristocratic, has an interest in a specific topic. He wishes to learn from an elder, a customary practice for transferring knowledge in ancient Athens. He begins a conversation with Socrates who in turn guides him, but not in the manner as someone imparting information through a lesson. Socrates' method was to engage his interlocutor in an elenchus, which was an examination of a topic whereupon a given perspective was analyzed, perhaps refuted and then questioned. Socrates asks the young man what he thinks about a given topic – like virtue in Plato's Meno – without himself claiming he knows what it is:
But Meno...what do you yourself say virtue is? Speak and do not begrudge us so that I may have been deceived by the most fortunate falsity when I said that I had never happened upon anyone who knew, if you and Gorgias appear to know. (Plato, Meno, 71d).
When asked what is virtue, the young man usually responds confidently. Virtue seems easy to define:
But it is not difficult, Socrates, to state. First, if you wish to hear of a man's virtue, it is easy to say since this is the virtue of a man: to be able to manage the affairs of the city sufficiently, and while doing these things to do well for friends and harm enemies. And if you wish to hear the virtue of a woman, it is not difficult to relate, since there is need for her to manage the household well, preserving it and its internal affairs and listening to her husband. (Plato, Meno, 71e).
Socrates questions his interlocutor's answer.
I seem to enjoy great luck, Meno, if seeking one virtue I have found some swarm of virtue is yours. But, Meno, in keeping with this image of the swarms, if I were asking you about the being of a bee what is it, you were saying that they are many and variegated, what would you have answered me, if I asked you “Do you say that there are many and varied kinds and that they differ from each other, as far as being a bee? Or do they not differ in this way, but in some other way, like in beauty or size or some other thing of these sorts?” Say what would you answer if asked in this manner? (Plato, Meno, 72a-b).
His interlocutor, here Meno, is expected to respond honestly and in the most intelligent way he knows.
I would say this, that they do not differ, in the manner that they are bees, one from another. (Plato, Meno, 72b).
In the case of Meno, the response is an indication of some kind of commonality that exists between the manifestations of virtue, which is to say that one example of virtue must compare with all others in quality or some other aspect. When the commonality becomes apparent, the interlocutors are able to say something about each of the examples:
Recall the pattern-seeking that seems to be inherent in the literature of the ancient Greeks. Socrates is looking for patterns with his interlocutors. These patterns lead them to general statements about a given topic, like virtue, but the topic could just as well be matter, or time, or space. This pattern-seeking is a fundamental part of scientific inquiry, and the give and take resulting from Socrates' interactions with someone is the process of an interlocutor being relieved of his doxa, or opinion.[xviii] Socrates must first show that the opinion his interlocutor has is not true or perhaps in some way flawed. He usually accomplishes this task by demonstrating that should the young man accept his own view, he would inevitably fall into a contradiction or some other difficulty. When asked, Meno says that he believes all of the manifestations of virtue have some kind of commonality.
The idea here in Plato's Meno as elsewhere is that the doxa, or opinion, is reliant upon observations and belief alone, in other words becoming, which means that for Plato such a perspective is unstable. Naturally, the more stable thought comes from reasoning. Here reasoning is a development of observations and beliefs into a consistent, reliable argument that has a universal application. In other words, Socrates guides Meno through the topic of virtue by attempting to arrive at the eidos of virtue, or at least as close as possible to it. Remember that eidos means “form”, and the word itself derives from a verb that means “to see.” Eidos is linked to the mind through the sight organ, and mathematics is closer to eidoi. It is the most fixed way to make determinations as far as Plato is concerned. Employing this method, Socrates guides his interlocutors into a way of thinking about the topic that is at once philosophical and open-ended. He and the given young man investigate through argumentation and continued refutation until they have come to an informed place of ignorance. They do not claim to know the topic, but they have a greater understanding of it because they admit their ignorance, especially because they have engaged in a dialogue about it. Such a state is called Socratic ignorance. The question that arises for us is how this process of question and answer relates to scientific endeavor when its topic is immaterial, science being decidedly about material objects and forces in the universe.
Ancient induction is reasoning from the particular to the universal. One begins with discernible instances of their topic and proceeds to examine them for common characteristics. Ancient deduction is reasoning from the universal to the particular. One begins with a universal claim about a topic and then applies it to given instances. The very structure of an inductive logical examination is inherent in the kind of conversation that Socrates has with his interlocutor. When the pair has found the similarity that makes specific examples the same, even though they are individually different, they seem to have found that which is common to them. This commonality Plato believes to be a deeper and more fundamental reality of a given thing. It has been discovered through reason and it is, for Plato, an immaterial part of each thing as it becomes. This thing that Socrates and his interlocutor have found is perhaps one eidos, which is the same fundamental aspect of material reality that underlies the monologue in Plato's Timaeus, but the examples are employed as the beginning of reasoning. Because the most fundamental and firm foundation is achieved through a connection with the mind, reasoning is sturdier than observation and comparison, for Plato at least. Deductive reasoning, therefore, is the most important of all activities of the mind for Plato, and deductive reasoning is fundamentally involved in the question and answer process in the dialogue. The examples employed give the interlocutors an arena in which deductive reasoning is able to reveal in some measure that absolute, the eidos, without which any one thing in the universe cannot become manifest. Remember that the eidos is accessible to the mind. It is deductive reasoning that reveals what a thing really is to the reasoner, its being. Thus, inductive and deductive reasoning are integral to the process, but deduction is the most significant.
One contribution of Plato to science that has lasted is the continued questioning and the finding of better answers in the context of examples leading through induction to deductive conclusions. Other thinkers before Plato had certainly reasoned. They had certainly used logic, but Plato puts the discussion on display. He demonstrates the reasoning that has eliminated certain opinions and arrived at much better insights through a rigorous process of induction and deduction. There is nothing more scientific than this kind of interplay. The structure of argumentation has not changed fundamentally, except that modern science seeks patterns in nature rather than an immaterial Platonic eidos, and modern logic is a different configuration of the same forms found in Plato's dialogues. The rules and structures have evolved, but inductive and deductive reasoning remain a fundamental part of scientific progress. So, a fundamental aspect of scientific reasoning is on display in the literary expression of Socrates' conversations with his interlocutors:
(Plato, Meno, 73e1-74c7).
A particular manifestation of a shape is no less a shape than any other and a particular color is no less a color than any other. The color-ness of color and the shape-ness of shape are the forms, or eidoi, of the subjects. Again, what Socrates and Meno want to comprehend is the absolute of each of these topics, which is attainable via deductive reasoning alone. Particular manifestations are not real; what the mind apprehends is. Additionally, Socrates has a slave determine the length of the side of a square. Part of the process is such investigation and re-investigation, but the actual reasoning of the geometric shape is deductive:
(Plato, Meno, 84a4-c9).
Socrates has the slave work out the area of a two-foot square:
(Plato, Meno, 82b8-d4).
A two-foot side of a square produces an area of four feet. But then Socrates guides the slave into a difficulty. He makes the square four feet on each side:
(Plato, Meno, 82d5-e2).
The slave does not know that once the size of the square has been doubled, the numbers used in the equation for determining the area of the square have changed. A square that has sides of two feet will have an area of four feet because the sides must be multiplied. The result is the same as if one added the numbers together, but when the sides have been increased to four feet, the product of four and four is the area, which is sixteen and not eight. The kind of reasoning involved in this equation is deductive. That is, deduction in the modern sense. If one knows the lengths of the sides of a square and calculates properly, then they will come to an inevitable conclusion.[xix] Double the length of one of the sides of a two-foot sided square makes an area of sixteen:
(Plato, Meno, 82e12-83c3).
The conclusion is inevitable. Now, Socrates uses this example to prove another point. Like the geometric atoms, the argument here reveals something absolute about the answer. In other words, knowledge for Plato has a decidedly deductive aspect to it, which is linked to the certainty of the eidos and the form of a given thing. In this case, the form is that of a geometric figure. This certainty remains a part of the progress of scientific knowledge in the form of mathematics. Plato's Meno is especially important for our purposes not only because it is dialogue on display, but also because it demonstrates a fundamental part of Plato's notion of reality. The slave recalls eidoi that are part of his intellective activity. He does so while reasoning about a geometric shape, which is itself a mathematical figure, it being presentable in numbers. The calculation for a triangle can be thought, it can be written (A²+B²=C²). If the universe were to fall into disorder and all its objects were to combine into one homogeneous mass, the formula for a triangle would still be accurate, as we have seen. Plato takes the triangle's seeming permanence as a sign of its immaterial and undying quality revealed through deduction. We will see that mathematicians like Euclid employ a more pure form of deduction for their calculations,[xx] and these pure deductive calculations evolve into theoretical aspects of an hypothesis that is verified by tests. The effort at deduction in Plato has survived in modern science, but it has survived as a kind of theoretical certainty.
Furthermore, Plato's method of examination is a kind of ignorance that leads to knowledge, yet remains ignorance. This kind of investigation and re-investigation has spread to all the scientific disciplines in the form of articles and books on topics in specific fields that are influential, questioned and reexamined constantly. A worthy scientist will admit his ignorance and listen to others in his field and so the Socratic process is part of the progression of scientific knowledge. Socrates uses examples, like men having power and women being obedient, to examine what would be fixed, or stable, about each of the instances. The use of examples is the process of reasoning from the particular to the universal, which is basic inductive reasoning of the ancient kind and a seeking of repeatable patterns in nature. When one makes use of the examples in an argument, one reasons toward a universal, which is then used to construct an argument that is more fixed than the one arising from the examples. Once Socrates and his interlocutor have reasoned deductively, they have arrived at a better account than the examples. The examples are the becoming and the deductive reasoning is the effort at finding some aspect of being.
An examination of Plato’s arguments is in order so as to understand how he develops his notion of material reality, but one must here be very careful. Plato reasoned as others do, but he understood that paradox is part of reasoning and he combined his reasoning with drama. We are examining only some of his arguments in his Timaeus and Meno. We are not claiming that our analysis of his argumentation is a universal part of his dialogue form. These are, as in the case of other thinkers, instances that come from particular works. Ordinarily, his argument would be not merely one interlocutor's assertions but two or more. In Timaeus Plato's reasoning is relatively straightforward. He gives the above account of the interactions of the geometric atoms, which can be taken as his theory of chemical interactions. His way of thinking can be broken down to a few relationships between categories of things found in the universe.
We have simplified Plato's reasoning in order to understand the essence of his argument, and one can see a certain consistency in his thoughts, but the premises are not necessarily true. There is a good deal of imaginative thinking in the assertion that minuscule bits of reality are shaped like icosahedrons, cubes, octahedrons, and tetrahedrons. This notion is merely conjecture, but if you follow and believe in Plato’s reasoning, the rest of his assertions follow.
If it is true that all elements that are interchangeable are elements that transform into one another and the above is true, then fire, air and water are elements that transform into one another. Like other ancient thinkers, Plato is making universal statements about aspects of material reality without qualification, other than his assertion that he is talking about a “likely” genesis of the universe.
Once we have accepted the argument that Plato makes about the size and shapes of the elements, how they combine and break one another down follows. All one must do is observe the geometric shapes that supposedly comprise the elements and learn into what they are likely to separate. If the ultimate piece of material reality is a triangle, then one wonders why an element is not further broken into the most fundamental shape, but one can also see the logic employed in Plato’s reasoning.
Plato's Meno offers another aspect of argumentation, one that is likened to his eidos. Plato argues in the form of geometry, which is mathematical in nature. The measurement of the square is specific and incontrovertible. When one establishes the units that measure the square, one has a kind of starting place, or reference point. The measurement is a certainty that will allow Plato to make a clear and absolute assertion about the area of the square. The two foot square:
The area of a square with sides of two meters each is four, because the product of two and two is four. There is no escaping that conclusion. It is not an imagined belief, nor is it a set of particulars that supposedly lead to a universal conclusion. The measurement of the square's area will always be four meters. When the slave makes a mistake and doubles the side of four meter's length, he demonstrates that he does not understand what is the calculation. When he reexamines the four meter square and realizes that he must take the product of the side and the other side, he understands the inevitable conclusion: the area of a four meter square is sixteen meters, not eight.
The four foot square:
The lower corner is representative of how all of the squares are sectioned, so each square that comprises the larger square possesses four squares, resulting in sixteen squares. This is deductive reasoning in the form of mathematical measurement. One must come to the conclusion, given what information is available; it is inevitable. The absolute nature of the inevitable conclusion is the kind of conclusion Plato seeks in all his reasoning, which is why he is attracted to the idea of an eidos and mathematics. A similar kind of deductive reasoning is the theoretical and mathematical reasoning that remains in modern science. Enclosed in its own system, a calculation is not incorrect as long as it follows the rules of logic or mathematics, but its application may be incorrect. In other words, an equation may not fit what moderns call objective reality, and so when tested an equation may have its own consistency, but not apply to actual things. When Socrates and Meno seek virtue in its most elemental and essential form, they seek the certainty of a universal virtue. Here Plato applies mathematical methods to the search for an absolute moral. Yet, this kind of certain system may be applied to any subject, and such application is what remains of his method in scientific progress and reexamination. Science is, in part, a dialogue.
© 2018 Kirk Shellko All rights reserved.
[i] Plato may not have been responsible for as much science as once thought to be the case. For an argument against his being the one who presented a scientific method employed in scientific circles, see Zhmud. Plato may also not have been responsible for the principle of “saving the appearances”, fundamental to Greek geometry (See Lloyd), and customarily attributed to him. Still, Pirmin Weithofer asserts that for Plato abstract or mathematical structure is necessary for pure knowledge, both of which are necessary elements of science. There is no shortage of debate.
[ii] Interest in Timaeus during the twentieth century has declined and resurged. It is presently taken seriously as philosophy, if not physics. Some see Plato's Timaeus as metaphorical, or mythic, while others see it as more literal. For the mythical interpretation, see A.E. Taylor. For the more literal, see Cornford. This text treats Timeaus as mythical with literal elements. Most of the contentions are taken literally. Donald Zeyl treats the history of commentary on Timaeus.
[iii] Charles Kahn reads Timaeus as “a constructive account of phenomenon within the field of perception and change”, which means that expectations for accuracy as to what comprises the universe ought to be low. Plato must contend with opinion and becoming – not to mention matter – and thus his account will necessarily be unstable. For more, see Kahn, Cornford, Taylor.
[iv] For Francis Cornford Plato's sense-perception is “unreasoning” and does not grant full understanding. He quotes Proclus when he states that sense tells us apples are a specific color and shape, but reason tells us they are apples. Sense-perception will also never alert us to the fact that the sun is not a small ball in the sky, but reason will. Reason coupled with sense-perception comprise our good judgments. For more on the relationship between reason and perception, see Cornford.
[v] Dialectic (dialogue in Plato's works) is more precise and fundamental than even mathematics. Timaeus is not dialogic and so its exposition exists on a more superficial plane.
[vi] For more on the relationship between being and becoming and its debate, see Cornford, Kahn, Johansen.
[vii] T.J. Haarhoff points out that the modern conception of atoms is closer to Plato's conception of atoms than Democritus' conception. The regular solids depict atoms as regular mathematical entities; modern math and science do the same thing. For more, see Haarhoff, Cornford, Pohle, Cleary.
[viii] For an account of flux in Timaeus, see Mohr. For a thorough account of the elements as causes, see Broadie. For an account of the harmony, or commensurablility, of the elements and their “chemical” interactions, see Brisson and Meyerstein who believe that Plato proves systematic verification was not rejected because the chemical theory of the elements works in common interactions.
[x] For a brief but sophisticated view of the forms in Timaeus, see Ostenfeld. For an explanation of why the theory of forms is a part of Timaeus, see Ferber and also Kahn (1985).
[xi] Some scholars see the demiurge as metaphorical; some see it as literal. For the metaphorical view, see Cornford.
[xii] For a good, basic explanation of the relationship between forms and modern physics, see Joad.
[xiii] The role of number in Platonic Forms is somewhat ambiguous. While they play an intermediate role between being and becoming in Plato's Republic, some scholars believe that number is a fundamental part of the theory of Forms: the one, unlimited and limit – called the indefinite dyad – generates the Forms.
[xiv] These circles are the celestial spheres on which heavenly bodies rotate.
[xv] If there is a need to explain material becoming in spite of the geometric atoms, then the atoms themselves cannot be becoming. They must be being or the intermediate kind of being-becoming. Such a need means that the geometric atoms are akin to eidoi. For more on their geometric composition, see Johansen, Cornford, Kahn.
[xvi] Images reside in the world of becoming for Plato. The world of being actually is, and what receives the Forms as an image is the receptacle. Some see the receptacle as potential, like Aristotle's prime matter; some (Kahn) see it as three-dimensional space. It is likened to a base for perfume and clay as well as water reflecting an image. For more on this issue, see Cornford, Kahn, Sedly, Johansen, especially Miller.
[xvii] One must differentiate between mere questioning and Plato's dialectic; these are not the same things. Dialectic, sometimes called dialogic, is a qualitative seeking and a process that does not believe its practitioners know, while mee questioning may come from scholars who through modern mathematics or logic do believe they know.
[xviii] It is important to note that Plato sought the most absolute answers to philosophical questions. Modern science has given up on the formation of universal answers in the form of absolute resolutions. Science now thinks in terms of probabilities, whereas ancient philosophy sought absolute, universal answers. Thus, Plato sought something fundamentally different from what science now seeks. The difference is one of the possible differences between philosophy and science.
[xix] Not surprisingly, modern mathematics questions even the certainty of deductive reasoning, but that is another – most interesting – issue.
[xx] Plato thought that there was a more refined method than mathematics: dialectic (See Mueller). For more on how closely Plato's proof in Meno resembles Euclid's axiomatic method, see Haarhoff, Szabo, Kahn, Wolfsdorf,
© 2018 Kirk Shellko All rights reserved.
Life and Works
Aristotle was a student of Plato who lived from 384 to 322 BC. He was arguably the most scientific of ancient thinkers, but not because he followed scientific methods. Aristotle developed a system of analysis based upon his metaphysics, employing some aspects of what would eventually give rise to scientific method. Having traveled to Pella in 343 B.C., he began tutoring Philip of Macedon's son, Alexander, who later came to be Alexander the great. He founded his own school at Athens in 334/335 B.C., called the Lyceum, where he enlisted the aid of students in making observations of practically everything that exists. His students were known as peripatetics, and he had with Herpyllus (a slave or his wife) a son named Nicomachus, to whom his Nicomachean Ethics is allegedly addressed. He felt compelled to depart Athens because of anti-Macedonian sentiment, which arose shortly after Alexander's death in 323 B.C., leaving Athens for Chalcis where he died of natural causes. His and his students' investigations supplied the raw material for many of his “scientific” works. Some that survive are Physics, Metaphysics, On the Heavens, On Generation and Corruption, On the Parts of Animals, Categories, Topics, On the Soul, Posterior Analytics, Prior Analytics, Nicomachean Ethics and Meteorologia. The topics of his surviving works range from biology to metaphysics, but many or even all of these are works that likely were not meant for publication. Indeed, the works that were meant to be read seem to have been lost, including some dialogues on Homer and poetry in general, which were said to be even more beautiful than the works of Plato. The breadth and depth of his thought continues to impress scholars [i] and some consider Aristotle to be the greatest polymath of the ancient world. Aristotle's metaphysics leads him to observe and thus to analyze empirically physical objects themselves instead of merely reasoning about them. As a result he created an empirical and logical system of thought that covered not only all of the humanities but all of what moderns call the hard sciences. In fact, Aristotle's contribution to western thinking is so immense that we must limit ourselves to only a few facets of his thinking and a few texts. [ii] Aristotle did not discover causality or causes. Rather, as we have seen, many thinkers before him sought causes of things in nature. These were the first physicists from the western world who had remarkable insights into material reality and who many times expressed themselves in literary form. Aristotle believed in a form of interaction with these predecessors that was similar to the dialogue form Plato used in his dialogues. He critiques his predecessors' work by discussing and critiquing their arguments and thus builds upon earlier insights, but in prose style.
It is difficult to discern what comprises a first principle for Aristotle. His first principles can be thought to be a construct of things: elements, causes, motion and a material eidos. [iii] He was the first to create independent sciences and disciplines for various subjects and that is only one reason why articulating his first principle(s) is difficult. Aristotle was heavily influenced by Plato and so Plato's eidos is an intimate part of his thinking. Yet, Aristotle shifts the Platonic eidos from the immaterial reality of Plato to the material realm. [iv] He claims that the matter making up the manifestation of some creature or thing and the form, or eidos, are inextricably bound. Still, he retains the Platonic eidos as part of a metaphysical construct of things that move internally and externally and so become. This metaphysical eidos Aristotle then bound up with matter and motion. In other words, Aristotle thought that something has a nature if it moves into its own principle by becoming what it is:
Of existing things, some exist by nature, others through other causes: by nature animals and their parts exist, and plants, and the simple bodies, like earth, fire, air and water (for these things and things of this sort we say exist by nature), and all of these things seem to differ from the things not put together by nature. For each of these in itself possesses a source of movement and rest, in place or by growth and decrease, or by change; but a bed or a cloak, and other sort of thing that exists, by the manner in which it has happened upon each designation and to the extent that it exists from art, has no in-grown impulse of alteration at all. But by the manner in which they happen to exist of stone or earth or a mixture of these, they possess an impulse, and to that extent, since nature is a source of something and a cause of being moved and of being at rest in that to which it belongs principally, in virtue of itself and not accidentally, <they possess an impulse>. (Aristotle, Physics II, 192b8)
What he meant was that things that grow in and of themselves have a natural movement that can be likened to a particular principle, while things that must be crafted do not have this natural movement and thus no natures. A tiger has a nature, possessing an innate impulse to strive and survive as a tiger, as does a human. Beds and tables possess no nature because they have been crafted by someone; they lack the impulse to come about on their own. Yet, beds and tables exist by nature because they are crafted of things and made by men. Still, what comprises the table may have a nature or be a nature, depending on out of what the table is composed. In this manner Aristotle's perspective differs from Plato who thought that practically anything has an eidos, or what might be thought of as something like a nature, or some kind of organization. It is important to note that Aristotle deliberately avoids making a material element one of his first principles; he consciously avoids making a Platonic eidos a first principle as well, seemingly because he believed that any material principle or metaphysical construct would itself need another piece of matter or principle which would in turn need another first principle. There is then for Aristotle an infinite regress called the third man argument. Aristotle also acknowledged the four elements of Empedocles (earth, air, fire and water), but their status as first principles was for Aristotle dubious and they were not the principle he sought (again one significant reason is because of the third man argument). Some scholars claim that Aristotle argued himself into what is called prime matter, an underlying indefinite pure potentiality that inhered in the largest as well as the smallest and most fundamental of things.
It seems that Aristotle set out to explain how creatures and things arise in the world and what exactly is the relationship between the form, or organization, of something and its matter. Explaining matter and form as becoming in terms of potency and act seems to have been central to Aristotle's effort at understanding what it is that is the motion of a thing. Things that have an internal movement must be more than merely arrangements of the fundamental matter that comprises them. Otherwise, anyone could arrange matter in the correct way and create anything, god or creature. Aristotle sought what it is in matter that is yet more than mere material – that which produces itself, things and creatures yet remains en-mattered. In other words, he sought the things responsible for what comes to be, but these motions are not the mechanical activities modern science understands as causes. Aristotle set about to deepen human comprehension of the universe through investigation into productive activities that he called aitia, the Greek term meaning “causes.” Causes and things had different kinds of motion to them:
...there is motion only in respect of what kind, how much, and where, for in each of these there is contrary. Now let motion in respect of what kind be alteration, for it has been joined with this common name. But I say the what kind is not the thing in the substance of any thing (since then even the difference is a quality), but it is a passive attribute by which a thing is said to be affected or unaffected. Motion in respect of how much lacks a common name, but in each instance it is growth and decrease, the one tending to a thing's complete magnitude is growth, the other tending from it decrease. Motion in respect of place both in common and in particular is nameless, but let it be called generally change of place, although those things alone are said to be carried along when not on their own coming to a stand is possible for those things changing place. (Aristotle, Physics V, 3 226a).
This motion was intimately involved in what something is and how it manifests:
There is that which exists in completion, but also that which exists in completion and potentially: a thing being this thing, being this much, being this kind, or similarly for the other ways of categorizing being. Relation to something is said to be what exceeds or falls short, or what exists according to acts and being acted upon, and generally what moves (another thing) and what is moved: for the mover is a mover of something moved, and the moved is moved by something moving it, and no motion exists besides [this] in things. For what changes always changes either in substance, or in quantity, or in quality, or in place, and there is no commonality to take from these...and with respect to which is neither...a this, nor a this much, nor a what kind, nor any other kind of being: so that neither motion nor change will be anything apart from the things mentioned, since there exists nothing besides the things mentioned. (Aristotle Physics III, 1).
The word for motion that Aristotle uses is kinesis. He links kinesis to energeia, a term used to express the being-at-work [v] that is the continued actualization and also potential:
With a distinction having come about according to each kind of being between the complete and the potential, the completion of a thing existing by potential – by the manner in which this sort of thing exists – is motion: of the alterable, qua alterable, it is alteration; of the grow-able and its opposite, what can decrease (no name is common to the two), it is increase and decrease, of the generable and destructible it is coming to be and passing away, and of the mobile as far as place it is change of place. (Aristotle Physics, III, 1).
Kinesis of something for Aristotle is a kind of activity belonging to a thing that makes what the thing is a what. In other words, something coming to be moves into what it is, as itself being acted upon and acting in various ways:
And that this is motion, is clear from the following. For whenever the buildable – by which we say this sort of thing exists as itself – exists completely, it is being built, and this thing is building. Similarly learning, healing, rolling, leaping, ripening, aging. And because some of the same things exist potentially and completely, not at the same time or not according to the same thing, but like something fully hot and potentially cold. Many things will act and be acted upon by each other; for in every case they will be act-able and able to be-acted-upon. The result is that the moving is naturally moved, since each such thing moves the thing being moved and itself. (Aristotle, Physics III, 1).
A thing also moves from itself and into itself as it continues its being-at-work. It is necessary to understand that the energeia or the being-at-work of something is for Aristotle the moving organization that is its being, that being which for Plato is called an eidos. Aristotle contends that such activity is in the world as something material: being-at-work. It is not in another, allegedly deeper and immaterial part of reality, as for Plato. That activity has its own motion into itself, but it also may exist as potential for another thing. As a result, things have in and as themselves a kind of potential, for Aristotle. They possess motions that not only make them what they are, but make them possibly other things, and they are not necessarily always the same motion:
But the fullness of what exists potentially, whenever existing completely and working, not as itself but just as movable, is motion. I mean by “just as” the following. Bronze is a statue potentially, but similarly it is not the fulfillment of bronze by which it is bronze that is motion. For it is not the same thing to exist as bronze and to exist as some potentiality, since if they were the same thing simply and by definition, the completion of bronze as bronze would have been motion. But they are not the same, as has been stated. This is clear in contraries. For to be potentially healthy and to be potentially ill are different, since being healthy and being ill would be the same thing. The underlying thing both being sick and being healthy, whether humour or blood, is one and the same. Since they are not the same thing, just as color is not the same as being visible, the fulfillment of the potential, as potential, is motion. (Aristotle, Physics III, 1).
Causation and Cosmology
Aristotle understood causality as the thing (that material eidos) acting in and for itself, and thus his metaphysics, along with the third man argument seen above, compelled him to refrain from establishing a simple, material first principle. [vi] In other words, Aristotle establishes four causes in order that observations of particular causality may lead to a deeper understanding of a manifestation of anything. When one articulates the causes, the depth of the what of any thing emerges:
For since our labor is for the sake of knowing, and we think we do not thus far know each thing until we have taken hold of the why of each thing (and this is the taking up of the first cause), it is clear that we must do this about both coming to be and passing away and about all natural change, so that, once we know sources, we may try to lead back to them each of the things that are sought. (Aristotle, Physics II, 3).
Aristotle's analysis is not restricted to matter or to empirically verifiable aspects of things, and in this sense his method does not resemble scientific method. That is, a cause may be the skill or knowledge someone has or the arrangement of something and its purpose, which differs greatly from the sort of cause associated with modern science. His perspective on causes one may call metaphysics. One Aristotelian cause is the material that comprises a something. It is something without which the movement, or nature, cannot manifest.
One manner of cause is said to be, then, that out of which something comes into being, while being an underlying presence in it, as bronze of a statue or silver of a bowl, and other kinds of these things. (Aristotle, Physics II, 3).
The bronze of a statue is a kind of material from which a statue may be made. There are other substances from which a statue can be made, of course, but they must have certain qualities: resilience, aesthetic beauty in themselves and perhaps the ability to melt, or at least the ability to be formed into some shape or organization. One way to understand matter here is that it is a kind of being-at-work itself that possesses another characteristic of potential as well. The continued whatness of bronze does not change but the motion of the potential for being statue acts through its being-at-work. Matter may be interpreted here as another being-at-work, or a mysterious part of the universe that cannot be articulated; there is also prime matter, which has been interpreted to be an indeterminate mass of pure potentiality. [vii] Another of Aristotle's causes has been called the formal cause. It is the organization of what something is or the what.
In another manner [cause] is the form (eidos) or pattern, and this is the reasoning of the “to be what it is”, and the kinds of this thing (as of the two-to-one ratio octave, and broadly number), and the parts that are in the reasoning. (Aristotle, Physics II, 3).
One will recall that the octave is the two-to-one numeric ratio applied by Pythagoreans to the lengths on a musical scale. Certain distances on one chord produce similar sounds. Without the specific distance of the chords and without the particular material in specific arrangement comprising a string that produces a musical sound there would be no similarity of vibration that produces a sound. Each thing that is necessary to produce the sounds in the length relationship of two-to-one or one-half is itself an arrangement needed to form the octave. These differing but harmonized arrangements are the producers of the octave and thus formal causes, or formal producing activities. Thus, there is a relationship and order to the formal cause and it has an arrangement to it, but [viii] Aristotle is not making the formal cause numeric in nature, as we have seen, other than to state that there is an organization, which may in turn be analyzed mathematically or logically. [ix] A thing's nature has a kind of activity that produces a fullness, one that the assessment of its eidos by mathematicians lacks. The fullness is articulated for Aristotle in its motion. One can see aspects of the modern conception of geometry and calculation as abstractions that must be proven here. Aristotle thinks of geometry as not numbers spatially extended, but as the outline of things that must be filled. What fills it is motion of a specific kind. So, a natural thing in movement is the fullness of something actually moving as itself. [x] A third cause is what may be called the motive or the efficient cause:
...it is that from which the initial source of change or of rest is, as the legislator is a cause, or the father of a child, or broadly the making of what is being made, or the changing of a thing being changed. (Aristotle, Physics II, 3).
The motive cause is similar to the modern notion of cause, as that which produces or acts upon another thing in the first place. There is a generative quality to this cause as in all Aristotelian causes, but the motive, or sometimes-called efficient, cause may be more properly called the initial interaction between cause and caused or perhaps the physically active cause. It is the initial motion and interaction between things that makes up this cause in the sense that there are things causing and being caused externally at the beginning of change. The last cause, sometimes called the final cause, explains why something has come to exist, or its purpose. Its end is not necessarily to produce a human purpose for something, but to produce some kind of activity that can be defined as the end or goal of something:
...it (cause) is meant as an end. This is the “for the sake of which”, as health is of walking around. Why is he walking around? We say “so that he becomes healthy,” and in speaking in this manner we think to have given the cause. (Aristotle, Physics II, 3).
This cause, coupled with the formal cause, is more completely representative of the Platonic eidos. One may even think of the formal and final causes as the same but immersed in different acts of what becomes. When something attains its fullness, it attains such fullness as what it already was and what it was and is to become. Such is the final cause. One can think of human DNA as analogous. When matter goes into the design of the creature whose DNA guides it, the end goal is that pattern or organization that is the DNA, or the structure of what the creature will be. We call it genotype. The actual manifestation is the phenotype. The genotype and the phenotype are inextricably fused for Aristotle, not only in creatures and men, but in anything that possesses the impulse of nature. The final cause is that towards which the impulse of motion directs itself in order to become what it is by nature. The Platonic eidos is sometimes interpreted as an ideal, or as something to which to strive. The final cause is similar, but of course it is in the matter whereas the Platonic eidos is immaterial and “outside” of becoming. We must keep in mind that though we have seen roughly what are the causes for Aristotle, his work contradicts itself perhaps, or at least there is an effort to explain and then reexplain that makes his thoughts twist and turn on the page one reads. Perhaps it is the case that Aristotle recognized how fused together are these causes such that no one articulation suffices, but elements of each enter into different being-at-works:
For cause is spoken of in many ways, and of those of the same form, one prior and one later, as in [the cause] of health is the doctor and also the craftsman, and [the cause of] the octave [is] the double and also number, and always all-encompassing things as related to particular ones. (Aristotle, Physics II, 3).
And all of them, both those things said naturally and those said by chance, are some potential things and other things working , as of building a house, either the builder or the builder building. And similarly it will be articulated for what things the causes are causes for the things that have been said, as [cause] of this statue or a statue or generally an image, and of this bronze or of bronze or generally of material, and the same with the things having happened by chance. Further, these things weaving themselves together (and being woven) [with] those will be said, such as not Polycleitus nor a sculptor but the sculptor Polycleitus. (Aristotle, Physics II, 3).
Aristotle examined becoming and thus sought to understand Plato's eidos in terms of how things bring about things. He was compelled to accomplish this task by looking at the things around him and then reasoning. At the same time that he developed an observational method, and probably as a result, Aristotle noted that there exist rules for thinking. The effort at articulating the rules for logic seems to be linked to the effort at finding causes. At the same time, Aristotle's logic rules make clear how his predecessors reasoned and allowed future thinkers to analyze the arguments of the tradition, as seen in the arguments given above for ancient thinkers. These efforts at explaining things in terms of cause make Aristotle an authentically analytic thinker, which is to say that an essential aspect of his thinking is based upon observation and then later reasoning over the various observations of animals, of substances, of politics, and of other objects and creatures one finds in the universe. That he is analytic is not to say that he concentrates on logic and the rules of logic alone. Aristotle's metaphysics compel him to observe and then reason. One seeks to understand what the causes are, and so the eidoi, of a particular thing's motion. Observation and thought were already fundamental aspects of Greek thinking, but Aristotelian causes articulated precisely what to observe and how to reason in order to make sound arguments about the general aspects of each thing as it is being-at-work. It is for these reasons that we can call Aristotle a scientific thinker, but be careful to know that he is no scientist. He never would have thought that one first needs an hypothesis that was well-thought, a theory that was followed by empirical verification. While observation was an essential aspect of his thinking, the control methods and supposedly objective verification process of science were virtually unknown to him. In fact, Aristotle may not even have possessed any awareness of object reality in the scientific sense. The notion that there exists an objective state of things that may be assessed by reason and reached through the senses whereby an empirical verification of a theory may take place was not a part of his observational method at all. In fact, he would have been averse to empirical verification, other than the use of observations. Again, Aristotle observed and assessed through reason, but never did he or his students empirically verify an already-systematized theory. [xi]
Aristotle articulated his cosmology in various works, but the most complete actual cosmology was his De Caelo, or “On the Sky”, which was accepted as true for more than 18 centuries. There he articulated a world composed of the four elements: earth, air, fire and water. These naturally move up or down in a sublunary realm with fire being the most light and earth heaviest. Objects are composed of the different elements and they are imperfect because their elements are displaced, the natural places for the elements being where they tend to go. Fire ascends and earth drops, for example. Bodies move thus naturally and we have already seen how a natural motion is a part of all things. Stars and planets are more exalted bodies that move in circles and earth lies in the center of the universe with the celestial bodies circling it. The initial motion of bodies was begun by a prime mover that is itself desired [xii] by all the other bodies. The prime mover manifests itself in no way because otherwise it would be imperfect; celestial bodies are moved by the prime mover and the movement radiates downward into other spheres that surround earth; the matter that comprises the celestial sphere is the eternal aether. These spheres number 55, explaining and predicting the motions of the stars, as the celestial bodies were affixed to each of the spheres. The principles that govern the celestial sphere are not for Aristotle the same as the ones that govern the sublunary sphere. The celestial sphere and its occupants are eternal while the sublunary sphere, with its elements, are mortal. This universe has always existed because it is perfect, and it will always exist. Obviously, Aristotle made use of observations in creating this model. Aside from the fact that most of his assertions are wildly inaccurate, it is notable that his 55-sphere system did somewhat accurately predict celestial motions. Aristotle himself recognized the speculative nature of his musings and expressed hope that another, more informed, theory may emerge.
Keeping Aristotle's perspective in mind, we are able to re-articulate his rules of logic that we discussed in the first part of this blog. A re-articulation will make clear how his metaphysics, his logical rules and his analytic method work together. Aristotle noticed certain things about how human beings reasoned. He formulated the first rules for argumentation, guidelines for the formation of good reasoning. Good reasoning involves not only observations that are true, or what moderns may call facts. Observations must be understood such that they have a connection to one another, and the more certain the relationship between specific observations, the more firm foundation will be the thought about a specific topic. [xiii] Aristotle began with propositions, which are statements taken to be true. If there exists no connection between the propositions, then no conclusion can follow. But if there is a connection between the propositions, then new information and insight may follow:
If then one wants to prove syllogistically A of B, either as an attribute of it or as not an attribute of it, one must assert something of something else. If now A should be asserted of B, the proposition originally in question will have been assumed. But if A should be asserted of C, but C should not be asserted of anything, nor anything of it, nor anything else of A, no syllogism will be possible. For nothing necessarily follows from the assertion of some one thing concerning some other single thing. Thus we must take another premiss as well. If then A be asserted of something else, or something else of A, or something different of C, nothing prevents a syllogism being formed, but it will not be in relation to B through the premisses taken. Nor when C belongs to something else, and that to something else and so on, no connexion however being made with B, will a syllogism be possible concerning A in its relation to B. For in general we stated that no syllogism can establish the attribution of one thing to another, unless some middle term is taken, which is somehow related to each by way of predication. For the syllogism in general is made out of premisses, and a syllogism referring to this out of premisses with the same reference, and a syllogism relating this to that proceeds through premisses which relate this to that. But it is impossible to take a premiss in reference to B, if we neither affirm nor deny anything of it; or again to take a premiss relating A to B, if we take nothing common, but affirm or deny peculiar attributes of each. So we must take something midway between the two, which will connect the predications, if we are to have a syllogism relating this to that. If then we must take something common in relation to both, and this is possible in three ways (either by predicating A of C, and C of B, or C of both, or both of C), and these are the figures of which we have spoken, it is clear that every syllogism must be made in one or other of these figures. The argument is the same if several middle terms should be necessary to establish the relation to B; for the figure will be the same whether there is one middle term or many. (Aristotle, Prior Analytics I.23).[xiv]
If an argument has a good connection, it can be said to be valid. Yet, validity only means that there are statements possessing some set of things in common with one another and because of that commonality, one can make a new statement about them. This new statement is called an inference. It is information derived form the propositions one is using that was not present until the argument came about.
This argument has good form, and so it is valid because if we accept that all quadrupeds are wild animals and that all tigers are quadrupeds, we can come to a reasonable conclusion that all tigers are wild animals because they share a set or category: quadrupeds. This commonality allows us to draw a new inference from information we already possessed. Yet, it may not be the case that all quadrupeds are wild animals, and thus the argument would not be true even though it makes proper connections. The information in the premises must say something true about the world. Here is the observation that one makes in order to form good, sound arguments. Recall some of the statements of earlier thinkers. These assertions may be true or not true, but many assertions of the early Greek physicists are mere observations that are taken as universals. That they are not universal, or perhaps even not true, is what seriously complicates an argument. One wants, of course, to make good inferences, so arguments must possess a commonality that allows good form, but the premises (observations) must also be true. If the argument has true premises and good form, then it is said to be sound. So, while validity in argumentation is necessary, soundness of argument is the true goal.
Aristotle talks of argumentation in terms of sets of things. Remember that he thought the eidos was in the material comprising something. An eidos in fact is a kind of general category of existing things that possesses a kinesis and an energeia. We have seen that these terms assist in explaining being-at-work or what-it-is-to-be [xv] for a thing, or its nature. So, when he observed things in nature, Aristotle was seeking the eidos of some particular thing, its four causes. He was looking not just for what mechanical cause made something happen, but he sought also the “why” and the arrangement of it. Thus, he looked for formal and final causes. [xvi] When he made observations about things, he was able to say of them that they always or never had certain characteristics or that some of them had and some had none of certain characteristics. He was looking for the part of something that made it was it is as well as that part of it that made it potential for becoming something else, or a part of something else. When there was a connection between two things, he was able to say something about the eidos of what he examined:
All of the set of quadrupeds belongs to the set of land-based creatures. So, nothing that is a quadruped is not a land-based creature. Because we know something about all quadrupeds (Let's assume this statement is true), we can say something about any other thing that belongs to this group. These a mathematician may call sets of things. When sets overlap, we can make statements about that overlapping, like the statement “All Tigers are land-based creatures.” What Aristotle sought in part was the connection that demonstrated how beings-at-work were motions in common, and these motions in common were causes in the sense that all causes had a kind of motion to them, which in turn was in part the formal and final cause and so the material eidos as well. Aristotle needed some kind of premise with which to make his arguments and he thus needed to observe first; he reasoned later with the observations made. The significance of Aristotle's contribution is that his metaphysics compelled him to create a method of observation and reasoning. He observed and sought what is the motion of a given thing and then reasoned in order to understand it more. The observation was for Aristotle a direct link to the nature of a given thing. Given the titanic influence Plato had on the ancient world, such an observational method concentrating on things that become rather than the being of things was a radical step towards what we now call science.
There were four kinds of statements that said something about the eidos, or the sets of different things. These are called categorical propositions. Notice the word “categorical.” One seeks categories of different things in the world and statements about them that are true. When these statements are made, they form sets of things that may be related to one another. The relation to one another reveals something new, another inference, about the subjects in the statements. One kind of statement made a claim about all of the members of a specific set of things. These are A statements:
All X are Y.
Every X is in the category of Y. The statement “All tigers are land-based animals” is just one such statement. All of the set of tigers are creatures in the set of land-based animals. Another statement made a claim about the exclusivity of the two sets. These are E statements:
No X are Y.
This statement means that of the set of Ys there are no Xs in the set of Y. This statement also means that none of the set of Y are in X because if there is no connection between Xs and Ys, then there must be no connection between Ys and Xs. These sets, or categories, are exclusive. “No tigers are sea-dwelling creatures” is one such statement. A third kind of statement means that some of one set of things is in another. These are I statements:
Some X are Y.
This statement means that there is at least one X, and it is in the same set of things as Y. There may be more than one X in the set of Y, but we are justified in saying that there is at least one. “Some tigers are animals friendly to humans” is one such statement. The fourth kind of statement is a negation of the third kind. These are O statements:
Some X are not Y.
These assertions mean that there is at least one X, and that X is not in the set of Y. “Some tigers are not animals friendly to humans” is one such statement. One can see how observation of the natural world is imperative to making claims about certain things. When a statement is made, it means that there has been no reflection upon this natural occurrence other than to observe how one natural thing relates to another. The fixed relationships between the statements can be summed up by the traditional square of opposition, which shows the truth values of the four statements with respect to one another:
Universal statements are on the top and particulars on bottom. Universals are the whole of a set of things. The particulars are simply one or more of a set of things but not the whole set of things; they are, however, not necessarily specific manifestations of things, or examples. Affirmatives are on the left and negatives are on the right. The statements that run diagonal to one another are contradictories, which means they always have opposite truth values. “Some tigers are gentle friends” is contradictory to “No tigers are gentle friends.” It cannot be that the set of tigers has within it no gentle friends and some gentle friends at the same time. Given that any observational proposition is true, this relationship is a constant one. In other words, there is a certainty about the relationship between the statements given the existence of no gentle tigers as opposed to some gentle friends that are tigers. That certainty reveals something about the eidos of tiger and the eidos of gentle friends. The same kind of certainty is true for the relationships between other statements. Contraries are only on the top of the square; they cannot both be true at the same time. The statement “All tigers are quadrupeds” and the statement “No tigers are quadrupeds” obviously cannot be true at the same time. In fact, if one is true, then the other must be false. Contraries can, however, both be false at once. The statement “All tigers are fierce and dangerous” and the statement “No tigers are fierce and dangerous” are both false, if it is true that some tigers are fierce and dangerous. Notice that either statement can be true, given certain circumstances, but those statements together cannot both be true. Again, there are sets and sub-sets of things in the world that in some way coincide with one another and thus have the same or interlocking relationship, a simple example being that all tigers being fierce also means that some tigers are fierce. One end of being a tiger is fierceness and without fierceness in the tiger, a tiger would not have the arrangement necessary to be a tiger. Strikingly, sub-contraries are the opposite of contraries. Always one of the statements is true. If you know that one is false, then the other must be true. Yet, they may both be true. The statement “some tigers are fierce and dangerous” and the statement “some tigers are not fierce and dangerous” can both be true, but they cannot both be false. There must exist or not exist this particular combination of characteristics in tigers. In sub-alternation the truth falls down, so if an A statement is true, an I statement is true. If all Tigers are Cats, then surely some Tigers are Cats. Additionally, if I is false, then A is false as well. If it is false that some Cats are Turtles, then it must be false that all Cats are Turtles. Yet, vice versa is not the case. If I is true, then A may be true. If some Apes are orange Creatures, then perhaps all Apes are orange Creatures, but we cannot make that determination based on the knowledge that some Apes are orange Creatures. Alternately, if it is false that all Cats are Tigers, then it is not necessarily false that some Cats are Tigers. True universal categorical propositions and false particular categorical propositions are the most potent statements. If one knows that a universal proposition is true or a particular proposition false, one is able to derive the truth of all the other statements. Yet, false universal statements and true particular statements only grant the truth value of the contradictory. The expectation when comparing these statements seems to be to find universal statements about the interlockings of what things are, or more precisely how they move together. When we find the interlocking aspect of the comparison, we have identified a cause, like a formal and final cause of tigerness is ferocity. It is only one cause of tigerness, but we know something about them by means of observation and argumentation. In this sense, Aristotle's method is observational, but it is not properly a scientific method. Again, he creates no hypothesis that he then proves true by empirical observation. Aristotle existed at a time when most thinkers presupposed the world to be static, or constant in some sense. Change existed, but change was alteration and re-alteration rather than development or permanent alteration or dissolution. So, all things with natures have an unchanging eidos that moves themselves to it. Alteration seems to have involved the growth and decay from and into their own eidos in particular manifestations. Aristotle would not have believed that animals and plants and living organisms of all kinds evolve in order to survive. Perhaps this perspective limited the followers of Aristotle, yet his method was potent indeed. These statements cover a great deal of the observed natural world if used properly, and Aristotle understood their constant [xvii] relationships to one another, as there are supposedly regular and reliable relationships of the things about which they speak a truth.
What results are some informative relationships between things in the world revealed through observations and articulated in language. In other words, there are aspects of these statements that are themselves kinds of constant relationships that can be relied upon and so are constant relationships between things.
Statements and thus propositions must have specific elements to them in order to draw good inferences. The first element we will discuss is that of distribution. Distribution is of vital importance if one is to make that essential connection between sets of things. When one argues with syllogisms, one must know something about an entire set, or category, of at least one class of things. A set of things is distributed when a statement makes a claim about all of its members. Such certain knowledge of an entire set is necessary because an argument needs to have a foundation on which to stand. There must exist some kind of fundament in order for new information to emerge, and such certainty is that of the knowledge about some aspect of an entire set of things.
One is able to say something about all tigers because they all belong to the set of fierce animals and the set of fierce animals, all of them, belong to the set of dangerous creatures (given the truth of the statements). One would not be certain that there exists some characteristic of tiger called “dangerous” unless there existed some certainty that all of them are fierce. Fierceness is a formal cause and a final cause to tiger only because there is a discussion about every tiger. If one were able to talk about only some tigers, then no necessary connection would be possible, only possible connections with an unknown amount of probability.
The relationships between specific statements always have particular facets. Given that the statements are true, one can establish certain relationships between things. There is a kind of clarity and certainty established by the working out of the specific relationships between categorical propositions, but it is important to recall that it is language-based. That is, the logic that emerges from the relationships between statements is mathematical in nature in the sense that there is knowledge of an entire set of things, but language expresses the observations rather than a mathematical formula. The reduced outline of something and the precision of mathematics is not Aristotle's aim. In fact, just as he complained about the Pythagoreans he would critique modern scientific method should he be alive today. Mathematics does not reveal that fullness of being-at-work in the universe of things. Later, mathematical reasoning will take a primary place, but for Aristotle's method logic and the observing of the overlapping of sets of things is language-based. Put simply, when one compares statements, one can see that Aristotelian propositions demonstrate certainties. There are three kinds of relationships worthy of note: conversion, obversion and contraposition. Each alteration of statements reveals specific relationships. Each relationship is one form of a constant, which is an important element in scientific (especially mathematical) reasoning.
Conversion means to switch the subject term with the predicate term. The conversion of an E statement or an I statement gives a new statement that is logically equivalent to the given statement.
E = “No tigers are fierce animals” becomes “No fierce animals are tigers.”
I = “Some tigers are fierce animals” becomes “Some fierce animals are tigers.”
The above statements mean essentially the same thing. Yet, converting an A or O statement does not necessarily make a logically equivalent statement. These converted statements may not have the same truth value as the given statement. So, given the truth or falsity of the first statement one cannot always determine the truth or falsity of the second statement.
A = “All tigers are fierce animals” is not the same truth value as “All fierce animals are tigers.”
O = “Some tigers are not fierce animals” is not the same truth value as “Some fierce animals are not tigers.
Note that only the subject and predicate have been exchanged. Otherwise, the statement is the same, yet the relationship between the existing things is not the same at all. Always these statements will have specific and reliable relationships to one another, given the truth of the observations. This reliability is one of the foundations of the logic that will in turn lead to inferences, which tell us something about the world that we did not know. Mathematics uses a form of these kinds of constants when it employs specific measurements like sine, cosine, tangent, π, πr² and the like. The truth value of a conversion or an A or O statement depends upon the content of the statement:
“All bachelors are unmarried men” becomes the equivalent “All unmarried men are bachelors”.
“Some apples are not oranges” becomes the equivalent “Some oranges are not apples.” Because of these given relationships, logically speaking, the converse of an A or O has undetermined truth value. Conversion may be used to provide the certain relationship between the premise and the conclusion of an argument:
Always will one be able to conclude the above, given the truth of the statements. That is one certainty of the Aristotelian system, though it is not true that Aristotle achieves absolute certainty or truth. The premise of each argument form necessarily has the same truth value as the conclusion. So, if the premise is true, the conclusion is necessarily true. On the other hand, one is able to prove that an argument is invalid:
If one says “All Cats are cuddly Creatures”, one cannot definitively make the statement “All cuddly creatures are Cats.” All of the set of Cats is in the set of cuddly Creatures, but that statement by itself does not say anything more than that. So, one cannot say something about all cuddly Creatures. The same is true for the O statements. Simply because some Dogs are not Creatures fond of cats says nothing necessarily about Creatures fond of cats. It only says that some Dogs are not in the set of Creatures fond of cats. Such a statement does not even mean that some Creatures fond of cats are not Dogs.
Obversion is another transformation of categorical propositions that Aristotle noticed have specific relationships. One changes the quality of the statements without changing the quantity, and one then changes the predicate with its term complement. The complement of a set is everything that exists outside the set. It is ordinarily expressed by using the prefix “non.” The term complement of “Tigers” is “non-Tigers.” The quality of a statement changes it from affirmative to negative or from negative to affirmative. So, “All Tigers are Animals” becomes “No Tigers are non-Animals.” One can see the relationships between the statements using a Venn diagram, which is a visual representation of the sets. Each circle represents a set of things. The “S” set is tigers and the “P” set is fierce animals. The colored areas are those areas where there is nothing in that section. The X represents at least one member of a given set; perhaps more but at least one.
One notices immediately that all the statements are logically equivalent, here represented by the shading of the same parts of the circles. One may suspect that obversion in itself says little or nothing about the expressed sets, but obversion provides one with immediate inferences that are as certain as other relationships between statements. Thus, the following statements are valid:
These are simple, straightforward statements easily seen to be true, and that is their virtue. They provide the foundation for building a complex argument that reveals something about the world one did not before know: the inference. Again, Aristotle's system is language-based, rather than mathematics based, yet there exists in it an element of mathematical certainty because of the reliable relationship between the propositions.
Contraposition is our final transformation of statements. In order to contrapose a statement, one switches the subject and predicate terms, then replaces both the subject and predicate terms with their term complements. The statement “All tigers are fierce animals” becomes “All non-fierce animals are non-tigers.” In half of these transformations the truth value is equivalent. In the other half, there come about significant changes. The Ss are the tigers and the Ps are the fierce animals.
The following conclusions can be determined by the contraposition of A and O statements:
Again, these statements are simple and they immediately reveal something about the universe, given that they are true. Contraposition also allows us to know when a statement is illicit. Each of the following are fallacies:
In the first example, one says something about some S, but nothing about non-P. One cannot say that because something is not P, it must be not S. The same is true for the second statement. The statements that remain transform into new assertions. “No S are P” contrapositions into “No non-P are non-S”, which means that all things outside the set of P are inside the set of S. The statement “Some S are P” contrapositions into “Some non-P are non-S”, which means that some things outside the set of P are also outside the set of S. All of these statements tell us something about the universe and assist in determining causes and thus reveal the being-at-work and what-it-is-to-be of different things. We will see that Euclid and Archimedes employ a similar kind of reasoning. They create systems of thought based on what they believe to be certainties. In the case of Aristotle, there are certainties of relationships between statements. In the case of the mathematicians there is certainty of mathematical propositions. But we need to cover the form of the syllogisms and some of the rules for logic that Aristotle articulated. These are intended to create guidelines for the most certain relationships that lead to clear and true inferences.
First, there is such a thing called a syllogism. A syllogism is a set of statements like we have seen repeatedly. The first two assert something about the universe and the third uses the commonality between the two to assert something new about the universe. Here is one of our examples:
We have two sentences that articulate observations about certain characteristics of animals, specifically tigers. The third, as we have seen again and again, makes use of their relationship to one another. One proceeds from an observation one believes to be true to new information. Such an argument is a categorical syllogism. A categorical syllogism is an argument that contains categorical propositions, each of which is one of the four types of propositions we have already seen:
A: All x are y.
E: No x are y.
I: Some x are y.
O: Some x are not y.
The standard categorical syllogism consists of two premises and a conclusion, each of which is one of the above forms. We have already seen many of them, but here is an example:
Note that the statements in this example happen to all be of type A. There is no necessity that a syllogism have all statements of the same type; any combination is possible. There are specific parts of a syllogism that need to be in specific places, and one must know the parts of a syllogism in order to form one properly. An improper syllogism will not provide the certainty of the relationships between the statements.
The Middle Term—The middle term is a word or a phrase that occurs in both premises. (The letter “M” represents it in the diagrams below).
The Minor Term---The minor term is the subject of the conclusion.
In a standard-form syllogism, the premise that contains the minor term (called the minor premise) is part of the second premise (as in Ex.1). In order to determine if a syllogism is in standard-form, one starts with the conclusion, checking to see if the premise containing the subject of the conclusion is in the second premise. If it is, the syllogism is in standard-form; if it is not, one must place the syllogism in standard-form by reversing the order of the premises (“S” represents the minor term in the diagrams below). The minor term must always be the subject of the conclusion in order to set the syllogism in standard form, but the minor term may appear as either subject or predicate in the minor premise.
The Major Term—The major term is the predicate of the conclusion. In a standard-form syllogism, the premise that contains the major term (called the major premise) is part of the first premise (see Ex. one). If the premise containing the predicate of the conclusion is part of the first premise, the syllogism is in standard-form; if it is not, one must configure the syllogism in standard-form by reversing the order of the premises (“P” represents the major term in the diagrams below). Although the major term is always the predicate of the conclusion, this term can appear as either subject or predicate in the major premise.
The Parts of a Syllogism Placed in Standard-Form
The above argument has the following structure or argument-form:
No P are M
All S are M
No S are P
P is the major term (in this case “tigers”), M is the middle term (in this case “creatures with gills”, and S is the minor term (in this case “fish”). The logical validity of a syllogism [xviii] does not depend on its content but only on its logical form, and the form of the argument is what gives it certainty. That is, there are specific relationships between these statements and a specific interlocking, or non-interlocking, that bring out clear and certain conclusions. This logical form is the certainty that Aristotle found in the relationships between these specific statements, similar to the relationships between the converted, obverted and contrapositioned above; these statements have certain relationships and that certainty is needed in order to form valid and sound arguments. One can see how standard form is important, if specific statements have specific relationships to one another. The form must be of a certain kind in order to make the connection. Logical form is akin to mathematical precision, and determination of the logical form of a categorical syllogism depends on two aspects of a syllogism, the mood of the syllogism and the figure of the syllogism.
The Four Figures of a Syllogism help determine the validity of a syllogism. These are structural parts of the argument without which the certainty of the relationship between the statements cannot exist. The figure of a syllogism refers to the positioning of the middle term. As we have seen, the middle term is the term that appears in both premises. There are four kinds of figures (i.e. four possible configurations of the middle term in a syllogism):
Notice that there is a shirt collar pattern below made by the placement of the middle term when the four figures are presented in order, 1-4. This pattern helps to determine what figure is what. In other words, in Fig. 1, the middle term slants down and to the right, i.e. it is the subject of the major premise and the predicate of the minor premise. In Fig. 2, the middle term is the predicate of both premises; in Fig.3 the middle term is the subject of both premises; and finally, in Fig., 4 the middle term slants up and to the right, i.e. the middle term is the predicate of the major premise and the subject of the minor premise.
The mood of a syllogism is the list of the types of categorical propositions that appear in a syllogism. Remember that the four types of categorical propositions are the following:
A All X are Y.
E No X are Y.
I Some X are Y.
O Some X are not Y.
The mood of example one is AAA (i.e. it has all A-type of statements as its component statements).
All humans are living beings.
One can determine the mood and figure of any syllogism with relative ease. Here are some other examples:
No spoons are wombats.
Some cosmonauts are grass-eaters.
No gorillas are dogs.
One indicates the figure of a syllogism by placing its number right after the three letters that represent that syllogism’s mood. Again, syllogisms must be in standard form in order for the determination of validity to take place. One cannot make a good determination of the relationship between the statement-types unless the statements are arranged properly. The premise that contains the minor term is called the minor premise and the premise that contains the major term is called the major premise.
Distribution of Terms As we have seen, a term is said to be distributed in a given proposition if that proposition makes an assertion about all of the members of the set denoted by the term. In the statement “All cats are animals”, the term “cats” is distributed, because we are saying something about all cats; but the term “animals” is not distributed because we are not saying something about all the animals. In “No dogs are cats” both terms are distributed; we are saying of all dogs that they are not cats, and also we are saying of all cats that they are not dogs. In the statement “Some cats are healthy animals” neither term is distributed. We are not saying anything about all cats, but only about some of them; likewise we are not saying anything about all healthy animals, but only about some of them. In the statement “Some cats are not healthy animals” only the predicate term, “healthy animals” is distributed; we are saying something about all healthy animals here; we are saying of all healthy animals that there are some cats that are not any of them. We want to find the connection between the statements and that connection refers to all of a specific set of things. Distribution makes this connection possible. One distinguishes valid from invalid arguments in part by knowing the distribution of terms:
A All S are P. ONLY THE FIRST TERM IS DISTRIBUTED
E No S are P. BOTH TERMS ARE DISTRIBUTED
I Some S are P. NEITHER TERM IS DISTRIBUTED
O Some S are not P. ONLY THE SECOND TERM IS DISTRIBUTED
Once we have understood these aspects of categorical syllogisms, we are able to use observations to form arguments and thus make inferences about the things in the world. There are five rules for determining the validity or invalidity of syllogisms according to Aristotle. [xix]
Rule one: No valid syllogism has two negative premises. Any syllogism possessing two negatives premises is necessarily invalid because there is no connection between the sets of things. Thus, the following syllogisms possessing the following moods are invalid: EEE, EEA, etc., EOA, EOE, etc. OOA, OOI, etc. One cannot draw an inference from these specific combinations of statements.
Rule two: If a valid syllogism has a negative premise, it must have a negative conclusion; if it has a negative conclusion, it must have a negative premise. All syllogisms violating this rule are invalid, so one can see immediately that syllogisms with the following moods are invalid: EAA, OII, AIE, IAO, etc. Here a similar relationship holds true. If there is a negative relationship, then the syllogism is saying that there is no connection, so it must conclude that there is another lack of connection between sets of things.
Rule three: The middle term of a valid syllogism must be distributed at least once in one of the premises. A syllogism with a mood of IIA, IIE, etc., will be invalid, because the middle term cannot be distributed in a syllogism that has two “I”-statements as premises. “I” statement-types distribute none of their terms, and so there is no interlocking or connection between the sets of things it discusses. The figure and mood strictly determine if the middle term is distributed. Their strictness is mathematical in that, again, a firm connection must emerge between sets of things. If the middle term does not describe an entire set of things, then one cannot say something about all of them. A numeric quality to the word makes language potent in an argument. The following syllogism-form is AAA-2:
The middle term is distributed in neither premise, because the first term alone in an “A” statement is distributed. So, the middle term is not distributed. This syllogism-form commits the Fallacy of Undistributed Middle, and so is invalid. Here is an example:
This statement says nothing about the set of all animals, and so there is no necessary reason that because cats and dogs are animals they are the same animals. The necessary reason would be present should the term that makes their connection certain be distributed. The middle term must be distributed only one time for the necessary connection to be present. Thus, AAA-1 is a valid syllogism form; an AAA-1 syllogism distributes its middle term in the major premise:
This syllogism is valid because it makes a statement about an entire set of things, humans, that bridges the gap between the other statements. All men are humans, but all humans are animals so all men is a subset of all humans and all humans is a subset of all animals. All men must be a subset of all animals. Here is the certainty, and that certainty is reached only when the syllogism has a specific form to it. One bases a conclusion on the observation inherent in the first premise.
Rule four: Any term distributed in the conclusion must be distributed in a premise. This rule concerns the major and minor terms. If they are distributed in the conclusion, they must be distributed in the premises as well. The same reason applies. If one says something about the whole set of things in the conclusion but they did not make an assertion about the whole set of things in a premise, then there is an unjustified leap of reasoning because the whole set of things is the connection between terms. One cannot say that some tigers are fierce and then suddenly say all tigers are fierce. Thus, a syllogism with an “E” statement conclusion will only be valid if both the major and minor terms are distributed in the premises as well. Should the conclusion of a syllogism be an I-statement, clearly one would not need to concern oneself about this rule, since neither term is distributed in its conclusion. Should the conclusion of a syllogism be an A-statement, and thus the minor term is distributed in the conclusion, one would be compelled to determine if it is distributed in the minor premise as well; should the conclusion be an O-statement, and thus the major term is distributed in the conclusion, one would be compelled to determine if that term is distributed there as well. Should a syllogism have a term that is distributed in the conclusion but not in the premise in which it appears, that syllogism would be necessarily invalid. Two fallacies may arise from violating this rule: Fallacy of Illicit Major and Fallacy of Illicit Minor.
Any EAE-4 syllogism we know has the following structure:
Both terms are distributed in the conclusion, so the conclusion says something about all of the set of S and P. A valid syllogism of this kind, according to rule four, must possess distributed S and distributed P in order for the connection to be necessary and strong. S is not distributed and P is distributed, so EAE-4 commits the Fallacy of Illicit Minor. It is invalid. If we look at a syllogism of the form AEE-1, we see that a slightly different problem arises.
Both terms are distributed in the conclusion, but the major term P is not distributed in the major premise because we are not talking about the whole of its set. So AEE-1 commits the Fallacy of Illicit Major. It also is invalid. Again, in order for Aristotle's kind of logic to be correct, it must say something about whole sets of things. If no whole set of things is assumed, then the argument makes no necessary conclusion and one cannot draw proper inferences. Thus, the certainty of the needed connection takes on a mathematical quality not only because it makes an assertion about an entire set of things, but also because the validity of the argument relies on the necessary connection resulting from statements that are certain. The statements must be categorical indeed in order for basic Aristotelian logic to operate correctly.
Rule five: One cannot draw a particular conclusion from two universal premises without making further assumptions in addition to the two premises. This much more modern [xx] rule relies upon the assertion that universal statements are interpreted to make no claims one way or the other about the actual existence of the things referred to by their terms. If one asserts that “All sphinxes are unnatural creatures”, according to this interpretation of the meaning of universal claims, one really only says “If anything is a sphinx, it is an unnatural creature”, and one is not actually asserting that sphinxes exist. But if one makes a particular affirmative or particular negative (an “I” or an “O”) assertion, one is claiming that something exists; “Some S are P” implies that at least one S and P exists, and “Some S are not P” implies that at least one S exists. Rule five thus asserts that one cannot draw a necessary inference from premises that make no assumptions about existing things. In other words, if one says something about things that do not necessarily exist, then one cannot say in the conclusion that they exist. This rule asserts that syllogisms with the moods AAI, AEO, EAO, etc., cannot be unconditionally valid. Such syllogisms have conclusions that imply the existence of some members of the sets denoted by their terms, but premises that by themselves do not authorize such assertions. These syllogisms commit the Existential Fallacy. Still, if a syllogism does not violate the other four rules but only violates rule five, it can be considered conditionally valid–valid if one assumes that the relevant sets denoted by the specific terms actually exist. For example:
This syllogism is AAI-1. It commits only the existential fallacy and so it is conditionally valid, valid on the condition (in this case) that leopards and big cats really exist. If the last premise had been “All leopards are living beings” the argument would have been unconditionally valid because, as interpreted here, universal statements do not make claims about existence; “All big cats are living beings” only means “if there are any big cats, they are living beings.” In that case the rule five would not be violated. If one chooses to interpret universal statements as making claims about existing things, we would be following Aristotle more closely, who in his discussion of the square of opposition asserted that one can validly infer “Some x are y” from “All x are y”. But Aristotle assumes that the ‘x’’s and ‘y’s in question exist. Rule five asserts that if one proceeds from universal claims [like A- or E-type statements] to a particular conclusion [an I- or an O-type statement] then one attempts to infer the existence of at least one thing from premises that do not assert the existence of anything. Yet, in Aristotle’s square of opposition the A- and E- type statements do assert the existence of things. We will follow the ancient tradition because we are talking about ancient science, but it is appropriate to know the modern rule in order to understand how in some instances the ancient kind of thinking differs from the modern.
Here are the five rules for valid syllogisms, including the existential fallacy:
Rule 1: No valid syllogism has two negative premises.
Rule 2: If a valid syllogism has a negative premise, it must have a negative conclusion; if it has a negative conclusion, it must have a negative premise.
Rule 3: The middle term of a valid syllogism must be distributed at least once.
Rule 4: Any term that is distributed in the conclusion must be distributed in a premise.
Rule 5: You cannot draw a particular conclusion from two universal premises (without making further assumptions in addition to the two premises).
Again, if a syllogism does not violate a single rule, it is unconditionally valid. If a syllogism does violate a single one of the first four rules, it is invalid. If a syllogism conforms to rules 1-4 and only violates rule 5, then it is conditionally valid. Such a syllogism could be valid if certain assumptions are made regarding the existence of the sets discussed. One may see easily how Aristotle's rules and his system work by looking at Venn diagrams, which are visual representations of arguments (already seen above). Venn diagrams were invented by John Venn more than one-hundred years ago, but they are so effective that they remain in use today. Each circle represents a set of things in the universe.
The above circle represents all the cats that exist in the universe, but the set of cats is quite large.
So, one needs to separate the sets in order to see what set belongs to what category. Each circle, or set, represents something about the set of cats.
The set of tigers, for example, belongs completely inside the set of cats, as does the set of house-cats. Yet, clearly there are differences between them. So, one needs to determine where they overlap in order to demonstrate the difference.
When there is nothing in a set of things represented by the circles, that area is shaded. The above set of circles indicates that no house-cats are big cats. One can use the circles to represent any set of things.
When there are some or at least one in a set of things, an “X” represents that particular in a set. In the above set of circles, some house-cats are tabby-cats.
One may then use three circles to represent the three sets of things in any argument. So, the above circles indicate “Cats”, “Dogs” and “Animals.” All of the set of cats and dogs are in the set of animals and none of the set of cats are in the set of dogs nor is the set of dogs in the set of cats, but not all animals are in the set of cats or the set of dogs. Notice that each of the circles represents a number of things that may exist in the universe. One can represent simple arguments with these circles, and how things relate to one another becomes clearer.
The above diagram illustrates the following argument:
The above diagram illustrates the following argument:
When only some of a particular set find themselves in another particular set, the overlapping has an X in the space where the two sets overlap. The above diagram illustrates the following argument:
For our purposes the circles represent not only sets of things and therefore they have a mathematical element to them, but they also represent the eidos of Aristotle, so to speak. In other words, they are different sets of things that share a kind. There is some aspect of a cat, for example, that transcends the set of house-cats. It is some arrangement of the creature: four legs, fur, a specifically shaped head, a tail. Cats have specific ends, which is to say that cats become, well, cats. They move toward the end of being cat. As many sets of things as one wishes to examine are as many sets that may be formed into an argument. But each argument represents a number of sets of things. Aristotle's rules of logic demonstrate how sets of things interlock and interact with one another. Additionally, Aristotelian argumentation is a classification of things based upon observations. When the sets of things have specific relationships, like we saw when we compared how each statement-type relates to other statement-types, one can draw an inference about them. One knows more about the universe. Thus, Aristotle had a decidedly inductive aspect to his methods. [xxi] He observed and used observations in order to make inferences about things that have a nature. Then, he argued about their function. What he discussed was biological formation of animal parts, or even the reproductive organs of animals. Again, he was the first biologist.
So, Aristotle explained the physical processes of animals and things in part through causes. These causes were the material, motive, formal and final causes. Each one of these causes can be interpreted as influenced by the Platonic eidos. The Platonic eidos was a deeper aspect of reality, one which had no spatial or temporal “being” to it. Aristotle makes his formal cause the arrangement or organization of what something is. An organization of something is what a thing with a nature moves into, which is itself. Movement is thus an aspect of something's being, a kind of being-at-work accomplishing the task of becoming itself. Its work-being may be called energeia, which the term from which we derive the word energy. It literally means “the work within” or “in the work.” This activity is a kind of actuality that in turn is a potential to be something else. Thus, bronze is an activity that makes bronze into itself, and bronze is the potential to be a sword, or a statue, even a candle holder. So, the formal cause moves into itself in order to attain its end, which is the final cause. The final cause is the goal or end of something, which again is itself. The goal of the bronze is becoming bronze, or the activity that leads to the becoming that comes to be and continues to be bronze. In this way the formal cause and the final cause are the same, since the arrangement that makes something itself is its own end, at least for Aristotle. The motive cause is the most scientific cause that Aristotle articulates. It is the activity of one thing that begins the activity that is another thing. A man reproduces and creates another human through this cause. The act of one boulder striking another is a motive cause of an avalanche. These are the activities that come from each thing doing what it is that it is, like a tumbling boulder.
Aristotle had almost no interest in this cause, and it is this cause that most closely resembles the sense of scientific cause so prevalent today. In fact, the formal cause, the final cause and even the material cause are virtually non-existent in the discussions engaged in by the scientific community. The material cause may be interpreted in several ways, but it is essentially necessary that if something has an activity that makes it what it is, and if what it is ultimately has a formal and final cause to it, then even material has a formal and final cause to its becoming. In that sense, material itself is form, and because the eidos of Aristotle is a part of form and arrangement, all of Aristotle's causes have an aspect of eidos to them. The eidos is investigated by the observations that lead to inferences, which in turn present new and deeper insights about things in the world. So, the investigation into how sets of things overlap and interlock is the investigation of the eidos of things and thus their causes. Aristotle has developed a method that brings out the activity of something that explains not only how it influences objects around it, but also what it is and what purpose it has. It is in this sense that Aristotle's thought is scientific; it is analytic. His inductive method means that we are able to say something about how the different sets of things are part of, potential for, subject to or even responsible for one another. The certainty of a properly ordered argument is the kind of certainty Plato sought in the immaterial world. It is the eidos of Plato put into matter, and that is what moderns call metaphysics and not science proper. The deductive aspect of Aristotle's thought is the clarity of being-at-work and what-it-is-to-be of a given thing. We have not only found what something is, but we have found what-it-is-to-be, or its potential by making use of certain statement relationships as well as specific forms of propositions (premises). The resulting insight grants human being enormous power over what it analyzes, and it gives much more precise analysis to those who seek to control and manipulate nature. Such is the potency of his analytic method. In other words, we can be certain of some partial relationships. We can be certain of other complete relationships. We can make universal statements about things in the world.
Science answers the question how something has come to be and Aristotle is scientific in the sense that he is analytic, but he also has included the standard sense of cause in his investigation. In this sense, Aristotle's analytic method is more broad and ambitious than science. Many hard scientists believe that philosophers want to obtain absolute answers, and some do. But the most worthwhile of thinkers are philosophical in the sense that they admit to ignorance. They do not claim to know. Aristotle's analytic system is on the whole desultory. It seems as if he sometimes does not conform to his own expectations because he was able to classify his topics in a more systematic way, but he does not appear to do so. Additionally, what remains of his corpus is a mess of writings that may not have been intended for anyone to read. So, we do not know the extent to which Aristotle's work was systematized. There is much that has been lost. There may well have been long works detailing the causes of different animals, political systems, being and other topics that have fallen into oblivion through neglect or deliberate destruction. Whole scores of Aristotelian followers may have amassed observations that transformed into notions of how things work. We do not know fully what the Aristotelian school produced. Yet, what we have tells us that his analytic system was capable of classifying practically any topic, which is precisely what science has done with virtually any human inquiry today. We will see other thinkers, a new kind of mathematician, make still stronger arguments, but they do not analyze nature. They analyze a framework that leads to hypothesis, which will eventually be empirically verified. Aristotle investigated in a more full sense what is there.
[i] See Glanville Downey.
[ii] It is important to note that Aristotle's surviving work is sometimes contradictory and corrupt, resembling the desultory state of a rat's nest. What scholars possess are parts of his esoteric works, or works that were not meant for what we now call publication. Some scholars believe these writings to be notes, perhaps for purposes of lecture or student notes. The state of his work makes interpretation quite difficult, and how one interprets Aristotle depends largely on what one thinks he must have meant. Our interpretation concentrates on his causes and the genesis of his logic rules. Like Plato, more thought is always necessary when interpreting Aristotle, no matter how much one has already contemplated his work.
[iii] Aristotle thought that one could argue to first causes, which may mean that all of the above-listed parts are necessary in order to determine a first cause for a given science. That is why we talk about many aspects of his thought, in order to understand why there is one sense in which he has no first principles and in another sense he asserts many. Jaakko Hintikka puts it succintly, saying a particular science “is characterized by its subject matter, that is, by the genus of objects it is about.” Aristotle even states that “a single science is none whose domain is a single genus.” (Post. An. I, 28, 87a38). For more, see Wilson, Irwin, Hintikka.
[iv] De Santillana and Reiche describe how the Platonic eidos is shifted from its metaphysical place in Plato to an actualization that manifests potential in Aristotle. Rihill believes, perhaps rightly, that ancient thinkers “did not so much stand on their predecessor's shoulders as knock them down, step over them, and go elsewhere.” Perhaps it was not quite so eristic. Loux and Lewis articulate some interpretations as to how form and matter combine in particulars. Marjorie Grene points out correctly that Aristotle's use of eidos and genos is chaotic.
[v] The phrase being-at-work is taken from the Joe Sachs translation of Aristotle's Physics.
[vi] He did, however, accept Empedocles' four “roots” as elements of the moving universe, and Aristotle did believe that one can argue toward first principles (See Irwin). Matter seems to have as part of its structure a play of opposites, which is one way of interpreting earth, air, fire and water. Matter may also be thought of as a substratum, which may also be called prime matter (Patrick Suppes), but the nature of something, as a material eidos, is more its “what” than the matter comprising it. For a good review of the Aristotelian causes, see Falcon.
[vii] One must be careful here to note that interpreting prime matter as an indeterminate mass of potentiality is the traditional view. A new view that demands prime matter to be of some kind of characteristic is emerging. For the traditional view, see Luyten, H.M. Robinson, For the emerging and other views, see Barrington Jones, Margaret Scharle, Sarah Broadie (who claims that substantial change does not require prime matter). M MacKinnon suggests how this belief came to be. Traditionalists try to demonstrate how prime matter is a something, like spatial extension; it is able to move and cease motion. See also Charlton and Fine. One must acknowledge Aristotle himself: “Matter qua matter is purely potential and without attributes.” (Metaphysics, 1029a19) and “the matter of the heavy and light, qua their matter, would be the void.” (Physics, 217b22).
[viii] Mohan Matthen and R.J. Hankinson describe what amounts to the fullness in Aristotle when they talk about Aristotle's non-reductionist philosophy; the “parts are ontologically and causally subordinate to wholes.” His way of analyzing things recognizes not only the constitution of the whole by the part, but material causation is subordinate to formal causation. Such subordination is another way of making the eidos primary. E. Drabkin touches on the same issue when he states that conceptually force remains a qualitative conception, though Carteron correctly points out that forces cannot be entirely subordinated because they represent an early attempt at quantitative formulation. So, principles like inertia do not receive the bare-bones treatment needed for a proper theory of dynamics in Aristotle. Aristotle thus theorizes that a force constantly applied to an object will merely continue that objects motion rather than continually accelerate it (see Drabkin and Carteron). For these scholars Aristotle did not sufficiently abstract from the particular cases sufficiently to think in terms of the primary actors without all of the accidents, like elements that interfere with motion. One must always acknowledge Aristotle: “...a body must either remain at rest or be moved ad infinitum unless something stronger obstructs it.” (Physics IV, 8, 215a 19-22). He rejected the idea.
[ix] Richard Pettigrew talks of a special way that geometry belongs to the sensible realm. His interpretation differs from a more traditional view, like that of Edward Hussey and Ian Mueller. Because Aristotle writes in a dialectical manner, there is much debate about quantitative and qualitative reasoning and many other issues. Joe Jones explains what may be thought of as intelligble matter as it relates to magnitude and thus mathematics. The issues of math, matter and magnitude are by no means clear-cut in Aristotle. For a better understanding of the differences between Aristotle's syllogisms and mathematical reasoning, see Kneale, Hintikka, Mueller, Wilson.
[x] One can see, then, Aristotle's adaptation of the Platonic eidos as something moving from and into its own manifestation through and as matter. Its manifestation is itself a material eidos, but it is an eidos that moves into potential and actuality. Plato called this act of becoming not being, but participation. Participation is matter taking on an eidos as its fundamental reality, matter not really being anything other than that primordial formless and formable mass. In some ways the Platonic eidos is form, in some ways shape, in some ways motion, but it is not well articulated in any of these phrases. In sum, Aristotle brings the Platonic eidos into matter and becoming and so makes of it something empirically analyzable and graspable. In brief, the Aristotelian eidos is much more scientific.
[xi] For an investigation into the phenomenological tone of Aristotle's observations, see Sean Kirkland, Dialectic and Proto-phenomenology in Aristotle's Topics and Physics.
[xii] De Santillana and Reiche have an interesting take on the desire for the prime mover. It is the Good of Plato. The celestial sphere is perfect, as Lear vehemently asserts when he claims that mathematical properties are perfectly instantiated among the satellites.
[xiii] Andrea Falcon points out the ambitious project of Aristotle, which attempts to establish the correct connections between things. These are rightly called explanatory connections and the whole of the universe is causally related as it is interconnected with form, matter and relationships. Nature is a topic of study materially, but not merely materially, if we are to trust Falcon. Patrick H. Byrne articulates the stages of Aristotle's scientific investigations.
[xiv] This translation is A. J. Jenkinson's. Found here: http://libertyonline.hypermall.com/Aristotle/Logic/Prior-Analytics.html
[xv] The phrase “what-it-is-to-be” is taken from Joe Sachs translation of Aristotle's Physics.
[xvi] Aristotle explores formal and final causes of biology in his Parts of Animals. He explores material and motive, or efficient, causes in his Generation of Animals. He seems to have been the first biologist, but strangely he did not make use of his own logic rules and observations in order to create a systematic account of a given topic, as one would expect. Perhaps he left such work to his students or perhaps the systematic works that he wrote have all perished.
[xvii] Aristotle not only established rules for logic but he also gave to Euclid many of the methods found in Euclid's Elements. The constants that appear in Aristotle's logic are mathematical in the sense that they are reliable relationships whereas mathematical constants are abstractions that are always true. There is debate on the differences between the two men's theories. Solmsen, Barnes and Ross argue over the origin of the theory of demonstration. H.D.P. Lee claims that Aristotle's definitions correspond to Euclid's definitions, Aristotle's axiomos to Euclid's common notion, and Aristotle's primitive existence claims (hypotheses) to Euclid's P1-P3. Yehuda Rav points out that modern mathematicians do not work with axioms even though they may be necessary and the division between axioms and definitions is not marked. There even exists a work that put Euclid's first six books into Aristotelian syllogisms: Analyseis Geometricae Sex Librorum Euclidis (Herlinus and Dasypodius). For a detailed explanation as to why the Aristotelian syllogism cannot be used strictly in a Euclidean proposition, see McKirahan. For more on this issue in general, see Taisbak, Maziarz and Greenwood, Russo, Bloom, Lear. It is best to remember that logic is not monolithic, but malleable and Aristotle's and Euclid's systems are one of many permutations of logic.
[xviii] Remember the difference between logical validity and logical soundness. A syllogism can be logically valid even if it has false premises or a false conclusion; indeed, it can be logically valid even if it contains nonsensical statements, provided only that if the premises were true, the conclusion would have to be true. Recall too that all sound arguments are valid arguments having all true premises.
[xix] Although Aristotle realized the importance of the concepts of figure, mood and distribution and understood most of these rules, the following formulations of them are taken from Patrick Hurley’s AConcise Introduction to Logic. Aristotle's articulation of logical relationships and logic rules is arranged quite differently and what we here review is a distilled version of his logic analysis. See Aristotle's Prior Analytics and Posterior Analytics for his much more nuanced understanding of logic.
[xx] Aristotle is not responsible for rule five. Rule five came about as the result of modern logicians analyzing logical inferences. Aristotle would have believed that statements about entire classes of existing things are just that: statements the extend over an entire, existing class. Modern science is inductive. It does not make absolutely universal claims about nature and the laws of nature. Ancient thinkers, however, did.
[xxi] See introduction to Generation of Animals. Aristotle, Generation of Animals, Harvard University Press, 1979. Trans. A.L. Peck. Loeb Classical Library.
Life and Works
Not much is known about Lucretius' history. He was an Epicurian poet who wrote in dactylic hexameter and a pioneer in the writing of philosophical poetry in Latin. His extant work, De Rerum Natura,i is beautiful and unusual though he may not have put the final touch to it. Its primary topic is physics and Lucretius is known for his account of the clinamen, or the swerve, which has come to be a part of the discussion about ancient atomism and physics in general. He was born perhaps around 94 BC and died in 54 or 51 BC, but the sources of this information are not reliable. Cicero's claim that Lucretius has “flashes of genius”ii tells us that he wrote his poem in the first century BC. Early Christians vilified Lucretius because he was not a believer in any religion, one saying that he was driven mad by a love philter and that he died by his own hand,iii but later scientists and philosophers were influenced by him.iv Four fundamental themes of De Rerum Natura are universal causal explanation, the elimination of fear of the world through reason, free will and the soul's dissolution after death.v We concentrate here on universal causal explanation. Lucretius calls the material causes of things semina and primordia rerum and also materies, but he does not use the term “atom”, which would have been available to him. Semina means “seeds” and primordia rerum “the first-beginnings of things”. Materies comes from the Latin for “mother”. We will use the term primordia to signify all of these for simplicity and convenience, but know that Lucretius uses different terms and alternately phrases in order to indicate the same minuscule bits of material reality.
Following Democritus, Leucippus and Epicurus, Lucretius asserts that being always existed and always will exist, similar to Parmenides and Empedocles. Nothing comes into being out of nothing:
No thing ever is produced from nothing by a god. (DRN, 1.150). ...for if things came to be from nothing, every kind of thing would be able to be born from all things, and nothing would require a seed. Men would be able to sprout first from the sea and the scaly stock from the earth, and birds, cattle and herds out of the sky. (DNR, 1.159-62). Naturally, where there would be no generative bodies for each, what would be able to stand as a fixed mother to things? But as it is, since each thing is created from fixed seeds, from there each thing is grown and goes out into the bounds of light where matter and the first bodies of each inhere. (DRN, 1.167-71). ...whatever is created discloses itself in its own time when fixed seeds of things have flowed together, the seasons are at hand...(DRN, 1.176-78). Since if they (roses, grain, vines) came to be from nothing, they would suddenly arise at an unfixed place and during strange parts of the year, naturally there would be no first beginnings which would be able to be shut out of a generating union at the wrong time. (DNR, 1.180-83).
Nothing is broken down into nothing and motion, conflict and force govern eternally-existing bodies:
...if anything were mortal in all its parts, each thing would perish from our eyes, suddenly snatched. For no use would exist for a force capable of arranging the separation of the parts of each and able to dissolve its bond. But now, since each thing consists of eternal seed, until an opposing force that cleaves the thing with a blow or penetrates inside through empty spaces and disunites it, nature allows the destruction of nothing to appear. (DRN, 1.217-224). Finally, the same force and cause would sweep away everything everywhere, unless they were held together by eternal matter entangled more or less in its own bond with itself. (DRN, 1.238-240).
...the wind...rages furiously and becomes fierce with a threatening crash.... [T]here are concealed bodies of wind which scour the sea, the lands, and finally the clouds of the sky and carry them suddenly, tossing them in a spinning wind. (DRN, 1.275-279). (W)e sense at a distance the different smells of things, but nonetheless never do we see them coming to our noses. (DRN, 1.298-99). Finally, clothes suspended on the wave-breaking shore grow wet, but spread out in the sun they dry. (DRN, 1.305-6). Thus into small parts the fluid scatters, parts eyes are in no way able to see. (DRN, 1.309-10). Indeed year upon year a ring on a finger grows slenderer beneath by wearing, the fall of trickling liquid makes a stone hollow, the iron hook of a farmer's plow diminishes unnoticed in fields and we observe the stone-spread paths now worn down by the feet of a crowd. (DRN, 1.311-16).
Nor are bodies held everywhere compressed through all nature, for there is in things a void. (DRN, 1.325-30). If void did not exist, in no manner would things be able to be moved. For the function of the body that stands out, that which is the natural role of body, to stand before and obstruct, would be present at each time for all things. Nothing then would be able to advance, since no thing would grant a beginning of retreat. (DRN, 1.335-39). If void did not exist, these things (oceans, lands, heavens)...would in no manner have been brought forth at all, since everywhere compressed matter would be inactive. (DRN, 1.343-45).
Objects possess empty space because of their void:
For if there is so much of body in a ball of wool as in a ball of lead, just so much of body is there equal to hang, since the function of body is to press everything down. But by contrast void's nature remains: to exist without weight. Thus the thing which is equal in magnitude but appears lighter...makes clear that there is more of void. Yet again the heavier thing affirms there is in it more body and that within it has much less empty space. It is thus...that this thing we seek exists, mixed with another thing, that what we call void. (DRN, 1.360-69).
Everything that seems to have existence of its own is composed of these two aspects of material reality:
Moreover that which exists always through itself either will act on something or will have obligation to be engaged by other things acting upon it, or it will exist in such a way that in it things are able to exist and to be carried. But there is no thing capable of acting or of being engaged without a body, nor again able to provide a place unless it is void and empty. Therefore besides void and bodies no third nature by itself is able to remain in the number of things that would fall under our senses at any time, nor anything anyone with calculation of mind would be able to reach. For the things which are always spoken of you will discover are joined together with these two things, or you will see their accidents. (DRN, 440-450).
There do exist inseparable or accidental properties of bodies and void, but they are joined (coniuncta) and incidental outcomes (eventa).
Whatever is nowhere able to be divided and sundered without a ruinous disintegration is a thing joined together, as weight to rocks, heat to fire, fluidity to water, tangibility to all bodies, and intangibility to void. (DRN, 1.451-54).
Slavery and poverty and wealth, freedom, war, harmony, and the rest by arrival and departure of which the nature of things remains whole, we are accustomed to call...accidents. (DRN, 1.455-58).
The minute parts of any given object are tiny and imperceptible. Science calls them atoms, but Lucretius called them primordia rerum or the indestructible primordia of things:
Furthermore, bodies are in part the primordia of things, in part those which consist in the bond of their origins. But the things which are the primordia of things no force is able to extinguish; for with dense body they at length conquer. (DRN, 1.483-86).
These first-beginnings Lucretius also calls semina or “seeds”viii and sometimes materiesix or “matter.“ These first-beginnings are uncut, everlasting, yet they posses parts that would not exist except for the combination that they are, which in turn makes the first-beginnings:
For wherever always space is vacant, what we call void, in that place no body exists; where far off body holds itself, there in no manner is empty void. Thus first bodies are dense and without void. (DRN, 1.507-10)
Then further, since there is always an outermost extremity of that body (the atom), which our senses are now unable to discern that...is without parts, and consists in the smallest nature, and neither did it ever exist cut from itself, nor will force exist for it to be cut, since it is on its own a part of the other (body) both first and unified, and then some and other similar parts in rank fill the nature of the body (of the atom) in a condensed moving collected multitude; parts which, since they are unable to exist separately, must necessarily adhere to that from which they cannot in any manner be torn away. (DRN, 1.599-608).
In other words, the first-beginnings are themselves indestructible, but if we were able to break them apart, their constituent elements themselves would vanish. Lucretius does go on to say that these primordia rerum do not have parts:
For although fundamentally the height is infinite, nevertheless the bodies which are smallest will equally be composed of infinite parts. But since true reason cries out and denies that the mind is able to trust it, it is necessary to grant that there are finally things furnished with no parts and come from the smallest nature. (DRN, 1.620-26)
What he seems to mean is that these elements of things are the smallest of things that can have generative properties, like connections, weights, blows, concurrences and motions:
Lastly, if nature the creator of things had been accustomed to compel all things to be unbound into their smallest parts, the same nature would be completely unable to retrieve again anything out of them, since because they are composed of no parts they cannot have what generative principles must have, various bindings, weights, blows, assemblages, motions, through which each thing is produced. (DRN, 1.628-34).
The modern tendency is to call them “atoms”, but the term “atom” comes from Democritus and Leucippus, which means “uncut” or “uncuttable” as we have seen. Since what we call atoms in Lucretius' system are fundamental and unbreakable elements that themselves have parts and since they can be separated, they are not the Democritean atoms. They are similar to our atoms in that they are fundamental pieces of the universe that themselves are not the most fundamental part of material reality,x and in one sense these “atoms” are singular and unified ever smaller pieces and in another sense they are the fundamental, unbreakable pieces of material reality that comprise all bodies. One may make an analogy between modern atoms and sub-atomic particles that comprise them, as with Plato's reception of the Democritean atom.
Body and void exist together with these primordia as the objects we see and make up the dynamic that allows for outside and inside forces to change things:
In addition, unless matter were eternal, before now each thing would have returned absolutely to nothing, and from nothing would be born again whatever we see. But since...nothing can be brought forth from nothing and what has come about is unable to be restored to nothing, primordia must exist immortally in body into which each thing may be dissolved at its last moment, so that matter may be upcoming for the remaking of things new. (DRN, 1.540-47).
Primordia are part of the procreative matrix of the universe, yet they have their own generative nature. They are the pieces that make and the parts that are made into things from the connections, weights, blows, concurrences and motions of the universe. The universe for Lucretius is infinite, extending infinitely with what we call gravityxi as a fundamental force of change:
In addition, if all space in the universe together stood enclosed in fixed boundaries on all sides and were limited, by now the abundance of matter would by its dense weight have flowed together from all parts to the lowest place, nor could anything be accomplished under the cover of sky, and sky would not exist at all nor light of sun, because...all matter would have been lying already piled up from settling down through infinite past. (DRN, 1.984-91).
The universe possesses infinite matter mixed with void:
Furthermore, so that the sum of things is not able to arrange itself nature stands guard, nature who compels what is body to be bounded by void and what is void by body so that by alternates she delivers an infinite universe, or as yet one or another, if one did not bound the other, would yet simply open itself without limit. (DRN, 1. 1008-13).
Causation and Cosmology
The first-beginnings move constantly and through movement they change the things composed of them:
For since the primordia of things wander through the void, it is necessary that all be born either by their own weight or by the accidental blow of another. For when roused and having met they often collided, it comes about that having separated they leap apart suddenly; and no wonder, since they are hardest in their dense weight and nothing opposed them from behind. (DRN, 2.83-8). ...there is no lowest limit in the sum of things nor do the first bodies have place to stand, since space is without end or limit, it extends boundless from all sides in all parts, a thing shown to be demonstrated by many things and by certain reason. No wonder no rest is granted to the first bodies throughout the deep void, but rather agitated by constant and varied motions, some pressed together then leap back along wide distances, some even in small spaces are shaken by a blow. (DRN, 2.91-8).
Somewhat stable arrangements of primordia make up the objects we perceive and our own bodies:
And those heaped together in a more condensed union leap back through narrow spaces, caught fast in their own interwoven shapes, these constitute the vigorous roots of stone and the savage bodies of iron and the rest of their kind. (DRN, 2.100-04).
These arrangements are in motion, though we do not perceive it.xii
...since all the primordia of things are in motion, the sum seems nevertheless to stand in supreme rest, except for anything that may give motion with its particular body. For the whole nature of first things resides far beneath our senses; on account of which where you cannot see them themselves they necessarily keep their motions from your sight, especially when things that we are able to perceive do often hide their motions when separated a great distance. (DRN, 2.309-16).
The void and matter present in objects account for their density; Lucretius believed that because body and void mingle in objects, each thing has its own weight. No body can move upward unless it is affected by some force.xiii Bodies fall downward of their own accord, but they do not fall without a somewhat arbitrary interruption. If they did, then there would be no collision and thus no coming to be of anything. Gravityxiv is a fundamental causal agent in the matrix of the universe, but it cannot produce objects on its own because blows and collisions are needed to form things, as we have seen. There exists a force that deviates falling primordia from a certain and unbroken path. The force in the universe that causes this deviation Lucretius calls the clinamen,xv or “swerve.”:
...while the first bodies are carried down by their own weight straight through the void, altogether at uncertain times and in uncertain places, they turn aside a little from their course, just so much as you may call a changed motion of weight. For if they were not accustomed to bend aside, all would fall downwards as drops of rain through the deep void, no collision would come to be and no blow would be caused among the primordia: thus nature would have produced nothing. (DRN, 2.217-224).
All the primordia fall at the same speed through the void and so in order to create some kind of difference that begets a collision, which in turn gives things their properties, a swerve from their uniform fall to the earth becomes necessary. Thus, the clinamen.
Primordia exist in as many kinds of shape as exist in the parts of any species. Each shape has a different ability and performs a different function when they are conglomerated.xvi We can see how they function in ordinary experience.
...through a colander we see wine suddenly...strain; but to the contrary olive oil sluggishly hesitates, either, no wonder, because it has larger elements, or they are more hooked and folded among themselves, and for that reason it comes about that the individual primordia are unable to be so detached unexpectedly and penetrate individually through the opening of each. (DRN, 391-97).
Qualities in things come from the shape and characteristics of the primordia. Lucretius describes how the shapes of the primoridia and their possible conglomerates make the substances that we find in nature.
Again, what appears to us as hardened and close, these things must consist of elements more hooked among themselves just as if held together deeply by branch-shapes. Among the first of this kind invincible stones,...stand in the front rank, accustomed to disregard blows; and granite and the firmness of unyielding iron and bronze bars that shout while resisting the bolt. Those others, the liquids that consist of fluid body, must be comprised of elements lighter and more spherical. (DRN, 2.444-52).
The shapes of the primordia are limited in number. If they were not limited, some would be limitless in size.
Increase of body follows novelty of shape. On account of which it is not the case that you are able to believe different seeds to exist in infinite shapes, otherwise you must compel certain of them to be of immeasurable magnitude.... (DRN, 2.495-99).
The primordia of any shape are infinite in amount. Otherwise, the sum of the bodies would be finite and the manner of combination of finite primordia would not be possible.
Truly, since the difference of shapes is finite, it is necessary that what shapes are alike be infinite, or that the sum of matter be finite, the thing which I have proved not to be. With verse I will show that small bodies of matter hold the sum of things from infinity with a continuous succession of blows from all parts. (DRN, 2.525-31).
If I should assume that the bodies generative of one thing and scattered in everything were finite, from what place, where, under what force, having come together in what way will they assemble in such a sea of matter, in such a strange disorder? (DRN, 547-50).
Lucretius is talking about the generative aspect of the primordia. They are not merely inactive elements that other forces of the universe manipulate, but rather they actively participate in becoming. Nothing consists of one kind of primordia; Lucretius uses earth as an example:
There is nothing of those things in nature plainly that consists only of one kind of principle, nor anything that does not consist of various intermingled seeds; and whatever thing possesses many forces and more abilities in itself, so it shows there are within it most generative things and various shapes. First, the earth holds the first bodies in itself from which springs, turning, the cold around assiduously renew the immense sea and contains that from where fires arise. … then further, earth contains means to lift up bright fruit and joy-bringing trees for mankind, that source from which it is able to produce rivers and leafy branches and glad pasturages for the mountain-ranging stock of feral beasts. (DRN, 2.583-97)
The primordia of the earth are generative because they produce different things from themselves. They must be different, each one possessing a different ability to produce. Primordia are not all conjoinable with one another, certainly not arbitrarily. Otherwise, horrible creatures and objects would result. As different as things are from one another, so different are the primordia. Lucretius claims that such difference is necessary:
For just as all things made are completely different from each other, so it is necessary that each consist of primordia differently shaped; not that very few are endowed with a similar shape, but that commonly not all are composed equally to all. Since, further, seeds differ, there is need that intervals, paths, connections, weights, blows, assemblages, motions differ, which not only separate animal bodies, but sever the lands and the whole sea and hold back all sky from earth. (DRN, 2.720-29)
Otherwise, all of the primordia would come together in one homogeneous mass. The primordia do not have the same qualities that arise from them. They lack heat, sound, moisture and smell.
The primordia of things must not add their own smell to things in the making, nor sound, since they are capable of emitting nothing from themselves, and for a similar reason no taste at all, not coldness, nor heat besides that nor tepid vapor or the rest. … it is necessary that all these be separated from the primordia, if we wish to establish an immortal foundation for things upon which the sum of vigor may rest: or else you find all things pass back absolutely to nothing. (DRN, 2.854-64).
It would seem as if primordia are similar in function to the parts of the primordia themselves. They emit nothing and have no qualities in themselves, but when combined in certain ways they make the qualities of things in the universe. The parts of the primordia themselves cannot exist without the combinations that make them, yet the primordia themselves have their own cohesion and unity. Just so much do the primordia not have smell or taste or sense and yet combinations of them give rise to these qualities.
The universe that we know is not the only universe because of the infinite number of elements interacting. When the possibility of some cosmos to exist or thing to arise is there, there is a generation of things; innumerable primordia lead to innumerable creatures and objects. The variety and number of creatures and objects is proof of this assertion:
Now, it must not be supposed that this is the only circuit of lands and sky that have been made since limitless space lies open everywhere, and since seeds innumerable in number and the sum of depth fly about in many ways driven by eternal motion, nor that so many of those bodies of matter outside do nothing: especially since this place was made by nature, and seeds themselves by their own volition pushed and thrown together in many ways, at random, without aim, to no purpose, have cultivated those which are suddenly joined so as to become always the origins of great things... (DRN, 2.1052-62).
Because nature generates creatures and things itself there are no gods.
If you have these things thought well in mind, straightway nature is seen to be free, rid of arrogant masters, herself doing everything by herself spontaneously, and she is seen to have no part of the gods. (DRN, 2.1090-92).
We have seen that Lucretius believes the nature of things and how they act are determined by the shape and the nature of the primordia of things. He also seems to believe that shapes determine actions of larger objects; these same shapes are present as causes for qualities of objects, as we have seen.
But that which necessarily is movable must consist of seeds extremely rounded and minutest possible, that they are able to be moved when incited by little motion. For water is moved and flows with so little motion, obviously made of small circling shapes. But on the other hand the nature of honey is more cohesive, its fluids more inactive, and its impulse more delayed; for the whole abundance of its matter clings to itself more closely, no wonder because it does not consist of bodies so smooth or so fragile and round. (DRN, 3.186-95).
Once the primordia have found their proper place, they are then able to perform their function.
Finally, why are the rational mind and plans never produced in the head or feet or hands, but they cling to individual positions and fixed region for all, if not because fixed places are imparted to each thing for growing and somewhere where it is able to endure when made...? (DRN, 3.615-19). … (a) tree cannot exist in the sky, nor clouds in the deep ocean, no fish live in fields, nor blood in sticks or sap in rocks. It is a certain and set forth thing where each thing grows and inheres. (DRN, 3.784-87).
It is important to note that Lucretius uses examples of everyday growth and flourishing to support his claims. Here is an inductive element, an example that demonstrates his willingness to observe and then reason. Thus, perception is prominent in his thinking.
Lucretius trusts the senses, which always grant the truth. What is seen comes to the perceiver through simulacra, which are images flowing to the eyes and ears through the air. What is most important about Lucretius' view of senses is that they are to be trusted, even when they present illusions.xvii The philosopher who does not trust the senses but rather only reason is in a conundrum. The only means through which reason may develop its stance is through senses, yet some say the senses are not trustworthy. Reason would then have no foundation for itself. Lucretius asks if reason will be able to refute the senses when they are completely derived from the senses. In this manner, he infuses a strong empirical element into his system.xviii He praises and employs reason, but only once the senses have given it what it needs to fulfill its nature. Equal credit is to be given to each sense for what it senses. Sounds reverberate because some of the pieces of the sound fall into the ears while others bounce about on rocks and other objects. In fact, as we have seen, the whole of Lucretius' cosmos is composed of blows that come from the primordia striking one another. Senses are those parts of human being that receive these blows and later interpret them. All of the universe is in this sense material and explainable through physical interactions of material things. These interactions seem to be primary and that is why the senses must be trusted. They receive what is fundamental to the whole of the order of things. Lucretius interprets hearing and taste through this same lens and there thus arises the need to explain why we can hear sounds when there are obstructions, like walls, that ought to prevent any contact between the origin of the sound and the perception. His system is in this sense one-sided, most of his reasoning proceeding from a few insights. Perhaps these are the “flashes of genius” about which Cicero wrote.
In contrast to thinkers like Plato and Aristotle Lucretius believes the mind to be not only material but a physical interaction of a narrow kind. Everything, including the movements of the mind, is material, and how something comes about depends heavily on the shape and substance of the primordia.
...many likenesses of things are moving about in many ways and all around, thin, which easily join themselves in the air when they encounter one another, as if a web of leaf-gold. Naturally, these are much more thin in structure than those which seize the eyes and excite vision, since these things penetrate through the mesh of the body, stir the thin nature of mind within, and stimulate sense. Thus it is we see Centaurs, and the limbs of Scylla, and faces of dogs like Cerberus.... For certainly no image of a Centaur comes to be from one living, since never a living thing of this nature existed. (DRN, 4.725-739).
In this sense Lucretius anticipates the method of modern science in that for each and every topic available to him he sees a material explanation, in some cases radically simplifying the process of explanation – like the viscosity of water as opposed to honey. Lucretius seems to anticipate Darwin's theory of evolution by claiming that combinations, blows and recombining of primordia create creatures and objects until there exists a stable, survivable species.
For so many primordia of things with so many manners, from infinite time now excited and roused by blows and their own weight they have been accustomed to be carried and to assemble in all manner of ways, and to attempt all things, whatsoever they, having come together, were able to produce, that it is no wonder if they fell into such arrangements as well, and came into such courses, by what nature this sum of things now shows in the renewal of their making. (DRN, 5.185-94).
He claims that water and air and other elements of the earth perish and again are born. It is the constant rearrangement of things that provides an empirical example of how the universe is composed of bodies that come to be and perish. Becoming, then, is his main focus as evidence for a material universe:
And since the body of earth and fluid, the light currents of air, and hot steam from which this sum of things is seen to consist, all remain in a body born and mortal, the whole nature of the world one must suppose to be of the same thing....(DRN, 5.235-39). ...and since beyond doubt she is seen to be at the same time the parent of all and the common sepulcher, therefore you see that the land is lessened and once increased regrows. (DRN, 5.258-60).
He continues to provide examples of objects breaking apart and combining again into new objects. Lucretius speculates on the material causes and movements of heavenly bodies, always taking as the primary cause a material interaction, yet he asserts only suggestions in some places, like alternate explanations for the movement of the sun and the light that shines from or upon the moon. He claims that there are many possible causes for given events, and multiple causation seems to be an appropriate element of his system, since he believed that objects and creatures arose randomly as the primordia combined and dissolved only to recombine. It is this same perspective that makes him believe that there are infinite worlds. He says that creatures “by fixed law preserve the distinctions of nature.”xix Lucretius does not appear to question deeply what the consequences are of his insights. He promotes his assertions with examples, yes, but how there is such a fixed law of nature he does not adequately address. He ends his poem with explanations of natural phenomena, like clouds and lightening and evaporation. All of these forces he explains in material terms, insisting that reason be employed in order to learn the nature of things. He seems to believe that the laws of nature remain the same everywhere. The universe is a vast order and men are insignificantly tiny parts of it:
…for nature herself demands to be at every part like itself. (DRN, 6.542)
It must be that in these affairs deep and far you look, and distinguish all parts broadly, so that you may remember that the sum of things is vast, and you see the sky as one, a tiny and trifling part of the whole sum, not so much a part as one man is of the whole earth. (DRN, 6.647-52)
Substances adhere to one another or repulse one another depending on the amount of void in each object along with their shapes, as we have seen. He thus explains the nature of the magnet and also the reason why certain substances attract and mix while others do not. Oil does not mix with water but wine does. Tin can adhere bronze to bronze. The void in an object and the body in an object together enable the combinations of things. Each of the primordia have a different kind of productive power. It seems then that variety of primordia is responsible for varying functions and causes in the universe. Multiple causes exist as a result of this perspective and his universe seems to have as many causes as it does variety of bits of reality. Consistent with this view is the idea that there exists a finite variety of primordia but an infinite number of them.
One thing remains to examine in Lucretius' view of things. We have seen that there is an empirical element to his reasoning in his use of examples, yet there is no verification process. Lucretius speculates as to the process inherent in the example; he does not control the process and verify that the example is precisely what he claims. Also, his verification of things by no means is rigorous or sophisticated; speculation is a very powerful element of his partially empirical system. In order to place his thinking in the context of the other thinkers we have examined, an analysis of some of the above arguments is in order. We may then see how his manner of reasoning is part of a system similar to other ancient systems. These other systems have as part of their structure a scientifically logical framework and a comparison may then be made. First, we will examine his view that the universe is composed of the minuscule primordia that have a generative ability. Second, we will examine his view that void exists and that void is mixed with body. Third, we will look at one argument for the generative ability of the primordia and how Lucretius argues that these bits of reality make the objects and creatures around us. We will see that Lucretius' thinking is valid if not sound. His reasoning is not as vigorous as Plato or Aristotle nor as deep as Heraclitus or Parmenides, but it possesses a fair logical structure coupled with the empirical verification we have already seen.
First, Lecretius argues that there are minuscule pieces of material reality, alternately called primordia rerum, materies and semina. Particles of water, metal and stone wear down gradually as humans use them. These are common experiences that seem to occur regularly enough to claim universality. A modern scientific interpretation of these occurrences would claim that there is a degree of probability that, say, the metal of a ring will wear down with use. The degree of probability is high enough that we can claim a virtually universal law for all practical purposes. Thus it is with scientific discovery of material law today. Lucretius seems to believe that the laws of the interactions of material are in fact universal, largely because of their reliability. He thus makes a universal claim that seems credible, yet his assumption is that these interactions of matter will always remain the same. His logic follows the same path, though his verification comes from what ancients like Plato thought of as unstable change. He assumes that matter remains the same before he argues that it is fixed and so he implies universals:
One can substitute the wearing down of metal or the erosion of stone by water for the water gatherings and find the same form of argument. The material processes are likened to one another by the size of the elements involved and the wholly material relationships. The logic of the argument is sound and so Lucretius makes good his reason, yet he assumes that the processes are material only. He also seems to assume that the elements in his system are constant, a kind of eternal materialism. He has discovered something correct about the material universe without the use of implements like microscopes and mathematical speculation, but his correct assertion is qualified by speculation. There are tiny bits of material reality that constitute objects and creatures, but they do not possess the nature that he assumes. A more rigorous system of control and manipulation like we saw at the beginning of this text would demand at least more argumentation and defense of problematics that emerge. How precisely do the primordia of water depart from the clothing on the shore would be a question worthy of almost an entire book's answer. Lucretius does not concern himself with these details; remember that he lists various causes for different events.
Lucretius also asserts that void exists, as we have seen, seeming to assert that void exists in two ways. He claims that void is part of the physical matter that exists in the universe and that void provides the space for movement. Really, these are the same function of void. The proof of void in matter we look at first.
We cannot make the same conclusion as Lucretius because his premises make assumptions about the nature of material reality that he does not prove nor argue. It is not necessarily the case that things that press down are made lighter by void as much as it is that things press down because of how they are constituted. A complete void does not exist nor is it part of objects and Lucretius does not argue rigorously for a complete void, nor does he put forth convincing examples for its existence. Void and its affects are assumed in many ways; he believed that the only reason one object could be lighter than another is that it has void in it. Yet, Lucretius uses examples in order to argue his case and an element of more direct empiricism has found its way into his reasoning, as we have seen. His is not the categorization of Aristotle nor the speculation of a Parmenides or a Plato. Natural processes as they exist are a large part of Lucretius' reasoning, but he speculates on what precisely are the processes without further or repeated examination. His lack of empirical verification leads him astray as speculation mislead others before him, even though he trusts in the senses to give him efficacious examples.
Lucretius also claims that the shapes of these bits of material reality have some effect on how they act in everyday experience. More than one type of primordia exist and they combine to form things that have more than one ability. Each has its own power and thus naturally when an object is composed of more than one primordium, it has more than one power. These bits of material fall naturally to earth and the collisions that result from their knocking against one another are the motions that compose objects and creatures around us. Thus, we recall Lucretius' clinamen. He reacts to a criticism that if the primordia fall to earth they ought to fall in specific and unwavering trajectories, in which case a rigid and unalterable causation would dictate how things come to be; there would be no randomxx element that allows for the differing direction of things and variable change. Free will would not exist as well because our minds are merely material objects like other things. The clinamen, or swerve, is the answer to the criticism. It lets Lucretius assert that no absolute trajectories dictate becoming. One form of his arguments are as follows:
Now, this argument claims that hard things are things accustomed to despise blows, which means they are tough. These tough things are all things that have as a commonality, or a category, the physical property of being held together by branch-like shapes. The shapes are then the determining factor in the toughness of the substance and so shapes are determinant in the production of physical quality. Lucretius applies shape to diamonds, iron and bronze. Iron and diamond are obviously composed of different shapes because they have different qualities, but the category of hardness comes from a similar shape. The argument is valid in that the form is good, but the premises come from a speculation that itself arose from an observation. Since Lucretius trusts observation, he proceeds no more deeply, though more observation may prove that his assertion is not as universal or correct as it seems. Lucretius argues further that no object is composed of one simple kind of element:
Again, Lucretius has rendered a valid argument that is not sound. The sound argument must have true premises along with good form. His argument here has merely good form, but again the empirical dimension to Lucretius' argumentation is speculative guess-work that has itself a low degree of probability. He thinks to have found an explanation of how things come about without examining more closely in order to verify his perspective. Lucretius further argues that causation arises from what we call gravity coupled with a deviation from objects' path or paths. The deviation from the path causes the collisions that themselves create the objects and creatures of the universe.
Gravity in the object, and for Lucretius density, is a fundamental force of the universe not only because it pulls objects downward but it is one aspect of causation without which collisions and thus objects themselves would not be possible. Lucretius argues his case fairly well, as we can see from his valid arguments, yet, again, he does not pursue his case rigorously. He contradicts himself when he claims that the primordia have no characteristics themselves and he also states that these fundamental bits of material reality also have shapes that produce specific characteristics. He had no sense that he ought to prove his assertions rigorously with evidence perhaps because of the tendency for influential thinkers like Plato and others to distrust experience in general; specifically were ancient thinkers distrustful of particulars. Many centuries passed before a rigorous empirical proof became a requirement for verification of scientific theory, and so Lecretius' views can have only a partial veracity to them. He came up with many insights that proved to be correct in their essence, if not precisely what he imagined. Imagination for Lucretius was an intimate part of his insight and he was able to go no further without new tools, yet he does not seem to have out-imagined himself in the way that Empedocles did. His imaginative view of the invisible primordia did not run amok so much as it reached beyond the grasp of his senses.
iDe Rerum Natura is sometimes seen as an epic (Donahue). Lucretius seems to want to change the reader's mind (See, Solomon, Gale, Volk).
iiCicero, Letters to his Brother Quintus CXLVII (Q, FR III, 1).
ivDe Rerum Natura influenced such notables as Newton, Dalton, Maxwell and Kelvin. For more on the influence of Lucretius on science, see Downs, Fisher. Kenneth Rexroth talks about the correctness of Lucretius view, saying “no sounder view of nature was to appear for almost two thousand years. We should understand that the atom of Lucretius is not our atom, much less our molecule, but...today it is the foundation of our physics of ultimate particles.” Gassendi (See Fisher, 191) follows the ideas of Lucretius most closely.
vSome scholars believe that Lucretius' main aim is to eliminate fear; the seemingly cavalier attitudes Epicureans take toward discovering precise causes is one possible piece of evidence for this stance. For more on this interpretation, see Mayer, Wasserstein. Diogenes Laertius (DL, X, 79) expressly disagrees with Lucretius, saying that there is nothing in knowledge of eclipses, solstices or risings or settings that makes human beings happy. The Epicurean perspective was far from universal, and it is important to note that the seeking after causation was not rigorous for the Epicureans. They may have been more concerned with the affects of a reasoned life than with reasoned argumentation.
viLucretius seems to have interpreted nature as in itself powerful, in itself without a master, like a god. There is a random element to that ability, but it is sometimes viewed as both random and necessitated. Monte Ransome Johnson calls this element an ability of nature “as a proposition in natural science, the result is that nature of its own accord behaves in a way that it would otherwise be compelled to do by law” (Johnson, 105). The behavior of nature for Johnson then is something that possesses “law-like regularities”, which means that the same patterns of nature science seeks are the patterns of universals. Lucretius gives a kind of antidote to the metaphysical and spiritual speculations of a Plato or an Anaximander (See Kevorkian). For more, see Johnson, Fowler. For a different perspective, see Long.
viiThese speculations are based upon observation, an inductive method. Lucretius employs observation in a serious manner, and that is one scientific aspect of his thought. Still, he lacked rigor and the control methods that he desperately needed.
viiiDe Rerum Natura, 1.500-2.
ixDe Rerum Natura, 1.56-60.
x W.T.L. talks about how Lucretian atoms themselves have parts, or minima of the material world. It is the perspective of this text that Lucretian atoms are not technically uncuttable. For more on how atoms have parts and yet are not separable, see W.T.L. For a clear and basic comparison of the ancient and modern atom, see Stocker.
xiLesage notes how likely it is that if the Epicureans had possessed geometric knowledge and some cosmographical knowledge of the day, they would have discovered the law of gravitation and its cause. Ritchie points out that Lucretius shows no signs of scientific knowledge of the ancient world. Lucretius' induction and empiricism is quite strange, if he bases it on no contemporary theory or method.
xii C.B. Believes that should Lucretian atoms exist large enough to see, they would be unseen regardless because they do not possess qualities. Here is another strange and problematic aspect of Lucretian atomism. For more, see C.B.
xiiiDe Rerum Natura 2.184-213.
xivIt is important to note that ancients had no conception of gravity in the sense that we do, yet they knew that rocks fall from heights and that leaping cats ultimately come down. Saying that ancient peoples had no notion of gravity is false, but saying that they understood gravity in a scientific and thus sophisticated way is also false. The point is that even a dog understands some aspect of gravity. The ancients knew many practical things about this force of the universe and thus had an understanding of it.
xv The clinamen is a contentious issue with scholars. Some explain the swerve in terms of motion and deflection, or simply motion; some claim that it is a random element incorporated into Lucretius' philosophy in order to solve certain problems and thus it recalls modern physics; others claim that the notion of the atom has always had in it a deviation. There is some debate as to the direction and inclination of the swerve. Cicero mocks the clinamen and St. Augutine calls it “the soul of the atom”. Epicurus is said (Pullman) to have thought up the clinamen in order to answer problems that arose because of the generative force of gravity (an iron-like grip on causation would result if all falling objects fell in straight lines). Lucretius is said (Pullman) to have contributed little besides a reworking of Epicureanism and so did not think it up. One of the most significant problems for ancient thinkers concerning gravity's affect on human culture was that it seemed to destroy free will and thus morality and ethics along with it (Cicero, St. Augustine among others). Englert asserts that swerves follow volitions. Sedley claims that swerves are caused by volitions from the top down; volitions are emergent properties able to obtain leverage on atoms. It seems that Epicurus thought of atoms in three ways: weight, the clinamen and collisions. Lucretius follows Epicurus and adds little. It was perhaps a reaction to Aristotle that brought about the clinamen (see O'Keefe). Wasserstein claims that the problem of the clinamen is of ethical origin moreso than physical. It is the result of needing a problem, and so the problem is contrived. For more on its interpretation and thus function, see Clark, Winspear, Masson, Purinton, Bailey, Fowler, Englert, Pullman, Furley, Johnson, Wasserstein. For a thorough analysis, see Greenblatt.
xviFor more on the properties of atoms that produce specific qualities, see Winspear.
xvii Reason and the senses seem to be inextricably linked for Lucretius. Humans ought to trust their senses, he claims, but he also states that reason must assist in the assessment of what is perceived. For more on the relationship between the senses and reason, see Clark, Lehoux, Sharrock. Lehoux talks especially about the relationship between reason and sense perception. We may take Lucretius' view as scientific in the sense that perception is absolutely needed but that a strong element of reasoning must accompany our senses, perhaps a precursor to objective analysis. The inability of the senses to give certain truth may be the result of atoms being counter-intuitive: they have no color, smell, temperature and they are thus difficult to pinpoint perceptibly. For more on images involving perception, see Dyson.
xviii Some scholars believe that Lucretius posited a “theory of radical induction” based upon observations that in turn were combined with reason in order to discover a truth about nature. Lucretius is supposed to have anticipated Bacon's Novum Organon by many centuries. Perhaps “radical induction” is overstated, but one can at least rest assured that Lucretius' manner of reasoning possessed a strong empirical and thus inductive element. Alban Winspear remarks that “it was only in dealing with small and circumscribed problems, where an idea could be brought immediately to the test of practice, that the ancient man of science can be said to have developed a scientific method.” p. 84. Crew claims that there were two methods for the ancients: philosophical and mathematical. Galileo made a third: experimental. Vavilov asserts that “the poem of Lucretius is the first attempt preserved to us...to explain nature completely on the basis of physical principles.” p. 24. For more, see Winspear, Stocker.
xixDe Rerum Natura 5.922-25.
xx Lucretius inserts in his physics a strong element of randomness, which was antithetical to most ancient thinking. For more on randomness in Lucretius' work, see Clark.
Life and Works
Euclid was a geometer, mathematician and perhaps a Platonisti whose floruit occurred between 325 and 250 B.C., but what information scholars posses about Euclid's life is at best unreliable. He likely taught in Alexandria Egypt during the reign of Ptolemy Soter (323-285 B.C.) and has been considered a great schoolmaster as well as a man of the utmost honesty.ii The most prominent mathematician of the Graeco-Roman period, he is best known for his work Elements, and largely relegated to the status of geometer. Other preserved works are Optica, Catoptica, Sectio Canonis and Phaenomena, all of whom together demonstrate that Euclid knew the whole range of mathematics and mathematical topics of his day.iii Indeed he seems to have culminated all that came before him in his work. Though he covered other material in other works, his Elements is far from simply geometry. Euclid's aim in Elements was to make geometric proofs using only a compass and a ruler and to establish a theory of numbers; thus with limited tools, he produced a highly sophisticated work.iv Geometry was meant to be simple at its root and therefore beautifully structured, which reflected the Greek sense of virtue in general. Euclid may have made certain proofs on his own, but he also compiled the work of earlier thinkers into Elements, proving their theories perhaps more vigorously. Thinkers were not obliged to cite sources in the way modern scholars do, and so Elements is perhaps the legitimate culmination of geometry and some mathematics, but it is not the work of a titanic genius, as may be supposed. It is at least the work of a genius compiler who proved theories that already existed, theories that had not been proven as vigorously and in conjunction with one another. Make no mistake; Elements is a work of genius. Yet, Euclid's contribution must be put into perspective. Not only is he not responsible for all of the proofs in Elements, but he may not be responsible for the majority of them. There is no way to extract fully what Euclid adds to what already existed. Still, scholars have some idea who contributed to his work and basically what that constituted. One must think of Elements as an enormously successful textbook,v sometimes mistakenly giving readers the impression that Euclid discovered a titanic multitude of proofs and truths.
The method Euclid employs is an axiomatic one,vi and that means he makes use of fundamental, atomic propositions that he uses to build a very powerful system, beginning with the easier material and progressing methodically into complicated proofs.vii Atomic propositions are statements considered to be self-evident, like the old metaphor of building blocks. Once one acknowledges these statements as true the rest of Euclid's system follows because he reasons in a deductive manner. His proofs are composed of long combinations of accepted truths and logical connections. These connections make propositions. In fact, Euclid makes a proof for one proposition and uses that as an element for another, later proof. His way of making an argumentviii is thus thorough and meticulous, sometimes seemingly tedious, but all of the pieces need to be arranged in order to build a strong edifice.ix And his construction is so strong that it lasted until late in the nineteenth century when alternate geometries gained traction in the academic community.x One anecdotexi from the ancient world speaks to the rigor and potency of Euclid's work. He was asked by Ptolemy the king if there was not a shorter road to geometry than through his Elements. His response was that there is no royal road to geometry. Another anecdote speaks to what one may obtain from studying geometry. Euclid was asked by one of his students what profit there is to gain from the study of geometry. He called a slave and told him to give the student a coin or two, since he must profit from what he learns. Euclid's Elements may seem sometimes tedious, but it is beautiful and true, profitable and useful in itself – powerful indeed.
It is appropriate here to acknowledge the work of the thinkers and scholars on whose work Euclid built his famous text. They are, after all, responsible for his success in a most significant way. Elements is composed of thirteen books,xii beginning with simple proofs and graduating to more difficult ones. Euclid drew from Aristotle at least one of his most fundamental axioms, but he may have used a great deal more.xiii Hippocrates of Chios whose floruit comes c. 440 B.C. wrote the first known Elements of geometry a century before Euclid wrote his Elements. Hippocrates attempted to “square the circle” and portions of his Elements have been preserved in the work of Proclus, a Greek philosopher. Perhaps the whole idea of a book of geometric elements came from Hippocrates. The Pythagorean Theatetus who lived from 417-369 B.C. contributed a great deal of material from which Euclid wrote the later books of his treatise: XI–XIII three-dimensional figures, XI intersections of planes, lines, and parallelepipeds (solids with parallel parallelograms as opposite faces), and XIII the construction of the five regular Platonic solids (pyramid, cube, octahedron, dodecahedron, icosahedron) in a given sphere. Eudoxus of Cnidos whose floruit occurred c. 395/390–342/337 B.C. was another significant contributor. Book XII of Euclid's Elements consists in part of Eudoxus’s method of exhaustion, which Euclid used to prove that areas of circles are to one another as the squares of their diameters. Also Euclid used Eudoxus' work to prove that the volumes of spheres are to one another as the cubes of their diameters.
Book I of Elements lists twenty-three definitions, five unproved axioms and five more unproved assumptions. Euclid called the axioms “postulates” and the unproved assumptions “common notions”, but it is best to define our terms in order to be as orderly as possible. An axiom is a self-evident truth about something, an assertion that everyone agrees is true. It needs no proof. The term comes from the Greek axioma, which means “worthy.” A postulate is simply asserted as true but it lacks the self-evident nature of an axiom. It is as if Euclid were to say that he wants his readers to take this assertion as true, and he employs that assertion as a truth. There is no need to prove a postulate, though perhaps a proof will make a postulate stronger. These assertions are sometimes intuitive, and sometimes simply asserted for purposes of making a proof, but without doubt they are necessary as atomic elements of each line of reasoning. We will see that there are some proofs of things that we would take as truth without proof, but Euclid knew that he needed to prove every item. Again his attention to detail of this kind leads some to see his work as tedious, but the so-called tedium is quite necessary.xiv Many of the postulates are self-evident, but most are provable.xv First, here are Euclid's definitionsxvi from book I of Elements, which make clear what terms mean what.
A point is that of which no part exists.
And a line is length with no breadth.
And the extremities of a line are points.
A straight-line is whichever one lies evenly with points on itself.
And a surface is what has length and breadth alone.
And the limits of a surface are lines.
A plane surface is whichever lies evenly with the straight-lines on itself.
And a plane angle is the bending of the lines to one another, when two lines in a plane meet one another, and are not lying in a straight-line.
And whenever the lines containing the angle are straight, the angle is called rectilinear.
And whenever a straight-line stood upon (another) straight-line makes adjacent angles (which are) equal to one another, each of the equal angles is a right angle, and the former straight line is called a perpendicular to that upon which it stands.
An obtuse angle is an angle greater than a right-angle.
And an acute angle (is) an angle less than a right-angle.
A boundary is that which is the limit of something.
A figure is that which is contained by some boundary or boundaries.
A circle is a plane figure contained by one line [which is called a circumference], (such that) all of the straight lines falling towards [the circumference of the circle] from one point among those lying inside the figure are equal to each other.
And the point of the circle is called the center.
And a diameter of the circle is some straight-line, drawn through the center, and terminated in each direction by the circumference of the circle, (and) whichever (straight line) also cuts the circle in half.
And a semi-circle is the figure contained by the diameter with the circumference cut off by it (the diameter). And (the) center of the semi-circle is the same thing as (the center of) the circle.
Rectilinear figures are the ones (figures) contained by straight-lines: trilateral figures (being) the ones contained by three straight-lines, quadrilateral by four, and multilateral by more than four.
And of the trilateral figures, an equilateral triangle is the one having three equal sides, an isosceles (triangle) the one having only two equal sides, and a scalene (triangle) the one having three unequal sides.
And further of the trilateral figures, a right-angled triangle is the one having a right angle, an obtuse-angled (triangle) the one having an obtuse angle, and an acute-angled (triangle) the one having three acute angles.
And of the quadrilateral figures, a square is the one which is right-angled and equilateral, a rectangle the one which is right-angled but not equilateral, a rhombus the one which is equilateral but not right-angled, and a rhomboid that having opposite sides and angles equal to each another which is neither right-angled nor equilateral. And let quadrilateral figures besides these be called trapezia.
Parallel lines are straight-lines which, being in the same plane, and being extended to infinity in each direction, meet with one another in neither (of these directions).
Notice that Euclid is making clear what he means when he uses specific terms that could themselves be defined in other ways, sometimes differently nuanced. So, definitions are not accepted truths as much as they are ways of using specific terms. Also, notice that a point is a most fundamental unity, or whole in that it is “that of which no part exists.” It is atomic and number as a unit is also this uncut part of reality. It has no length or breadth because it is simply at some spot that technically cannot be measured. A point is a rendering of a whole or unity, which is for Euclid likened to number. In fact, Euclid defines number in a similar manner, and from this definition he creates a spatial extension of number. So, in some sense he is a Pythagorean influenced strongly by Plato. Second, here are the postulates from Book I of Euclid's Elements. Euclid writes “let the following be postulated.”
To draw a line from any point to any point.
To produce a finite straight line continuously in a straight line.
To draw a circle with any center and distance.
That all right angles are equal to one another.
That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if extended to infinity, meet on that side of which are the angles less than the two right angles.xvii
And here are the axioms from Book I of Euclid's Elements. Euclid calls them “common notions.”
Things which are equal to the same, or to equals are also equal to one another.
If equals be added to equals, the sums are equal.
If equals be taken from equals, the remainders are equal.
Things which coincide with each another are equal to each another.
The whole is greater than the part.
The definitions, postulates and axioms function together as the foundation of Euclid's axiomatic proofs, but it is important to note that each book – with the exceptions of 8, 9, 12 and 13 – has definitions. One will remember that Pythagoras talks about the relationship of sections on a chord that produce a similar sound. The ratio of ½ is an eighth, but one must begin at a particular point in order to determine the ratio. One cannot begin at 0. Once the point on the line of the chord is determined, one is able to determine where are the other eighth tones because the point on the line that lies a specific distance from another point has a similar sound. Yet, one must begin at some point in order to make the determination. Beginning at 0 leaves no possibility to make a ratio because there is no measure. In other words, ½ of zero is zero, so one moves to no other point on the chord. The definitions, postulates and axioms of Euclid operate in a similar fashion in his Elements. They supply the reference points for an analysis of figures in two and three-dimensional space, a starting point that allows the system to begin and founds it. Notice as well that such systems of reasoning are relative. They, like all reasoning, need a reference point from which to understand the system. Otherwise, like beginning at zero there is nothing from which one proceeds.
Euclid's system of measurement and logic produces quite strong proofs of the certainty of many geometric figures and shapes, though some moderns have found fault with his methods and proofs. His method is fully deductive and it is not until the nineteenth century that logicians and geometers realize they are able to begin a new system of reasoning by beginning at different reference points. Euclid's proofs were taken as certain for that long, and it is important to remember that his proofs are not proofs in the sense that they make reasoned hypotheses and later prove them with empirical evidence in the traditional sense.xviii They are more pure reasoning and “prove” something in the sense that they demonstrate how specific aspects of figures and shapes can be realized with mathematical precision and logical certainty. A proof will demonstrate that one can be certain about what amounts to outlines of objects encountered in the universe. They demonstrate precise measurement and relationships between lines, points and space. There are no more clear arguments, though perhaps Euclid's geometric system is not the only way to measure figures.
Causation, Argumentation and Cosmology
Book I of Elements proves elementary theorems about triangles and parallelograms, culminating in Euclid's proof of the Pythagorean theorem. Book I also contains the famous pons asinorum,xix beyond which good students may pass but past which bad students cannot. Recall that Aristotle's syllogisms individually are three statements long. Each syllogism may be coupled with other syllogisms in order to form longer arguments, but the fundamental arguments are composed of three lines. Euclid's arguments may be three lines, but they may also be much longer, and that longer proof may become an element of another, more sophisticated, argument. Our concern will be showing how the various proofs can be used together. We concentrate on Book I, which culminates in Euclid's proof of the Pythagorean theorem. His proof of the Pythagorean theorem is itself comprised of propositions 4, 14, 31, 41 and 46.
Euclid begins Book I by demonstrating how it is possible to create an equilateral triangle: Book I, Proposition 1
On a given finite straight line to construct an equilateral triangle. Let the given straight line be AB. There is need then to construct an equilateral triangle on the straight line AB. Let the circle BCD be drawn with center A and radius AB [Post. III], and again let the circle ACE be drawn with center B and radius BA [Post. III]. And let the straight lines CA and CB have been joined from the point C where the circles cut each other to the points A and B (respectively) [Post. I]. And because point A is the center of the circle CAE, BC is equal to BA [Def. 1.15]. But it was shown that CA is equal to AB. Thus, CA and CB are both equal to AB.
Things equal to the same things are equal to each other, and CA then is equal to CB. Thus, the three CA, AB, BC are equal to each other. Therefore, the triangle ABC is equilateral, and has been constructed on the given finite, straight line AB, the very thing there was need to do.
Notice that postulates 3 and 1, definition 15 and axiom 1 are each pieces of the proof. Each has been taken as an accepted truth, and each supports the overall demonstration of how one may be certain to construct an equilateral triangle. The certainty of the radius of a circle is a part of constructing the triangle within the circle. In fact, the triangle is merely the radius of two interlocking circles, and this certainty is deductive in the modern sense. That is, if one accepts the premises of the proof, one must accept the conclusion. Here is clear, confident certainty, and practical application of the shape into spatial extension is possible on multiple levels. So, Euclid's proof is both deductively certain and utilitarian.
Proposition 4 demonstrates that if two triangles share a corresponding side angle and side, then they are congruent: Book I, Proposition 4. If two triangles possess two sides equal to two sides, each (equal) to each one, and possess angles contained by equal straight lines equal, then they will possess the base equal to the base, and the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles, each (equal to) each one, under which the equal sides will stretch. Let ABC and DEF be two triangles having the two sides AB and AC equal to the two sides DE and DF, each one (equal to) each: AB to DE and AC to DF. And let the angle BAC (be) equal to the angle EDF.
I say that the base BC is equal to base EF, and the triangle ABC will be equal to triangle DEF and the remaining angles will be equal to the remaining angles, each one (equal to) each, under which the equal sides will stretch: ABC (equal) to DEF, and ACB (equal) to DFE. For when triangle ABC is applied to triangle DEF and point A placed upon point D and the straight line AB on DE, point B will coincide with point E on account of the equality of AB to DE. With AB coinciding with DE, the straight line AC will coincide with DF as well, on account of the equality of BAC to EDF. The result will be that the point C will coincide with the point F again on account of the equality of AC to DF. But, point B coincided certainly with point E, so that the base BC will coincide with the base EF. For if B coincided with E and C with F, and the base BC will not coincide with EF, two straight lines will encompass an area, the very thing that is impossible [Post. 1]. Therefore the base BC will coincide with EF, and will be equal to it. The result is the whole triangle ABC will coinicide with the whole triangle DEF and will be equal to it. And the remaining angles will coincide with the remaining angles and will be equal to them: ABC (equal) to DEF and ACB (equal) to DFE. Therefore if two triangles possess two sides equal to two sides, each (equal) to each, and possess the angle(s) enclosed by equal straight lines equal to (their) angles, they will have their base equal to their base and the triangle will be equal to the triangle. And the remaining angles will be equal to the remaining angles each (equal) to each, under which the equal sides stretch, the very things that it was necessary to demonstrate.
Again, the proof consists of other pieces from Elements: postulate 1, axiom 4 etc. The notion that the triangle is congruent seems obvious, but Euclid demonstrates precisely how it is true that they coincide based upon his definitions axioms and postulates.
Proposition 4 has an element of indirect proof but proposition 6 we include because it is at its core an indirect proof. Indirect proofs have been a large part of logical and mathematical reasoning for many centuries. The idea is that if one assumes the opposite of what one wishes to prove and if a contradiction follows, then the opposite of what one has assumed must be true. If we want to prove that when the snow falls Charles will shovel our sidewalk, then we assume that Charles does not shovel when the snow falls. But what we know of our contract, the premises of the argument, is that we pay Charles to shovel snow; Charles is reliable and always comes when the snow falls; all is in order in our contract with Charles. These are all the things in the universe that exist as far as Charles shoveling snow is concerned. So, there are two possibilities in our snow-shoveling universe: Charles will shovel snow, Charles will not shovel snow. If we assume that Charles will not shovel snow, then we arrive at an absurdity, or in other words a contradiction, because everything about our universe indicates that Charles will shovel snow. We have a contract; he is reliable; snow has fallen; all is in order with the contract. So, given the contradiction we can assume that our belief that Charles will not shovel snow is incorrect. The conclusion flies in the face of all the things that we know about the situation. Thus, the opposite must be true: Charles will shovel the snow. This kind of proof seems initially perhaps not as rigorous and perhaps a bit too indirect, but it has been used and is now used regularly when a proof of a problem is perhaps too cumbersome, or even if the route through a direct proof does not seem possible.
Book I, Proposition 6: If the angles of a triangle are equal to each another, then the sides stretching under the equal angles will be equal to each another as well. Let ABC be a triangle having the angle ABC equal to the angle ACB. I say that side AB is equal to side AC as well. For if AB is unequal to AC then one of them is greater. Let AB be greater. And let DB, equal to the lesser AC, have been cut off from the greater AB [Prop. 1.3]. And let DC have been joined [Post. 1]. Therefore, because DB is equal to AC, and BC (is) common, the two sides DB, BC are equal to the two sides each to each, and the angle DBC is equal to the angle ACB. Therefore, the base DC is equal to the base AB, and the triangle DBC will be equal to the triangle ACB [Prop. 1.4], the lesser to the greater.
The very notion (is) absurd [C.N. 5]. Therefore, AB is not unequal to AC. (It is) thus equal. Therefore, if the angles of a triangle are equal to each another, then the sides stretching under the equal angles will be equal to one another as well, the very thing it was necessary to demonstrate.
Proposition 14 proves that if a straight line extends from two lines and the angles formed are congruent to two right angles, then the lines must be in a straight line with one another. Book I, Proposition 14: If two straight lines, not lying on the same side, make adjacent angles (the sum of which is) equal to two right-angles with some straight line, at a point on it, then the two straight lines will be straight on (with respect) to each another. For let two straight lines BC and BD, not lying on the same side, make adjacent angles ABC and ABD (the sum of which is) equal to two right-angles with some straight line AB, at the point B on it. I say that BD is straight on with respect to CB.
For if BD is not straight on to BC, then let BE be straight on to CB. Therefore, because the straight line AB stands on the straight line CBE, the (sum of the) angles ABC and ABE is thus equal to two right angles [Prop. 1.13]. But (the sum of) ABC and ABD is also equal to two right angles. Therefore, (the sum of angles) CBA and ABE is equal to (the sum of angles) CBA and ABD [C.N. 1]. Let (angle) CBA have been subtracted from both. Thus, the remainder ABE is equal to the remainder ABD [C.N. 3], the lesser to the greater. The very thing is not possible. Therefore, BE is not straight on with respect to CB. Similarly, we will demonstrate that neither (is) any other (straight line) than BD. Therefore, CB is straight-on with respect to BD. If, therefore, two straight-lines, not lying on the same side, make adjacent angles (the sum of which is) equal to two right angles with some straight-line, at a point on it, then the two straight-lines will be straight on (with respect) to each another, the very thing it was necessary to demonstrate.
This proof becomes one of the primary pieces of Euclid's proof of the Pythagorean theorem. It is another that seems obvious, but Euclid demonstrates that – given the acceptance of his postulates, definitions and axioms – it is always true, and it is the “always true” that matters in the construction of proofs. If proposition 14 were sometimes true, then obviously we would have a foundation that sometimes did and sometimes did not function properly.
Proposition 31 proves that with any line and any point, one can always construct a parallel line through the given point. Again, the truth of this proposition seems obvious, but that it is always true is essential to constructing Euclid's system. Book I, Proposition 31: To draw a straight line parallel to a given straight line, through a given point. Let the given point be A, and the given straight line BC. It is necessary to draw a straight line through the point A parallel to the straight line BC. Let the point D have been taken randomly on BC, and let AD have been joined. And let (angle) DAE, equal to angle ADC, have been constructed on the straight line DA at the point A on it [Prop. 1.23]. And let the straight line AF have been produced in a straight line with EA.
And because the straight line AD, (in) falling across the two straight-lines BC and EF, has made the alternate angles EAD and ADC equal to one another, EAF is therefore parallel to BC [Prop. 1.27]. Therefore, the straight-line EAF has been drawn parallel to the given straight-line BC, through the given point A, the very thing it was necessary to do.
Proposition 41 states that if a parallelogram and a triangle have a base in common and are on the same parallel lines, then the parallelogram is congruent to double of the triangle. Here is a less obvious true statement about the relationship between to shapes. If we have determined that a parallelogram and a triangle have a base in common and are on the same parallel line, then we know something certain about the triangle and the parallelogram.
Book I, Proposition 41: If a parallelogram possesses the same base as a triangle, and is between the same parallels, then the parallelogram is double (the area) of the triangle. For let parallelogram ABCD possess the same base BC as triangle EBC, and let it be between the same parallels: BC and AE. I say that parallelogram ABCD is double (the area) of triangle BEC. For let AC have been joined. The triangle ABC is equal to triangle EBC, since it is on the same base, BC, as (EBC), and in the same parallels, BC and AE [Prop. 1.37].
But, parallelogram ABCD is double (the area) of triangle ABC. For the diagonal AC cuts it in half [Prop. 1.34]. The result is that parallelogram ABCD is also double (the area) of triangle EBC. Therefore, if a parallelogram has the same base as a triangle, and is in the same parallels, then the parallelogram is double (the area) of the triangle, the very thing it was necessary to demonstrate.
Once more, the certainty is what matters. Unfortunately, there is not enough space for it, but Book I, Proposition 46 proves that with any given line, one can always construct a square. Again, a proposition that proves one can draw a square seems hopelessly unnecessary, but that such figure has been proven possible is most important. The end-point of book I is a proof of the Pythagorean theorem. In fact, book I of Euclid's Elements culminates in this proof: Proposition 47.
Book I, Proposition 47: In right-angled triangles, the square on the side stretching under the right angle is equal to the (sum of the) squares on the sides containing the right angle. Let ABC be a right-angled triangle possessing the angle BAC, a right angle. I say that the square on BC is equal to the (sum of the) squares on BA and AC. For let the square BDEC have been described on BC, and (the squares) GB and HC on AB and AC [Prop. 1.46]. And let AL have been drawn through point A parallel to either of BD or CE [Prop. 1.31]. And let AD and FC have been joined.
And because each of angles BAC and BAG are right angles, then two straight lines AC and AG, not lying on the same side, make the adjacent angles with some straight line at the point A on it, (the sum of which is) equal to two right-angles. Therefore, CA is straight on to AG [Prop. 1.14]. On account of the same (reasons), BA is also straight on to AH. And because angle DBC is equal to FBA, for each is a right angle, let ABC have been added to both. Therefore, the whole (angle) DBA is equal to the whole (angle) FBC. And because DB is equal to BC, and FB (is equal) to BA, the two (straight-lines) DB, BA are equal to the two (straight-lines) CB, BF, each (equal) to each. And angle DBA (is) equal to angle FBC. Therefore, the base AD [is] equal to the base FC, and the triangle ABD is equal to the triangle FBC [Prop. 1.4]. And parallelogram BL [is] double (the area) of triangle ABD. For they have the same base, BD, and are in the same parallels BD, AL [Prop. 1.41]. And square GB is double (the area) of triangle FBC. For again they have the same base, FB, and are in the same parallels, FB and GC [Prop. 1.41]. [And the doubles of equal things are equal to one another.] Thus, the parallelogram BL is also equal to the square GB. Similarly, AE and BK being joined, the parallelogram CL will be shown (to be) equal to the square HC. Thus, the whole square BDEC is equal to the (sum of the) two squares GB and HC. And the square BDEC is described on BC, and the (squares) GB and HC on BA and AC (in turn). Therefore, the square on the side BC is equal to the (sum of the) squares on the sides BA and AC. Therefore, in right-angled triangles, the square on the side stretching under the right angle is equal to the (sum of the) squares on the sides surrounding the right [angle], the very thing it was necessary to demonstrate.
The geometric shapes Euclid fashions are precise and exacted. Euclid seems to have believed he was proving something fundamental and unchanging about spatial relationships and about the relationships between lines. In fact, his notion of spatial relationships was taken to be true about space for thousands of years. It still is true to a large degree. Certainly the reliability and continued use of these proofs is testament to something that is always true about them. In fact, the Pythagorean theorem is used similarly to π and e=mc². It is itself a constant that can be relied upon as an atomic element in other calculations. Again, one cannot formulate an argument – mathematical or otherwise – without a reference point, and these constants provide the reference points needed.
It is important to recall that Euclid's proofs are not evidence-driven. These proofs are axiomatic arguments that demonstrate how specific figures come to be in consistent and repeated patterns. He neither found them in the universe, nor claim that he somehow was able to perceive these figures in their apparent perfection. They are deductive, mathematical “entities”, and so each proof can be relied upon to be true on every occasion, given specific conditions. There is no need for any of the proven shapes to exist as an object – or in other words become something that is born grows and perishes – in order for the proof about that figure to be true. There is something genuinely fixed and permanent about Euclid's figures, which was most probably why Plato had so much interest in them. Coming much later than Plato, Euclid takes two-dimensional figures and uses them in conjunction with one another to craft three-dimensional figures. This method seems simple, but it speaks to the unity of the second and third dimension in spatial relationships. They cannot do without one another. Euclid demonstrates the deductive organization of plane figures, lines and their relations and three-dimensional figures. He makes no outward claim about metaphysics or about organization of objects, not about movement of objects, nor about atoms. One is able to take Euclid's geometry as a more pure geometry and one is not forced into a specific theory. While Euclid himself may have believed in the material universe of Plato's Timaeus in some form, his system may have a great deal or nothing to do with Plato's reality. Euclid's system can be applied to many different conceptions of material or metaphysical reality. Such is its strength. Numbers as well as shapes and their relationships to one another Euclid uses in order to measure abstract things. These abstractions have practical value, but they have not been solidified into any one thing. That is important, because number and shape then become things that are more like tools and less metaphysical parts of material reality. In that sense Euclid is modern, though he lived during the time of Ptolemy Soter. Euclid's reality-divorced system divorced can at the same time be applied quite practically to material reality, any metaphysics notwithstanding. He and his system thus reside in the realm of hypothesis. His way of formulating an argument remains with us in the form of kinds of logical maneuvers, like the indirect proof, and in the form of an attempt at the greatest certainty in theory that may then be applied to material reality in the controlled environment of the laboratory. Thus, Euclid's Elements resides in one half of scientific method, the theoretical. Though there are calls for Euclidean geometry to be less central in school curricula, his Elements remains a staple of educational systems across the globe. The way Euclid structured and proved his Elements was perhaps what made mathematicians and thinkers believe his treatise is the definitive work on space and spatial relations. Its most significant influence, however, is as an axiomatic and deductive framework that was itself copied, and indeed significantly modified, through millennia. Elements is the theory of math made manifest, and science needs theory in order to make the falsifiable test. The structure of Euclid's Elements continues in the format of almost all scientific reasoning.
iA few scholars say that some mathematicians are Platonists while others are “constructivists.” For more on the debate as to Euclid's philosophy, see Maziarz and Greenwood, Zeitler.
iiProclus and Pappus respectively.
iiiSome of which include calculations about prime numbers, incommensurability and others that have since been approached by means of algebra. For more on this topic and an excellent review of portions of Elements, see Gjersten.
ivOne number theory incorporated into Elements is the Fundamental Theorem of Arithmetic, which states that every number is either a prime number or a product of prime numbers in one way only. For an articulation of this theorem, see C. M. Taisbak in Tuplin and Rihill. Greek mathematics is static, as opposed to dynamic. Modern goemetry is dynamic in that it contends with relative space while ancient geometry thought of itself as finding eternal figures. For more on precisely how numbers and geometry are synthesized in Elements, see Unguru, Kline. Hilbert (1918) talks about the interrelations of logic and mathematics with respect to science in general.
vElements was so successful that it was copied down and most probably edited to its detriment. The confusion of axioms as opposed to postulates is one possible result. Axioms are things that are a part of our thinking, without which we cannot reason. Postulates are suppositions that things are a certain way, but not necessarily integral to our very kind of reasoning. An 1884 article claimed that Euclid as a textbook was used nowhere but England. One wonders how Euclid made a return, as he must have done. For more on this issue, see Cornelius Lanczos, Ziwet.
vi It is interesting to note that mathematicians do not employ the same methods. While Euclid's method is mathematical it is also logical. These two cannot be separated, really, but the form of proof varies as does the form of logic. Logic is not monolithic in the sense that there is one way to pursue the practice, and mathematics is similar. A famous mathematician, Gauss (quoted in Rav 135-6), said that types of proofs and methods varied greatly by scientific discipline. P.A.M. Dirac (1931) claimed that mathematics will shift as progress is made, a view consistent with A. Szabo who claimed that the axiomatic method was connected to the dialectical method. Logic and math are versatile indeed and Elements can still be used as a means for understanding proofs (McClure). For more on the similarities and differences between logic and mathematics as well as the nature of mathematics, see Rav, Cellucci, Hintikka, Feferman, Langevin, Jeans, Eves and Newsom, Aley, Court.
viiThe style and rigor Euclid employs may coincide with the author and work he is currently proving. B.L. Van der Waerden says that “when Euclid is guided by a first rate author, such as Theatetus or Eudoxus, he is himself excellent; but when he copies from a less eminent author, his standard goes down.” Maziarz and Greenwood point out that there are three steps to a Euclidean demonstration: enunciation, the proof and the conclusion. For more on the structure of the proofs, see Maziarz and Greenwood, Russo, Adler and Wolff, McKirahan, Lee. Lee and McKirahan talk in detail about precisely how Aristotle influenced Euclid. Kneale among others points out that Aristotle does not employ if...then statements for good reason. Euclid uses if...then statements regularly; indeed these kinds of statements are thought to be more mathematical. For the alternate idea that Euclid is not axiomatic, at least in Book I, see A. Seidenberg. J. Hintikka believes that axiomatic reasoning is fundamentally not mathematical. For more on the definition of axiomatics, see Blanche.
viiiEuclid was influenced by Plato as well as Aristotle in how he reasoned. For more on this issue, see Maziarz and Greenwood. Russo details some of the ways Elements is specifically structured as Plato and Aristotle required. Howard Bloom gives a good basic review of the layout of Euclid's Elements.
ixSome scholars point out that empirical elements find themselves in Elements. A.D. Ritchie and E.A. Milne point out that there are certain assumptions found in proofs, including assumptions about spatial relations, physical bodies and light rays. Euclid was perhaps not radically divorced from physical processes. For more, see Ritchie and Milne,
xLater thinkers were to find that Euclid's theory of space simply does not work with relative systems of thought. Jefferson Hane-Weaver says that “Euclid's geometry is fine for a slate or a sandy beach where points stay put at least some of the time but it is of no use in outer space where there is no central reference point against which position coordinates can be compared. …. Space must be augmented with time and change before it makes any sense in a physical context.” Willem deVries further asserts that no axioms, in the modern scheme of thinking, are adequate to ground empirical knowledge and Robert Downs claims that over small areas the difference between Euclidean and non-euclidean geometry is slight, but covering large areas requires new geometries. These new geometries were created by Nikolai Lobachevsky and Bernhard Reimanian. Reimanian geometry was later used by Einstein in his theories and the new geometries inspired art, like Lewis Carroll's Alice in Wonderland and HP. Lovecraft's madness and horror. The discovery of non-euclidean geometry gave mathematicians a deeper sense of the relative rather than the absolute. For more on the new geometries and how Euclidean geometry is “the beginning of mathematical physics”, see Freely, Downs, Bardi, Carnap. Later thinkers also found subconscious use of assumptions, most importantly postulate 5 used to prove itself. For more on the fall of Euclid's Elements and the debate that surrounds it, see Stillwell. Gjersten. If George Sarton is to be believed, Euclid's genius demonstrated itself merely in postulating I.5, since there is no proof of it. The evolution of scientific thought would not have been possible without Euclid's geometry being superseded by non-euclidean geometry. The transition from Newtonian physics to relativity theory was based upon such new ideas, which were themselves impossible without treatises like Elements (See Moore).
xiiFor a good summation and review of all thirteen books, see Artmann. One ought to keep in mind that works prior to Elements were not complete, sometimes with too long proofs. Others left out significant aspects, like proportion and some even started with introductions that merely attacked rivals. These prior texts contained much unnecessary material.
xiiiThe idea that equals subtracted from equals are equal, for example, Euclid takes from Aristotle's Prior Analytics (22). H.D.P. Lee claims that Euclid's definitions correspond to Aristotle's definitions, Euclid's common notions to Aristotle's axioms and Aristotle's primitive existence claims to Euclid's PI-PIII.
xivAristotle believed that first principles of geometry ought all to be self-evident, so axioms and postulates ought to be self-evident.
xvOne notable exception is the fifth postulate. Remarkably, for many centuries geometers occupied themselves trying to prove it. This postulate is pivotal to the construction of the whole of Euclid's Elements. In the nineteenth century, when it was denied, or multiplied, the resulting alteration created new geometries, and we continue to construct new geometries now.
xviHeiberg, J.L., Euclid's Elements of Geometry, trans. Richard Fitzpatrick, from Euclidis Elementa, edidit et Latine interpretatus est I.L. Heiberg, in aedibus B.G. Teubneri, 1883–1885, 2007. Translations of Elements are derived from this text unless otherwise stipulated.
xvii Postulate 5 has a long history of attempted proof. It seems self-evident, yet it is sophisticated enough to require a
proof. It has yet to be proven and may not be provable.
xviiiJohn Mumma makes the case that the diagrams of Elements play a vital role in the proof itself. There is perhaps a very empirical role played by diagrams that make a sensible demonstration of some aspect of a proof. If the diagrams are empirical in nature, then there has long been a vital role for empirical aspects of reasoning in Euclid's Elements. The traditional view (Leibniz, somewhat Mueller) sees the figure drawn as independent of the fore of the demonstration. In this view the axioms and theorems comprise the reasoning.
xix“Bridge of Asses.”
Life and Works
Archimedes was almost a modern scientist, though he lived from 287 to 212 B.C. He was born and lived in Syracuse, but he may have traveled to Alexandria Egypt in order to study geometry and mathematics. Archimedes was friend to king Hieron of Syracuse and perhaps his family was somehow connected to the royal household. Whatever his connection to royalty, Archimedes enjoyed wide latitude and freedom to pursue his interests. There are more than a few stories about him, and as usual these tales are suspect, but they point to what his life may have been. King Hieron asked him to determine if a crown made by a possibly unscrupulous blacksmith was constructed from pure gold. The king suspected another metal had been mixed into the crown and consecrating a gold and silver crown as gold would be an insult to the gods. Thus, the issue was spiritual and politically important. Archimedes is famous for the method by which he determined the composition of the crown as well as his reaction when he discovered the solution. Ancient sources claim that he was in a bath and he noticed that when his feet entered the water, he displaced a certain volume of the liquid. Without concerning himself to don clothing he ran from the bath naked, saying “I have found it” or Eureka in ancient Greek. Archimedes is said to have been killed during the capture of Syracuse by the Roman general Marcellus, who wanted Archimedes to be taken unharmed, but perhaps because of the eccentric mathematician's dismissive disposition he died at the hands of a soldier. There are conflicting anecdotes about his death. One is that a soldier came in order to fetch him for Marcellus and when Archimedes saw a shadow on the proof he currently worked, he demanded the soldier get out of his way. Another story [i] tells of how a soldier came to Archimedes who was so absorbed in his work that he had not noticed Syracuse had fallen to the Romans. The soldier said to Archimedes that Marcellus awaits and Archimedes replied that he wanted to finish his proof first. In either case, the soldier grew angry and slayed the septuagenarian mathematician. It is likely that we will never know precisely how Archimedes died, but he almost certainly perished in the taking of Syracuse.
Several works of Archimedes survive extant, but a new codex has recently come to light and perhaps more of his work awaits rediscovery. On the Sphere and Cylinder is the work he was most proud to create, including a calculation for the surface area of a sphere and that the volume of a sphere is two-thirds that of the cylinder in which it is inscribed. [ii] Measurement of the Circle is a section of a longer work that has in it a calculation of π, still a mathematical constant. On Conoids and Spheroids determines the volumes of segments of solids in a conic section, which is a precursor to calculus. On Spirals deals with tangents to and areas related to the spiral of Archimedes.[iii] On the Equilibrium of Planes [iv] concerns centers of gravity of various figures; it establishes the “law of the lever.” Quadrature of the Parabola asserts that the area of any segment of a parabola is 4/3 of the area of a triangle having the same base and height as that segment, among other things. The Sand Reckoner attempts to express very large numbers in an effort to remedy Greek numerical notation. Method Concerning Mechanical Theorems describes mathematical discovery, demonstrating how Archimedes used a “mechanical” method in some of his primary discoveries. On Floating Bodies concerns hydrostatics; it determines positions solids will take while floating in fluid. Archimedes wrote other works that do not survive, but what scholars possess is formidable indeed. One, called the Stomachion, is a game consisting of a square cut into fourteen pieces.
Archimedes was an eccentric, seemingly interested only in his calculations and his not insignificant insights into material reality, and so his main preoccupation was pure mathematics. One may be justified in calling him a theoretical mathematician, except that the specific modern difference between theoretical and applied mathematics did not exist during his lifetime, and Archimedes used theoretical mathematics to make discoveries about empirical aspects of material reality; he used material reality for insights into mathematics. He, like modern physicists, used mathematics and geometry to explain the “how” of material reality. Archimedes explains the geometric aspects of the effect of gravity upon a lever in Equilibrium of Planes, for example. Occasionally, he paused from his work in order to invent some practical device. These practical devices are the things for which he was popularly known in the ancient world: the water screw, possibly a grappling hook that was a siege-engine of war, a method for determining the density of an object placed in water. Archimedes, like Euclid, does not explicitly state his first principles. Numbers for Archimedes seem to have a spatial significance, though he was likely not a Pythagorean in the traditional sense. He applies geometry to relations between physical objects and motions and in this sense his work anticipates what is called classical mechanics in physics. Archimedes' lack of application of applied mathematics tells us that he was not necessarily seeking god or the origin of the universe in his work, but rather he seems to be observing and working out mathematical observations and relationships of physical happenstance. He may be thought of as having a first principle of mathematics as it is applied to motion, objects and physical relationships. He studied what was there and sought regular patterns that could be explained upon each manifestation through numbers. Seemingly unconcerned to articulate first principles, he believed in the ability of mathematics to explain how things operate and that is a fundamental aspect of modern science.
Causation and Cosmology
Archimedes may have believed in a heliocentric universe, but we may correctly suspect that he did not believe in the traditional pagan gods in the manner that his contemporaries did. His Sand Reckoner contains what may be a clue to Archimedes' s cosmology: a description of a heliocentric system and an attempt to determine the sun's diameter. Causation for Archimedes seems to lie at least partially in gravitation as well as in differences between shapes of objects interacting with one another in space, and not necessarily time. Thus, explanation of how objects operate in the universe is a fundamental aspect of what may be called his cosmology, which is the same as saying that he is a precursor to those early modern physicists who defined and labored over classical mechanics, classical mechanics being the concern over the physical laws that make up motions of objects that have been influenced by forces of the universe. The study of motion was not new, even to Archimedes, but he studied motion without the hindrance of seeking god or the prejudice of a preconceived notion of some kind of transcendence. He simply sought to explain how what is there operates physically. One cannot call his perspective objectivity, but one may want to say that he carried with him less preconceived notions than earlier thinkers. He is an applied mathematician and geometrician. Thus, the mechanical objects he discovered and his breakthroughs in geometrical relationships are manifestations of his sense of causation and how the universe around us is composed. Thus, he does not articulate a cosmology, and if he did he would not have adhered to a traditional one, thought it may have had the veneer of traditional belief.
Archimedes' water screw [v] is a simple mechanism still employed to drive water up from depths where perhaps it otherwise may be impossible to obtain. The device is rather simple in construction. It is a tube with blades lining the inside in the shape of a screw.
As the tube turns it drives water up or down depending on the direction of the rotation. The tube can be stationary with the screw-blade inside moving or the apparatus can be constructed such that the whole of the tube moves and thus the position of the blades changes. In either case, the result is the raising or lowering of liquid. In some instances the screw is used for moving dirt or other material. Able to be constructed of any length and capable of relocating any kind of liquid, the device has myriad applications and remains in use today. It is a simple construction that makes use of geometric shapes to form an ergonomic device. If Archimedes did not design the water screw, then someone with a practical knowledge of geometry did.
Another anecdote that speaks to Archimedes' talent for practical invention is the verifiable story of king Hieron II when he asked Archimedes to determine if a crown was genuinely constructed from solid gold or not. There is some debate as to how Archimedes evaluated the density of the crown, but it is likely he made use of his Law of Buoyancy [vi]:
On Floating Bodies I, Proposition 5: Whatever solid is lighter than a fluid in magnitude will, once set in the fluid, be submerged in such a way that as much as is the weight of the solid, just so much is the weight of the submersion; let it have an equal heaviness to the whole magnitude. Let the same things be distributed, and let the fluid not move. Let the solid be EGHF, and let BGHC be the portion of it immersed when the fluid is at rest. Conceive a pyramid with vertex O including the solid, and another pyramid with the same vertex continuous with the former and equal to it. Suppose a portion of the fluid STUV at the base of the second pyramid to be equal and similar to the immersed portion of the solid; and let the construction be the same as in proposition 3. If therefore the fluid is not moved, similarly the parts of it itself placed inside are pressed equally.
Then, since the pressure on the parts of the fluid at PQ, QR must be equal in order that the fluid be at rest, it follows that the weight of the portion STUV of the fluid must be equal to the weight of the solid EGHF. And the former is equal to the weight of the fluid displaced by the immersed portion of the solid BGHC. It is clear then, that as much as is the weight of the fluid, so much is the part of the magnitude of the solid submerged and it has equal heaviness of the entire magnitude.
and his Law of the Lever [vii]:
6: Commensurable magnitudes establish an equilibrium at lengths proportionally corresponding to the magnitudes.
Let A, B be commensurable magnitudes, and let A, B be their centers. Let DE be a straight line divided at C so that as A is to B, so is DC to CE.
It must be shown that, if A is positioned at E and B at D, then C is the center of the two. Because A, B are commensurable, so are DC, CE. Let N be a common measure of DC, CE. Produce DH, DK each equal to CE, and EL (on CE) equal to CD. Then EH = CD, because DH = CE. Thus, LH is cut in half at E, as HK is cut in half at D. Then LH, HK must both contain N evenly. Let there be a magnitude O so that O is measured as many times in A as N is measured in LH, and from that as A is to O, so LH is to N. But as B is to A so is CE to DC and HK to LH. The result is, correspondingly, as B is to O, so is HK to N, or B is measured by O as much as HK is measured by N. Then, O is a common measure of A, B. Cut LH, HK into parts each equaling N, and cut A, B into parts each equaling O. The parts of A will equal those of LH in number, and the parts of B will be equal those of HK in number. Set one part of A at the middle of each of the parts N of LH. The center of the parts of A, placed equal distances on LH, will be at E, the middle point of LH [Prop. 5, Cor. 2], and the center of the parts of B, placed equal distances on HK, will be at D, which is the middle of HK. A is then applied at E and B applied at D. Yet the configuration formed by the parts O of A and B together is a configuration of equal magnitudes even in number and set at equal distances along LK. And because LE is equal to CD and EC is equal to DK, then LC is equal to CK so that C is the middle of LK. Therefore, C is the center of the configuration along LK and thus A operating at E and B operating at D balance around C.
7: Incommensurable magnitudes establish an equilibrium at lengths proportionally corresponding to the magnitudes.
Let the magnitudes be incommensurable, and let them be (A + a) and B. Let DE be a straight line divided at C so that (A + a) as to B is equal to DC as to CE. If (A + a) positioned at E and B positioned at D do not balance around C, (A + a) is too large to balance B, or not large enough.
Let (A + a) be too large to balance B. Subtract from (A + a) a magnitude smaller than the subtraction that would make the remainder balance B, but so that the remainder A and the magnitude B are commensurable. Because A, B are commensurable and A to B is less than DC to CE, A and B will not balance [Prop. 6], but D will be depressed. But this is impossible, because a-subtracted was insufficient reduction of (A + a) to produce equilibrium; the result was that E remained depressed. Therefore, (A + a) is not too large to balance B; and likewise it may be proven that B is not too large to balance (A + a). The result is that (A + a), B together have their center at C.
One suspends the crown [viii], which amounted to a wreath made of supposed gold, on one end of a scale and balances it with an equal mass of gold on the other end. One immerses the crown and gold along with the apparatus into a container of water. If the crown and the gold balance on the scale while submerged, then the two have the same volume, and the crown has the same density as the gold. If the scale tilts toward the gold, then the crown has a greater volume and its density is less. The crown would then be an alloy of gold and some other, lighter metal. Scholars do not know precisely how Archimedes determined the weight and density of the crown, but it is likely that these two propositions were integral to solving the problem. If one understands the laws of proportions, even in a purely mathematical way, then one can see the affects of gravity upon physical objects clearly. As long as a geometric rendering of the physical interaction is possible, Archimedes was able to find it. In other words, the spatial relationship of the objects on the lever and in the water corresponds to the relationship of the weight of each object and that weight's affect on the other objects. A physical interaction can be reduced to a geometric formula and so pure math renders how material reality interacts. Here is a wholeheartedly modern method that discovers something empirically verifiable by purely mathematical means. As modern and incredible as these discoveries seem historically, they are practical inventions that were probably only afterthoughts on Archimedes' part, if one takes the surviving anecdotes as evidence of his attitude.
Other insights were much more theoretical, and to his liking. These were not known widely, but Archimedes shared them with like-minded thinkers. Thus, Archimedes also engaged in more pure mathematical exercises. For instance, he found a way to determine the value of π. π is a mathematical constant, the ratio of the circumference of a circle to its diameter. It is the same ratio no matter what the size of the circle and like the Pythagorean theorem it is true whether or not there exists some shape that has come into existence. It has been used by mathematicians to found many systems and proofs. Remember that Euclid needs axioms and postulates in order to begin arguing for a certain point. Aristotle's conversion, contraposition and obversion of different statements are the same logical conception. These are reliable truths that philosophers, logicians and mathematicians can employ as initial certainties, or at least givens. A sound proof, argument or proposition cannot exist without these foundations. Archimedes' method for determining the value of π is somewhat complicated, and so it is not included in this section, yet his basic idea is simple, possessing both mathematical and geometric aspects. He constructed a polygon consisting of equilateral triangles inside of a circle. He then constructed another polygon with the same number of sides outside of a circle. The polygons have six sides initially. The polygon inside of the circle is inscribed; the polygon outside the circle is circumscribed.
iObtained from http://www.math.nyu.edu/~crorres/Archimedes/Crown/CrownIntro.html.
If we want to know how to calculate the relationship between the circumference and the diameter for the six-sided hexagon, then we divide the circumference by the diameter. Since Archimedes constructed the hexagons from equilateral triangles, the first calculation is simple. The diameter of the inscribed hexagon is two where the circumference is six. 6/2 = 3 is the first calculation of π, because it shows the relationship between the circumference and the diameter of the hexagon, but it is not very accurate because the hexagon merely resembles a circle. One needs a shape that is much closer to the circle in order to find the better approximation. One can simply measure the perimeter of the hexagon, but then the calculation is limited by one's tools, and π is an irrational number. So, 3 cannot be π. It is merely an approximation. Archimedes constructed another set of polygons, one inscribed in the circle and another circumscribing the circle. These new polygons were dodecagons, each possessing twelve equal sides. He thus doubled the number of sides of the polygons and they both now resemble more closely a circle.
He then found the length of each new side, the sides being equal for each polygon, and when he multiplied the new length of each side by the number of sides of the new polygon he arrived at a number he was able to divide by the diameter of each shape in order to more closely approximate π. Yet, there remains a difficulty. Archimedes needed to know what the length of the new side was in order to calculate more accurately what the ratio is of the new figure. A more elegant and easier way to understand what Archimedes was doing is by using one polygon inside of a circle instead of his two. The Pythagorean theorem can then be used to determine the sides of the polygon. The inscribed polygon is the more elegant and more modern way of employing Archimedes' method. It works just as well, and it is easier to use. Remember that Archimedes used the calculations of triangles inside and outside the circle. He employed parts of Euclid's Elements as well as aspects of the Pythagorean theorem. The method below is a refined version of his method. One can say that it is derivative of Archimedes' method.[ix]
The first triangle one can determine because one knows the lengths of the two sides. The triangle is equilateral, so the sides are all the same length. The sides near the circle are cut in half (here in the upper right corner of the image), so if the diameter of the circle is 1, then each one of those sides is ½. We know by the Pythagorean theorem that a = √1- (S₁/2)² and b = 1 – a. We can see that S₂ = √b² + (S₁/2)². The Pythagorean theorem one can use to find the value of a and from that the value of b. Once the value of b is attained, the value of S₂ follows because we again use the Pythagorean theorem to determine its length. S₂ is the length of the side that we need. S₂ is multiplied by the number of sides in order to attain the perimeter, which is then divided by 2. 2 is the diameter of both the circle and the hexagon. Thus, a continually refined value of π can be found by increasing the number of sides until the sides of the polygon resemble more closely a circle. Archimedes was thus able to calculate the ratio of the circumference to the diameter each successive time he added sides to the figures inside and outside of the circle. He performed this same task repeatedly, each time bisecting the sides of the new polygon until he created a 96-sided figure. The 96-sided figure was a much closer approximation of the value of π because Archimedes used an accepted practice of exhausting the space between the figure whose sides he could determine and and the shape of the circle. In other words, the more sides created on a polygon, the more circular the polygon becomes. His astounding task was to find measurements of curved objects. Remember that ancient Greeks were able to determine the areas of things with triangles, but triangles do not fit neatly into curved shapes. Archimedes refines the method of exhaustion of Eudoxus.
One exhausts the space between the two shapes. Using two polygons, Archimedes approximated π to between two numbers: < 3 1/7 and > 3 10/71. That is why one polygon circumscribes the circle and another is inscribed. One must remember that p is a constant, but it is also an irrational number. Its calculation will continue indefinitely, if one continues to seek greater accuracy. Irrational numbers are strange in that they seem to exist as both odd and even numbers. They are like points on a line that can never be found and yet they are present. So, calculating p to its exact measure is not possible, but Archimedes approximated it with only a circle and a polygon of 96 sides. Archimedes' calculation of π is another example where math allows the thinker to make a determination of something that would not be possible merely with human senses. Math thus reaches into a part of reality where perception goes dark. One will remember as well that the Pythagorean theorem is an integral part of scientific development. It is useful in determining p with Archimedes' method. Not only does it tell us something about most triangles, but it is a constant, like π, that can be used in other proofs and it has practical applications. Again, when one is attempting to determine what the side of a new polygon is, one is able to use the Pythagorean theorem to determine its length because one knows the lengths of the other two sides of a right-angled triangle that is inscribed within the polygon. Again, one must keep in mind that Archimedes used two polygons, one inscribed in and one circumscribing the circle.
Archimedes, like any good mathematician, takes things that he knows, like constants and formulas taken from other mathematicians, and he uses them to find the things that he does not know. These inferences are really no different fundamentally than the ones of earlier mathematicians and philosophers, except they are geometrically or mathematically based rather than language based. The spirit of science resides in all of these insights and arguments about material reality. Archimedes' use of the Pythagorean theorem is only one such example. As we have seen, he also determined the area of a circle, πr² , which the ancients called squaring the circle. Euclid knew that there was some constant that allowed one to determine the area of a circle, but he did not know nor calculate what that constant was. When the value of p was found, the calculation became possible. One may recall the discussion earlier in this text about sine, cosine and tangent. These are all constants used in calculating formulas and proofs in mathematics. They are aspects of triangles themselves, like the Pythagorean theorem. One may also recall Plato's eidos, which is a kind of constant ideal in which something participates. These geometric formulas are very much like a Platonic eidos in that they are constant calculations that will exist for each triangle or circle or whatever shape, no matter whether there is a manifestation of them or not. In that sense modern science is built on something that closely resembles a Platonic form, or eidos. Scientists may not want to admit to being Platonists, but some fundamental aspects of their very way of arguing relies upon notions that inspired Plato's theory of eidos.
i Plutarch's Parallel Lives: Marcellus.
ii Archimedes requested that a representation of this discovery be placed on his tomb, which it was.
iii The locus of a point moving with uniform speed along a straight line that itself is rotating with uniform speed about a fixed point. (You found this precise phrase here: https://www.britannica.com/biography/Archimedes)
iv Scholars suspect that much of this book is a spurious creation of Archimedes, as it contains reworkings below the standard of Archimedes. Perhaps its main concept was conceived by thinkers who came before Archimedes.
v Some scholars believe that Archimedes did not invent the water screw. It is supposed that he perhaps refined the design on a trip to Egypt or that others, amazed at his ability to construct workable devices, attributed it to him mistakenly. Perhaps Archimedes was merely alive at the time the screw was introduced to the Greeks and he was given credit for its invention because of his fame. For more, see Oleson, John Peter (1984), Greek and Roman Mechanical Water-lifting devices. The History of Technology, Dordecht: D. Reidel.
vi Parts of this translation were taken from Heath, Thomas. 1897. The Works of Archimedes. Cambridge, Cambridge University Press. The section is On Floating Bodies.
vii Portions of this translation were taken from the above text. The section is Equilibrium of Planes.
viii Obtained from http://www.math.nyu.edu/~crorres/Archimedes/Crown/CrownIntro.html.
ix One can watch a working out of p here: https://www.youtube.com/watch?v=_rJdkhlWZVQ