Ancient Pieces of Science


© 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.

Example one:

  • All cats are quadrupeds. Assertion
  • All lions are cats. Assertion
  • Therefore, all lions are quadrupeds. Inference

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.

  • Some humans are owners of sports cars. Assertion
  • Some owners of sports cars are safe drivers. 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.

  • All S are P.
  • Some S are P.

These are the subject and the predicate of the given statement. There are also middle terms, indicated by the “M” in the examples below.

  • All M are S.
  • Some P are M.

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.

  • All cats are quadrupeds.
  • All lions are cats.
  • Therefore, all lions are quadrupeds.

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.

  • Some cats are quadrupeds.
  • Some lions are cats.

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.

  • All paper towels are men with socks.
  • All unicorns are paper towels.
  • Thus, all unicorns are men with socks.

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.

  • All cats are quadrupeds.
  • All lions are cats.
  • Therefore, all lions are quadrupeds.

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:

  • Seven individuals came down with explosive diarrhea while eating in the
  • same restaurant. They all ate different food, but used Momma's Sweet
  • Red-Hot sauce. The hot sauce was the only commonality. Food scientists
  • concluded that the ailment came about because of the hot sauce.

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]

Table 1.png

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:

  • Five students performed worse on exams at Sultan University. A list of several
  • present and absent factors were examined as possible causes. Among eight likely
  • causes, only one was absent for all five students: their cell phone.

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:

Table 2.png

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:

  • Eight members of the Men's Club of America contracted a rare form of plague.
  • Hearing about it, a doctor flew to their clubhouse with a serum that was thought
  • to be a cure. When the doctor arrived, only four of the men would accept the
  • serum, but all eight had previously treated themselves with various club remedies.
  • After a short time the four who received the serum recovered while the other
  • four did not. Among those who recovered, no single club remedy had been
  • given to all; and among those who did not recover, every club remedy had
  • been given to at least one. The doctor concluded that the serum was a
  • cure for the disease.[x]

Here is a table representing the Double Method of Agreement:

Table 3.png

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:

  • Two identical white mice in a controlled experiment were given identical amounts
  • of four different foods. In addition, one of the mice was fed a certain drug. A short
  • time later the mouse that was fed the drug became nervous and agitated. The
  • researchers concluded that the drug caused the nervousness.[xi]

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.

Table 4.png

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:

  • Two identical white mice in a controlled experiment were fed identical diets. In
  • addition, both were given vitamins A, B and C. One of the mice was also given
  • vitamin D while the other was not. The mouse that was not fed vitamin D
  • developed rickets, but the other one did not. The researchers concluded that
  • the lack of vitamin D caused the rickets.[xiii]
Table 5.png

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:

  • George, who exercised regularly, took vitamins, and got plenty of rest,
  • contracted a rare disease. Doctors administered an antibiotic and the disease
  • cleared up. Convinced that the cure was caused by either the exercise, the
  • vitamins, the rest, or the antibiotic, the doctors searched for analogous cases.
  • Among the two that were found, one got no exercise, took no vitamins, and got
  • little rest. He was given the same antibiotic and was cured. The other person,
  • who did the same things George did, was given no antibiotic and was not cured.
  • The doctors concluded that George was cured by the antibiotic.[xiv]
Table 6.png

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.

  • ABC is coincident with abc
  • AB + C is coincident with ab + c
  • AB – C is coincident with ab – c

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.

  • [i] Pullman, Bernard. 1998. The Atom in the History of Human Thought. New York, Oxford, 6.
  • [ii] Lloyd, G.E.R. Demystifying Mentalities. Cambridge, 1990, 29.
  • [iii] French, R.K. Ancient Natural History. London, 1994, xiii.
  • [iv] Aristotle claims as much in various places: “there is a knowledge of being qua being…” (Metaphysics IV, 1), “substance is the subject of our inquiry” (Metaphysics XII, 1), “Nature is a principle of motion and change and it is the subject of our inquiry” (Physics III, 1), et alia.
  • [v] For a review of induction and deduction in modern science, see Sambursky, S. 1956. The Physical World of the Greeks. New York.
  • [vi] The section that concerns causality comes from Hurley, Patrick J.A Concise Introduction to Logic.
  • [vii] The following conditional statements are taken from Hurley's A Concise Introduction to Logic, 487. The examples that follow are modified versions of those given in A Concise Introduction to Logic.
  • [viii] These are John Stuart Mill's methods. We use these explanations of cause and effect because they fit more easily into the ancient framework of thinking. Modern methods are more complicated and in some ways better, but Mill's methods make clear the connection the ancient world has to the modern one. What we review are not precisely the same methods Mill produced, but they will give us a better connection to the ancient way of reasoning. We may thus better compare and understand how ancient thinking is “scientific.” There is no claim that the whole of scientific reasoning is here represented, but these methods provide some contextualization of modern science and contrast with ancient inquiry.
  • [ix] The following tables are all taken from Hurley’s Concise Introduction to Logic.
  • [x] This example is a modified version of one that comes from Hurley’s Concise Introduction to Logic, p 493.
  • [xi] This example was taken from Hurley’s Concise Introduction to Logic, p 493.
  • [xii] Hurley, p. 495.
  • [xiii] Hurley, p. 495.
  • [xiv] Hurley, p. 496.

© 2017 Kirk Shellko All rights reserved


© 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.

First Principles

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.

  • [i] A.W.H. Adkins, 17. F.R. Earp is likely correct that the poets were responsible for transmitting the ideas of the gods to the Greeks. The poets thus transmit the idea of powers or forces of the universe. It is from that conceptual point that the philosophers and protoscientists begin to speculate. According to A.A. Long, “divine intervention <cannot> be simply removed from the poems to leave a kernel of sociological truths.” Grube thinks of the gods as those who “have the capacity to see everything and to know everything when they care to use it. There are no other powers beyond or behind but only below them. Where Homer uses more abstract terms such as Fate, Moira, Aisa and the like they are but another way of referring to the power of the gods, or else they represent subordinate functions.” 63. O. Primavesi talks about the “decoding” of the Homeric gods. He asserts correctly that “Homeric gods represent the basic entities of the physical universe, like elementary qualities or the elements themselves. Thus, the traditional anthropomorphic design of these gods is redefined as a mere surface under which a deeper, physical level of meaning has been hiding all the time.”, 257.
  • [ii] For a basic understanding of epic creation of gods in human images, see A.W.H. Adkins. For Adkins the gods are the powers that overcome one another and control things; these are the hierarchy of power in human relationships that are applied to the universe; even beggars are involved in power relationships. See also Grube.
  • [iii] G.E.R. Lloyd properly states of epic that “while these pre-philosophical myths contain a variety of stories of origins, none of them constitutes a cosmology in the strict sense.... None of them...presents a comprehensive account of the world as we know it as an ordered system.”, 147.
  • [iv] Il. 17, 425, Il. 5, 504, Od 3, 2, Od 15, 329 and 17, 565. These are all passing references to the sky as iron or bronze.
  • [v] Unless otherwise stated, all translations are mine.
  • [vi] Plato may be less than serious and one must always remember that Aristotle does not always render his rivals fairly. Nevertheless, the idea that Oceanus is the origin of all things seems to have been popular.
  • [vii] Turkeltaub, Daniel. 2007. “Perceiving Iliadic Gods.” Harvard Studies in Classical Philology 103:56.
  • [viii] For Jasper Griffin “a god who watches is normally a god who intervenes, a patron and an avenger.” So, Homeric gods are actors in the scheme of things. Griffin goes on to say that “we must accept everywhere in Homer that aspects of the divine which seem to us disparate or irreconcilable are in fact inseparable.” Griffin, Jasper. 1978. “The Divine Audience and the Religion of the Iliad.” 2, 18.
  • [ix] Aristotle, Poetics 9.1448b4-17.
  • [x] For a good explication of Aristotle's view of mimesis, see Golden “Mimesis and Katharsis.” For Goldstein Aristotle claims that “mimesis means that the method or process of art imitates the method or process of nature” (570), and art comes from knowledge of its topic. Knowledge of nature is needed for art as well as science because each discipline is lodged in human ability to discern patterns. For more on Aristotle's view of poetry (mimesis) as universal, see C.O. Brink, Viviene Gray, Malcolm Heath, Katherine Gilbert, John Armstrong, Leon Golden, G. Hagberg and especially Silvia Carli. Armstrong rightly points out that mimesis is the representation of a single action, meaning one action type. This means that poetry–epic–represents something that is repeated: a pattern. Gilbert asserts that “art, then, is human making in the image of divine making, for art emulates the energetic processes of nature, and God is the Prime Mover of nature.” 560. Longinus (On Sublimity 22.1) states that “imitation of the activities of nature” is achieved best by the writers who imitate what men really do, adapting their language to their characters. James Redfield writes that plot is “ terms of relations between...causes and consequences” and conveys “some universal pattern of human probability or necessity”, 56. Plato had a radically less material perspective on mimesis. McKeon says of Plato's perspective on mimesis that “the word 'imitation' indicates the lesser term of the proportion of being to appearance: if God is, the universe is an imitation; if all things are, shadows and reflections are imitations....” In Plato's Statesman (293e) there is one state governed by wisdom and justice and all others imitate that state; later (300c) laws are said to be imitations of truth and still later (300e) it is said that if humans imitate without knowledge, they imitate badly, but if they have knowledge they transcend imitation. In his Timaeus (49a) the universe itself is an imitation of an eternal universe. Significantly, in Timeaus (28a-b) it is said that when one uses the unchanging universe as a model, the result is beautiful whereas using any other model produces non-beautiful copies. J.A. Philip concisely reviews Plato's perspective: “Mimesis for Plato suggests the creative activity by which the higher realities are sensibly embodied; it also implies a real transcendental relation which does not..exclude the immanence implied by the presence of a like element in exemplar and example. Indeed homoiousthai (the verb for “make similar"), often regarded as a typical immanence word, is regularly used as a synonym for mimeisthai (the verb for “imitation”), usually regarded as a typical immanence word.”, 465. Parentheses mine. Cornford and Havelock interpret mimesis as at times “impersonation” and at times “representation.” Plato is here mentioned in order to demonstrate that some form of imitation had an important role in acquiring knowledge. McKeon states that “the copy-model conception of the universe...has the important consequence of awarding a significant status to mimesis as a means for clarifying and apprehending reality itself.” 150. There are myriad other references to mimesis in Plato, too many to list here. For a review of several ways Plato uses the term mimeisthai, see P. Vicaire. The concept was vital to his philosophy. Even Charles Darwin claims that "the principle of imitation is strong in man....", 76-7. It is important to note that the conception of mimesis is a later conception of the relationship between the knower and the known. Early Greek conception of knowledge linked perception and knowing much more intimately. Francis Cornford (1957) and James Lesher point out how Greeks conceived perception. For Lesher Homer's Odyssey demonstrates that sense perception and thought were not differentiated, and their union was taken paradigmatically as representative of all cognition. The ancient Greeks may have thought there was no distinction between perception and thought, but their actual assimilation and processing of perception into knowledge entailed a more sophisticated cognition: mimesis. The paradigmatic shifts of Xenophanes and Heraclitus were awakenings to what thought and perceptions are; their mimesis was a subconscious part of their literature and later a conscious part of their thought.
  • [xi] For more on the organization of art and science, see Harvey Goldstein and Richard McKeon. Northrop Frye seems to have believed in mimesis as art's dependence on reality. This is a much more traditional view. For more, see Frye.
  • [xii] Goals may have much more to do with imitative behavior than mere mimicking. For more, see Meltzoff and Decety. Also, empirical studies suggest that imitation may not be innate. For more see Nicholas Shea. R.W. Byrne suggests that imitation may not be as innate as has been thought. G.R.F. Ferrari claims that “imitative behaviour appears to emerge out of the infant's acquisition of different kinds of knowledge and motor, cognitive skills.” 2325. Susan S. Jones agrees that imitation is an “emergent system.” No researcher asserts that imitation does not exist, but rather only the time of inception or nature are points of contention.
  • [xiii] Marco Iacoboni and others claim that two areas of the brain are used: the inferior frontal area and the parietal lobe area. This means that “imitation may be based on a mechanism directly matching the observed action onto an internal motor representation of that action ('direct matching hypothesis').”, 2526. The point here is that science is in the process of proving that imitation is a function of human learning.
  • [xiv] For more of what imitation produces, see Meltzoff and Decety.
  • [xv] Stefan Willer claims that “it is not the act of signification which is at the heart of language, but the desire of the speaker to actually be like the world around him. This desire is the original impulse of imitation.”, 204.
  • [xvi] See Stevens, Jung, 1994. See also Niko Tinbergen, H.F. Harlow, M.K. Harlow and Konrad Lorenz. J.A. Philip says that “here the craftsman makes his craft product like a model, as the actor makes himself like his model; but the craftsman's model is an archetype, a form of the product to which the particular instance imperfectly conforms. It is a like element in form and craft object that shapes the object. The craftsman knows his form and makes his product like that form. This process of making in the likeness of the form is called mimesis.”, 463.
  • [xvii] Anthony Stevens justifies his apology of Jungian archetypes by claiming that “very similar ideas to Jung's have become current in the last forty years in the relatively new science of ethology (that branch of behavioural biology which studies animals in the natural habitats). Every animal species possesses a repertoire of behaviours. This behavioural repertoire is dependent on structures which evolution has built into the central nervous system of the species. Ethologists call these structures innate releasing mechanisms, of IRMs. Each IRM is primed to become active when an appropriate stimulus–called a sign stimulus–is encountered in the environment. When such a stimulus appears, the innate mechanism is released, and the animal responds with a characteristic pattern of behaviour which is adapted, through evolution, to the situation.”, 36.
  • [xviii] Leon Golden asserts rightly that “we remember that learning for Aristotle involves a movement from the particular to the universal and that it reaches its climax in an insight or inference.” Golden, Leon. 1969. “Mimesis and Kaharsis" 146.
  • [xix] ,28-32.

© 2017 Kirk Shellko All rights reserved


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© 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.

First Principles

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.

  • Bad witnesses are eyes and ears for human beings who possess minds foreign to understanding. (Fr. 107, Sextus adv. math. VII, 126).

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:

  • Sea is the most pure and most soiled water; for fish it is potable and preserving, for humans non-potable and destructive. (Fr. 61, Hippolytus Ref. IX, 10, 5).
  • A path up-down is one and the same. (Fr. 60, Hippolytus Ref. IX, 10, 4).
  • Sickness makes health sweet and good, as hunger does for satiety and toil for rest. (Fr. 111, Stobaeus Anth. III, I, 177).
  • The same thing exists within, the living and dead, the wakeful and sleeping, the young and old. For these things here having changed are those there and those there having changed again are here these. (Fr. 88, [Plutarch] Cons. ad Apoll. 10, 106E).

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:

  • The thing setting itself against [itself] brings together and from different things the finest arrangement comes, [and all things have come to be through strife = B 80]. (Fr. 8 Eth. Nic. Q 2. 1155b4).
  • It is necessary to know that war is common to all and justice is strife. And all things have come to be through strife, it being necessary. (DK B80).

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:

  • The growth of things is accustomed to conceal itself. (Fr. 123, Themistius Or. 5, p. 69).
  • An agreement unapparent is stronger than an apparent one. (Fr. 54 Hippolytus Ref. IX, 9, 5).

All things are for Heraclitus in constant change. His famous river image [ix] gives the metaphor:

  • Upon those stepping into the same rivers different things and different waters do run. (Fr. B12, Arius Didymus ap. Eusebium P.E. XV, 20).
  • It scatters and collects, stands together and departs; it is nearby and away. (Fr. 91, Putarch de E 18, 392B).

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).

  • And some say not that some of existing things move themselves and others do not, but all things move themselves and always do they do so, but that this escapes our senses. (Aristotle Phys. Q3, 253b9).[x]
  • Changing, it rests. (Fr. 84A, Plotinus, Enn, IV 8, 1 [n. B 60]).

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.

  • All subjective comprehensions are common apprehensions of things.
  • All perceptions of things are subjective comprehensions.
  • All perceptions of things are common apprehensions of things.

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:

  • All things misapprehended if taken privately are possible unfit witnesses of underlying truth.
  • All common apprehensions of things are things misapprehended if taken privately.
  • All common apprehensions of things are possible unfit witnesses of underlying 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:

  • All paths up are paths down.
  • All paths down are paths up.
  • All paths down are paths down.

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.

  • All paths down are paths up.
  • All paths up are paths down.
  • All paths up are paths up.

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.

  • [i] For an insightful method on how to interpret Heraclitus's cryptic aphorisms, see Reames.
  • [ii] Diogenes Laertius IX, 5.
  • [iii] Herbert Granger correctly points out that “the theology of the early Greek philosophers goes hand-in-hand with their physics, since they had inherited a conception of divinity and nature that intertwines the two, and thus the study of the one topic brings along the other.” 163-4. For more, see Granger,
  • [iv] For a sense of logos in Heraclitus as objective reason and representative of universal law, see Hülsz. Hülsz believes that logos is representative of a measure of objective reality – itself a kind of proportion. Daniel Graham (2013) writes of logos as an expression of subjectivity and its implications, which is part and parcel to logos as measure of a private interpretation of a common universe. Emlyn-Jones claims that Heraclitus prefers images in place of logical explanations, and so logos may be interpreted as a kind of semi-metaphor. Wilcox points out that the Greeks made a direct connection between sight and knowing that compelled them to trust in perception too much. Heraclitus's logos seems to be a reaction and warning to that paradigm. Minar gives a review of logos and claims it signifies the proportionality, harmony and rhythm of the Pythagoreans. Thus, logos is variously interpreted, as Heraclitus seems to have intended.
  • [v] Ancient Greeks generally disdained the politically private in favor of the politically public, but here that inherent prejudice seems to take on an epistemological tone. The division between a more objective assessment and more subjective assessment seems obvious, but Heraclitus may have meant something other than objectivity entirely or he may have had a nascent awareness of objective reality. Also linked is the notion of opinion for ancient Greeks in general and Plato in particular. Opinions were unstable, unreliable private assessments that had no place in the comprehension of an ordered universe that had as its foundation universal elements or some kind of static basis.
  • [vi] The PreSocratic Philosophers, 190.
  • [vii] We must here be careful because in some ways the same thing elicits not merely different descriptions or characteristics, but the same thing actually is those different aspects. Opposites and their coincidence are part of Heraclitus' implicit structure of things and such structure is the whole object, not merely different aspects.
  • [viii] For a discussion on the unity of opposites as harmony, see Dilcher, Kahn. McIntosh-Snyder seems to think that unity comes from agreement, not so much tension, but the agreement may come only through the tension. Rabinowitz and Matson contend that Heraclitus expresses a “reality of pattern or order” and that is his “hidden attunement”. For Moravcsik claims that “to remain the same living thing requires constant change, and systematic replacement of parts, in order to remain the same entity” and he states that “ challenging us to accept a radically new conception of reality in which the dynamic underlies even what on the surface looks like static.” 559. Moravcsik also believes Heraclitus to be a precursor to Plato. It is easy to see how Heraclitus sees a kind of motive attunement in things (and so he is not Aristotelian but Aristotle is Heraclitean), but that it is an underlying order like Plato's eidos seems suspect if only because Heraclitus suggests his conclusions; he does not develop his theory formally.
  • [ix] There are other so-called “river fragments”. B49: Into the same rivers we step and do not step, we are and are not. B91: (a) Into the same rive it is not possible to step twice. (b) It scatters things and again gathers them...and approaches and recedes. A6 (Plato, Cratylus 402a8-10): Heraclitus says, I think, that all things pass and nothing remains, and comparing things that exist to the running of a river, he claims that you cannot step twice into the same river.
  • [x] Aristotle also comments on static constancy through change at Politics 1276a34-39 and On Generation and Corruption 33 6b28-33.
  • [xi] Heisenberg claims that “modern physics is extremely near to the doctrine of Heraclitus. If we replace “fire” by the word “energy” we can almost repeat his statements word for word from our modern point of view. Energy is in fact that from which all elementary particles, all atoms,...are made; and energy is that which moves. ...Energy is a substance,.... Energy can be changed into motion, into heat, into tension. Energy may be called the fundamental cause for change in the world.” Haarhoff quotes Heisenberg further that “a little later he speaks of the transmutation into matter. Fragment 22 of Heracleitos reads: 'All things are exchanged for fire [i.e. energy] and fire for all things.' This idea of change and exchange, reminding us of the phenomena of radioactivity, is part of modern science. In the lecture referred to Heisenberg says: 'Energy is merely the force which keeps everything in perpetual motion; the “fire” of Heraclitus, the fundamental substance of which it consists.” 128-9. Daniel Graham states that “if measures or portions of the world are alternately being kindled and being quenched, the world is not fire in the sense of having the properties of fire. What happens of course is that the world is constantly changing, and in that sense it is firelike. But Heraclitus calls the world fire, as if to identify the nature of the world with its leading component; and in B31a he talks of the turnings of fire, as though it were fundamental in a way the other stuffs in the series are not.” 145-6. Graham continues by saying that “in summary, the evidence we have seen so far allows us to view Heraclitus as construing the transition from one substance to another in the series of basic substances as instances of generation. On this view there is no continuing element or subtratum, but a change of identity in which one substance dies as another is born. The relationship between the two is not a strict identity but an equivalence of value in which a given quantity of the first substance corresponds to a different, but proportionate, quantity of the second. As a quantity of the first substance turns into the second, a proportionate quantity of the second turns into the first. On the other hand, B30 and B31a give a pride of place to fire, and seem to assert that it remains when all else changes. Is there indeed an underlying reality that is preserved through all changes? Heraclitus's analysis of elemental changes tells us that there is not. Fire is fundamental just by being symbolic of the constant change that the elements undergo.” Graham further states that “the basic substances of the world are constantly undergoing reciprocal transformation in a lawlike way: i. Each portion of a given basic substance that turns into another substance is replaced by an equivalent portion form another basic substance which turns into the first substance, ii. Hence the total amount of each basic substance in the world remains constant. We have the beginnings of a law of conservation, not precisely of mass or of matter, but of material proportions. 167-8. Karl Popper articulates it a bit differently but with a similar understanding. He claims that “in reality a material thing is like a flame; for a flame seems to be a material thing, but it is not: it is a process; it is in flux; matter passes through it; it is like a river. Thus all the apparently more or less stable things are really in flux; and some of them – those which indeed appear stable – are in invisible flux.” 387 G.S. Kirk articulates the same kind of concept when he defends the unity of opposites theory. He states that “the unity depends not only on [opposites] alteration one into the other, but also on the quantitative regularity of this alteration. If the total amount of old age in the world begins greatly to exceed the total amount of youth, then the succession will eventually fail. If the total amount of heat and dryness I summer begins to out-weigh the total amount of cold and wetness in winter, or vice versa, first the crops will fail and eventually the earth will be overcome by one of those catastrophes of fire or flood. If the balance of processes is destroyed then the underlying unity of the cosmos fails, and this, for Heraclitus, was unthinkable. And this balance depends on metron.” 37 Finally, Vlastos adds that “for Anaaximander's equilibrium of elements he (Heraclitus) substitutes equilibrium of processes of change.” 359 For more see Haarhoff, Graham, Popper, Kirk, Moravcsik, and Vlastos (1955).
  • [xii] According to Daniel Graham there are five doctrines that form the ancient tradition of Heraclitus interpretation: (1) Fire is the origin and ruler, material cause, of the universe, (2) The universe periodically is consumed and then reborn, (3) All is in flux, (4) Heraclitus's opposites are identical and (5) Heraclitus violates the law of non-contradiction. Modern interpretation assaults all of these interpretations. For arguments against (2), (4) and (5), see Burnet. For arguments against all five, see Reinhardt, Kirk and Marcovich. For arguments for flux theory, see Vlastos, Popper, Mondolfo, Guthrie, Jones, Stokes and Barn es. Graham claims that Kahn, Mackenzie and Wiggins attempt “to break out of the limits of previous interpretations. Moravcsic looks at the history of interpretation. Kahn talks of identity preserved through alteration (168). Graham has the most comprehensive and thought-provoking interpretation, in the humble opinion of this text.
  • [xiii] Aristotle criticizes Heraclitus for this reason in Metaphysics IV, 1005b 17-20 and Metaphysics A6, 1015b 36. Plato discusses corollaries related to this issue at Theatetus 152 c-e.

© 2018 Kirk Shellko All rights reserved.


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  • Sider, D. and D. Obbink (eds.). 2013. Doctrine and Doxography: Studies on Heraclitus and Pythagoras, Berlin: De Gruyter.
  • Stokes, Michael. 1971. One and Many in Presocratic Philosophy. Washington: Center for Hellenic Studies.
  • Taran, L. 1999. “Heraclitus: the River-fragments and their implications,” Elenchos 20: 9- 52. Repr. In id. Collected Papers (1962-1999).
  • Vlastos, Gregory. 1952. “Theology and Philosophy in Early Greek Thought,” Philosophical Quarterly 2: 97- 123.
  • _____________. 1955. “On Heraclitus.” The American Journal of Philology 76, No. 4: 337-368.
  • Wiggins, David. 1982. “Heraclitus' Conceptions of Flux, Fire and Material Persistence,” in Language and Logos: Studies Presented to G.E.L. Owen. M Schofield and M.C. Nussbaum (eds.) Cambridge: Cambridge University Press.
  • Wilcox, Joel. 1993. “On the Distinction Between Thought and Perception in Heraclitus.” Apeiron: A Journal for Ancient Philosophy and Science 26, No. 1: 1-18.


© 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]

First Principles

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.

Musical Harmony III.jpg

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.

RIght-angled triangle.GIF

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 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 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.

  • All things measurable by formulations of number are things representative of all parts of reality.
  • All right triangles are things measurable by formulations of number.
  • All right triangles are things representative of all parts of reality.

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.

  • [i] See Geoffrey Lloyd.
  • [ii] Diog. L. VIII, 3.
  • [iii] Plato, Rep. 600 A-B.
  • [iv] Aristotle, Met. A5, 986a 30 and Rhet. B23, 1398b 14.
  • [v] Iambl. V.P. 199 (DK 14, 17).
  • [vii] 1 was mind; 2 opinion; 3 the number of the whole; 4 justice; 5 Marriage; 6 ?; 7 the right time; 8?; 9 justice; 10 perfect. Aristotle reviews these but could not determine 6 or 8. For more, see Cornford (1922 and 1923).
  • [vii] Specifically Philolaus, Archytas and Eudoxus. The works of these men are lost, but referred to in later commentaries and other works. Some believe that Pythagoras deserves credit for creating a kind of science. Reviel Netz seems correct in saying that at least Pythagoras combined disciplines that would not otherwise have been connected: math and philosophy. His influence was keenly felt no matter his actual contribution. For a review of the problems that resulted from a possibly overstated reputation of the Pythagoreans and the source materials, see Huffman. For more, see Cornford, Jaeger.
  • [viii] Leonid Zhmud claims that the Egyptians helped the Greeks with practical geometry, or “land measurement”, but they did not influence the ancient Greeks even as much as the ancients believed. Pythagoras would then perhaps be one of the Greeks who began the abstract study of arithmetic and number. S. Luria (in Zhmud) points out that the Egyptians did influence the Greeks; their influence was not vital; the proofs were the result of Greek genius. Walter Burkert adds to this perspective that the Greek study of mathematics was not Pythagorean, but Greek. He claims as well that the Pythagoreans would not have been amazed at their discoveries had they been actual mathematicians. Francis Cornford seems to have believed that Parmenides was influenced by the Pythagoreans, and there is some possibility that Parmenides was a former Pythagorean. Otto Neugebauer agrees that the “Pythagorean” theorem had been used for centuries in Babylon, but the Babylonians never evolved into scientific thinkers and Euclidean mathematics and axiomatic methods were exclusively Greek developments. For an ancient account of the Pythagorean discovery of musical intervals, see Boethius De Institutione Musica I, chaps. 10-11.
  • [ix] The interval sounds are in the plucking of an open string. It is the depression of the string at the correct place that brings out the sound. All musicians of Pythagoras's time would have known the phenomenon. For a succinct and clear explanation of how the intervals operate, see Ferguson. Van der Waerden thinks that Pythagoras himself ought to be credited with the mathematical understanding of the harmonic ratios, which were already known. Van der Waerden also thinks that Pythagoras ought to be credited with understanding that sound comes from vibrations in air.
  • [x] Whitehead claims that Pythagoras began a long tradition of examining the importance of definite numbers in the make-up of the world and even of god. Yet, J.A. Philip points out that there is no evidence for mathematical advancement among the Pythagoreans before Archcytas, a student of Pythagoras. For more, see Maziarz and Greenwood.
  • [xi] There are other ways to interpret this insight. Some critics say that Pythagoras and his followers thought of numbers as extended things and not as ratios, for instance. J.A. Philip claims that the harmonia of the one is the balance of odd and even and thus the “number-thing” of what something is is this balance. For an explication of the limit and unlimited, see Maziarz and Greenwood,
  • [xii] Proclus claims that the Pythagoreans claimed the “how many” subsists in itself or must be related to another, and the “how much” or quantity moves or remains in place. Thus Pythagoreans asserted that math arithmetic is the quantity that subsists itself, but music is related to another and geometry understands continued quantity in so much as it is not moveable, Proclus in Friedlein, p. 35. Marziarz and Greenwood claim the Babylonians taught Pythagoras that the constellations have two characteristics: amount of stars and shape of configuration. This concept and the ratio of the musical scale suggested number for everything, including ethics and religion. They also claim that the Pythagoreans attempted to find a deeper explanation of reality than Milesians like Thales. Numbers were stripped of particulars and thus “could be taken as the real constituents...of the world.” 14 It is notable that the Pythagoreans most likely did not agree among themselves.
  • [xiii] See Plutarch (Diels, Dox. 96), Stob. Ecl. Phys. I.I. 10, p22 in Cornford (1923).
  • [xiv] There are also ten vertices in the five-pointed Pythagorean star and ten pairs of opposites (principles): limit/ unlimited, odd/even, one/many, right/left, at rest/moving, straight/curved, square/oblong, light/darkness, male/female, good/bad. For a clear explanation of the Pythagorean use of the number ten, see Anglin.
  • [xv] We will see later that the Pythagoreans may have been quite shaken, and understandably so, when they discovered irrational numbers.
  • [xvi] George Allman correctly points out that “according to Plutarch, the Egyptians knew that a triangle whose sides consist of 3, 4, 5 parts, must be right-angled” 29. If the Egyptians were aware of the practical application of what we call the Pythagorean theorem, then Pythagoras perhaps made a proof of the theorem but he did not discover its mathematical truth. His not discovering this truth does not mean that he made no contribution, but his real contribution is perhaps less than has been estimated. Euclid's proof of the Pythagorean theorem is Euclid, I, 47. For an explication of a proof, see Maziarz and Greenwood.
  • [xvii] Aristotle Met. A5, 985b23.
  • [xviii]Aristotle Met. A5, 986b2.
  • [xix] Kirk and Raven point out that the unit (1) is indivisible and so the relationship between the odd and even numbers is visualized by the above gnomons. The unit in ¼ is a relationship of one unit out of a total of four. The unit would seem then to be divisible by repetitions of itself and thus be divisible, but only in a sense. The unit is preserved in the fraction while at once it is divided.
  • [xx] For more on the limit and unlimited, see Maziarz and Greenwood,
  • [xxi] See also Aristotle Met. M6, 1080b16.
  • [xxii] Kirk and Raven, The PreSocratic Philosophers, 246-50.
  • [xxiii] This conception may have been borrowed from Anaximander. For more on this fundament of the universe, see Cornford (1923).
  • [xxiv] There are later refinements of this kind of argument for the magnitude of numbers, some creating squares and then cubes. The end in any case is the same. It seems to have been argued that number is precisely what allows configurations of point, line, plane, solid to come about and so the kind of shape and then solid would not be as important, except that certain shapes were more important than others. The Pythagoreans were compelled to reconfigure their arguments after Parmenides' criticism of their association of number with objects. There are, thus, pre-Parmenidean and post Parmenidean Pythagoreans and such arguments are not necessarily post Parmenidean.
  • [xxv] Catherine Rowett points out that the Pythagoreans took the cosmology of Anaximander and applied mathematical ratio to it. Anaximander's ratios were arbitrary, but the Pythagoreans applied their notion of number in a much more logical manner. She claims that Pythagoreans appealed to “patterns of numbers” and worked out a more logical harmony of opposites than Heraclitus, and thus taking Heraclitus and Anaximander seriously, but not the Pythagoreans, is inconsistent and incorrect. See Burnett, Heath and Maziarz and Greenwood.
  • [xxvi] Aristotle, Physics IV.6, 213b22. Charles Kahn points out that the One is technically not a number, since it is prior to numbers; the numbers divide into odd and even, and the One itself is even and odd. This seeming paradox is part of the Pythagorean universe and itself may suggest why the Pythagoreans may not have had the crisis of irrational numbers thought so long by scholars to have taken place. For more on the cosmogony and the crisis, see Kahn and Zhmud.
  • [xxvii] No definitive proof exists, but some scholars show that the only evidence we have of deductive proof early in Greek history are the arguments of Parmenides and Zeno. The Pythagoreans did the same, perhaps. We do not know. For more, see Szabo.


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© 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).

First Principles

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.

  • All things being are existing things.
  • No things not being are existing things.
  • No things not being are things being.


  • All forces that break continuous being into segments of time are forces allowing being to arise from non-being.
  • No existing forces are forces allowing being to arise from non-being (No things not being are things being).
  • No existing forces are forces that break continuous being into segments of time.


  • All forces needed for becoming and perishing are forces that break continuous being into segments of time.
  • No existing forces are forces that break continuous being into segments of time.
  • No existing forces are forces needed for becoming and perishing.


  • All forces that divide being are forces needed for becoming and perishing.
  • No existing forces are forces needed for becoming and perishing.
  • No existing forces are forces that divide being.


  • All forces that create dissimilarities of being are forces that divide being.
  • No existing forces are forces that divide being.
  • No existing forces are forces that create dissimilarities of being.

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 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.


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© 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.

First Principles

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.

  • No things possessing absolute fullness are things that have no being.
  • All things that have being are things possessing an absolute fullness.
  • No things that have being are things that have no being.

“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.

  • All true beings in the universe are beings made invisible by their size.
  • All full beings are true beings in the universe.
  • All full beings are beings made invisible by their size.

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.

  • All places where motion takes place are void-spaces.
  • No absolute fullnesses are void-spaces.
  • No absolute fullnesses are places where motion takes place.
  • All beings without void-spaces are beings that move into void-spaces.
  • All absolute fullnesses are beings without void-spaces.
  • All absolute fullnesses are beings that move into void-spaces.

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:

  • All interchanges and exchanges of atoms are changes in material objects.
  • All changes in material objects are comings-into-substance and dissolutions.
  • All interchanges and exchanges of atoms are comings-into-substance and dissolutions.

Changes come from the void space and the lack that void makes possible by its nature:

  • All complete lacks of space and being are voids.
  • All places where motion is possible are complete lacks of space and being.
  • All places where motion is possible are voids.


  • All motions are changes made possible from void-space.
  • All comings-into-substance and dissolutions are motions.
  • All comings-into-substance and dissolutions are changes made possible from void-space.

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:

  • All determiners of atom function are aspects of atoms that allow conglomeration or separation.
  • All shapes and sizes of atoms are determiners of atom function.
  • All shapes and sizes of atoms are aspects of atoms that allow conglomeration or separation.

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:

  • All spaces in which motion may occur are spaces in the universe responsible for alteration or cohesion.
  • All void-spaces in material objects are spaces in which motion may occur.
  • All void-spaces in material objects are spaces in the universe responsible for alteration or cohesion.

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.


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© 2018 Kirk Shellko All rights reserved.

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.

First Principles

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: 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.

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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.

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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.

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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.

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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.

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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.

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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:

  • If then I said after these things: tell me this very thing accordingly, Meno: in what thing do they not differ but quite all are the same. What do you say this thing is? Would you be able to tell me?
  • I would.
  • So it is truly the same case concerning the virtues: if they are many and varied, quite all of them have some one and the same form through which they are virtues, into which it is a fine thing to have looked for the one being asked by a questioner to demonstrate that thing which happens to be virtue. (Plato, Meno, 72b-c).

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.

  • [Is] it...concerning virtue only, Meno, that there is one virtue of man, another of woman and of the others, or concerning health and size and strength is the case just the same? Does it seem to you that there is one health of man and another of woman? Or is there the same form in each case, if health is present whether in a man or in any other whatsoever?
  • Health seems to me to be the same both for a man and for a woman.
  • And what about size and strength? If a woman is strong, will strength exist with respect to the same form and the same strength? I am saying this as far as “the same”: strength does not differ in respect of being strength, whether in a man or in a woman. Or does it seem to you there is some difference?
  • It doesn't seem so to me. (Plato, Meno, 72d-e).

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:

  • Is [justice] virtue, Meno, or some virtue?
  • What do you mean?
  • As with any other thing. For example, if you wish, I would say concerning roundness that it is some shape, but not simply that it is shape. I would speak of these things in this way because there are other shapes.
  • You speak rightly indeed, since I myself say that not only is justice a virtue but there are other virtues.
  • What are they? Do say, as I could mention other shapes to you if you urge me to do so. Tell me yourself then about the other virtues.
  • Courage seems to me to be a virtue, and moderation, wisdom, and magnanimity, and quite many others.
  • Again, Meno, we have experienced the same thing: we have again found many virtues while seeking one, albeit in another way. But one that is present through all of these we are not able to find.
  • I am not able to find as yet, Socrates, what you seek, one virtue for them all, just as in the other cases.
  • That is likely indeed , but I will be eager, if I have the ability, to lead us forward. For you understand that it is like this about everything. If someone should ask you what I mentioned just now: “What is shape, Meno?” If to him you said that it is roundness, if he said to you the very things I did, “Which of two is it? Is roundness shape or is it some shape?” You would doubtless have said that it is some shape.
  • I certainly would.
  • ... So, too, if again about color he asked you the same thing what it is, and you said it is white, and after these things your questioner took up [again], “Is white color or a color?” you would have said that it is some color, because there happen to be other colors as well?
  • I would.

    (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:

  • At first [the slave] did not know what the basic line of the eight-foot figure [square] was; just as not even yet does he know, but then he thought he knew it, and readily answered as if knowing, and he did not think himself lost, but now he does think himself at a loss, and even as he does not know, he does not think he knows.
  • You speak the truth.
  • So he is now better positioned with respect to the matter he does not know?
  • This seems to me to be.
  • Have we harmed him, having made him at a loss and numb like a the ray fish does?
  • It does not seem so to me.
  • Indeed, we have probably done something useful toward discovering how matters stand, for now, not knowing he would gladly seek to find out, but before he thought to speak well about the double figure [square] to many persons and repeatedly how there is need to have a [base] line double in length.
  • So it seems.
  • Do you think that before he would have tried to seek or understand that thing which he thought he knew while not knowing, before he fell into confusion and then believed he did not know and longed to know?
  • It does not seem so to me, Socrates.
  • Once numbed, has he benefited?
  • It seems so to me.

(Plato, Meno, 84a4-c9).

Socrates has the slave work out the area of a two-foot square:

  • Tell me, boy, do you know that a square figure is this sort of thing?
  • I do.
  • A square then is a figure having all these equal lines, being four?
  • Yes indeed.
  • And does it not also have these [lines] through the middle equal?
  • Yes.
  • And such a figure could be larger or smaller?
  • Yes indeed.
  • If then this side should be two feet, and this other side two feet, how many feet would the whole be? Consider it this way: if it were this direction two feet, and that direction only one foot, would the figure have been once [times] two feet?
  • Yes.
  • But when it is two feet that way, it comes to be twice two feet?
  • It comes to be [that].
  • … How many is twice two feet? Tell me once you have reasoned it.
  • Four, Socrates.

(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:

  • Now another figure twice the size of this one could come about of this sort, having all equal sides just as this one does.
  • Yes.
  • How many feet will that be?
  • Eight.
  • Come on, try to tell me how long each side of that will be. The side of this one here is two feet. What is the side of that one which is its double?
  • It is clear, Socrates, that it will be double.

(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:

  • you say that a figure double in size comes to be from a line double in length? I am talking about this sort of thing: not long on one side and short on the other, but let it be equal in every part like this one, and double the size – eight feet. See if it still seems right to you that it will be based off of a line double the length.
  • It does.
  • Now the line becomes double its length if we add another of the same length from this quarter?
  • Yes indeed.
  • And the eight-foot square will be based on it, if four lines come to be of that length?
  • Yes.
  • Well, let us draw from it four equal lines. Is this here the other one which you say to be the eight-foot square?
  • Very much so.
  • And within this figure are four squares, each of which is equal to this one, the four-foot square?
  • Yes.
  • How much then does it come to be? Does it not come to be four times as much?
  • What else [would if be]?
  • Then is the square that is four times as much double?
  • No, by Zeus.
  • How many times as much is it?
  • Four times.
  • Then, my boy, not double but quadruple is the figure that has come to be based on a line twice the length?
  • You speak the truth.
  • And four times four is sixteen, no?
  • Yes.

(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.

  • All elements that transform into other elements are elements that will pass into another shape.
  • No elements of earth are elements that will pass into another shape.
  • No elements of earth are elements that transform into other elements.

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.

  • All elements that will pass into another shape are elements that are interchangeable.
  • All elements called fire, air and water are elements that will pass into another shape.
  • All elements called fire, air and water are elements that are interchangeable.

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.

  • All elements that are interchangeable are elements that transform into one another.
  • All elements called fire, air and water are elements that are interchangeable.
  • All elements called fire, air and water are elements that transform into one another.

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:

Plato 7.png

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:

Plato 8.png

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,


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© 2014 by Kirk A. Shellko a.k.a. Lucian Whyte