Eric Bleys
Early Greek Philosophy and Science, Classics
11/18/2016
William Mullen Bard College
Why does Pythagorean thought see mathematics as something capable of describing the empirical world, as well as the realms of the moral and the metaphysical? Perhaps it is because Pythagoras did not have the idea of a separation between these two realms. So therefore he did not see a reason to describe them differently. An interesting subject that is relevant to the entirety of philosophy and intellectual history is the question of which forms of language are appropriate for describing particular aspects of reality and which communication forms are needed to answer particular questions? It is true that mathematics, as a language, or form of communication, is more appropriate for describing the exact dimensions of a building than is written language. But it is also true that many physical patterns in the visual arts are themselves mathematical structures, and that these same works of art often testify to moral and spiritual ideals. So in some sense because math is predicated about art, and art is predicated about morality, so therefore math, in this particular case, is predicated about morality. However this doesn’t seem to be the typical situation in regards to the way in which we express our moralistic ideals. Certainly ethics and religion are expressed primarily through written and spoken language. And if one were to express one's moral disagreement with a given person's behavior, one would not express this through a math problem but through spoken or written language. Or perhaps Pythagoras believed in the effectiveness of math for describing these two realms because of the innate clarity and lack of ambiguity within the nature of mathematics itself? This paper will explore why and how Pythagoras thinks math can describe both realms. And the paper will also explore the similarities between Pythagoras’s views in regards to math and Hericlitus’s idea of the logos. The respective roles of each way of thinking will be described.
McKirahan summarized Pythagoras’s views on math’s ability to describe the world in the following way, “The essence of the order in the world, the Pythagoreans believed, is located in the connections of its parts, that is, kosmos depends on harmonia, especially on harmonia based on number. This doctrine first applied to musical harmonia but was later extended more widely” (Philosophy Before Socrates, 92). What is meant here by essence? How can we define the term essence and determine its most fitting definition in this given context? One of Merriam-Webster dictionary definitions of essence is “the permanent as contrasted with the accidental element of being.” If we applied this definition to the description of Pythagoras's views then we would reach the understanding that he viewed the permanent structure behind the order of the world as mathematical. So the mathematical structures would not themselves be temporal but would be the unchanging source of the order which is empirical and temporal. This concept would foreshadow Platonic forms, a concept which posits the existence of abstract entities of which the empirical world is merely a shadow. The order consists in the way in which objects relate to one another. This is similar to the way in which laws of mathematical physics describe relationships between objects. What they can describe is not the nature of objects within a vacuum, but of the relational harmony between objects.
Let us take note that the way in which Pythagoras’s position is described in the previous paragraph actually posits a metaphysical nature for mathematics even as it is being posited as something which describes the empirical world. This understanding presents a cohesive unity between the idea of math as empirically descriptive and yet metaphysical. This is because the metaphysical stability which produces the order of the empirical world is itself mathematical. So then the empirical world testifies by its own patterns to a metaphysical order. And that metaphysical order is itself a mathematical structure. These ideas of Pythagoras are described by Aristotle’s metaphysics in the following way, “In numbers they thought they observed many resemblances to the things that are and that come to be… such and such an attribute of numbers being justice, another being soul and intellect, another being decisive moment, and similarly for virtually all other things...since all other things seemed to be made in the likeness of numbers in their entire nature” (Philosophy before Socrates, quotation 9.19).
What is meant by the idea that numbers resemble things that are and that come to be? Time is conceptually divided into the past, the present, and the future. Things that are refers to existent things as opposed to non existent things. There is an issue in the philosophy of time as to whether or not the past still exists. So there is the possibility of interpreting the statement as meaning that the term “things that are” would refer to the present and the past. However in popular usage the term “things that are” would refer to the present. The passage implies that numbers bear an innate similarity to the present as well as things that will come to be. And so therefore numbers bear a similarity to objects across all times, or at least to the present and the future. The ability to predict the future is predicated on the notion that at least to some extent or at least in certain ways the future will resemble the present which we have observed. We believe the sun will rise after the darkness which will come tonight because after every night which we have experienced the sun rose in the morning. We believe that these experiences of the past give us insight into the future because they attest, we believe, to a pattern which is innate and fundamental. We believe that the future will conform to this pattern. And therefore we believe that the pattern transcends the realm of immediate experience. Because we expect the pattern to tell us something about something which we do not yet have direct empirical access to. So in some sense the pattern itself could be called metaphysical. In some sense it exists “beyond” the physical because the pattern contains information about something which does not yet exist.
The rest of the description from Aristotle describes numbers as things which resemble all things, which include justice, the soul, as well as the intellect. Justice, as a concept is abstract in a different sense in comparison to the mathematical patterns behind empirical events. Justice is a concept which is related to the rightness and the wrongness of behaviors as well as the just consequences for these actions. Whereas the mathematical order of empirical events need not have ethical insight. However, there is a similar quality which is the same between mathematical descriptions of empirical patterns and the concept of justice which is they are both irreducible to the empirical reality which is immediately present. The intellect itself is a notion which bears witness to the idea of intellectual order. Or, in other words, patterns of thought. For how can we conceive of the intellect without thought? A mathematical algorithm is a step by step process for solving a problem. Mathematics is a method of thought. Mathematics itself testifies to methods of thought and in this way testifies to intellect. There are functions of the mind that we do not typically associate with thought, such as dreaming. However, it is difficult to deny that the notion of thought does not imply some sort of intellect. Because one of the most common and essential definitions of the word intellect is the capacity for intelligent and rational thought.
What about the notion of the soul? It is the immaterial or spiritual element of a person. So what does it mean to say that numbers resemble the soul? What aspect of the soul do they resemble? We can say that there are mathematical truths that exist across all times. We can know that 2 + 2 = 4 in the same way as someone from two thousand four hundred years ago can know this to be true. And we cannot conceive of the possibility that one day it would not be understood to be true within a human mind. The meaning of the number 2 means the same thing across these ages. Because it does not exist in any one material place but exists in human minds across thousands of years, it resembles the spiritual. Because the spiritual exists within people but it does not exist in any particular physical place. Perhaps the reason for the Pythagorean view on the significance of Mathematics is because Math possesses this great quality of being able to have qualities similar to the spiritual while also being capable of describing the empirical world. Perhaps Mathematics is itself the intellectual link between the physical and the spiritual world.
Interestingly, Aristotle claims that the Pythagorean reason for calling numbers the substance of all things is because the limited and the unlimited are not distinct substances but are predicated of one another. If the limited and the unlimited are A) metaphysically united with one another, and B) mutually predicate things about one another, then what understanding can we infer from these two positions? First of all let’s clarify what can be meant by the unity between the limited and the unlimited. A finite object can be thought of as a part of an infinite set. And this means that they can exist within one object. It also means that both the infinite and the finite in this case can tell us things about one another. Because the finite would be a sub pattern within a larger pattern as well as an example of the nature of the larger pattern. It is also true that the larger pattern would be able to tell us the structure of the finite pattern. The two concepts are composed of one another without possessing exact equivalence.
It is important to analyze the text in which Aristotle makes this claim about Pythagorean thought, “The Pythagoreans similarly posited two principles, but added something peculiar to themselves, not that the limited and the unlimited are distinct natures like fire or earth or something similar, but that the unlimited itself and the one itself are the substance of what they are predicated of. This is why they call number the substance of all things” (Philosophy before Socrates, McKirahan, quotation 9.27). So this passage indicates that the limited and the unlimited are not distinct from one another in the same way in which fire and earth are distinct from one another. Fire and earth do not share the same quality of being the same substance and we do not say that one exists within the other. Studying earth does not give us direct access to the nature of fire. Whereas if we study one composition of something of which fire is built then we do have access to the nature of fire from understanding that smaller component. Therefore what is being explained is that the relationship between the limited and the unlimited is not the relationship between two things in which one is not composed of the other. Instead the unlimited is at least partially composed of the limited. And hence they predicate things about one another. And because they predicate things about one another, then it is likely true that a medium of thought and expression which is held in common between the two can be used to understand them both.
This understanding presented in the preceding paragraph supplements and does not contradict the view that mathematics is the intellectual bridge between the spiritual and the material worlds. Because the spiritual is understood to be unlimited whereas the material world is understood to be limited. On one hand Mathematical thoughts and numbers are rational truths which are understood within the human mind. But they are also capable of describing the dimensions of a building. The ability of math to describe both, from a Pythagorean perspective, is rooted in the view that mathematics resembles both realms. And in fact, as we discussed early, the order of the empirical world itself (for Pythagoras) testifies to an abstract reality which is the essence behind the events.
Here is another relevant description of Pythagorean thought in regards to mathematics which describes his views on the generation of the physical world, “It is absurd to construct an account of the generation of things that are eternal, or rather it is an impossibility. There is no need to doubt whether or not the Pythagoreans construct such an account, since they say clearly that when the one had been constructed - whether from planes or surfaces or seed of from something they are at a loss to specify- the nearest parts of the unlimited at once began to be drawn in and limited by the limit. But since they are constructing a kosmos….” (Philosophy Before Socrates, quotation 9.29). This passage means that after the unlimited began to be itself in some sense, limited, a process of constructing a kosmos began. So there's no need to describe the eternal past before the kosmos because it would be describing the non temporal with a temporary events. The kosmos itself is the derivation of the infinite. It is the particularization of something which existed infinitely. What does this imply for the philosophy of mathematics? It means that the metaphysical and the physical are fundamentally united. That the physical is a derivation of the metaphysical. Because both the unlimited and the limited are composed of numbers mathematics is a study which can give us knowledge of both realms. And because one intimately originates from the other and the limited is part of the unlimited we can conclude that both have the potential to provide us understanding of one another. Knowing the limited will help us understand the unlimited. And knowing the unlimited can help us understand the limited. Because math helps us to understand both it can serve as an intellectual bridge between the two. And understanding the infinite then is a way of understanding the origin of the empirical world. And so therefore the patterns of the empirical world can be understood through the abstract mathematical laws which themselves describe the eternal. The past, present, and future all originate from the infinite. And so it is highly conceivable that eternal mathematical laws could help us to understand and predict empirical patterns.
Heraclitus posited a concept which plays a similar role to the position of math in Pythagorean thought. He posits the existence of the logos. An all encompassing, rational order, which is composed of great diversity, which is also fundamentally rational. The concept is given many of the same traits as math in Pythagorean thought, “Likewise, the word “logos” has connotations of rationality, not irrationality, and is linked with other concepts of positive value, notably justice but also law (which preserves things from anarchy) and soul (which is responsible for life, a condition with positive value)” (Philosophy Before Socrates, P 137). These three concepts, of order, of justice, and of the soul, are all said to resemble numbers in Pythagorean thought. Furthermore rationality is playing a similar role to math because it is a method of thought which itself is a tool for understanding all things. So this rationality exists metaphysically as well as in the human mind. It is a way in which the mind can apply itself to understand existence.
The logos is described as a special language with the unique capability of describing the world. If language is deeply linked to understanding then the language which we use must be presented as an essential aspect of one's epistemology. Just as describing a thought requires a particular kind of statement, in the same way, a particular kind of language is needed for the purpose of understanding. Ambiguity is the enemy of understanding. The need to think and articulate thought in the clearest possible way is a core part of the epistemological enterprise. Ultimately, in the same way in which a statement without context losses much of its meaning, a language which does not describe the totality of things does not describe the particular thing accurately. And this is the significance of the way in which Heraclitus and Pythagoras use the concept of the total and the divine in relation to the particular. The total or the unlimited provides a context for the particular in such a way as to illuminate the particular and provide a deeper understanding of its position in the world. The ability to predict the future exists within this framework. Because if we know the place of the particular within the pattern of the eternal framework then we can know how the pattern will cause the future of that particular.
The need to transcend ordinary insight is likened to a growing similarity to the divine, “In contrast to the normal human state of ignorance and unbelief, the divine has knowledge and insight (10.28) and is the only truly wise being (10.30) Neither this claim nor the observation that we are like babies in comparison with god (10.29) means that we must remain wholly ignorant any more than the thesis that understanding is common to all (10.31) means that we all possess the very insight Heraclitus denies we have (10.28). Rather, as Children grow into maturity, we may grow in insight. Our ultimate goal is thinking (10.31), self-knowledge and thinking rightly (10.32). To the extent that we attain this insight and wisdom we transcend the human and resemble the divine (10.28, 10.30).” (Philosophy Before Socrates, P. 127). In a way similar to Plato’s allegory of the cave in the Republic, or Paul’s description of the journey to spiritual maturity in first Corinthians, Heraclitian thought posits the existence of a higher level of thought and reality which we must obtain through the expansion of our minds in a way which is tied together with moral and spiritual growth. Correct thinking, for both Pythagoras and Hericlitus, is more than just a process of understanding concrete facts about the world. In contrast to logical positivism, moral and metaphysical insight has logical value and is an essential realm of inquiry for the application of rational thought. Ultimately the contemporary world view often lacks these same insights. Mathematics is deeply associated not with morality and metaphysics but with science and technology. Logical positivism's position that all propositions which can be deemed true must have direct empirical grounding ultimately fails to search for the same kind of ultimate context for things which are temporal. The search for meaning and the search for the positioning of the temporal world in relation to a grand abstract eternal reality possesses the quality of a key religious impulse. The impulse is the search for ultimate meaning and the search for the eternal amidst the temporal. This is unsurprising considering Pythagoras’s role as a religious leader. The key similarity between Hericlitus’s and Pythagoras’s views on a method of thought describing all things is that they both posit the need for a highly sophisticated and refined method of rational thought which expresses itself in a special form of language which is both spiritual, and temporal, as well as both moral, and empirical.