Evaluating Pythagorean Mathematical Realism

by existential Calvinist on 2007年10月11日 10:25 AM

@ Home / Essays / ESSAY19 (edit, history)

Mathematics clearly occupied a central position in the thought of the Pythagoreans; however, there remain doubts about exactly how they saw mathematics working within their fundamental ontology. Did the Pythagoreans consider mathematics to be a kind of ontological “stuff” that forms the building blocks of the world, in a manner similar to how other pre-Socratic philosophers thought about the ἀρχή? Or, did the Pythagoreans merely consider mathematics to be the most useful tool for investing the nature of the ultimate, in a manner similar to how, e.g., Buddhist sages have explained the benefits of meditation? Since in interpreting the maxim, “all is number” either reading can potentially be made to fit, we need first to determine a means by which we can reliably attribute views to the Pythagoreans before we can understand what they thought.

There are two basic ways to approach these questions. First, we can gather all relevant historical evidence and then construct and defend an interpretation of their philosophy on the basis of its coherence with the sources. Second, we can perform our own investigation of the issue at hand beginning with the same concerns that motivated Pythagoreans and then ascribe to them the view that seems most substantially plausible to us on the grounds of charity in interpretation. Of course, any complete analysis will combine elements of each approach, but this paper will focus primarily on the latter. With this in mind, our investigation must begin with two more questions: First, given the nature of mathematics is it even possible to regard it as the basis of substance, or can mathematics merely describe the actions of substances a posteriori? Second, is mathematics independently existing or is it merely the product of human invention?

Thus, to answer these questions, we must first go back to the source of Pythagorean philosophy and re-examine why mathematics was of special interest to them and many generations of later natural philosophers as well. Next, we must consider the conditions that allow mathematics to take on these fascinating properties, namely the conditions of “computation.” We will then examine the numerous difficulties that arise out of the definition of computation and relate these difficulties to the fundamental difficulty with treating “mathematics” as a unitary phenomenon attended by a perfectly defined set of truths. Having examined these many issues, we will at last be in a position to return to our original questions and ascribe to the Pythagoreans (and to ourselves) a tentative position on the nature and use of mathematics in the composition of the physical world.

Contents

  1. 1. Why is math interesting?
  2. 2. Mathematics as computation
  3. 3. Difficulties inherent in treating math as ontologically primary
  4. 4. Conclusions
  5. 5. Bibliography
    1. 5.1 Comment:

1. Why is math interesting?

In order to reconstruct the Pythagorean view of connection between mathematics and the world, we will first examine which properties of math were so attractive to Pythagoreans in the first place. Clearly, the quality of math that first attracted attention in the ancient world is its usefulness. The Babylonians used math because it allowed them to keep complicated business records. The Egyptians used math because it allowed them to calculate the area of land flooded by the Nile each year (Anglin 1). The Greeks inherited and extended their forebears’ insights. Thales’ famed prediction of a solar eclipse was in all likelihood made possible by Babylonian mathematical analysis of astronomical records (KRS 82). Mathematics was originally investigated for what appear to have been purely practical reasons.

Pythagorean interest in math, however, extended beyond the merely practical benefits of its use. If, as Merrill Ring asserts in in Beginning with the Pre-Socratics, “Pythagoras was fundamentally a religious teacher whose chief concern was in the ‘right way of life,’ ” (40) then it stands to reason that Pythagoras’ interest in mathematics was not only a matter of mere practicality. There are any number of useful arts that could have entertained the Pythagoreans’ attentions, from farming and cooking to navigating and cartography, and all of these did engage the Pythagoreans to a certain extent, but none of them became as central to the Pythagorean identity as math. It is not purely its utility that made math seem worth of distinctive honor among the sciences to the Pythagoreans. Rather, it was math’s apparent claim to what seem to be “eternal truths” that are rigorously demonstrable. That is, math is capable of giving a hint at facts that, like moral laws, give guidance in a range of seemingly disparate situations. By merely reasoning about shapes and numbers, one can discover surprising and novel “facts” about the mathematical “world,” which can be proven to others with perfect clarity. Though the Pythagorean theorem almost certainly pre-dates Pythagoras, it must nevertheless have been a source of inspiration to his followers that its truth could not only be physically demonstrated by the measuring of any well drawn right triangle no matter what its particular size or other angles, but also logically demonstrated through a well-reasoned proof. Thus, through the rise of the proof, mathematics seemingly promises to open our minds to an unbounded world of surprising but certain knowledge.

Of course, mathematics would not be the discipline that it is if it did not merge both of these elements, the practicality of its applications and the eternality of its proofs, into one body. Hence, perhaps the most interesting aspect of math for the Pythagoreans is that it seems to provide a bridge between what would come to be called after Plato the realms of eternal forms and the fleeting physical world. The reflection of one realm in another may be termed a “cosmological correspondence,” if the relations of objects in one realm perfectly mirrors the relations of objects in the other. Should this prove to be the case about our own universe, then rather than having to discover properties of the physical world through individual observation, we can merely do mathematical inquiry and in so doing reveal necessary properties of our own world. Thus, to understand what truly gives mathematics the power that it has, we must next focus our investigation on the method by which we bring truths down from the mathematical realm into our own in order to come closer to understanding under what conditions a truly cosmological correspondence between the mathematical world and the physical world would be possible.

2. Mathematics as computation

Post-euclidean mathematics takes as its basis the idea that through extrapolation from a limited number of self-evident axioms, an infinite variety of truths can be produced and demonstrated. As humans, however, the consequences of our axioms do not make themselves immediately apparent to us. Rather, we must work by starting out from our axioms algorithmically and catalog truths as we encounter them. For example, as children we are taught a process called long division by which we can reliably discover the exact ratio between two fixed length numbers, no matter what numbers may be. Thus, by learning one thing (long division), we absorb the key to knowing indefinitely many things (individual ratios). However, one algorithm, like that for long division, is only a small part of the larger sphere of possible computations. We will here define an algorithm as “a set of finite, well-defined rules governing the performance of some activity” and computation as “the process by which the fact that one system is rule governed is used to make reliable inferences about another rule governed system.” A rule governed system is “a system in which all possible states of the system may be reached through the repeated application of the algorithmic processes endemic to the system.” The reasons for these definitions will become more clear with examples.

Typically, our mental image of computation is dominated by the modern electronic and binary computer; however, the definition offered here encompasses a much broader range of activity. Thus, even though the adding machine was not invented until long after the age of the Pythagoreans, nevertheless, we can think of them as doing “computation” in a sense. Another way to explain the definition of computation given here is that computation is “the application of an analogy that works with a specific degree of precision.” For example, an abacus is governed by rules like “the beads do not merge together when you are not looking at them,” so we can use them to do finite math with natural numbers by calling the patterns of beads analogs of the natural numbers. Similarly, we can make inferences about how a plane will fly by using a wind tunnel and modeling the plane. If the rules of aerodynamics in a wind tunnel were not easily converted to the rules of real airplanes, we would not be able to use them to predict how actual planes will fly using the wind tunnel. These are examples of physical-to-non-physical and physical-to-physical computation respectively, but it is also possible for a computation to be entirely non-physical. For example, we might use facts known about geometry to make reliable inferences about algebra, or vice-versa. The salience of mathematics comes, of course, from the possibility of doing non-physical-to-physical computations, as for example, one infers the size of one part of an existing object from knowledge of the geometric ratios mathematically particular to it.

Thus, the importance of mathematics lies in the fact that it can become entangled with the world through computation in a three fold process. First, the physical world is assessed and associated with a mathematical structure by means of an interpretive analogy between our (presumably rule governed) world and the rule governed world of a mathematical system. For example, we see a physical triangle and measure two of its sides, assigning them each a number that represents its length. Next, an algorithmic transformation is performed on the numbers inside the mathematical world. Quite frequently, this processes is aided by sub-computation that uses a physical device that has been seen to work in a manner similar to the mathematical world. For example, an abacuses beads can represent numbers. Even more commonly, certain figures are written on paper to represent the numbers and aid the human mind as it tries to explore the mathematical landscape. In any case, an algorithmic behavior is applied according to some preset rules in order to transform the initial state (a pair of numbers representing the sides of the triangles) into the final state (a single number representing the length of the other side of the triangle). Finally, a last analogy is drawn between the resulting number and the length of the triangle in the physical world, and this analogy is what allow the interpreter to possess a reliable inference about the physical world by means of the earlier computations.

In any computation, the key is that the reliability of each of the systems ensures their joint predictive powers. Should the shapes of the objects being computed about change, or should it somehow be possible for the rules of mathematics to change during a computation, then the reliability of the inference would be shattered and the process made pointless. To learn about the physical world more easily, we use algorithmic processes to discover the structure of the mathematical world. To learn the structure of the mathematical world more easily, we use physical computers to discover what physical states imply one another. Both processes rest on the necessity that it is possible through computation to reliably convert facts from one world of discourse to another.

3. Difficulties inherent in treating math as ontologically primary

While the preceding definitions of the structure of mathematics are not ones that would have been given by the Pythagoreans themselves, nevertheless, it adequately formalizes certain strains of thought that seem to lie under the pre-suppositions of mathematics throughout the ages. However, these definitions also create difficulties that must be dealt with in order to understand in what senses it is possible that mathematics may be the ontological substance of the world.

The first difficulty to be faced is that even if we suppose that world is “just math,” then we are still not sure what particular part of math it is. Our previous definitions gave no particular preference for one kind of computation as being any better example of a computation than another. Any rigorously connected set of algorithms and world is as good as any other, even physical-to-physical computations. Thus, it is up to individual philosophical investigators of nature (φυσικός) to propose specific ways of correlating the non-physical algorithms of the mathematical world with our own world. For example, in Timaeus Plato has Socrates hypothesize that the world is composed of Platonic solids (Anglin 31). (This speculation is believed to show the Pythagorean influence on Plato.) However, this hypothesis can no longer be considered to cohere with empirical evidence, given the discoveries that have been made since then about the way that the sub-atomic world acts. Thus, in order to make productive use of the knowledge that the world is mathematics, we must also know precisely what kind of mathematics it is. Moreover, for the mathematics describing the world to be most helpful, we need to be able to find a precise formula or algorithmic process that generates reality. Such equation is the goal of modern physicists who seek the so-called “Theory of Everything.” (See Wolfram in the bibliography for a description of the process by which a contemporary thinker may sift through different algorithms in order to “find our own physical universe.”)

Second, even if it is true that the world is a particular equation, it is not clear what kind of mathematical system we should look for it in. For centuries, mathematicians and philosophers all accepted that the world is described by a Euclidean geometry, where for example, the interior angles of all triangles add up to 180°; however, Einstein’s theory of gravitation shows that this is not actually the case for our physical world. Thus, not only must we search for a preferred equation that describes reality, we must also search for a preferred mathematical system that can contain the equation! Of course, it is the hope of some students of foundational mathematics that all currently known mathematical systems such as algebra, geometry, probability, and so on can be perfectly described using only a few basic axioms for set theory together with definitions specific to the branch of mathematics under study. If this is the case, then ultimately whatever mathematical system the equation for the universe fits into, it will be merely one particularized expression of the one singular mathematical fundament. For example, since Giuseppe Peano created a formal definition of arithmetic which can be expressed in set theory, set theory can be said to encompass arithmetic. If set theory is in fact a fundamental theory of mathematics, then every other mathematical system is capable of being expressed in set theory. However, it is not necessarily the case that there is single meta-mathematical theory that encompasses all others. For example, there are already several variants of axiomatic set theory, and no clear way to determine which one is the “most fundamental” of them. If there is no single theory that can serve as the basis for all possible mathematics, then modern would-be Pythagoreans are faced with yet another issue to resolve before they can correlate the physical with the mathematical with perfect confidence.

A third difficulty arises in connection to Alan Turing’s revolutionary paper “On Computable Numbers, With An Application To The Entscheidungsproblem,” which proved that the facts calculable by finite means do not completely describe the truths inherent to a mathematical system. In his result, Turing extended the proofs of Kurt Gödel, whose “On formally undecidable propositions of Principia Mathematica and related systems” showed that certain propositions can neither be ruled in nor ruled out as members of sufficiently complex mathematical systems. Thus, even if the universe is a computation, there will remain certain facts that are not captured by our treatment of the system, which means that a study of mathematics will not be able to lead us to all possible knowledge of our own world, but only a limited (though perhaps highly useful) set of facts about our mathematical world. If the world is fundamentally mathematical, it may also be fundamentally unknowable.

A related hypothesis, the Church-Turing Thesis, holds that it is not possible in our own world to build machines theoretically capable of any more than is calculable by finite means. While this hypothesis has not been proven conclusively, if true, it would present a fourth difficulty for the conception of mathematics as the fundamental substance of the world, as even more facts that are inherent in our mathematical systems will remain forever outside our grasp, no matter how great our advances in mathematical or computational prowess. One key concern for modern Computer Scientists is the mapping of which rule governed systems may be reliably correlated with which other systems and with what costs. For example, it is not clear if digital electronic computers are capable of simulating quantum mechanics efficiently. It may be the case that for every additional quantum particle simulated in a digital computer, the processing costs rise exponentially or worse. Accordingly, currently there is an ongoing effort by industry to create a so-called “quantum computer” that is able to efficiently simulate quantum events by harnessing the quantum nature of particles that are under the researcher’s control. It is possible that there exists an efficient means of simulating these particle interactions with ordinary computers, but if it turns out that there are not, it has radical implications for what kind of computation might underlie the universe and where we should search for the equation behind reality.

Fifth, according to our prior definition, the process of computation always takes place with respect to some agent that interprets the transitions of one system in order to make inferences about the other system. It is not clear if it make sense to speak of “computation” in the abstract as the most fundamental process of the universe, if computation as we observe it is always an agent oriented interpretive process that is laid on top of the transformations that naturally occur in the base system in order to talk more clearly about the inferred system. Hence, it is not proper to speak about the totality of things being a “computation” per se. At best, we can only say that the universe can be understood as “like a computation,” but one for which the interpreter function is fulfilled in a manner quite different from that ordinary computation, if it is filled at all.

Sixth and finally, when we return to the process by which humans do mathematics, we find that humans themselves never enter directly into the rule-governed world of mathematics in itself. Rather, we enter into mathematics only phenomenologically as we observe the ways that our human minds work through the laws of mathematics during the process of computation and then we reify these observations into a picture of a frozen mathematical world. That is, our interaction with mathematics takes its relevance from our ability to correlate it with the physical world. However, for this reason, math is never grasped directly as math, but always indirectly as prior computations. The noumenal essence of math seems to remain forever just outside of our grasp, at least in non-mystical experience. As W.S. Anglin explains in Mathematics: A Concise History and Philosophy, there are three basic ways of answering the question of whether mathematics is independent of the human mind: Platonism, formalism, and intuitionism (217). Platonists “hold that numbers are abstract, necessarily existing objects, independent of the human mind” (218). Formalists hold that “mathematics is no more or less than mathematical language. It is a series of games played with strings of linguistic signs,” and, “really, mathematical terms do not have any exterior meaning or reference” (218). Intuitionists hold that, “mathematics is a creation of the human mind. Numbers… are merely mental entities, which would not exist if there were never any human minds to think about them” (219). This paper is not in a position to stake a claim about which of those interpretations best characterizes the being of mathematics, but the existence of these different position raises for us final doubts about whether it is proper or not to prioritize the existence of mathematics over and above human existence, since the very existence of a debate about the nature of mathematics is potential evidence that we do not know math in its essence, but only in the reflection of that essence in cognitive experience. Without noumenal access to mathematics in itself, we cannot speak to its being.

4. Conclusions

We are now in a position to return to the original questions that motivated our inquiry and evaluate them according to a principle of attributive charity. While we cannot know for certain what position the Pythagoreans held, we can infer based on our own investigations which answers would best cohere with their over all project.

Concerning the question of whether mathematics can be a substance or whether it can only describe the actions of substances by analogy after the fact, we will answer with the latter, but with some qualifications. Mathematics as we interact with it is always the hermeneutically derived interpretation of analogous systems as computation. As such, mathematics cannot be the fundamental substance in the sense of being the thing which imparts reality into formal objects, because phenomenal mathematics of itself does not perform a calculation in the absence of an observer, and the fundamental mathematical substance would need to be some equation or algorithm which is worked out through computation, rather than a mathematical system existing statically. The problem for mathematics as substance is that mathematics is at least partially disclosed to us (indeed, this disclosedness is a necessarily part of mathematics intrinsic fascinatingness), and in its disclosedness, mathematics runs into the same difficulties that any empirical element would encounter in claiming primordial status. Empirically done math is phenomenally derivative rather than fundamental.

Saying that mathematics as we know it does not have the sort of being proper to a reality imbuing substance does not suggest, however, that some other aspect of mathematics cannot be something independently existing, as is asked by our final question. It is possible that there is such a thing as “noumenal” mathematics which is the primordial working out of all possible rule-governed systems. Indeed, it may even be the case that noumenal mathematics is somehow involved in the reality imbuing process of the cosmos. However, this noumenal mathematics is only ever “reflected” to us in our computational dealings, rather than directly encountered as a thing in itself. Thus, to be most gracious in our interpretation of the Pythagoreans, we may allow them to be “Platonists” in their conception of mathematics as reflecting something which is prior to human reasoning, but at the same time, we will say that mathematics as computation is not the fundamental stuff of the universe. The utility and hence interestingness of mathematics comes from its correlation with the physical world, and this reveals that mathematics as computation is not itself the “Platonic form” of mathematics which would be at the heart of reality. Rather, the fundamental nature differs from ordinary, computational mathematics in its being able to be self-interpreting and self-selecting of the privileged equations underlying reality existing transcendently removed from our physical and mental worlds.

Thus, when we answer our original pair of questions, we should say that for the Pythagoreans mathematics as done by human beings is not the ἀρχή, rather it is a window into the nature of the ἀρχή, just as for the mystic philosopher, meditation is not an ἀρχή-like fundament, but rather the means by which the fundament’s nature may be probed and perhaps even directly experienced. The inherently fascinating quality of mathematics is that its seemingly bridges our world and the transcendent world beyond. However, we must recognize that in serving as the bridge between worlds, mathematics as we encounter it is not itself the other side of the bridge. Mathematics is the path to the other side, but those of us who remain on this side can only understand the other side metaphorically until we ourselves cross over the bridge. In ascribing the views extrapolated here to the Pythagoreans, we can allow them to keep their numerological impulses, their belief in the transcendence of proofs, and their religious concern for the right way of life without sacrificing any of their salience or importance as thinkers for the modern world.

5. Bibliography

5.1 Comment: