Philosophy of Mathematics

Dov M. Gabbay, Paul Thagard, John Woods, Dov M. Gabbay, Paul Thagard, John Woods

• Publisher: North Holland
• Year: 2009
• ISBN:
  9780080930589
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

One of the most striking features of mathematics is the fact that we are much more certain about the mathematical knowledge we have than about what mathematical knowledge is knowledge of. Are numbers, sets, functions and groups physical entities of some kind? Are they objectively existing objects in some non-physical, mathematical realm? Are they ideas that are present only in the mind? Or do mathematical truths not involve referents of any kind? It is these kinds of questions that have encouraged philosophers and mathematicians alike to focus their attention on issues in the philosophy of mathematics. Over the centuries a number of reasonably well-defined positions about the nature of mathematics have been developed and it is these positions (both historical and current) that are surveyed in the current volume. Traditional theories (Platonism, Aristotelianism, Kantianism), as well as dominant modern theories (logicism, formalism, constructivism, fictionalism, etc.), are all analyzed and evaluated. Leading-edge research in related fields (set theory, computability theory, probability theory, paraconsistency) is also discussed. The result is a handbook that not only provides a comprehensive overview of recent developments but that also serves as an indispensable resource for anyone wanting to learn about current developments in the philosophy of mathematics.-Comprehensive coverage of all main theories in the philosophy of mathematics-Clearly written expositions of fundamental ideas and concepts-Definitive discussions by leading researchers in the field-Summaries of leading-edge research in related fields (set theory, computability theory, probability theory, paraconsistency) are also included

LES LIAISONS DANGEREUSES

W.D. Hart

Mathematics and philosophy are roughly coeval in our historical imagination. Plato’s dialogues form the oldest surviving extended body of work in the canon of western philosophy. Euclid’s Elements is the oldest surviving intact monument in the evolution of our mathematics. Plato taught Aristotle, who died in 322 B.C., and Euclid’s floruit is around 300 B.C., so the gap from Plato to Euclid is like that from grandparent to gra—————zndchild, and from nearly two and a half millennia later, that gap looks small.

There was of course philosophy before Plato. We have fragments from the presocratics, and Plato made his teacher Socrates the star of most of his dialogues. There was mathematics before Euclid. He seems to have been as much an editor as a mathematician, and probably not the first. The Greeks had invented or discovered proof centuries before; just think of the Pythagorean theorem or the proof of the irrationality of the square root of two. In Plato’s day, Theatetus seems to have proved that there are exactly five regular solids, a gorgeous result that impressed Plato enough to give Theatetus a leading role in a major dialogue. As proofs proliferate, patterns start to emerge, and aficionados want to organize the profusion of arguments into a coherent whole developed logically from a minimal stock of assumptions. Doubtless there were such editions of geometry before Euclid, but his Elements is the work whose authority lasted through the centuries.

Aristotle seems to have been less impressed by mathematics than Plato. But Aristotle did begin the systematic study of logic. His account of syllogisms is now usually assimilated to our monadic quantification theory (and truth functional logic is usually credited to the later Stoic philosophers).¹ An interest in logic could have arisen from the effort to piece disparate proofs together into a unified and coherent system, though syllogistic is a pretty thin description of the reasoning deployed in ancient geometry. Still, we should not be impatient, since it was not until the nineteenth century that people like de Morgan² and Peirce began to work out a systematic understanding of relations, which was crucial in the logicist regimentation of mathematics.

But besides starting systematic logic, Aristotle also articulated a version, or a vision, of the axiomatic method. In the Posterior Analytics he describes a real body of knowledge as deduced by infallible logic from axioms. The axioms should have an immediate appeal, and the logic should transmit this appeal to the theorems. As we said, Aristotle wrote before Euclid. But he might have been trying to articulate an ideal he saw struggling to emerge from editions of geometry older than Euclid. And one wonders whether Euclid might have been struggling to realize an ideal earlier articulated in Aristotle.

This is our theme, the dangerous liaisons between mathematics and philosophy. They are not just coevel, like strangers or distant acquaintances who happen to have been born in the same town within a short span of time. They have also been bedfellows, sometimes strange even if not made so by politics. We will sketch some of their offspring. Some does not mean all; ours will not be a family tree, but a selection of hybrids. And because I am a philosopher who admires mathematics but does not claim to be a mathematician, most of these compounds will have more philosophical elements than mathematical.

In some ways, the axiomatic method can seem like proof writ large. To be sure, a proof aims to establish a single theorem, while in an axiomatic system we prove a sequence of theorems. In the heat of live mathematics, one does not practice axiomatically. One does not copy one’s premisses out of a constitution written down and approved by the founding mathematical fathers and mothers. One starts instead from what is clear, and clarity here probably means what one’s peers will accept without complaint. So one needs to be sensitive to one’s peers, and for pretty much all of us this requires being admitted to the community of peers through an education. But once the community has approved a body of proofs, some of the peers may set out to regiment it. This process includes collecting the clear starting premisses that passed muster, selecting from them some from which the rest can be derived, and so on until we have axioms from which a sequence of theorems follow, where of course some later theorems are deduced from earlier. Once such a system is established, incorporability of a new argument in it can become a standard for being a proof. Euclid set such a standard in geometry for centuries, and set theory (usually in Zermelo-Frankel form) did so for mathematics generally in the twentieth century.

This is a rather sociological description of axiomatization. Philosophers and mathematicians share a taste for long and abstract chains of reasoning, but they often differ in how they get started. Mathematicians seem to like their premisses to be shared, perhaps throughout their community, or as close to that as possible. That way the community can be expected to follow their reasoning. There are philosophers, like Aristotle and Kant, who seem not to want to frighten the horses, but they may be trying to calm things down after earlier philosophers like Plato and Hume have stirred them up by going where the reasoning led from premisses for which they may have claimed more popularity than was generally recognized. At any rate, philosophy looks more contentious than mathematics. But however much they disagreed elsewhere, Plato and Aristotle seem to have agreed that a version of the axiomatic method describes an ideal for knowledge.

Even if it is not perfectly clear whether this ideal starts life in mathematics or in philosophy, the axiomatic method is a mode of exposition that has become a tried and true device in the mathematical repertoire. It was exaggerated by philosophers into the ideal of foundations of knowledge, or this or that department of knowledge. A vivid example is Spinoza writing his Ethics in more geometrico. It is perhaps ironic to note that the Latin word “mos” from which “more” declines means custom or usage, which seems more sociological than Spinoza probably had in mind. (Anyone trying to formalize Spinoza’s system by modern lights is in for a bad time.) The basic philosophical idea seems to be that there is a right way to organize for justification truths, beliefs, or knowledge. This idea has gripped philosophical imaginations for centuries.

The ideal can be articulated in different ways. Sometimes the right order is the right order in which to justify our beliefs or knowledge. In Descartes’s urban renewal of knowledge we are to rebuild from clear and distinct ideas of indubitable certainty like the cogito. In more empiricist philosophers like Locke, Berkeley, and Hume we are to begin from sense experience, and increasingly their problem is whether we can get beyond our impressions without losing the certainty that made perception an appealing foundation.

It was this empirical spectre of skepticism that startled the horses and woke Kant from his dogmatic slumbers. To trace out firm foundations for knowledge, he looked to the surest systematic body of knowledge going, and from the Greeks on, mathematics had always been the best-developed system of the most absolute truth known with the greatest certainty. In Kant’s day and before, mathematics meant first and foremost geometry, and geometry meant Euclid’s system not just of planes but also of the space in which we live and move and have our being. The idea of other spaces is later and quite unkantian, and the mathematics of number (beyond elementary number theory like the infinity of the primes) achieves independence only in the nineteenth century. Kant does of course give sensibility a basic role in contributing to knowledge. But it is his conception of the character of geometrical knowledge that not only gets his critical philosophy going, but also sets an agenda for many later and rather unkantian philosophers.

To exposit this conception we need some distinctions.³ Assume the anachronistically labeled traditional analysis of knowledge as justified true belief. Epistemology is much more about justification than knowledge. Kant calls knowledge a posteriori when it is justified, even in part, by appeal to sense experience. Knowledge is a priori when it is knowledge but not a posteriori, that is, not justified even in part by experience. Kant thought that mathematics, that is, geometry, and logic are systematic bodies of a priori knowledge. We will consider an argument for this thought in a moment.

Consider next judgments. This is Kant’s usual term for mental states like beliefs (such as that grass is green or seven is prime) and thoughts. Around the turn of the twentieth century, G. E. Moore and Russell will replace judgments by propositions, which are platonic abstracta like numbers rather than mental. Frege’s thoughts are more like Russell’s propositions than Kant’s judgments. During the twentieth century, philosopher-logicians like Tarski will replace both judgments and propositions with sentences. Sentences are linguistic items where judgments and propositions were supposed to be independent of language. (Around 1950 J. L. Austin will try to replace sentences with statements thought of as actions performed using sentences.) Kant divides judgments into analytic and synthetic. Analytic judgments are reminiscent of Locke’s trifling propositions (not to be confused with russellian propositions) and Hume’s relations of ideas.

One way to move in on analyticity is through examples. An example of Moore’s is the claim that all bachelors are unmarried. The Social Science Research Council would be ill advised to fund a door-to-door survey in which bachelors are asked whether they are married, the results are tallied, and finally the bold hypothesis that all of them are unmarried is advanced. This would be a waste because, so the story goes, being unmarried is part of what it means to be a bachelor.

It seems clear that there is some sort of difference between the claim that bachelors are unmarried and the claim that bachelors are more flush financially than husbands. Controversy sets in when we try to articulate the difference. Kant gave two accounts of analyticity. On one, the predicate of an analytic judgment is contained in its subject. Note three points about this account. First, it seems to presuppose that all judgments are of subject-predicate form. Whatever grammarians may say, Russell was excited by the revelation in the logic reforming around him of other forms, especially quantificational, of judgment. We follow Russell, so Kant’s account may seem too narrow to us. Second, his account presupposes that judgments have subjects and predicates. That is, Kant seems to be reading sentence structure back into judgments. One role in which Kant’s judgments or Russell’s propositions or Tarski’s sentences are cast is as bearers of the truth values; these are the things that are true or false. Whether it is true that Socrates was snub-nosed depends in part on the man Socrates and what his nose ...