Penelope Maddy

Believing the Axioms. I

§0. Introduction. Ask a beginning philosophy of mathematics student why we believe the theorems of mathematics and you are likely to hear, “because we have proofs!” The more sophisticated might add that those proofs are based on true axioms, and that our rules of inference preserve truth. The next question, naturally, is why we believe the axioms, and here the response will usually be that they are “obvious”, or “self-evident”, that to deny them is “to contradict oneself” or “to commit a crime against the intellect”. Again, the more sophisticated might prefer to say that the axioms are “laws of logic” or “implicit definitions” or “conceptual truths” or some such thing. Unfortunately, heartwarming answers along these lines are no longer tenable (if they ever were). On the one hand, assumptions once thought to be self-evident have turned out to be debatable, like the law of the excluded middle, or outright false, like the idea that every property determines a set. Conversely, the axiomatization of set theory has led to the consideration of axiom candidates that no one finds obvious, not even their staunchest supporters. In such cases, we find the methodology has more in common with the natural scientists hypotheses formation and testing than the caricature of the mathematician writing down a few obvious truths and preceeding to draw logical consequences. The central problem in the philosophy of natural science is when and why the sorts of facts scientists cite as evidence really are evidence. The same is true in the case of mathematics. Historically, philosophers have given considerable attention to the question of when and why various forms of logical inference are truth-preserving. The companion question of when and why the assumption of various axioms is justified has received less attention, perhaps because versions of the “self-evidence” view live on, and perhaps because of a complacent if-thenism.

Taking naturalism seriously

Publisher Summary Quine has concentrated much of his naturalized epistemological attention on “the learning of language and the neurology of perception,” but he recognizes that more than this goes into the development of good scientific hypotheses and suggests that further counsel is available anecdotally in the history of hard science. The chapter discusses the counsel of some such anecdotes. It also focuses on the practice of a truly naturalized methodology and explores some implications for the philosophy of mathematics. The status of some fairly concrete statements about point sets that turn out to be independent of the current set-theoretic axioms are also considered. A scientific theory with generous portions of the theoretical virtues will often also include a good measure of mathematics. It is realized that compounds with different chemical properties sometimes analyze into the same elements in the same proportions. Conflicts between the results achieved by the various methods discussed in the chapter led Dumas to conclude that atomic theory should be banished from chemistry.

Philosophy of mathematics: Prospects for the 1990s

For some time now, academic philosophers of mathematics have concentrated on intramural debates, the most conspicuous of which has centered on Benacerrafs epistemological challenge. By the late 1980s, something of a consensus had developed on how best to respond to this challenge. But answering Benacerraf leaves untouched the more advanced epistemological question of how the axioms are justified, a question that bears on actual practice in the foundations of set theory. I suggest that the time is ripe for philosophers of mathematics to turn outward, to take on a problem of real importance for mathematics itself.

A SECOND PHILOSOPHY OF ARITHMETIC

This paper outlines a second-philosophical account of arithmetic that places it on a distinctive ground between those of logic and set theory. In a pair of recent books,1 I’ve proposed an austere naturalistic approach to philosophizing, characterized by the practices of an idealized inquirer called the Second Philosopher— an inquirer equally interested in all aspects of the world and our place in it, equally at home in physics, astronomy, biology, psychology, linguistics, sociology, anthropology, etc., and even logic and mathematics, as the need arises. The closest conventional classification is ‘methodological naturalism,’ described by one prominent taxonomist, David Papineau, as a family of views that ‘is concerned with the ways of investigating reality, and claims some kind of general authority for the scientific method.’2 I shy away from this succinct portrayal—and come at Second Philosophy roundabout, as what the Second Philosopher does—in order to highlight the fact that the Second Philosopher doesn’t think of herself as marching under any banner of ‘science method’; instead, she simply begins with ordinary perceptual experience, gradually develops more sophisticated means of observation and generalization, theory formation and testing, and so on. What she doesn’t do is attempt any overarching account of her ‘method’;3 when she turns her attention to the question of how best to investigate the world, she fully appreciates that her techniques of inquiry often end up needing revision and supplementation, an open-ended process that can’t be foreseen and corralled in advance.4 We describing her might use the rough label ‘scientific’ for her approach, but a true understanding the nature of Second Philosophy requires tracing her efforts in various particular cases and getting the hang of predicting how she would react in a new one. In Second Philosophy, in one of those particular cases, the Second Philosopher turns her attention to logic, to the question of what grounds logical truth. Toward the end of that book, and as the main event in Defending the Axioms, she investigates the proper methods for higher set theory,5 and the metaphysical and epistemological background that explains why these methods are the proper ones. The result is a sharp contrast between the robust worldly supports she identifies for logic, and the objective but metaphysically Received: July 15, 2013. 1 Maddy (2007, 2011). 2 See Papineau (2009). 3 This would be to provide what’s customarily described as a ‘demarcation criterion’ between science and nonscience. 4 For example, in 1900, it wasn’t unreasonable to think that the existence of entities like atoms could never be established, because they couldn’t be observed. Perrin showed otherwise, introducing a subtle new method of confirmation into physical chemistry. 5 By ‘higher set theory,’ I mean set theory that goes beyond number theory and into analysis, the study of reals and sets of reals, and beyond. From here on, I leave the ‘higher’ implicit. c

Does V. Equal L

Does V = L? Is the Axiom of Constructibility true? Most people with an opinion would answer no. But on what grounds? Despite the near unanimity with which V = L is declared false, the literature reveals no clear consensus on what counts as evidence against the hypothesis and no detailed analysis of why the facts of the sort cited constitute evidence one way or another. Unable to produce a well-developed argument one way or the other, some observers despair, retreating to unattractive fall-back positions, e.g., that the decision on whether or not V = L is a matter of personal aesthetics. I would prefer to avoid such conclusions, if possible. If we are to believe that L is not V, as so many would urge, then there ought to be good reasons for this belief, reasons that can be stated clearly and subjected to rational evaluation. Though no complete argument has been presented, the literature does contain a number of varied argument fragments, and it is worth asking whether some of these might be developed into a persuasive case. One particularly simple approach would be to note that the existence of a measurable cardinal (MC) implies that V ≠ L, 1 and to argue that there is a measurable cardinal. The drawback to this approach is that its implying V ≠ L cannot then be counted as evidence in favor of MC, as it often is. Indeed, there seems to have been considerable sentiment against V = L even before the proof of its negation from MC, 2 and this sentiment must either be accounted for as reasonable or explained away as an aberration of some kind.

Logic and the Discursive Intellect

The effort to fit simple logical truths—like ‘if it’s either red or green and it’s not red, then it must be green’—into Kant’s account of knowledge turns up a position more subtle and intriguing than might be expected at first glance.

Set Theory as a Foundation

The view of set theory as a foundation for mathematics emerged early in the thinking of the originators of the theory and is now a pillar of contemporary orthodoxy. As such, it is enshrined in the opening pages of most recent textbooks; to take a few illustrative examples: All branches of mathematics are developed, consciously or unconsciously, in set theory. (Levy, 1979, p. 3) Set theory is the foundation of mathematics. All mathematical concepts are defined in terms of the primitive notions of set and membership … From [the] axioms, all known mathematics may be derived. (Kunen, 1980, p. xi).

New Directions in the Philosophy of Mathematics

Mathematical axioms have traditionally been thought of as obvious or self-evident truths, but current set theoretic work in the search for new axioms belies this conception. This raises epistemological questions about what other forms of justification are possible, and how they should be judged.