Mathematical axioms were traditionally regarded as self-evident first principles. But set theory – which provides the standard foundation for contemporary mathematics – relies on axioms that cannot obviously be so regarded and contemplates the adoption of further axioms that obviously cannot. What, then, are mathematical axioms, and how can they be defended? More generally, what is mathematical knowledge, and what does the existence of such knowledge tell us about the nature of reality and the powers of the human intellect? These and related questions are engagingly discussed in Maddy's ambitious little book of just 137 pages. The book is very well informed about contemporary set theoretic practice, yet manages to be accessible to a far wider audience than just philosophers of mathematics.
As readers familiar with Maddy's earlier work will expect, the book provides a defence of her distinctive brand of philosophical naturalism, which she applies to the mentioned philosophical questions. The discussion is structured around the ‘transcendental’ question of what sets are like, given that the methods and practice of contemporary set theory are suitable for finding out about them. I shall concentrate my comments on three aspects of this key question, namely the naturalism that is underlying it; Maddy's characterization of the relevant methods and practice; and the two metaphysical views – Thin Realism and Arealism – that she proposes as equally good, and ultimately equivalent, answers.
Naturalism, according to Maddy, rejects any ‘special vantage point outside of science’ (39). But the naturalistic philosopher does this not by blindly deferring to scientists but by herself becoming ‘an active participant’ of science who reasons ‘entirely from within’ science (39). The naturalistic philosopher is thus not an alien to science but (in Quine's pun) becomes fully naturalized. At the opposite extreme to this naturalism, or ‘Second Philosophy’ as Maddy also calls it, lies ‘First Philosophy’, understood as an enterprise that proceeds entirely independently of science.
These characterizations prompt the question of what it means to proceed within science. Many philosophers who are not naturally classified as naturalistic nevertheless aspired to proceed scientifically and would no doubt insist that, just as biologists and mathematicians possess special expertise about their respective domains, so too do philosophers. Maddy does not respond directly to this challenge but insists that ‘no specific characterization of what counts as “scientific” is in play here’, other than ‘beginning from observation, developing more refined methods, and so on’ (39). Apart from a faint echo of empiricism, this threatens to deprive naturalism of most content. One is reminded of Russell's programme, during a brief foray into politics, to do politics rationally. One need not be an advocate of irrational politics to find fault with this programme. In this reviewer's option, talk about reasoning from within science should either be fleshed out into something more substantive or else be replaced by more neutral talk about proceeding with intellectual integrity.
Maddy's naturalism takes on more substance in her subsequent philosophical theorizing. In particular, she takes the correctness of successful scientific practice as a datum for philosophical theorizing rather than something susceptible to philosophical challenge. It is this methodology that leads to the key transcendental question mentioned above.
Any answer to the transcendental question must obviously begin by identifying the relevant methods and practice. Here Maddy's naturalism serves her well, and her discussion is rewarding. A prominent theme is the extent and significance of ‘extrinsic’ justifications in set theory, that is, broadly pragmatic justifications tied to considerations of mathematical need and fruitfulness. Such justifications contrast with more traditional ‘intrinsic’ justifications, which are tied to self-evidence and analysis of the relevant mathematical concepts. Extrinsic justifications were made explicit in Zermelo's pioneering 1908 article; although Maddy suggests there is a continuous tradition going back to Cantor and Dedekind, whose introduction of transfinite sets was driven by explanatory goals that arose within mainstream mathematics. She next provides a wealth of more recent examples from the study of so-called determinacy hypotheses in descriptive set theory, quoting extensively and effectively from the work of relevant mathematicians.
Having identified the prevailing set-theoretic methods – which, as a naturalist, she regards as proper – Maddy asks whether these methods ‘successfully track the existence of sets and their properties and relations’ (54). The transcendental question thus leads back to some traditional philosophical questions about the metaphysical nature of sets and our cognitive ‘access’ to them. When grappling with these traditional questions, Maddy's naturalism serves her less well and disposes her to accept answers that strike this reviewer as either inadequately explanatory or downright wrong. I shall give three examples.
First, concerning the metaphysics of sets, Maddy writes that ‘the Second Philosopher is naturally inclined to entertain the simplest hypothesis that accounts for the data: sets just are the sort of thing set theory describes; this is all there is to them; for questions about sets, set theory is the only relevant authority.’ (61) I find this view, which forms the core of Maddy's ‘Thin Realism’, puzzling. Set theorists study mathematically interesting properties of sets; beyond that, they are either agnostic or venture outside their area of expertise. It no more follows that sets have no non-set-theoretic properties than that rocks have no properties beyond those studied by geologists. (This observation also complicates Maddy's later rejection of the idea that sets are in some sense ‘constituted’ by us or our practices (66)).
Second, Maddy argues that the Continuum Hypothesis ‘has a determinate truth value, despite its independence from ZFC’ as follows: ‘“CH or not-CH” is a theorem, established by [our] best methods about [the set-theoretic hierarchy] V; therefore CH is either true or false there’ (62–3). What saves this from being an unwarranted inference of bivalence from the Law of Excluded Middle is presumably the assumption that ZFC is about a determinate structure, V, which it truly describes (cf. 80). However, this assumption also begs the question against the view that CH lacks a determinate truth value.
Third, in answer to the Benacerrafian question about how mathematicians ‘track’ sets, Maddy writes that the naturalistic philosopher ‘has no grounds to question the very procedures that do such a good job of delivering truths, so she concludes that the proper methods to employ […] are the ones that set theory itself provides; she concludes that the methods of set theory are reliable guides to facts about sets’ (55). At best, this provides a minimal answer to the question of whether the methods are reliable. But it is silent on the more interesting question of how these methods are reliable. Perhaps Thin Realism is supposed to help here. For the Thin Realist concludes from ‘her confidence in the authority of set-theoretic methods’ and the fact that ‘more familiar methods appear irrelevant […] that the call for a non-trivial, external epistemology is misplaced’ (63). But this conclusion would be unconvincing. Our confidence in the authority of biological methods and the irrelevance to biology of (most) other methods do not preclude a more substantive epistemology of biology. Our conviction that certain methods are uniquely reliable in a given domain does not make it misplaced to ask why they are so.
The second philosophical view of set theory developed in the book is Arealism, which seeks to avoid the assumption underlying Thin Realism that set theory is a body of truths and (therefore) concerned with certain objects. However, the two views are united in regarding set theory as ‘a spectacularly successful enterprise’ (89) and in insisting on the objective correctness of most set-theoretic reasoning.
The main challenge confronting Arealism is the Fregean argument – encountered here in a version due to Michael Liston – that the applicability of mathematics presupposes its truth. Maddy resists this argument, contending instead that the successful applications of mathematical methods are adequately explained by purely mathematical features (for instance, the reliability of Euler's method, by an equation's satisfying the Lipschitz condition (96)).
Surprisingly, Maddy insists that ‘Thin Realism and Arealism are equally accurate […] descriptions of the nature of pure mathematics. They are alternative ways of expressing the very same account of the objective facts that underlie mathematical practice’ (112). How can views committed to such different claims be ‘equally accurate’? Here Maddy invokes Mark Wilson's idea that there is often no fact of the matter about whether our concepts apply to a given phenomenon; for instance, it is indeterminate whether water frozen in a non-crystalline structure counts as ice. Likewise, Maddy claims, it is indeterminate whether pure mathematics should be regarded as a body of truths (111). However, Wilson's claim about ice is much less problematic than Maddy's claim about mathematics. The former may be a case of semantic indecision. But the latter requires that one and the same system of facts can be correctly described both as involving objects and as not, and that statements can be concerned with these facts without the question of their truth thereby becoming obligatory. Both ideas raise deep philosophical problems, which are not properly addressed.
There is much to admire in Maddy's book. Her knowledge of set-theoretic methodology is impressive, and her respect for it, salutary. She recognizes that her naturalistic framework leaves room for some traditional philosophical questions to be formulated. And, like many great philosophers before her, she doesn't take these questions at face value but seeks to replace them with clarified, more tractable questions. But despite this laudable goal, some of the successor questions that arise from Maddy's naturalism, and some of the proposed answers, are not, in my opinion, much clearer. For instance, Maddy writes that ‘my hope is to shift attention away from these elusive matters of truth and existence, and to direct it toward the type of mathematical objectivity [that she has] emphasized’ (1). However, is this ‘post-metaphysical’ notion of objectivity really any clearer than the notions of truth and existence (116)? If anything, it seems to me an adequate account of objectivity will presuppose the notions of truth and existence. Another example is Maddy's claim that ‘the be-all and end-all of mathematics isn't a remote metaphysics that we access through some rational faculty, but the entirely palpable facts of mathematical depth’ (137). This shift from self-evident facts about being to palpable facts about depth is not obviously helpful. Despite these misgivings, the book is an engaging contribution to an important philosophical debate and deserves to be read far beyond the ranks of philosophers of mathematics.
