Objectivity in Mathematics. One of the many things about the practice of mathematics. that makes the philosophy of mathematics so difficult, in fact

Transcription

1 Objectivity in Mathematics One of the many things about the practice of mathematics that makes the philosophy of mathematics so difficult, in fact maybe the leader in that troublesome company, arises from the pure phenomenology of the practice, from what it feels like to do mathematics. Anything from solving a homework problem to proving a new theorem involves the immediate recognition that this is not an undertaking in which anything goes, in which we may freely follow our personal or collective whims; it is, rather, an objective undertaking par excellence. Part of the explanation for this objectivity lies in the inexorability of the various logical connections, 1 but that can t be the whole story; if we try to treat mathematics simply as a matter of what follows from what, we capture the claim that the Peano axioms logically imply 2+2=4, that some set theoretic axioms imply the fundamental theorem of calculus, but we miss 2+2=4 and the fundamental 1 See [2007], Part III, for more on the status of logical truth.

2 2 theorem themselves. Another way of putting this is to say that we don t form our mathematical concepts or adopt our fundamental mathematical assumptions willy-nilly, that these practices are highly constrained. But by what? One perennially popular answer is that what constrains our practices here, what makes our choices right or wrong, is a world of abstracta that we re out to describe. This idea is nicely expressed by the set theorist, Yiannis Moschovakis: The main point in favor of the realistic approach to mathematics is the instinctive certainty of most everybody who has ever tried to solve a problem that he is thinking about real objects, whether they are sets, numbers, or whatever. (Moschovakis [1980], p. 605) Often enough, this sentiment is accompanied by a loose analogy between mathematics and natural science: We can reason about sets much as physicists reason about elementary particles or astronomers reason about stars. (Moschovakis [1980], p. 606) 2 In keeping with our close observation of the experience itself, it seems only right to admit that mathematics is, if anything, more tightly constrained than the physical sciences. We tend to think that mathematics doesn t just happen to be true, it has to be true. 2 Cf. Gödel [1944], p. 128: It seems to me that the assumption of such objects [ classes and concepts conceived as real objects existing independently of our definitions and constructions ] is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. They are in the same sense necessary to obtain a satisfactory system of mathematics and physical bodies are necessary for a satisfactory theory of our sense perceptions. Also Gödel [1964], p. 268: the question of the objective existence of the objects of mathematical intuition is an exact replica of the question of the objective existence of the outer world.

3 3 Now it s well-known that so-called Platonistic positions of this sort are beset by a range of familiar philosophical problems; 3 concerns, 4 for myself, I m more troubled by purely methodological but I won t go into those here as I want to focus instead on one prominent line of reaction to these difficulties. This springs from a sentiment famously expressed by Kreisel -- or perhaps I should say apparently expressed, as no clear published source is known to me. 5 Dummett s paraphrase goes like this: What is important is not the existence of mathematical objects, but the objectivity of mathematical statements. (Dummett [1981], p. 508) Putnam casts the idea in terms of realism: The question of realism, as Kreisel long ago put it, is the question of the objectivity of mathematics and not the question of the existence of mathematical objects. (Putnam [1975], p. 70) Shapiro makes the connection explicit: there are two different realist themes. The first is that mathematical objects exist independently of mathematicians, and their minds, languages, and so on. Call this realism in ontology. The second theme is that mathematical statements have objective truth-values independent of the minds, languages, conventions, and so forth, of mathematicians. Call this realism in truth- 3 The canonical reference is Benacerraf [1973]. 4 See [2007], pp Dummett [1978], p. xxviii, identifies the source as something Kreisel remarked in a review of Wittgenstein, but if the passage in question in the one pinpointed by Linnebo [200?] -- namely Kreisel [1958], p. 138, footnote 1 -- it s hard not to agree with Linnebo that it is rather less memorable than Dummett s paraphrase. (The relevant portion of the note in question reads: Incidentally, it should be noted that Wittgenstein argues against a notion of a mathematical object but, at least in places not against the objectivity of mathematics.)

4 4 value. The traditional battles in the philosophy of mathematics focused on ontology. Kreisel is often credited with shifting attention toward realism in truthvalue, proposing that the interesting and important questions are not over mathematical objects, but over the objectivity of mathematical discourse. (Shapiro [1997], p. 37) 6 On this approach, our mathematical activities are constrained not by an objective reality of mathematical objects, but by the objective truth or falsity of mathematical claims, which traces in turn to something other than an abstract ontology (say to modality, to mention just one prominent example). I bring this up because my hope today is to float an idea that would do Kreisel one better: an account of mathematical objectivity that doesn t depend on the existence of objects or on the truth of mathematical claims. To get at this in reasonable compass, I ll have to skate over many themes that demand more detailed treatment, but I hope what amounts to an aerial overview of a book-length argument might be of interest, nonetheless. 7 The goal, then, is to uncover the source of the perceived objective constraints on the pursuit of pure mathematics. The test case here will be my long-time hobby horse: the justification of set-theoretic axioms. What makes this axiom candidate rather than that one into a proper fundamental assumption of our theory? 6 See also Shapiro [2000], p. 29, and [2005], p The book in question is Defending the Axioms ([2011]) This paper was written first, and the two now overlap in various places. Interested readers are encouraged to consult the book for more complete versions of this material.

5 5 The plan is to approach this question from a broadly naturalistic point of view, so let me quickly sketch in the variety of naturalism I have in mind. Imagine a simple inquirer who sets out to discover what the world is like, the range of what there is and its various properties and behaviors. She begins with her ordinary perceptual beliefs, gradually develops more sophisticated methods of observation and experimentation, of theory construction and testing, and so on; she s idealized to the extent that she s equally at home in all the various empirical investigations, from physics, chemistry and astronomy to botany, psychology, and anthropology. Along the way this inquirer comes to use mathematics in her investigations. She begins with a narrowly applied view of the subject, but gradually comes to recognize that the calculus, higher analysis, and much of contemporary pure mathematics are also invaluable for getting at the behaviors she studies and for formulating her explanatory theories. (Here she recapitulates the mathematical developments from the 17 th to the 21 st centuries.) This gives her good reason to pursue mathematics herself, as part of her investigation of the world, but she also recognizes that it is developed using methods that appear quite different from the sort of observation, experimentation and theory formation that guide the rest of her research. This raises questions of two general types. First, as part of her continual evaluation and assessment of her methods of investigation, she will want an account of the methods of pure

6 6 mathematics; she will want to know how best to carry on this particular type of inquiry. Second, as part of her general study of human practices, she will want an account of what pure mathematics is: what sort of activity it is? What is the nature of its subject matter? How and why does it intertwine so remarkably with her empirical investigations? In this humdrum way, by entirely natural steps, our inquirer has come to ask questions typically classified as philosophical. Philosophy undertaken in isolation from science and common sense is often called First Philosophy, so I call her a Second Philosopher. 8 Given that the Second Philosopher will want to pursue set theory, along with her other inquiries, the most immediate problem will be the methodological one -- how am I to proceed? -- so it makes sense to begin there. To get a feel for the forces at work, let s review some concrete examples. I. Some examples from set-theoretic practice i. Cantor s introduction of sets In the early 1870s, Cantor was engaged in a straightforward project in analysis: generalizing a theorem on representing functions by trigonometric series. 9 Having shown that such a representation is unique if the series converges at every point in the domain, Cantor began to investigate the possibility of 8 For more, see [2007]. 9 See Dauben [1979], chapter 2, Ferreirós [1999], IV.4.3 and V.3.2, for historical context and references.

7 7 allowing for exceptional points, where the series fails to converge to the value of the represented function. It turned out that uniqueness is preserved despite finitely many exceptional points, or even infinitely many exceptional points, as long as these are arranged around a single limit point, but Cantor realized that it extends even further. To get at this extension, he moved beyond the set of exceptional points and its limit points to what he called the derived set : It is a well determined relation between any point in the line and a given set P, to be either a limit point of it or no such point, and therefore with the point-set P the set of its limit points is conceptually co-determined; this I will denote P and call the first derived point-set of P. (As translated and quoted in Ferreirós [1999], p. 143) Once this new set, the first derived set, P, is in place, the same operation can be applied again: with P, the set of its limit points is conceptually co-determined ; this P is the second derived set of the original P; and so on. Cantor then proved that if the n-th derived set of the set of exceptional points is empty for some natural number n, then the representation is unique. 10 Of course there had been talk of point sets before Cantor navigated this line of thought, but here for the first time a point set is regarded as an entity in its own right, susceptible to the operation of taking its derived set. From a methodological point of view, what s happened is that a new type 10 Ferreirós ([1999], p. 160) notes that uniqueness continues to hold if the -th derived set is empty for some transfinite ordinal, but Cantor apparently never made this extension.

8 8 of entity -- a set -- has been introduced as an effective means toward an explicit and concrete mathematical goal: extending our understanding of trigonometric representations. (ii) Dedekind s introduction of sets Around the same time, Dedekind also made several early uses of what we now recognize as sets. The first came in algebra, in his theory of ideals, where he elected to replace the ideal number of Kummer, which is never defined in its own right, but only as a divisor of actual numbers by a noun for something which actually exists. (Dedekind writing in 1877; see Avigad [2006], p. 172, for translation and references.) This something which actually exists, the ideal number, Dedekind identifies with the set of numbers Kummer would have taken it to divide. By side-stepping the computational algorithms central to Kummer s treatment, Dedekind was able to demonstrate that the theory could be developed nonconstructively, and to explain why the properties of ideal numbers didn t depend on the details of how they were represented. Here again, sets are being introduced in service of explicit mathematical desiderata -- representation-free definitions, abstract (non-constructive) reasoning -- though Dedekind s vision is broader than the above-cited example from Cantor: he introduces a promising new style of reasoning whose mathematical fruitfulness was dramatically demonstrated as abstract algebra went on to thrive in the hands of Noether and her successors See McLarty [2006].

9 9 The same drive toward new numbers as actual objects with representation-free characterizations is on display in Dedekind s theory of the real numbers. Here Dedekind s goal is to provide a perfectly rigorous foundation for the principles of infinitesimal analysis, 12 and in particular, to remove the geometric evidence [that] can make no claim to being scientific. Since the calculus deals with continuous quantities, he reasons it should be founded on an explanation of this continuity, and he sets out to secure a real definition of the essence of continuity. The result, of course, is his elegant definition of continuity and construction of the real numbers. Competing theories of Weierstrass and Cantor begin from particular convergent series or sequences of rationals, identifying many equivalent such items with a single real; in contrast, Dedekind appeals to a cut, simply an infinite set of rationals. 13 This approach wipes out all detailed series or sequence structure, yielding one cut for each real, and the abstract characterization allows for broad generalization. So here again we see Dedekind preferring definitions that aren t tied to particular representations (like series or sequences), while pursuing broader mathematical goals (a general theory of continuity). 12 All quotations in this paragraph come from Dedekind [1872], p He had already defined the integers and rationals in terms of natural numbers (see Ferreirós [1999], p. 219).

10 10 Another important mathematical goal, also clearly present in this work on real numbers, is the pursuit of rigor: In science nothing capable of proof ought to be believed without proof (Dedekind [1888], p. 790). This declaration opens Dedekind s account of the natural numbers, a third venue for his appeal to sets. Here he officially lays out his background set theory and goes on to develop his account of the natural numbers. In all these cases, we find Dedekind introducing sets in the service of explicit mathematical goals: a representation-free, non-constructive abstract algebra; a rigorous characterization of continuity to serve as a foundation for analysis and a more general study of continuous structures; a rigorous characterization of the natural numbers and resulting foundation for arithmetic. (iii) Zermelo s defense of his axiomatization Turning from the introduction of sets to the adoption of axioms about them, we find Zermelo in 1908 with a range of motives. Locally, he hopes to quiet the controversy over his proof of the well-ordering theorem from the Axiom of Choice. 14 More globally, he sees himself as contributing to the logical foundations of all arithmetic and analysis (Zermelo [1908b], p. 200). He despairs of finding a compelling and fruitful definition of set on which to base the subject -- something comparable, say, to Dedekind s definition of continuity and its role in founding analysis -- and opts instead to analyze the 14 See Moore [1982], pp

11 11 practice of set theory and seek out the principles required for establishing the foundations of this mathematical discipline (Zermelo [1908b], p. 200). Of particular interest for our purposes are his reflections on the proper methods for justifying axioms. Presumably their foundational success counts in favor of his axioms as a whole, but when pressed on the Axiom of Choice in particular, Zermelo distinguishes evidence of two sorts. The first is intuitive self-evidence, which we might now describe as being implicit in the informal concept of set. Zermelo argues that Choice must enjoy this sort of subjective obviousness on the grounds that so many set theorists have used it, often without noticing. But, as we ve seen, he despairs of defining the set concept with a precision adequate to the development of set theory. Instead he appeals to a second standard of evidence that can be objectively decided, namely whether the principle is necessary for science (op. cit.). Here he lists various outstanding problems that can be resolved on the assumption of Choice, and concludes So long as the relatively simple problems mentioned here remain inaccessible [without Choice], and so long as, on the other hand, the principle of choice cannot be definitely refuted, no one has the right to prevent the representatives of productive science from continuing to use this hypothesis -- as one may call it for all I care -- and developing its consequences to the greatest extent principles must be judged from the point of view of science, and not science from the point of view of principles fixed once and for all. (Zermelo [1908a], p. 189)

12 12 This mode of defense goes beyond the observation that his axioms allow the derivation of set theory as it currently exists and the foundational benefits thereof; Zermelo here counts the mathematical fruitfulness of his axioms, their effectiveness and promise, as points in their favor. Gödel also recognized the importance of such evidence, for example, in this well-known passage: Even disregarding the intrinsic necessity of some new axiom, and even in case it has no intrinsic necessity at all, a probable decision about its truth is possible also in another way, namely, inductively by studying its success. Success here means fruitfulness in consequences, in particular in verifiable consequences, i.e., consequences demonstrable without the new axiom, whose proofs with the help of the new axiom, however, are considerably simpler and easier to discover, and make it possible to contract into one proof many different proofs. There might exist axioms so abundant in their verifiable consequences, shedding so much light upon a whole field, and yielding such powerful methods for solving problems that, no matter whether or not they are intrinsically necessary, they would have to be accepted at least in the same sense as any well-established physical theory. (Gödel [1964], p. 261) It has become customary to describe these two rough categories of justification as intrinsic -- self-evident, intuitive, part of the concept of set, and such like -- and extrinsic -- effective, fruitful, productive. (iv) The case for determinacy To round off this list of examples, we should consider a contemporary case. Determinacy hypotheses came in for serious study beginning in the 1960s 15 as part of a broader search for new 15 See, e.g., Kanamori [2003], 27.

13 13 principles that might settle the problems in analysis 16 and set theory 17 left open by the now-standard descendent of Zermelo s system, Zermelo-Fraenkel with Choice (ZFC). In his 1980 stateof-the-art compendium on the subject, Moschovakis observed that no one claims direct intuitions either for or against determinacy hypotheses, that those who have come to favor these hypotheses as plausible, argue from their consequences (Moschovakis [1980], p. 610). At that time, he concluded: At the present state of knowledge only few set theorists accept [determinacy] as highly plausible and no one is quite ready to believe it beyond a reasonable doubt; and it is certainly possible that someone will simply refute [it] in ZFC. On the other hand, it is also possible that the web of implications involving determinacy hypotheses and relating them to large cardinals will grow steadily until it presents such a natural and compelling picture that more will succumb. (Moschovakis [1980], pp ) Here Moschovakis displays impressive foresight, as more have succumbed in recent decades, on the basis of new discoveries. In telegraphic summary, the current evidence for determinacy falls roughly into four classes. 18 First, it generates a rich theory of projective sets of reals with many of the virtues identified by Gödel. 19 Second, Moschovakis s web of implications relating [determinacy] to large cardinal hypotheses has indeed grown steadily. In the decade following 16 E.g., the Lebesgue measurability of projective sets. 17 E.g., of course, the Continuum Hypothesis. 18 See Steel [2000], Koellner [2006]. 19 And AD L( ) is necessary for this theory: it s actually implied by its consequences for definable sets (see Koellner [2006], pp. 170, 174).

14 14 Moschovakis s book, Martin, Steel and Woodin, building on work of Foreman, Magidor and Shelah, showed that determinacy follows from the existence of large cardinals; indeed it is now known to be equivalent to the existence of certain inner models with large cardinals. 20 Third, a striking phenomenon in terms of consistency strength has emerged; in John Steel s words, any natural theory of consistency strength at least that of [determinacy] actually implies [determinacy] (Steel [2000], p. 428). Given the longstanding foundational goal of set theory and the open-endedness of contemporary pure mathematics, we have good grounds to seek theories of ever-higher consistency strength; if all reasonable theories past a certain point imply determinacy, this constitutes a strong argument in its favor. Fourth, in the presence of large cardinals, forcing cannot succeed in showing a question about projective sets to be independent. 21 This means that if any question about projective sets is left unresolved by determinacy, this can t be shown by forcing; the independence involved would have to be a new and unfamiliar variety. Given that we want our theory of sets to be as decisive as possible, within the limitations imposed by Gödel s theorems, this so-called generic completeness would appear a welcome feature of determinacy theory. 20 See Kanamori [2003], 32, Koellner [2006], for discussion and references. 21 If there is a proper class of Woodin cardinals, then L( ) is elementarily equivalent to the L( ) in any forcing extension. (See Koellner [2006], p Cf. Steel [2000], p. 430.)

15 15 In short, the current case for determinacy has blossomed so impressively that many would agree with Hugh Woodin s assessment: determinacy is the correct axiom for the projective sets (Woodin [2001], p. 575). II. Proper set-theoretic method Assuming these examples are typical, the Second Philosopher hoping to undertake an investigation of sets has access to a rich array of methods, both for introducing sets in the first place and for determining their extent and their properties thereafter. In broad summary, these rest on the pursuit of various mathematical goals, from relatively local problem-solving to providing foundations to more open-ended pursuit of promising mathematical avenues. Given what set theory is intended to do, relying on considerations of these sorts is a perfectly rational way to proceed: embrace effective means toward desired mathematical ends. At the same time, she begins to appreciate the extent to which these methods differ from her familiar observation, theory-formation and testing: for example, she isn t accustomed to positing entities to increase her expressive power (as in Cantor) or rejecting a theory because it produces less interesting consequences (as with the alternative to determinacy s theory of projective sets that results from Gödel s Axiom of Constructibility). She might reasonably wonder if her more familiar, tried-and-true methods could be called upon to supplement or even correct these new approaches.

16 16 On examination, though, she concludes that the answer here is no. Ordinary perceptual cognition is most likely involved in our grasp of elementary arithmetic, 22 but she recognizes that this connection to the physical world has long since been idealized away in the infinitary structures of contemporary pure mathematics. Though Quine has argued that mathematical claims are empirically confirmed by a less direct route, this position appears to her to rest on accounts of science, mathematics and the relations between them that don t accurately reflect the true features of these practices. 23 Though she appreciates that providing tools for empirical science remains one of the central goals of pure mathematics, she also realizes that science no longer shapes the ontology or fundamental assumptions of mathematics as it once did in the days of Newton or Euler. 24 Finally, cases like group theory -- which was considered useless and nearly dropped from the curriculum at Princeton just years before it entered physics as an essential tool such cases convince her that any effort to reign in the broad range of goals pursued by pure mathematicians would be unwise. So she s faced with an array of new justificatory methods that appear to be both rational and autonomous. 22 See [2007], IV.2.ii. 23 See [1997], II.6-II.7, [2007], pp , for discussion. 24 See [2008]. 25 See [2007], pp , 347, for discussion and references.

17 17 If all she ultimately cared about were answering questions of the first type -- what are the proper set-theoretic methods? - - she d now be done, but our Second Philosopher will also ask questions of the second type, beginning with the stark: are these methods reliable? Do they successfully track the existence of sets and their properties and relations? Of course she s familiar with questions of this form: she investigates how ordinary perception gives her information about the medium-sized objects in the world around her; she examines the efficacy of our instrumental means of detecting the small parts of matter; she devises double-blinds to reduce the risk of misleading experimental results, and so on. In all these familiar cases, she employs her usual methods to evaluate how humans, as described in biology, physiology, psychology, evolutionary theory, and so on, come to know the world, as described in physics, chemistry, geology, astronomy, and so on. The case of set theory is the same: she s observed the methods of set theorists and now wants to know whether or not they successfully track the truth about its subject matter. This raises the prior question: should set theory be understood as describing a subject matter, as attempting to deliver truths about it? Now, as we ve seen, the Second Philosopher differs from Quine in rejecting the idea that the mathematics used in application is justified by ordinary empirical evidence along with the physical theory in which it is embedded. If she s to conclude that pure mathematics is a body of truths, her case for

18 18 this will presumably rest more loosely on the way it is intertwined with empirical science. For now, I d like to leave a bookmark at this point, to return to it later. For now, let s assume that the Second Philosopher is justified in regarding set theory as a body of truths, and since she has no reason to take its existence claims at other than face-value, 26 she s also justified in believing that sets exist. Though she s viewing the practice from her external, scientific perspective, as a human activity, she sees no opening for the familiar tools of that perspective to provide supports, correctives or supplements to the actual justificatory practices of set theory. She 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, the operative supports and correctives, are the ones that set theory itself provides; she concludes that the methods of set theory are reliable guides to the facts about sets. III. Thin realism To this point, then, the Second Philosopher has determined that set-theoretic methods are rational, autonomous and generally reliable. To explain why this is so, she must now delve more deeply into questions of the second type, about the nature of the human practice of set theory -- she s now faced with the 26 I don t have in mind here any general case for the reliability of surface syntax, e.g., of the sort proposed in Wright [1992] (see [2007], II.5, for further discussion and references). It s just that the Second Philosopher sees no reason to think that set-theoretic claims say anything other than what they appear to say.

19 19 challenge of explaining what makes these methods reliable, what sets must be like in order for this is to be so. Under the circumstances, 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. Various familiar conclusions fall out of this bare suggestion. Since set theory tells us nothing about sets being dependent on us as subjects, or enjoying location in space or time, or participating in causal interactions, it follows that they are abstract in the familiar ways. John Burgess sums up this particular sentiment nicely, One can justify classifying mathematical objects as having all the negative properties that philosophers describe in a misleadingly positive-sounding way when they say that they are abstract [acausal, non-spatiotemporal, etc.]. But beyond this negative fact, and the positive things asserted by set theory, I don t think there is anything more that can be or needs to be said about what sets are like. 27 Let me call this Thin Realism. 28 What s happened here is that the second-philosophical Thin Realist begins from her confidence in the authority of settheoretic methods when it comes to determining what s true and false about sets, and draws from this a metaphysical conclusion about the nature of sets, about their thinness. For this sort of realism, there is no troubling epistemological problem: sets 27 Personal communication, 24 April 2002, quoted with permission. 28 The intended contrast is with robust versions of realism, like Gödel s, that involve rich metaphysical and epistemological theories going far beyond the positive things asserted by set theory.

20 20 just are the kind of thing we can find out about in these ways. There s also no confounding worry about the determinacy of the Continuum Hypothesis: set theory is describing the set-theoretic universe V, and CH or not-ch is a theorem. This is not a version of neo-kantianism -- set theory doesn t tell us that sets are constituted by our practices or any such thing -- nor is it a version of Carnapianism -- a decision about a new axiom isn t a merely pragmatic choice of a new linguistic framework, it s guided by reliable set-theoretic methods, a new discovery about V. Now despite these attractive features of Thin Realism, I think it would be disingenuous to ignore a nagging worry that it s all too easy, that it rests on some sleight of hand. Connecting sets and set-theoretic methods so intimately continues to invite the suspicion that sets aren t fully real, that they re a kind of shadow-play thrown up by our ways of doing things, by our mathematical decisions. The position would be considerably more compelling if it offered some explanation of why sets are this way, but any step in that direction, in the direction of an underlying account of sets that explains this fact, seems to lead us inevitably beyond what set theory tells us about sets. In fact, I think something can be offered that draws the sting from this nagging doubt, but it won t take quite the form expected. What we want is a sense of what sets are that explains why these methods track them. What I think we can get, from the Thin Realist s perspective, is a sense of an objective reality

21 21 underlying both the methods and the sets that illuminates the intimate connection between them. Perhaps this will be enough. Let me come at the question by asking what objective reality underlies and constrains set-theoretic methods, what objective reality it is that set-theoretic methods track. The simple answer, of course, is that they track the truth about sets, but our goal is to find out more about what sets are, without going beyond what set theory tells us, and our hope is that asking the question this way might help. So, what constrains our methods? Part of the answer lies in the ground of classical logic, 29 but our interest here is in the mathematical features. To get at these, let me draw a brief compare-andcontrast with Kant on geometry. According to Kant, the concept of a triangle is defined by us, so we can know what belongs to it, that is, we can know trivial analytic truths like all triangles are three-sided. In contrast, no amount of meditating on the concept of triangle will reveal to us that the three interior angles of a triangle are equal to two right angles; for this we need to construct a triangle -- in our imagination or on the page -- draw a line through the apex parallel to the base and reason from there (cf. A716/B744). How does this process take us beyond the concept to something synthetic? Kant s answer is that the constructions involved here are shaped by the structure of our underlying spatial form of sensibility, either in pure intuition (when we 29 For discussion of the ground of logical truth, see [2007], Part III.

22 22 construct in our visual imagination) or in empirical intuition (when we draw an actual diagram). Because of this shaping, the argument tracks more than just what s built into the concept; the derivation is also constrained by the nature of space itself, which, as we know, Kant thought to be Euclidean. Of course this picture of geometric knowledge hasn t survived subsequent progress in logic, mathematics and natural science, but I think it provides a helpful analogy for what I want to suggest in the case of set theory. Kant is out to explain what underlies the proof of this geometric theorem, what makes it a proof; his answer is: not just the concept of triangle, not just logical consequence, but also the nature of the underlying space. We re out to explain what underlies the justificatory methods of set theory, what makes considerations of the sort we ve sketched into good reasons to believe what we believe. What takes us beyond mere logical connections and allows us to track something more? And what is this something more? We re looking for the counterpart to Kant s intuitive space. Before trying to answer these questions for set theory, let s first consider another type of case in which we go beyond the logical, namely, in mathematical concept-formation. In the logical neighborhood of any central mathematical concept, say the concept of a group, there are innumerable alternatives and slight alterations that simply aren t comparable in their mathematical importance. Logic does nothing to differentiate these one from

23 23 another, assuming they are all consistently defined, but group stands out from the crowd as getting at the important similarities between structures in widely differing areas of mathematics and allowing those similarities to be developed into a rich and fruitful theory. In ways that the historians of mathematics spell out in detail, group effectively opens the door to deep mathematics in ways the others don t. 30 So what guides our concept-formation, beyond the logical requirement of consistency, is the way some logically possible concepts track important mathematical strains that the others miss. Of course there are stark differences between group theory and set theory, because the two pursuits have different goals. Group theory aims to draw together a wide variety of diverse structures that share mathematically important features; it d be counter-productive to require that all groups be commutative (or not), because there are deep structural similarities between commutative and non-commutative groups that it s mathematically fruitful to trace. Set theory, on the other hand, aims at least in part to provide a single foundational arena for all classical mathematics, so it strives to develop a unified theory that s as decisive as possible (see [2007], pp ), for example, that settles the Continuum Hypothesis. Still, there are over-arching similarities. Set-theoretic concepts are formed in response to set-theoretic goals just as the concept group was formed in response to algebraic goals. 30 See, e.g., Wussing [1969] or Stillwell [2002], chapter 19.

24 24 In large cardinal theory, for example, we can trace the conceptual progression from the superstrong cardinal to the Shelah cardinal to the Woodin cardinal, which turned out to be the optimal notion for the purposes at hand, 31 or the gradual migration of the concept of measurable cardinal from its origins in measure theory to the mathematically rich context of elementary embeddings. 32 Of course the set-theoretic cases we ve been concerned with involve not definitions but existence assumptions -- like the introduction of sets in the first place or the addition of large cardinals -- and new hypotheses -- like determinacy -- but in these cases, too, far more than consistency is at stake: these favored candidates differ from alternatives and near-neighbors in that they track what we might call the topography of mathematical depth. This topography stands over and above the merely logical connections between statements, and furthermore, it is entirely objective: just as it s not up to us which bits of pure mathematics best serve the needs of natural science, just as it s not up to us that it would be counterproductive to insist that all groups be commutative, it s also not up to us that appealing to sets and transfinite ordinals allows us to capture the facts about the uniqueness of trigonometric representations, that the Axiom of Choice takes an amazing range of different forms and plays a fundamental role in many different areas, that large cardinals arrange themselves 31 See Kanamori [2003], p See Kanamori [2003], 2 and 5.

25 25 into a hierarchy that serves as an effective measure of consistency strength, that determinacy is the root regularity property for projective sets and interrelates with large cardinals, and so on. These are the facts that play a role analogous to Kant s Euclidean space, the facts that constrain our set-theoretic methods, and these facts, unlike Kant s, are not traceable to ourselves as subjects. A generous variety of expressions is typically used to pick out to the phenomenon I m after here: mathematical depth, mathematical fruitfulness, mathematical effectiveness, mathematical importance, mathematical productivity, and so on. I m using such terms more or less interchangeably. One point worth emphasizing is that the notion in question is not being offered up as a candidate for conceptual analysis or some such thing. To begin with, I doubt that an attempt to give a general account of what mathematical depth really is would be productive; it seems to me the phrase is best understood as a catch-all for the various kinds of special virtues we clearly perceive in our illustrative examples of concept formation and axiom choice. 33 But even if I m wrong about this, even if something general can be said about what makes this or that bit of mathematics count as important or fruitful or whatever, I would resist the claim that this something general would provide a more fundamental justification for the mathematics in 33 This is why I spend so much time rehearsing these various cases, to give the reader a feel for what mathematical depth looks like.

26 26 question; our second-philosophical analysis strongly suggests that the context-specific justifications we ve been considering so far are sufficient on their own, that they neither need nor admit supplementation from another source. It also bears repeating that judgments of mathematical depth are not subjective: I might be fond of a certain sort of mathematical theorem, but my idiosyncratic preference doesn t make some conceptual or axiomatic means toward that goal into deep or fruitful or effective mathematics; for that matter, the entire mathematical community could be blind to the virtues of a certain method or enamored of a merely fashionable pursuit without changing the underlying facts of which is and which isn t mathematically important. This is what anchors our various local mathematical goals. Cantor may have wished to expand his theorem on the uniqueness of trigonometric representations, but if this theorem hadn t formed part of a larger enterprise of real mathematical importance, his one isolated result wouldn t have constituted such compelling evidence for the existence of sets; similarly the overwhelming case for Dedekind s innovations depends in large part on the subsequent successes of the abstract algebra they helped produce. The key here is that mathematical fruitfulness isn t defined as that which allows us to meet our goals, irrespective of what these might be; rather, our mathematical goals are only proper insofar as satisfying them furthers our grasp of the underlying strains of mathematical fruitfulness. In other words, the goals are answerable to the

27 27 facts of mathematical depth, not the other way round. 34 Our interests will influence which areas of mathematics we find most attractive or compelling, just as our interests influence which parts of natural science we re most eager to pursue, but no amount of partiality or neglect from us can make a line of mathematics fruitful if it isn t, or fruitless if it is. 35 Thus we ve answered our leading question: the objective something more that our set-theoretic methods track is these underlying contours of mathematical depth. Of course the simple answer -- they track sets -- is also true, so what we ve learned here is that what sets are, most fundamentally, is markers for these contours, what they are, most fundamentally, is maximally effective trackers of certain strains of mathematical fruitfulness. From this fact about what sets are, it follows that they can be learned about by set-theoretic methods, because set-theoretic methods, as we ve seen, are all aimed at tracking particular instances of effective mathematics. The point isn t, for example, that there is a measurable cardinal really means the existence of measurable cardinals is mathematically fruitful in various ways ; rather, the fact of measurable cardinals being mathematically fruitful in various ways is evidence for their existence. Why? Because of what sets are: repositories of 34 I m grateful to Matthew Glass for pressing me to clarify this point. 35 Here at last are grounds on which to reject the nihilism of footnote 9 on p. 198 of [1997], and even the tempered version in [2007], pp If mathematicians wander off the path of mathematical depth, they re going astray, even if no one realizes it.

28 28 mathematical depth. They mark off a mathematically rich vein within the indiscriminate network of logical possibilities. So there is a well-documented objective reality underlying Thin Realism, what I ve been loosely calling the facts of mathematical depth. The fundamental nature of sets (and perhaps all mathematical objects) is to serve as devices for tapping into that well; this is simply what they are. And since set-theoretic methods are themselves tuned to detecting these same contours, they re perfectly suited to telling us about sets. This, I suggest, is the core insight of Thin Realism. Let me sum up the Second Philosopher s journey so far: she comes to realize that contemporary pure mathematics is a vital part of her investigation of the world and to regard it as a body of truths; she recognizes that its methods are new and distinctive, sees no opening for correction or defense from her more familiar methods, and concludes, in particular, that settheoretic methods are rational, autonomous and reliable guides to the truth about sets; to account for this striking fact, she forms the simple hypothesis that sets are the sort of thing that can be investigated in these ways; and finally she discovers the source of this fact, namely, that sets simply are means for producing certain mathematically fruitful outcomes, and that settheoretic methods are expressly designed to track just these deep mathematical strains. Thus Thin Realism presents itself as an attractive answer to our second group of questions: settheoretic activity in the investigation of an abstract realm of

29 29 sets; its methods are reliable simply because of what those sets are; the whole enterprise answers to the objective topography of mathematical depth; the pursuit of new set-theoretic axioms and of a solution to the continuum problem are legitimate parts of this inquiry. IV. Arealism So we ve achieved a kind of objectivity here, but despite its non-traditional aspects, it still relies on the existence of abstracta and the truth of our claims about them. What I d like to do now is return to that point where we left the bookmark, the point where the Second Philosopher concluded that set theory is a body of truths but her grounds were left vague. Eventually I want to return to the question of what those grounds might be and the extent to which they re persuasive, but first let me sketch in the position that results if we take the other fork in the road at that point, if we conclude that whatever its merits, pure mathematics isn t in the business of uncovering truths. But if he s not uncovering truths, then what is the pure mathematician doing? For the case of set theory, we ve got a sense of the answer: among many other things, Cantor is extending our grasp of trigonometric representations; Dedekind is pushing towards abstract algebra; Zermelo is providing an explicit foundation for a mathematically important practice; contemporary set theorists are trying to solve the continuum

30 30 problem. 36 Just as the concept of group is tailored to the mathematical tasks set for it, the development of set theory is constrained by its own particular range of mathematical goals, both local and global. Mightn t the Second Philosopher rest content with this description? Set theory is the activity of developing a theory of sets that will effectively serve a concrete and ever-evolving range of mathematical purposes. Such a Second Philosopher would see no reason to think that sets exist or that set-theoretic claims are true -- her well-developed methods of confirming existence and truth aren t even in play here -- but she does regard set theory, and pure mathematics with it, as a spectacularly successful enterprise, unlike any other. 37 Let s call this position Arealism. Now we ve noted that whatever reason the Thin Realist may have to count pure mathematics as true, it must rest somehow on the role of mathematics in empirical science, so we need to ask: can the Arealist account for the application of mathematics without regarding it as true? There s a complex story to be told here, 38 but examination of the historical and scientific record leads the Second Philosopher to believe that contemporary pure mathematics works in application by providing the empirical 36 And, lest we forget, much of pure mathematics is still consciously aimed at the goal of providing tools for empirical science. 37 In particular, its complex interrelations with natural science mark it off from other human endeavors -- astrology, theology -- whose methods also differ from those usual to the Second Philosopher. See [1997], pp , [2007], pp , and more below. 38 See [2008].

31 31 scientist with a wide range of abstract tools. The scientist uses these as models -- of a cannon ball s path or the electromagnetic field or curved spacetime -- which he takes to resemble the physical phenomena in some rough ways, to depart from it in others; indeed often enough, in fundamental theories, we aren t sure exactly how the correspondence plays out in detail. The applied mathematician labors to understand the idealizations, simplifications and approximations involved in these deployments of his abstract structures; he strives as best he can to show how and why a given model resembles the world closely enough for the particular purposes at hand. In all this, the scientist never asserts the existence of the abstract model; he simply holds that the world is like the model is some respects, not in others. For this, the model need only be welldescribed, just as one might illuminate a given social situation by comparing it to a imaginary or mythological one, marking the similarities and dissimilarities. Assuming then that the truth (or not) of mathematics is irrelevant to explaining its role in scientific application, it appears that Arealism is open to our Second Philosopher: she notes that mathematics is successful on its own terms and immensely useful to science, but since it isn t confirmed by her usual methods, even by her need to explain the role it plays in her empirical theorizing, she concludes that she has no grounds on which to regard its objects as real or its claims as truths. In philosophical taxonomy, the standard term for someone who

32 32 doesn t believe in abstract objects is nominalist. If we limit attention to mathematical abstracta, the Arealist would seem to qualify, but, at least as nominalism is usually conceived in contemporary philosophy of mathematics, this way of talking seems to me to invite mis-understanding. To see how, recall that contemporary nominalism began with Goodman and Quine s annunciation of namely, a philosophical intuition that cannot be justified by appeal to anything more ultimate We do not believe in abstract entities. We renounce them altogether. (Goodman and Quine [1947], p. 105) In Burgess and Rosen s characterization: Nominalism (as it is understood in contemporary philosophy of mathematics) arose toward the mid-century It arose among philosophers, and to this day is motivated largely by the difficulty of fitting orthodox mathematics into a general philosophical account of the nature of knowledge. (Burgess and Rosen [1997], p. vii) To avoid nominalism, one must explain in detail how anything we do and say on our side of the great wall separating the cosmos of concreta from the heaven of abstracta can provide us with knowledge of the other side. (Burgess and Rosen [1997], p. 41) Various familiar ideas on the nature of knowledge in concrete cases, like the causal theory of knowledge and its successors, are floated to highlight the severe obstacles that stand in the way of such an explanation. These elements provide the raw materials for a perfectly general, in-principle argument against abstracta of all kinds.