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 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 Now it’s well-known that so-called Platonistic positions of this sort are beset by a range of familiar philosophical problems;3 for myself, I’m more troubled by purely methodological concerns,4 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. 365-366. 5 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 value. … The traditional battles in the philosophy of mathematics focused on ontology. … Kreisel is often credited with shifting attention toward realism in truth- value, 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. 6. 7 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 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 17th to the 21st 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 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 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.