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.

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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’.

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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’.)

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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.

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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

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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.

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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.

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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 non-

constructively, 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.11

11 See McLarty [2006].

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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. 767. 13 He had already defined the integers and rationals in terms of natural numbers (see Ferreirós [1999], p. 219).

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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. 143-160.

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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)

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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 1960s15 as part of a broader search for new

15 See, e.g., Kanamori [2003], §27.

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principles that might settle the problems in analysis16 and set

theory17 left open by the now-standard descendent of Zermelo’s

system, Zermelo-Fraenkel with Choice (ZFC). In his 1980 state-

of-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. 610-611)

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 ADL() is necessary for this theory: it’s actually implied by its consequences for definable sets (see Koellner [2006], pp. 170, 174).

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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 long-

standing 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. 171. Cf. Steel [2000], p. 430.)

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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.

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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 tool25 -- 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. 314-317, for discussion.

24 See [2008]. 25 See [2007], pp. 330-331, 347, for discussion and references.

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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

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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.

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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 set-

theoretic 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’.

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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

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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-and-

contrast 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

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

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. 351-355), 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

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 counter-

productive 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. 461. 32 See Kanamori [2003], §§2 and 5.

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

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

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. 350-351. If mathematicians wander off the path of mathematical depth, they’re going astray, even if no one realizes it.

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 set-

theoretic 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 set-

theoretic 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: set-

theoretic activity in the investigation of an abstract realm of

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

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. 203-205, [2007], pp. 345-347, and more below. 38 See [2008].

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 well-

described, 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

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

a philosophical intuition that cannot be justified by appeal to anything more ultimate …

namely,

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.

33

I hope and trust it’s clear that this is not a portrait of

the second-philosophical Arealist. She doesn’t come to her

investigations with an a priori prejudice against abstract

objects or with any preconceptions about what knowledge must be

like that would seem to rule out knowledge of sets. She doesn’t

argue that set-theoretic knowledge is problematic or impossible

on principle; she simply surveys the evidence at hand and

concludes that it doesn’t confirm the existence of sets or the

truth of our theory of them. So if Arealism is to be considered

a version of nominalism, it certainly isn’t what Burgess and

Rosen call the ‘stereotypical’ variety (Burgess and Rosen [1997],

p. 29).

V. Post-metaphysical objectivism

At this point, we have two apparently second-philosophical

positions in play. And, though they disagree sharply over truth

and existence -- the Thin Realist holding that sets exist and set

theory is a body of truths, and the Arealist denying both --

still they are indistinguishable at the level of method. On

grounds like those that motivated Cantor and Dedekind, both would

elect to introduce sets into their pursuit of pure mathematics;

both would regard Zermelo’s defenses of his axioms as persuasive;

both would follow the path of contemporary set theorists on

determinacy and large cardinals. This methodological agreement

reflects a deeper metaphysical bond: the objective facts that

underlie these two positions are exactly the same, namely, those

34

distinctive strains of mathematical depth. For the Thin Realist,

sets are the things that mark these contours; set-theoretic

methods are designed to track them. For the Arealist,

these same contours are what motivate and guide her elaboration

of the theory of sets; she can go wrong as easily as the Thin

Realist if she fails to detect the true mathematical virtues in

play. For both positions, the development of set theory responds

to an objective reality -- and indeed to the very same objective

reality.

What separates the Arealist from the Thin Realist, then,

doesn’t lie in their set-theoretic practices or what underlies

them. Where they differ is in their second-philosophical

reflections on the human undertaking called ‘set theory’. They

would agree precisely on what counts as proper grounds for adding

a new large cardinal axiom to the theory of sets; they would

disagree only on the Thin Realist’s added assertion that these

grounds confirm the existence of the large cardinal in question

and the truth of the corresponding axiom. Notice that it isn’t

an ordinary set-theoretic claim of existence or truth that’s at

issue here: the Arealist like the Thin Realist will formulate

the axiom in existential form and call it ‘true’ in the sense of

holding in V. Their disagreement takes place not within set

theory, but in the judgments they form as they regard set-

theoretic language and practice from an empirical perspective and

ask second-philosophical versions of the traditional

35

philosophical questions, questions in the second group we’ve been

considering.

So how is the Second Philosopher to adjudicate between Thin

Realism and Arealism? This returns us at last to the problem

we’ve set aside twice: on what grounds does the Thin Realist

judge that set theory is a body of truths? Given that she

rejects the usual Quinean arguments, given that she endorses the

Arealist’s account of how mathematics works in application, the

Second Philosopher’s case for Thin Realism will have to rest more

loosely on the way mathematics is intertwined with empirical

science: she recognizes that pure mathematics arose out of a

subject very closely tied to our study of the physical world; she

regards the project of providing a rich array of structures for

the contemporary scientist as one of the over-arching goals of

mathematical practice; she well appreciates that contemporary

pure mathematics continues to find its way into scientific

applications, sometimes along deliberately anticipated paths, and

sometimes along wholly unexpected ones. Thus mathematics,

whatever its idiosyncrasies, appears as an integral part of her

overall enterprise (as opposed to astrology, theology, etc.,

which are idiosyncratic without playing a part in that

enterprise).39 On this picture, the Second Philosopher pursues

mathematics in a spirit continuous with her other inquiries:

some of its methods, like logical deduction and means-ends

reasoning, are familiar; others, like Cantor’s, Dedekind’s, 39 See [1997], pp. 203-205, [2007], pp. 345-247.

36

Zermelo’s, and the determinacy theorists’, are unfamiliar, but

taken to be rational and reliable along the lines we’ve been

following.

Thus the divergence between the second-philosophical

Arealist and the second-philosophical Thin Realist comes down to

this: as the Second Philosopher conducts her inquiry into the

way the world is, beginning with her ordinary methods of

perception and observation, theory formation and testing, she’s

eventually faced with the effectiveness of pure mathematics and

elects to add it to her ever-growing list of investigations; she

also recognizes that the appropriate methods are different and

that the objects studied are different; the point at issue hinges

on what she concludes from this. If the new objects seem a bit

odd -- non-spatiotemporal, acausal, etc. -- but still enough like

the old -- singular bearers of properties, etc. -- , if the new

methods seem a bit odd, but still of-a-piece with the old, then

she concludes that she’s made a surprising discovery, that the

world includes abstracta as well as concreta. If, on the other

hand, she regards the new methods and would-be objects as sharply

discontinuous with what came before, she has no grounds for

thinking pure mathematics is true, so she concludes that this new

practice -- valuable as it is -- isn’t in business of developing

a body of truths. So, which is it? Is pure mathematics just

another inquiry among many or it is a different sort of thing

that’s immensely helpful to the others? Are the grounds cited by

Cantor, Dedekind, Zermelo, and the determinacy theorists just

37

more evidence of an unexpected sort, or are they the trademarks

of a different sort of activity altogether?

It’s hard not to think that one must be right and the other

wrong, that either sets exist or they don’t, that set theory is a

body of truths or it isn’t, that either the considerations cited

by Cantor, Dedekind, Zermelo, and the determinacy theorists are

confirming evidence or they aren’t. But perhaps this tempting

position is in fact incorrect, perhaps our strong conviction

otherwise rests on what Mark Wilson calls, in his typically

colorful style, ‘tropospheric complacency’: we tend to think

that our concepts -- in this case ‘true’, ‘exist’, ‘evidence’,

‘believe’, ‘know’ -- mark fully determinate features or

attributes, that there is a determinate fact of the matter as to

where they apply and where they don’t, that this is so even for

questions we haven’t yet been able to settle one way or the

other. Wilson’s massive case against this picture, Wilson

[2006], rests largely on a wealth of fascinating and down-to-

earth examples. To get a feel for how these examples go, let’s

look at two of them.

First, consider ice. Surely we all know what ice is --

it’s frozen water -- but Wilson takes us in for a closer look:

Water, in fact, represents a notoriously eccentric substance, capable of forming into a wide range of peculiar structures. (Wilson [2006], p. 55)

He goes on to quote a recent textbook on the subject, which

describes ‘ice cousins’,

the clathrate hydrates … Like ice polymorphs, they are crystalline solids, formed by water molecules, but

38

hydrogen-bonded in such a way that polyhedral cavities of different sizes are created that are capable of accommodating certain kinds of ‘guest’ molecules. (Quoted by Wilson [2006], p. 55)

Wilson remarks that

The author doesn’t regard the clathrate structure as true ice … but is it clear that our everyday conception of ice requires -- as opposed to accepts -- this distinction? (I, for one, had never thought about such matters at all.) (Wilson [2006], pp. 55-56)

It gets worse: there are in fact more than a dozen ways that

water can form into a solid. In one case, if one cools water

quickly enough, the result lacks crystalline structure and more

closely resembles ordinary glass. Wilson asks

Should this glass-like stuff qualify as a novel form of ‘ice’ or not? Our chemist will presumably say ‘no’ because the stuff is not crystalline, but many of us would perhaps put a higher premium on its apparent solidity. (Wilson [2006], p. 56)

In fact other chemists do happily call this ‘an amorphous type of

ice’ (Caro [1992], p. 99).40 And so on.

Is there a right and a wrong answer here? Our everyday use

of the word ‘ice’ clearly correlates with an objective feature of

the world, the substance chemists call ‘ice Ih’ or hexagonal

crystalline ice. So ‘ice’ definitely doesn’t apply to liquid

water or to sand or to window glass. But does it apply to

amorphous ice -- is amorphous ice really ice? Wilson’s thought

is that nothing in our ordinary use or understanding of the term

‘ice’, indeed nothing in the underlying chemical facts that we

40 Wilson doesn’t cite this passage in his discussion of ‘ice’, but he does quote Caro’s book when he treats the relations between ‘water’ and ‘H2O’ (Wilson [2006], pp. 428-429).

39

subsequently discover about the many ways water can form into a

solid -- in short, nothing in our heads, in our language, or in

the world will force either answer to this question.41 And notice

that this isn’t a version of the well-known Kripkesteinian

challenge: what makes 1002 rather than 1004 the right

continuation of +2 after 1000? We have here not the hyperbolic

doubt of a radical skeptic, but real life cases ‘where the

underlying directivities seem genuinely unfixed’ (Wilson [2006],

p. 39).

A second example is more fanciful, but still quite

compelling for all that. Imagine the inhabitants of an isolated

island; imagine they’ve never seen an airplane until one passes

overhead and crashes in their midst. They might quite naturally

regard it as a bird, regard themselves as having learned,

unexpectedly, that the world includes a type of bird very

different from the ordinary birds they’re familiar with, a great

silver bird made of metal. Now imagine the story again, except

that this time the plane crashes undetected and the islanders

discover it in the jungle with the stranded crew taking shelter

in the fuselage. This time, the islanders might reasonably

41 A similar theme turns up in Austin [1940], pp. 67-68: ‘Suppose that I live in harmony and friendship for four years with a cat: and then it delivers a philippic. We ask ourselves, perhaps, “Is it a real cat? or is it not a real cat?” “Either it is, or it is not, but we cannot be sure which.” Now actually, that is not so: neither “It is real cat” nor “it is not a real cat” fits the facts semantically: each is designed for other situations than this one … Ordinary language breaks down in extraordinary cases … no doubt an ideal language would not break down, whatever happened … In ordinary language … words fail us. If we talk as though an ordinary [language] must be like an ideal language, we shall misrepresent the facts’.

40

regard it as a house, might well regard themselves as having

discovered a new and unusual type of house. Is there any

temptation here to think that one group is wrong and the other

right? It seems clear that nothing in their pre-airplane

concepts of ‘bird’ and ‘house’ or the corresponding worldly

resemblances is enough to determine this, that either option is

open to them as a consistent and defensible extension of the

earlier concepts, that their choice is determined by sheer

historical contingency. But notice:

neither set of alternative [islanders] has any psychological reason to suspect that they have not followed the pre-established conceptual contents of their words ‘bird’ and ‘house’. (Wilson [2006], p. 36)

Here we see the psychological force of tropospheric complacency

in its purest form.42

Could it be that a similar brand of complacency is at work

in the case of the Second Philosopher faced with pure

mathematics? Does the history and current practice of pure

mathematics qualify it as just another item on the list with

physics, chemistry, biology, sociology, geology, and so on? Do

honorifics like ‘true’, ‘exist’, ‘evidence’, ‘confirm’ --

indisputably at home in those other studies -- belong in pure

mathematics as well? Pure mathematics arose out of our empirical

study of the world; it remains intensely important as a tool for

that study, even in parts that weren’t expressly developed for

that purpose; it continues to be inspired by the descriptive and

42 See Wilson [2006], pp. 34-37, for more on the islanders, or [2007], pp 186-188, for a somewhat more complete summary.

41

inferential needs of the natural and social sciences. If all

this is taken to establish it as a body of truths, we’ve seen how

the Thin Realist explicates the ground of that truth and how

mathematical evidence manages to track it. But we’ve also seen

how the Arealist gives a plausible account of pure mathematics as

a deep and vital undertaking that happens not to aim at producing

truths.

What I want to suggest now, indeed at last to claim, is

that our central questions -- is pure mathematics of-a-piece with

physics, astronomy, psychology and the rest? is it a body of

truths? do its methods confirm its claims? -- that these

questions have no more determinate answers than ‘is amorphous ice

really ice?’ Once we understand the various ways in which water

can solidify, how these processes are affected by temperature,

pressure and other factors, how the various structures generated

are similar and how they’re different, there’s nothing more to

know; we can reflect these facts in either way of speaking, or,

to put it the other way around, neither way of speaking comes

into conflict with the facts. Some version of Wilson’s

tropospheric complacency -- our tendency to overestimate the

determinateness of our concepts -- might well leave us convinced

of the exclusive correctness of one or the other -- it must be

ice because it’s solid! it can’t be ice because it’s not

crystalline! -- but we’ve seen that this psychological confidence

often baseless, and also largely harmless.

42

Likewise, once we understand how pure mathematics

developed, how it now differs from empirical sciences,43 once we

understand the many ways in which it remains intertwined with

those sciences, how its methods work and what they are designed

to track -- once we understand all these things, what else do we

need to know? Or better, what else is there to know? Just as

robins are birds and bungalows are houses, physics and botany are

sciences, but this isn’t enough to settle the status of downed

airplanes and pure mathematics. Just as amorphous ice can be

classified as ice or as ice-like, mathematics can be classified

as science or as science-like -- and nothing in the world makes

one way of speaking right and the other wrong.

If this is right, then we, more self-aware than the

islanders, should recognize that there is no substantive fact to

which our decision between Thin Realism and Arealism must answer.

The application of ‘true’ and ‘exists’ to the case of pure

mathematics isn’t forced upon us -- as it would be if Thin

Realism were right and Arealism wrong -- nor is it forbidden --

as it would be if Arealism were right and Thin Realism wrong.

Rather, the two idioms are equally well-supported by precisely

the same objective reality: those facts of mathematical depth.

These facts are what matter, what make pure mathematics the

distinctive discipline that it is, and that discipline is equally

43 In case there’s any lingering doubt, I’m not assuming we have a characterization of ‘science’ or ‘empirical science’; I’m using the term as short-hand for the familiar list of activities we’ve been talking about.

43

well described as the Thin Realist does or as the Arealist does.

Once we see this, we can feel free to employ either mode of

expression, as we chose -- even to move back and forth between

them at will.

The proposal, then, comes to this: Thin Realism and

Arealism are equally accurate, second-philosophical 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. And here, at last, we have a

form of objectivity in mathematics that doesn’t depend on the

existence of mathematical objects or the truth of mathematical

statements, or even on the non-existence of mathematical objects

or the rejection of mathematical claims. This form of

objectivity is, as you might say, post-metaphysical. To return

to the phenomenology from which we began, I suggest that this

account of the objective underpinning of mathematics -- the

phenomenon of mathematical fruitfulness -- is closer to the

actual constraint experienced by mathematicians than any sense of

ontology or any extra-mathematical epistemology; what presents

itself to them is the depth, the importance, the illumination

provided by a given mathematical concept or theorem. A

mathematician may blanch and stammer, unsure of himself, when

confronted with questions of truth and existence, but on

judgments of mathematical importance and depth he brims with

44

conviction. For this reason alone, a philosophical position that

puts this notion center stage might be worthy of our attention.44

Penelope Maddy

44 This paper was delivered as the Sixth Annual Thomas and Yvonne Williams Lecture for the Advancement of Logic and Philosophy at the University of Pennsylvania and as a workshop presentation to the Southern California History and Philosophy of Logic and Mathematics Group. I’m grateful to audiences at both events, and especially to the organizers, Scott Weinstein and Thomas Williams, and Erich Reck, respectively.

45

References

Avigad, Jeremy [2006] ‘Methodology and metaphysics in the development of

Dedekind’s theory of ideals’, in J. Ferreirós and J. Gray [2006], pp. 159-186.

Benacerraf, Paul [1973] ‘Mathematical truth’, reprinted in P. Benacerraf and

H. Putnam, eds., Philosophy of Mathematics, (Cambridge: Cambridge University Press, 1983), pp. 403-420.

Burgess, John, and Rosen, Gideon [1997] A Subject with no Object, (Oxford: Oxford University

Press). Caro, Paul [1992] Water, P. Thickstun, trans., (New York: McGraw Hill,

1993). Dauben, Joseph [1979] Georg Cantor, (Cambridge, MA: Harvard University

Press). Dedekind, Richard [1872] ‘Continuity and irrational numbers’, W. Beeman and W.

Ewald, trans., in Ewald [1996], pp. 766-779. [1888] ‘Was sind und was sollen die Zahlen’, W. Beeman and W.

Ewald, trans., in Ewald [1996], pp. 790-833. Dummett, Michael [1978] Truth and Other Enigmas, (Cambridge, MA: Harvard

University Press). [1981] Frege: Philosophy of Language, second edition,

(Cambridge, MA: Harvard University Press). Ferreirós, José [1999] Labyrinth of Thought: a History of Set Theory and its

Role in Modern Mathematics, (Basel: Birkhäuser).

46

Gödel, Kurt [1944] ‘Russell’s mathematical logic,’ reprinted in his

[1990], pp. 119-141. [1964] ‘What is Cantor’s Continuum Problem?’, reprinted in

his [1990], pp. 254-270. [1990] Collected Papers, volume II, S. Feferman et al, eds.,

(New York: Oxford University Press). Goodman, Nelson, and Quine, W. V. O. [1947] ‘Steps towards a constructive nominalism’, Journal of

Symbolic Logic 12, pp. 105-122. Kanamori, Akihiro [2003] The Higher Infinite, second edition, (Berlin:

Springer). Kant, Immanuel [1781/7] Critique of Pure Reason, P. Guyer and A. Wood, trans.

and eds., (Cambridge: Cambridge University Press, 1997).

Koellner, Peter [2006] ‘On the question of absolute undecidability’,

Philosophia Mathematica 14, pp. 153-188. Kreisel, Georg [1958] Review of Wittgenstein’s Remarks on the Foundations of

Mathematics, British Journal for the Philosophy of Science 9, pp. 135-158.

Linnebo, Øystein [200?] ‘Platonism about mathematics’, to appear in the

Stanford Encyclopedia of Philosophy. Maddy, Penelope [1997] Naturalism in Mathematics, (Oxford: Oxford University

Press). [2007] Second Philosophy, (Oxford: Oxford University Press).

47

[2008] ‘How applied mathematics become pure’, Review of Symbolic Logic 1, pp. 16-41.

[2011] Defending the Axioms, (Oxford: Oxford University

Press). Martin, D. A. [1998] ‘Mathematical evidence’, in H. G. Dales and G.

Oliveri, eds., Truth in Mathematics, (Oxford: Oxford University Press), pp. 215-231.

McLarty, Colin [2006] ‘Emmy Noether’s “set theoretic” topology: from

Dedekind to the rise of functors’, in J. Ferreirós and J. Gray [2006], pp. 187-208.

Moore, Gregory H. [1982] Zermelo’s Axiom of Choice, (New York: Springer). Moschovakis, Yiannis [1980] Descriptive Set Theory, (Amsterdam: North Holland). Putnam, Hilary [1979] Mathematics, Matter and Method: Philosophical Papers,

volume 1, second edition, (Cambridge: Cambridge University Press), pp. 60-78.

Shapiro, Stewart [1997] Philosophy of Mathematics: Structure and Ontology,

(New York: Oxford University Press). [2000] Thinking about Mathematics, (Oxford: Oxford

University Press). [2005] ‘Introduction’ to Shapiro, ed., The Oxford Handbook of

Philosophy of Mathematics and Logic, (Oxford: Oxford University Press).

Steel, John [2000] ‘Mathematics needs new axioms’, Bulletin of Symbolic

Logic 6, pp. 422-433.

48

Stillwell, John [2002] Mathematics and its History, second edition, (New

York: Springer). Wilson, Mark [2006] Wandering Significance, (Oxford: Oxford University

Press). Woodin, Hugh [2001] ‘The continuum hypothesis, Parts I and II’, Notices of

the American Mathematical Society 48, pp. 567-576, 681-690.

Wright, Crispin [1992] Truth and Objectivity, (Cambridge, MA: Harvard

University Press). Wussing, Hans [1969] The Genesis of the Abstract Group Concept, A.

Shenitzer, trans., (Cambridge, MA: MIT Press, 1984). Zermelo, Ernst [1908a] ‘A new proof of the possibility of a well-ordering’,

reprinted in J. van Heijenoort [1967], pp. 183-198. [1908b] ‘Investigations in the foundations of set theory I’,

reprinted in J. van Heijenoort [1967], pp. 200-215.