Mathematics as Gameskeeping


Stephen Yablo, MIT

draft of August 6, 1999

comments very welcome

I. Introduction

Frege in “Notes for L. Darmstaedter” asks: “Is arithmetic a game or a science?” His own view, of course, is that it is a science, albeit a science of a very particular kind. Unlike physics and geology, its results are knowable a priori; unlike them too, it deals with a special sort of logical object. But an unusual science is a science nevertheless.

I say that Frege has asked the right question, but given the wrong answer. The picture of mathematics as a science has led to a stagnant, boring, and generally unsatisfying state of affairs from which the alternative picture holds out some hope of delivering us. Because the unsatisfyingness has essentially to do with the unequal treatment of pure mathematics and mathematics as it comes up in applications, let's start by talking about that.

II. Applicability

Frege is very clear about the reasons for his position. He tells us in the Grundgesetze that “it is applicability alone which elevates arithmetic from a game to the rank of a science” (vol. II, sec. 91). A game-like interpretation would be fine, it seems, except for the fact that arithmetic does so much work for us in our dealings with the material world.

If we stay within [the] boundaries [of formal mathematics], its rules appear as arbitrary as those of chess. [But] applicability cannot be an accident ( Grundgesetze, vol. II, sec. 89).
The suggestion here is that the explanation of why these rules are the ones we use is that these rules are the ones that track mathematical reality. If we take it that the tracking reality is the job-description of science, then here is an argument for the scientific interpretation he favors.

The funny thing is that, Frege notwithstanding, applicability is widely considered to create problems for the view of arithmetic as a science. Arithmetic qua science is a deductively organized description of sui generis objects entirely disconnected from the natural world. (Likewise the theory of real and complex numbers, and analysis, and set theory.) But now we have

THE PROBLEM OF APPLIED MATH: How can a story about objects disconnected from the natural world be so useful in natural science = the theory of the natural world?
This has also been called (by Wigner, a one-time student of Hilbert’s) the problem of "the unreasonable effectiveness of mathematics." By whatever name, it bids fair to be considered the most urgent problem in the philosophy of math. What have philosophers had to say about it?

There are really two issues here. One is, what have philosophers had to say in a constructive vein? What do they say by way of trying to solve the applicability problem? The other is, what do they say they say in a dialectical vein? What are the "morals" and "lessons" they're inclined to draw from the problem?

Now, you might expect that what is said dialectically would be based on what is said in a constructive vein. That is, we're to draw the following lessons from applicability, because they are the ones that emerge from our story about how applications in fact work.

But you'd be wrong. The "moral" that has been drawn is quite strong: applicability is taken to show that mathematics gives a literally true description of a special mathematical division of reality. Whereas when we look at what gets said in a constructive vein, we find that it is extremely weak. All that usually gets said is that standard mathematics had better be true , if its applicability isn't to seem a mystery. Even Frege appears to take a version of this line:

'How does [formal arithmetic] differ from a mere game?' Thomae ...allud[es] to the services it could render to natural science. The reason can only be that numerical signs have reference and chess pieces have not. There is no other ground for attributing a higher value to arithmetic than to chess (section 90) [1]
It is admitted, of course, that truth is only a necessary condition of applicability, not the full explanation. Most true statements, e.g., statements about what I had for lunch yesterday, are of no use whatever to scientists. To suppose that truth alone should make for applicability would be like supposing that random high quality products should improve the operation of random machines. This seems to be what the Dormouse believed in Alice in Wonderland; asked what had possessed him to drip butter into the the Mad Hatter’s watch, he says “but it was the BEST butter.” The best record of what I had for lunch won’t help science any more than the best butter will improve the operation of a watch.

But although no one quite maintains that truth is enough, not a lot has been written about about what more might be required. The assumption seems to be that any additional requirements will be particular to this or that application and of little overall philosophical interest. The most that can be said in general about how mathematics manages to apply is that mathematical terms refer and mathematical reality truly is as mathematics describes it.

III. Purity

A little while ago I said that applicability was the most urgent problem in the philosophy of mathematics. That may have surprised you. You may have had another problem in mind for the role:

THE PROBLEM OF PURE MATHEMATICS. Assuming that mathematics is true, what makes it so? What are the truthmakers for mathematical claims? (And how is it that we manage to be so extremely reliable about when the truthmakers obtain?)
Just about every philosopher of mathematics has started here, leaving applicability to be dealt with at a time to be named later. It's little wonder then that the agenda has been dominated by issues like the following:

is arithmetic true in virtue of

(a) the behavior of particular objects (the numbers), or
(b the behavior of ω-sequences in general, or
(c) the fact that it follows from Peano's axioms?

if (a), are the numbers sets, and if so which ones? how do we know that such things exist? how do we manage to be so reliable about their properties?

if (b), are the ω-sequences actual or can they be merely possible?
if the first, how do we know that such things exist? how do we manage to be so reliable about their properties? if the second, how does the proposal differ fron (c)?

if (c), are the axioms first-order or second-? if first-, what about the incompleteness theorems? if second-, is second-order logic to be seen as basic or as a branch of set theory? if basic, aren't we treating the (considerable) mathematical content latent in second-order logic as holding primitively and groundlessly? [2] if a branch of set theory, then doesn't the trilemma just recur in a new key? That is: is set theory true in virtue of

(a') the behavior of particular objects (the sets),
(b') the behavior of ∈-structures in general, or
(c') the fact that it follows from such and such axioms?
Having circled around these issues for many decades now, I am sure that we are all good and tired of them. But what is the alternative? A number of theories have, for better or worse, managed to bear up tolerably well under critical pressure – not well enough to make anyone happy, perhaps, but well enough to have earned their place in a standard menu of options. One can't just throw these theories in the waste basket and start over! And so the issue of what makes pure mathematics true continues to be what philosophers wrangle over.

IV. Why this is unsatisfying

You might feel edified by the decades of wrangling, or you might not. But one thing seems clear: the continued obsession with issues like the above means that not much time has been left for the problem of applications. One takes the occasional sidelong glance, to be sure. But this is mainly to reassure ourselves that as long as mathematics is true, there is no reason why scientists should not take full advantage of it.

Look at that claim again: as long as mathematics is true, there is no reason why scientists should not take advantage of it. I suppose this means that they should not feel guilty about taking advantage of it, no more than they would feel guilty about taking advantage of any truths. But then it begs a crucial question. Why should scientists want to take advantage of it? What good does it do them? What sort of advantage is there to be taken?

The reason this matters is that, depending on how we answer, the pure problem is greatly transformed. It could be, after all, that the kind of help mathematics gives is a kind it could give even if it were false . If that were so, then PURITY – which in its usual form presupposes that mathematics is true – is going to need a different sort of treatment than we have grown accustomed to.

V. First steps toward freedom

Where are we? I said that philosophy of mathematics has tended to emphasise PURITY over APPLICABILITY. The standard line on applicability has been that as long as mathematics is true, applicability shouldn't be much of a mystery. Also that beyond truth, there isn't a whole lot to say. And so we should feel free to go back to our true love: the pure problem, traditionally, the problem of whence the truth of mathematics derivies.

A notable exception to all these generalizations is the work of Hartry Field. Field begins his book Science without Numbers by noting that

most of the literature on the philosophy of mathematics takes the following three questions as central:

(a) How much of standard mathematics is true?....
(b) What entities do we have to postulate to account for the truth of (this part of) mathematics?
(c) What sort of account can we give of our knowledge of these truths?

A fourth question is also sometimes discussed, though usually quite cursorily:

(d) What sort of account is possible of how mathematics is applied to the physical world?

Now, my view is that question (d) is really the fundamental one. (vii)
Not only does Field see applicability as centrally important, he dissents from both aspects of the "standard line" on it. Where the standard line links the utility of mathematics on its truth, Field holds that mathematics (although certainly useful) is very likely false. Where the standard line offers little other than truth to explain usefulness, Field lays great stress on the fact that mathematical theories are conservative over nominalistic ones, ie., any nominalistic conclusions that can be proved with mathematics can also be proven (albeit often much less easily) without it. The utility of mathematics lies in the no-risk deductive assistance that it provides to the beleaguered theorist .

VI. Half-wayness

I think that this is on the right track. But there is something strangely half-way about it. I do not doubt that Field has shown us a way in which mathematics can be useful without being true. It can be used to facilitate deduction in special nominalistically reformulated theories of his own device: theories that are "qualitative" in nature rather than quantitative.

Interesting as this is, it leaves more or less untouched the issue of how mathematics does manage to be useful without being true. It's not as though the people who use mathematics in their scientific work are practitioners of Field's qualitative science. No, they practice regular Platonic science.

How without being true does mathematics manage to be of so much help to them? Field never quite says. He's quite explicit in fact that the relevance of his argument to actual applications of mathematics is limited and indirect:

[What I have said] is not of course intended to license the use of mathematical existence assertions in axiom systems for the particular sciences: such a use of mathematics remains, for the nominalist, quite illegitimate. (Or, more accurately, a nominalist should streat such a use of mathematics as a temporary expedient that we indulge in when we don’t know how to axiomatize the science properly.) (14)
And now one begins to wonder what Field sees himself as doing. Does he think that the role of mathematics vis-a-vis his the non-nominalistic theories that scientists in fact use is somehow analogous to its role in connection with his custom-built nominalistic theories? So that by explaining and justifying the one he explained and justified the other? If so, it is strange that he does not say more about how the analogy is supposed to work.

Or is his view rather that he has not explained (or justified) actual applications of mathematics -- but that's OK because, come the revolution, these actual applications will be supplanted by the Field-style applications of which he has already treated? This stands our usual approach to recalcitrant phenomena on its head. Our usual approach is to theorize the phenomena that we find, not popularize the phenomena we have a theory of!

VII. Indispensability and applicability

As you may have been beginning to suspect, these complaints have been based on a deliberate misunderstanding of Field's project. It is true that Field wants to know

(d) What sort of account is possible of how mathematics is applied to the physical world?
But (d) can mean either of two different things, depending on whether one is motivated by applicability, or indispensability.[3]

Applicability is, in the first instance, a problem: the problem of explaining the unreasonable effectiveness of mathematics. It is also, potentially, an argument for mathematical objects. For the best explanation may require that mathematics be true.

Indispensability is, in the first instance, an argument for the existence of mathematical objects. The argument is usually credited to Quine and Putnam. They say that since numbers are indispensable to science, and we are committed to science, we are committed to numbers. But, just as applicability was first a problem (for nominalism), second an argument (against nominalism), indispensability is first an argument against nominalism, second a problem. How do you nominalists propose to deal with the fact that numbers have a permanent position in the range of our quantifiers?

Once this distinction is drawn, it is clear that Field's concern is more with rebutting indispensability than applicability. His question is

(d1) How can applications be conceived so that they come out dispensable?
To this, Field's two-part package of (a) nominalistically reformulated scientific theories, and (b) conservation claims, seems a perfectly appropriate answer. But we are still entitled to wonder what Field would say about

(d2) How are actual applications to be understood, be they indispensable or not?
If there is a complaint to be made about Field, it is not that he has attempted and failed to answer (d2), but that he doesn't properly address (d2) at all, and nor do the resources he provides appear to be of much use with it.

Now, Field might reply that the indispensability argument is the important one. But that will be hard to argue. One reason, already mentioned, is that a serious mystery remains even if indispensability is established. How is the Fieldian nominalist to explain the usefulness-without-truth of mathematics in ordinary, quantitative, science?

Second and more important, though, suppose that such an explanation can be given. Then indispensability becomes a red herring . Why should we be asked to demathematicize science, if ordinary science's mathematical aspects can be understood on some other basis than that they are true?

Putting both of these pieces together: the point of nominalizing science is not achieved unless a further condition is met, given which condition the nominalization becomes unneeded. Since the nominalization would be superseded by a supplement that is required anyway, the nominalization serves no very important purpose.

VIII. Resting too much on conservativeness

That is my first worry about Field's approach. The second worry is related. One fault we found with the "standard line" on applicability is that it rested too much on the mere fact of a theory's truth. There are all kinds of (equally) true mathematical theories; why do some of them, e.g., ordinary arithmetic, come up again and again in science, whereas others, e.g., modular arithmetic, come up very infrequently? [4]

One can complain with equal justice that Field rests too much on the mere fact of a theory's conservativeness. If N is a theory of concrete objects, then just about any non-concrete theory M is going to be conservative over N. (This is a gross exaggeration, but not in a way that matters to the present argument.) Why then do the same few mathematical theories do the lion's share of the work? Field notices this but doesn't attempt to explain it:

Even standard axioms of number theory can be modified without endangering [conservativeness]; similarly for standard axioms of analysis. What makes the mathematical theories we accept better than these alternatives to them is not that they are true and the modifications of them are not true, but rather that they are more useful: they are more of an aid to us in drawing consequences from those nominalistic theories that we are interested in. (15)
This in a way just sharpens the question. Why is arithmetic so much more useful in drawing consequences than modular arithmetic? Also, how can the fact that number theory is more of an aid in drawing nominalistic consequences from Field-theories, which no one uses, help to explain the importance of number theory in connection with the theories people do use?

IX. Non-deductive usefulness

The third worry is this. Consider the kind of usefulness-without-truth that Field lays so much weight on; mathematics thanks to its conservativeness gives no-risk deductive assistance. It’s far from clear why this particular form of usefulness-without-truth deserves its special status. Does Field think that deductive assistance is the only kind of theoretical help that objects can give without having to go the trouble of existing? He gives no argument for this claim that I know of, but consider the following quote:

if our interest is only with inferences among claims that don't say anything about numbers (but which may employ, say, numerical quantifiers), then we can employ numerical theory without harm, for we will get no conclusions with numerical theory that wouldn't be valid without it. Indeed, numerical theory is not only harmless for the purpose of drawing nominalistic consequences, it is also rather useful for this purpose ... There are other purposes for which this justification for feigning acceptance of numerical theory does not apply, and we must decide whether or not to genuinely accept the theory. For instance, there may be observations that we want to formulate that we don't see how to formulate without reference to numbers, or there may be explanations that we want to state that we can't see how to state without reference to numbers... if such circumstances do arise, then we will have to genuinely accept numerical theory if we are not to reduce our ability to formulate our observations or our explanations ("Platonism for Cheap?" 161-2, italics added).
But, why will we have to accept numerical theory in these circumstances? Having just maintained that the deductive usefulness of Xs is not a reason to accept that Xs exist, he seems now to be saying that representational usefulness is another matter. One might wonder whether there is much of a difference here. I am not denying that deductive usefulness is an important non-evidential reason for making as if to believe in numbers. But it's hard to see why representational usefulness isn't similarly situated.

Representational usefulness will be the focus in what follows. But I don't want to give the impression that the possibilities end there. Another way that numbers appear to "help" is by redistributing theoretical content in a way that streamlines theory revision.

Suppose for example that I am working in a first-order language speaking of material objects only. And suppose that my theory says that there are between two and three quarks in each Z-particle:

(a) ( ∀z)[(∃q1)(∃q2)(q1 ≠ q 2 & q i∈z & ( ∀r1) ( ∀r2)((r1 ≠ r 2 & r j∈z) → (r 1 = q 1 etc.))].
Then I discover that my theory is wrong: the number of quarks in a Z-particle is between two and four. Substantial revisions are now required in my sentence. I will need to write in a new quantifier ' ∀r3'; two new non-identities 'r 1 ≠ r 3' and 'r 2 ≠ r 3'; and two new identities 'r 3 = q 1' and 'r 1 = q 2.' Compare this with the revisions that would have been required had quantification over numbers been allowed – had my initial statement been

(a') ( ∀z)(∀n)(n = #q (q∈z) → 2 < n < 3).
It would have been enough just to strike out the '3' and write in a '4.' Someone might say that the revisions would have been just as easy had we helped ourselves to numerical quantifiers ( ∃nx) defined in the usual recursive way:

(∃0x)Fx =df ( ∀x)¬Fx

(∃n+1x)Fx =df ( ∃y)Fy &( ∃nx)(Fx & x ≠ y).
But this just postpones the inevitable. Our theory might tell us that the stars have on average 2.5 planets. If we know the stars to number less than a million – without some upper bound there's no finite representation at all -- we can write this out as a huge disjunction:

(b) [( ∃2x)Sx&(∃5y)Py]v[(∃4x)Sx &( ∃10y)Py]v... [( ∃1000000x)Sx & ( ∃25000000y)Py]
Then we learn that it's actually 2. 4 planets that stars have on average. About 300,000 of (b)'s 500,000 disjuncts will now have to be scratched out, and all or almost all of the remaining numerical subscripts will need rewriting. Had we started out with

(b') #y (y is a planet) ÷ #x (x is a star) = 2.5
instead of (b) the job would obviously have been much easier. Again, someone might say that we get the same advantages as (b') by using numerical adjectives: there are 2.5-times-as-many planets as stars. The problem rearises though when we switch to cases of functional dependence. Suppose the ratio of planets to stars is thought to be 2.5 times the log to the base 10 of the # of planets). Well, maybe substitutional quantification would work here. But....And etc. But enough about that. let's leave other theoretical benefits to the side for now and go back to representational utility.

X. Numbers as representational aids

What is it that allows us to take our uses of numbers for deductive purposes so lightly? The deductive advantages that "real" Xs do, or would, confer are (Field tells us) equally conferred by Xs that are just "supposed" to exist. But the same would appear to apply to the representational advantages conferred by Xs; these advantages don't appear to depend on the Xs really existing either. The cosmologist need not believe in "the average star" to derive representational advantage from it, as e.g. in "the average star has 2.4 planets." Does the physicist need to believe in the real existence of numbers to derive expressive advantages from them, that is, to find it expressively helpful to couch her theory in numerical terms?

Suppose, for example, that our physicist has it in mind to record an infinite bunch of facts of the form

(i) a projectile fired at X meters per second from the surface of a planetary sphere Y kilograms in mass and Z meters in diameter will not escape its gravitational field
(ii) a projectile fired at X' meters per second from the surface of a planetary sphere Y' kilograms in mass and Z' meters in diameter will escape its gravitational field
Why not just say that

(iii) the escape velocity from a planet of mass M and diameter 2R is the square root of 2GM/R, where G is the gravitational constant?
Why not indeed? To express the infinitely many facts in finite compass, one brings in numbers as representational aids. One does this despite the fact that what we're trying to get across doesn't have anything to do with numbers, and could be expressed without them were it not for the finitude requirement.

The question is whether functioning in this way as a representational aid is a privilege reserved to existing things. The answer appears to be that it isn't. That (iii) succeeds in gathering together into a single content infinitely many facts of form (i) and (ii) owes nothing whatever to the real existence of numbers. All that's required is that we understand what (iii) asks of the non-numerical world , the numerical world taken momentarily for granted. How the real existence of numbers could help or hinder that understanding is difficult to imagine.

An oddity of the situation is that Field makes the same sort of point himself in his writings on truth. He thinks that "true" is a device that exists "to serve a certain logical need" -- a need that would also be served by infinite conjunction and disjunction if we had them, but (given that we don't) would go unfilled were it not for "true." No need then to take the truth-predicate ontologically seriously; its place in the language is secured by a role it can fill quite regardless of whether it picks out a property. It would seem natural for Field to consider whether the same applies to mathematical objects. Why couldn't it be that, just as truth is an essential aid in the expression of facts not about truth (there is no such property), numbers are an essential aid in the expression of facts not about numbers (there are no such things)? I am not saying that Field would go along with this; everything suggests he wouldn't. It's because I can't see why not to go along with it that I find the position strangely half-hearted.

XI. Premeditated objects

I don't want to overreach here. It is not as though whenever a theorist uses Xs for representational purposes, their existence or not is a matter of indifference. A classic if hackneyed example is the standard meter bar in Paris. Suppose a 19th century physicist says

(*) the wavelength of delta-rays is 2 times the length of the standard meter bar.
What role is the standard meter bar playing in this statement? We appeal to it for the help it gives in classifying objects as to length. Our topic is not how long the meter bar is but how large the wavelength. A reference to the meter bar is unavoidable, though, if we want the wavelength pinned down precisely.

Now in this case, it is certainly not a matter of indifference whether the meter bar exists as opposed to merely being supposed to exist. One reason has to do with descriptive precision. A merely supposed meter bar can be supposed to be any number of lengths, or rather, there is no very precise length such that the bar is supposed to be exactly that long . But then (*) construed in terms of a supposed meter bar fails to specify the wavelength of delta-rays very precisely.

Another reason why a merely supposed meter bar would not do is this. The supposed length of such a bar is liable to fluctuate over time. That is, what strikes us as the same length as our supposed meter bar today may strike us as longer or shorter than it tomorrow. The length of the actual meter bar is by comparison precise and stable -- which is good if the point of appealing to it is to point out a precise and stable property of delta-rays.

It is because the bar has features not predictable from our concept of it -- which features are needed for the application at hand -- that a merely supposed meter bar would not have been enough in this case. Of course, other cases can be imagined. Had our native grip on the notion of a meter been better, so that the meter bar, rather than providing a check on our visual judgments, was the party in check ("whoops, hold everything, looks to me like it's expanded to 1001 millimeters"), then a real meter bar would not have been needed. But as matters actually stand, or stood, the physical bar plays an indispensable role.

Compare in this respect the role that the number 2 plays in statement (*). Can we say in this case that the number 2 needs to exist to settle various questions not available from "what we have in mind" by 2? Our concept does indeed leave various things open: whether 2 contains 1, for instance; whether 2 numbers the bicycles in m basement; maybe even whether every number divisible by 2 is the sum of 2 primes (Goldbach).

But none of these are features that matter to 2's role in (*). As far as that statement is concerned, the features that matter are available already from my concept.. And so whether 2 exists or not is immaterial. Because its representational contribution trades only on aspects made just as determinate by our concept of 2 as they would be by the object itself, we can afford to be agnostic or even skeptical about it in reading "the wavelength is 2 times the length of the standard meter bar."

Where does this leave us? It's beginning to seem as though a distinction needs to be drawn between two sorts of (putative) object. Call an object or type of object X premeditated relative to a statement S if our conception of Xs is determinate enough that one of the following two statements is definitely correct and the other is definitely incorrect:


(i) assuming that the Xs as we conceive them exist, it is the case that S.
(ii) assuming that the Xs as we conceive them exist, it is not the case that S.

Numbers are premeditated relative to our claim S above, because, assuming numbers, there is no question but that the wavelength is 2 times the length of the standard meter bar. After all, it flows from our concept of number that if numbers exist , then A is 2 times as long as B iff A is twice as long as B. The standard meter bar is not premeditated, however; because, helping ourselves to no more about that bar than what is provided by our conception of it, there is just no telling how many times it would fit into the wavelength of delta-rays.

Pulling these various threads together, it appears then that to the extent that Xs are premeditated, they can play their representational role equally well whether they exist or not. Numbers are premeditated, so their existence or not is irrelevant to the issue of how (*) should be evaluated. The standard meter bar is not premeditated, so it needs to really exist if (*) wants to have a determinate truth-value. [5] The point remains, of course, that theoretical indispensability and/or utility is not itself an argument for existence. Our question should not be: how can these things be so useful if they don't really exist? But rather: how do these things make themselves useful and is that a way that calls on them to exist?

XII. The three worries revisited

The scientific utility of at least certain objects – the premeditated ones – can be explained in a way that remains neutral on the objects' existence. Numbers, sets, and the like, are premeditated, so their scientific applicability does not argue for their reality. Add to this that their applicability is the main thing that does argue for their reality, and it seems we ought to conclude that there is no reason to believe that these objects exist.

If they don't exist, though, then what is to be said about pure mathematics? That'll be our topic starting in the next section. First let's revisit our three earlier worries about Field.

Worry three was: Field recognizes only a single form of usefulness-without-truth, viz. no-cost deductive assistance. Our answer to worry three is that mathematics is useful not only for the deductive assistance it lends us but also the representational assistance.

Worry one was: Field doesn't give us an explanation of the usefulness-without-truth of mathematics in ordinary non-Fieldian science. Our answer to worry one is that mathematics provides representational assistance even in the context of ordinary, non-Fieldian science. [6] (E.g., we say how a physical system evolves over time by giving the equations of motion for a point in a made-up mathematical structure called phase space.)

Worry two was that the feature of mathematical theories that Field cites as explaining their applicability – their conservativeness w.r.t. nominalistic theories – does nothing to privilege "standard mathematics" over any number of other conservative-over-science theories that scientists rarely if ever appeal to. Why arithmetic, the theory of real numbers, analysis, and so on? One would like to be able to say that these theories are (is that for good and specific reasons) particularly well-suited to representing facts of the sort that need representing. [7] Different subject matters lend themselves to different sorts of fictinal representation. Much as the subject matter of genealogy invites representation in terms of trees, the subject matter of science invites representation in terms of numbers. More on this later.

XIII. Our opposite fix

To say it one more time, the standard procedure in philosophy of mathematics is to start with the pure problem and leave applicability for later. No surprise then that most current theories have a lot to say about what makes mathematics true, much less to offer about what makes it so useful in science.

The approach suggested here here looks to be in an opposite fix. Our theory of applications is not too shabby. But what are we going to say about pure mathematics? If the line on applications is right, then it becomes likely that arithmetic, set theory, and so on are largely untrue. At the very least then the problem of purity is going to have be reconceived. It can't be: in virtue of what is arithmetic true? It'll have to be: how is the line drawn between "acceptable" arithmetical claims and "unacceptable" ones? And it is very unclear what acceptability could amount to if it floats completely free of truth.

Just maybe there is a clue in the line on applications. Suppose that mathematical objects "start life" as representational aids. Some systems of mathematicalia will work better in this capacity than others; e.g., standard arithmetic will work better than a modular arithmetic in which all the operations are "modulo N." As wisdom accumulates about the kind(s) of mathematical system needed, theorists develop an intuitive sense of what is the right way to go and what the wrong way.

Norms are developed that soon take on a life of their own, guiding the development of mathematical theories past the point where natural science greatly cares. The process begins to feed on itself, as descriptive needs arise w.r.t.., not the natural world, but the of our system of representational aids itself. These needs encourage the construction of still further theory, maybe with further ontology, and so it goes.

You can see where this is headed. If the pressures our descriptive task exerts on us are sufficiently coherent and sharply enough felt, we begin to feel under the same sort of external constraint that is encountered in science itself. Our theory is certainly answerable to something, and what more natural candidate than the objects of which it purports to give a literally true account? Thus arises the feeling of the objectivity of mathematics qua description of mathematical objects.

XIV. Two ways of making-as-if

I can make these ideas a little bit more precise by bringing in some ideas of Kendall Walton's about making-as-if. The thread that links as-if games together is that they call upon their participants to pretend or imagine that certain things are the case. These to-be-imagined items make up the game's content, and to elaborate and adapt oneself to this content is typically the game's very point. [8] At least one of the things we are about in a game of mud pies, for instance, is to work out who has what sorts of pies, how much longer they need to be baked, etc. At least one of the things we're about in a discussion of Sherlock Holmes is to work out, say, how exactly Holmes picked up Moriarty's trail near Reichenbach falls, how we are to think of Watson as having acquired his war wound, and so on.

As I say, to elaborate and adapt oneself to the game's content is typically the game's very point. An alternative point suggests itself, though, when we reflect that all but the most boring games are played with props, whose game-independent properties help to determine what it is that players are supposed to imagine. That Sam's pie is too big for the oven doesn't follow from the rules of mud pies alone; you have to throw in the fact that Sam's clump of mud fails to fit into the hollow stump. If readers of "The Final Problem" are to think of Holmes as living nearer to Hyde Park than Central Park, the facts of nineteenth century geography deserve a large part of the credit.

Now, a game whose content reflects the game-independent properties of worldly props can be seen in two different lights. What ordinarily happens is that we take an interest in the props because and to the extent that they influence the content; one tramps around London in search of 221B Baker street for the light it may shed on what is true according to the Holmes stories.

But in principle it could be the other way around : we could be interested in a game's content because and to the extent that it yielded information about the props. This would not stop us from playing the game, necessarily, but it would tend to confer a different significance on our moves. Pretending within the game to assert that BLAH would be a way of giving voice to a fact holding outside the game: the fact that the props are in such and such a condition, viz., the condition that makes BLAH a proper thing to pretend to assert.

Using games to talk about game-independent reality makes a certain in principle sense, then. Is such a thing ever actually done? A case can be made that it is done all the time -- not indeed with explicit self-identified games like "mud pies" but impromptu everyday games hardly rising to the level of consciousness. Some examples of Walton's suggest how this could be so:

Where in Italy is the town of Crotone? I ask. You explain that it is on the arch of the Italian boot. 'See that thundercloud over there -- the big, angry face near the horizon,' you say; 'it is headed this way.'...We speak of the saddle of a mountain and the shoulder of a highway....All of these cases are linked to make-believe. We think of Italy and the thundercloud as something like pictures. Italy (or a map of Italy) depicts a boot. The cloud is a prop which makes it fictional that there is an angry face...The saddle of a mountain is, fictionally, a horse's saddle. But our interest, in these instances, is not in the make-believe itself, and it is not for the sake of games of make-believe that we regard these things as props...[The make-believe] is useful for articulating, remembering, and communicating facts about the props -- about the geography of Italy, or the identity of the storm cloud...or mountain topography. It is by thinking of Italy or the thundercloud...as potential if not actual props that I understand where Crotone is, which cloud is the one being talked about. [9]
A certain kind of make-believe game, Walton says, can be "useful for articulating, remembering, and communicating facts" about aspects of the game-independent world. He might have added that make-believe games can make it easier to reason about such facts, to systematize them, to visualize them, to spot connections with other facts, and to evaluate potential lines of research. That similar virtues have been claimed for metaphors is no accident, if metaphors are themselves moves in world-oriented pretend games:

The metaphorical statement (in its context) implies or suggests or introduces or calls to mind a (possible) game of make-believe...In saying what she does, the speaker describes things that are or would be props in the implied game. [To the extent that paraphrase is possible] the paraphrase will specify features of the props by virtue of which it would be fictional in the implied game that the speaker speaks truly, if her utterance is an act of verbal participation in it. [10]
A metaphor on this view is an utterance that represents its objects as being like so : the way that they need to be to make the utterance "correct" in a game that it itself suggests. The game is played not for its own sake but to make clear which game-independent properties are being attributed. They are the ones that do or would confer legitimacy upon the utterance construed as a move in the game.

XV. The kinds of making-as-if and the kinds of mathematics

Seen in the light of Walton's theory, our suggestion above can be put like this: numbers as they figure in applied mathematics are creatures of existential metaphor. They're part of a realm that we play along with because the pretense affords a desirable – sometimes irreplaceable – mode of access to certain real-world conditions, viz. the conditions that make a pretense like that appropriate in the relevant game. Much as we make as if, e.g., people have associated with them stores of something called "luck," so as to be able to describe some of them metaphorically as individuals whose luck is "running out," we make as if pluralities have associated with them things called "numbers," so as to be able to express an (otherwise hard to express because) infinitely disjunctive fact about relative cardinalities like so: the number of Fs is divisible by the number of Gs.

Now, if applied mathematics is to be seen as prop-oriented making-as-if, then the thought that immediately suggests itself is that pure mathematics should be conceived as content-oriented making-as-if.

Why not? It seems a truism that pure mathematicians spend most of their time trying to work out what is true according to this or that mathematical theory. [11] All that needs to be added to the truism, to arrive at the conception of pure mathematics as content-oriented make believe, is this: that the mathematician's interest in working out what is true-according-to-the-theory is by and large independent of whether the theory is really true – true not in the sense of being true-according-to-a-theory, but of fidelity to a realm of mathematical objects. [12]

XVI. "Objectivity" in pure mathematics

This brings us to the question that fictionalist accounts of mathematics have traditionally found most difficult. If the whole thing is just a game, why is it conducted in such a serious spirit -- a spirit more characteristic of a search after truth than the spinning out of a fairy tale? Why does mathematical theory-construction have the feeling of being answerable to objective external contraints?

A partial answer is that mathematics is no ordinary game. It's a game that was invented with prop-oriented applications in mind, and that, although it has taken on a life of its own, still derives a good deal of its authority from those applications. Our feeling of a right and a wrong way of going in pure mathematics is due to the fact that some ways of going really do yield a product better suited to the prop-oriented applications that started the ball rolling, and that keep it rolling today. Pure mathematics is an endeavor in which we trace out the contours of existing schemes of metaphorical representation and search for the best ways of elaborating those schemes.

A fuller answer would distinguish two contexts in which we debate mathematical truth. Some of the time we are doing "normal" mathematics; we are wondering what is the case according to this or that mathematical theory. But for all I have said, there is a perfectly objective fact of the matter about what is true according to Peano Arithmetic, or Zermelo-Fraenkel set theory. So no immediate problem of objectivity arises here. Where a problem does seem to arise is in the context of theory- development. Why do some ways of constructing mathematical theories, and extending existing ones, strike us as better than others? I have no really satisfying answer to this, but let me indicate where an answer might be sought. [13]

XVII. Metaphorical goodness

A distinction is sometimes drawn between true metaphors and metaphors that are apt. That these are two independent species of metaphorical goodness can be seen by looking at cases where they come apart.

An excellent source for the first quality (truth) without the second (aptness) is back issues of Reader's Digest magazine. There one finds jarring if not necessarily inaccurate titles along the lines of "Tooth Decay: America's Silent Dental Killer," "The Sino-Soviet Conflict: A Fight in the Family," and, my personal favorite, "South America: Sleeping Giant on Our Doorstep." [14] Another good source is political metaphor. When Calvin Coolidge said that "The future lies ahead," the problem was not that he was wrong – where else would it lie? -- but that he didn't seem to be mobilizing the available metaphorical resources to maximal advantage. Likewise when George Bush told us before the 1992 elections that "It's no exaggeration to say that the undecideds could go one way or another."

Of course, a likelier problem with political metaphor is the reverse: aptness without truth. The following are either patently (metaphorically) untrue or can be imagined untrue at no cost to their aptness. Stalin: "One death is a tragedy. A million deaths is a statistic." Churchill: "Man will occasionally stumble over truth, but most times he will pick himself up and carry on." Will Rogers: "Diplomacy is the art of saying "Nice doggie" until you can find a rock." Richard Nixon: "America is a pitiful helpless giant."

Not the best examples, I fear. But let's move on to the question they were meant to raise. How does metaphorical aptness differ from metaphorical truth? David Hills maintains that where truth is a semantic feature, aptness an aesthetic one: "When I call Romeo's utterance apt, I mean that it possesses some degree of poetic power... Aptness is a specialized kind of beauty attaching to interpreted forms of words...For a form of words to be apt is for it...to be the proper object of a certain kind of felt satisfaction on the part of the audience to which it is addressed" (119-120).

An immediate difficulty with this is that "apt" can be used in connection not with particular metaphorical claims but entire metaphorical frameworks, or types of them. It might be said, for example, that rising pressure is a good metaphor for intense emotion; that possible worlds provide a good metaphor for modality; or that war makes a good (or bad) metaphor for argument. What is meant by this sort of claim? Not that pressure (worlds, war) are metaphorically true of emotion (modality, argument). There's no question of truth because no metaphorical claims have been made. But it would be equally silly to speak here of poetic power or beauty. The suggestion seems rather to be that a make-believe game built around pressure (worlds, war) lends itself to the metaphorical expression of truths about emotion (possibility, argument). The game "lends itself" in the sense of affording access to lots of those truths, or to particularly important ones, and/or in the sense of presenting those truths in a cognitively or motivationally advantageous light.

Aptness then is at least a feature of prop-oriented make-believe games; a game is apt relative to such and such a subject matter to the extent that it lends itself to the expression of truths about that subject matter. A particular metaphorical utterance is apt to the extent that (a) it is a move in an apt game, and (b) it makes impressive use of the resources that game provides. The reason it is so easy to have aptness without truth is that to make satisfying use of a game with lots of expressive potential is one thing, to make veridical use of a game with arbitrary expressive potential is another.

Back now to the main issue: what accounts for the feeling of a right and a wrong way of proceeding when it comes to mathematical theory-development? I want to say that a proposed new axiom A strikes us as correct roughly to the extent that a theory incorporating A seems to us to make for an apter game – a game that lent itself to the expression of more metaphorical truths -- than a theory that omitted A, or incoporated its negation.

Take for instance the controversy over the axiom of choice. One of the many considerations arguing against acceptance of the axiom is that it requires us to suppose that the spheres of pure geometry decompose into parts that are reassemblable into enormously larger spheres. (The Banach-Tarski theorem.) Physical spheres are not like that, so we imagine, hence the axiom of choice makes geometrical space an imperfect metaphor for physical space.

One of the many considerations arguing in favor of the axiom is that it blocks the possibility of sets X and Y neither of which is injectable into the other. This is crucial if injectability and uninjectability are to serve as metaphors for relative size. It is crucial that the statement about functions that "encodes" the fact that there are not as many Ys as Xs should be seen in the game to entail the statement "encoding" the fact there are at least as many Xs as Ys. This entailment would not go through if sets were not assumed to satisfy the axiom of choice. [15]


Add to this that choice also mitigates the paradoxicality of the Banach-Tarski result, by opening our eyes to the possibility of regions too inconceivably complicated to be assigned a "size," and it is no surprise that choice is judged to make for an apter overall game.

Suppose we are working with a theory T and are trying to decide whether to extend it to T* = T + A. An impression I don't want to leave is that T*'s "aptness" is simply a matter of its expressive potential with regard to our original naturalistic subject matter: the world we "really believe in," which contains only concrete things. T* may also be valued for the expressive assistance it provides in connection with the mathematical subject matter postulated by T – a subject matter which we "take" to obtain in our role as players of the T-game. A "new" set-theoretic axiom may be valued for the light it sheds not on concreta but on mathematical objects already in play. So it is, e.g., with the axiom of projective determinacy and the sets of reals studied in descriptive set theory.

XVIII. Our Goodmanian ancestors

If mathematics is a myth, how did the myth arise? You got me. What we can do is construct a myth about how it might have arisen. The idea in other words is to tell a just-so story such that

(i) the story treats mathematical objects as pulled long ago out of thin air
(ii) everything is as if this really happened, and we had just forgotten.
My strategy here is borrowed from Wilfrid Sellars in "Empiricism and the Philosophy of Mind." Sellars asks us to

Imagine a stage in pre-history in which humans are limited to what I shall call a Rylean language, a language of which the fundamental descriptive vocabulary speaks of public properties of public objects located in Space and Enduring through time. ...What resources would have to be added to the Rylean language of these talking animals in order that they might come to recognize each other and themselves as animals that think, observe, and have feelings and sensations? And, how could the addition of these resources be construed as reasonable? (Sections 48-9)
I want us to go back to the same stage of pre-history; but since we are interested in the absence of abstracta (not private episodes) from the language, let us think of it not as a Rylean language but a Goodmanian one.

First Day , finite numbers . The original Goodmanian language has finite numerical adjectives --- three apples, five chairs -- but nothing else of a numerical nature. If we like we can think of the language as first-order logic with numerical quantifiers defined in the usual way:

∃0x Fx = df ∀x (Fx → x ≠ x)
∃n+1x Fx = df ∃y (Fy & ∃nx (Fx & x ≠ y))
Our ancestors can easily say things like "there are exactly two cats" and "there are exactly three hundred apples," and with the help of some simple truth-functional combinations, that "there are at least (at most) fifty houses" too.

Everything goes swimmingly until one day the Goodmanians realize that there are things they have no way of saying. An example is "there are exactly as many Fs as Gs." If the language had infinite disjunction, they could use

(∃0x Fx & ∃0x Gx) v (∃1x Fx & ∃1x Gx) v etc.
But they don't, and so more radical measures are required. They convene a conference and decide to institute a make believe game in which it's supposed that, in addition to the regular, "concrete," objects, there are new objects called "numbers." The point of the numbers is to serve as measures of cardinality; there is not supposed to be much more to the intrinsic nature of, say, 5, than that is suited to measure the cardinality of the Fs just in case there are five Fs. Their game is governed by the following simple rule:

(R1) when ∃kx Fx , say that there's a "number" k = (#x)Fx
With the help of this rule, that there are as many Fs as Gs can be put by "saying" that (#x)Fx = (#x)Gx. Of course they don't really believe in numbers but it's clear enough what the point of such an utterance would be; to call attention to the real-world condition that makes (#x)Fx = (#x)Gx sayable in the game. Other previously inexpressible contents now come within reach as well. That there are finitely many of cats can be put by saying that that some number k is the number of cats.

Second Day, addition. Our ancestors notice that whenever there are five Fs and seven Gs (and no Fs are Gs), there are twelve things that are either F or G. How to express this in complete generality, that is, without any use of schematic predicate letters? They call another conference and decide to add a second rule to the game

(R2) pretend that there's an "addition" operation on the numbers, such that #(F) + #(G) = #(F-or-G) (assuming that no Fs are Gs).
Using this made-up operation of addition, their observation can be put as follows: 7 + 5 = 12. Other previously inexpressible contents fall to the same method. No matter what the Fs and Gs (etc.) may be, if F ≈ F', and G ≈ G' , then F-or-G ≈ G'-or-F'. [16] (They use "≈" for "exactly as many as.") This can now by put by "saying" that ∀m ∀n (m+n = n+m). Another example: if F ≈ F', G ≈ G', and H ≈ H', then ((F-or-G)-or-H) ≈ (F'-or-(G'-or-H')). Addition lets them say it like so: ∀mnp ((m+n)+p) = (m+(n+p)).



Third Day , multiplication. Our ancestors notice that no matter what the Fs and Gs may be, the pluralities obtained by making #F copies of the Gs, on the one hand, and #Gs copies of the Fs, on the other, are equal in number. How to express this in complete generality, that is, without the resort to schematic letters or infinite conjunction? They call another conference and institute a third rule as follows:

(R3) pretend that there's a "multiplication" operation on the "numbers," such that #(F) × #(G) = #(K), the Ks being the result of making #(F) copies of the Gs.
Now they can access the desired content metaphorically; all they have to "say" is that ∀m∀n (m × n = n × m). Other previously inexpressible contents come within reach as well. For instance, what does ∀m∀n∀p ((m + n) × p = (m × p) + (n × p)) tell us?

Fourth Day , pure arithmetic . Our ancestors realize that their pretend game with numbers can be broken into two parts. One part set out autonomously, that is, without the distracting references to particular pluralities like the Fs and the Gs. This is pure arithmetic , axiomatized by Peano's Axioms (A1)-(A5). Another part consists of bridge principles linking the entities of pure arithmetic with particular pluralities. These include

(B1) F ≈ G iff #(F) = #(G)
(B2) F « G iff #(F) < #(G)
Fifth Day , numbers of numbers. Our ancestors note that among the things they can assign numbers to are pluralities of numbers, e.g., the # of numbers <10 is 10. This raises a problem, however. They may if they're lucky have numbers for all the pluralities they started with; that depends on whether there are infinitely many concrete things. But they certainly don't have numbers for all the pluralities they pretend are there, for they don't have a number for is the (finite) numbers themselves. They decide to enquire into the possibility of extending the game so that it becomes pretense-worthy that there is a number of all (finite) numbers whatsoever.

Sixth Day , functions. Our ancestors realize that if they are going to pretend there are numbers beyond the finite, they will need a way of disciplining talk about these numbers -- compare the way (R1) disciplined talk of finite numbers. They decide to pretend that there are, in addition to the natural numbers, ways of pairing things off, that is, one-one maps. This is done in two steps

(R4) for each x and y, pretend there's an ordered pair <x,y>, where the identity of a pair is fixed by its members and their order.

(R5) whenever you've "got" some ordered pairs, pretend they form a collection, where the identity of the collection is given by its members.
Functions, injections, and bijections – one-one maps -- can now be defined in the usual way. Three new bridge principles are introduced:

(B3) F≈G iff the Fs can be mapped 1-1 onto the Gs.
(B4) F «G iff the Fs can be mapped 1-1 into the Gs.
They notice that (B3) and (B4) cover some of the ground already covered by (B1) and (B2). But they satisfy themselves that no conflicts will arise: when the Fs and Gs are finite, the RHS of (B1), resp. (B2) is equivalent to that of (B3), resp. (B4). Note that (B3) and (B4) are meant to apply also when there are infinitely Fs and/or Gs. Rather than the numbers coming first, and being used to define cardinality relations (as in the finite case), cardinality relations will now be coming first, and numbers will be defined in terms of them.

Seventh Day , infinite cardinals : Our ancestors notice various things about « and ≈. They notice, for example, that if F « G « H, then F « H, and that if F ≈ G, then G ≈ F as well. How to express these facts in complete generality, that is, non-schematically? They make it part of the game that even when there are infinitely many Fs, there is a "number" #(F) that measures the Fs cardinality-wise. This is accomplished via the rule

(R5) pretend that there are finite and infinite numbers, all of them subject to the condition that #(F) <#(G) iff F « G, and #(F)=#(G) iff F ≈ G.
It is unclear as yet how many of these infinite numbers there are. The only one they can be sure of is ω = the # of finite numbers. They wonder, for instance, how many one-one maps there are from the finite numbers onto themselves – ω or perhaps more?

Eighth Day , rational numbers . Our ancestors now start thinking about numerical adjectives as they are used in the characterization not of cardinality but of length. Suppose that a rope R is of length k (k a finite number); that the n ropes R 1....Rn laid end to end are exactly as long as R; and the R is are the same lengths as each other. How are we to express the lengths L(R i) of the shorter ropes in terms of the length L(R) of the longer? Evidently L(R) will have to be n multiplied by L(Ri).But that equation is unsolvable for L(R i) unless we are lucky and k is divisible by n! They call another conference and make it part of the game that

(R6) there are fractions k/n, subject to the condition that the length of R' is k/n iff k ropes the length of R, laid end to end, are exactly the same length as n ropes the length of R'
These fractions do not allow our ancestors to express anything new but they are an essential step on the way to the introduction of the reals, as we see below. An autonomous axiomatic theory of the rationals is devised, by analogy with the development of Peano's axioms on Day Four.

Ninth Day , algebraic numbers : Our ancestors realize that the rationals don't provide a comprehensive enough system to represent how long one line is as compared to another. By Pythagoras, the hypotenuse of a right-angled triangle whose other sides are of length 1 is the square root of 2. And it can be proved that square root of 2 isn't rational. So the algebraic numbers are introduced, via a rule like

(R7) for every polynomial P(x) with integral coefficients and every a, b such that P( a) < 0 < P( b), say that there is a number c between a and b with P( c) = 0.
These "solutions" to P(x) = 0 are the algebraics. Our ancestors think of the positive square root of 2, for instance, as the "solution" to Q(x) = x 2 – 2 = 0 that lies between the positive rationals whose squares are less than 2 and the ones whose squares exceed 2.

Tenth Day , real numbers . Our ancestors realize that even the algebraics don't provide a comprehensive enough system of measures; the circumference of a circle, for example, is π times as long as its diameter, and π is not the root of any integral polynomial. So they bring in the reals:

(R8) whenever you've got a finite number k and some rationals each less than k, say that there is a number r at least as big as any of them, such that no rational q smaller than r is at least as big as any of them.
The reals are to be thought of as the "completion" of the rationals under least upper bound. This finally gives us all the length-measures that they want. (Infinitesimals are not much on their minds.) Functions on the reals are introduced to represent space and time, and dynamic processes taking place therein. Analysis is worked up to theorize these functions.



Eleventh Day , collections. Our ancestors notice that there are infinitely many Fs iff the Fs are equinumerous with the FGs, where not every G is an F. How to express this in complete generality, ie., without the predicates? They decide to generalize (R5) to

(R9) whenever you've got some things, pretend that they form a collection, subject to the usual identity-conditions.
This of course will need qualification down the road! Now they can put their point by saying that a collection C is infinite iff there's a one-one function from C to one of its proper subcollections.

Twelfth Day, sets vs. classes. Some bright ancestor realizes that contradictions ensue if we try to apply (R9) in full generality: some of the resulting collections are "too big." To restore order, the ancestor proposes that collections are limited to one of the following two cases. On the one hand we have those obtainable by starting with collections of "ordinary things," and then repeatedly "taking power sets" – repeatedly adding in the the collection of subcollections of things already on board. These will be called the sets. Instead of (R9) we're to use:

(R9') whenever you've got some sets, pretend that they form a class.

Every class is a set but not vice versa: for instance there's a universal class that contains all the sets there are, but that class is never obtained by the process of repeatedly taking power sets. Classes that aren't sets are proper classes, and this is the one remaining kind of collection.

For the same sort of reason – applied in full generality it leads to contradiction -- (R5) needs modification too. Say that the Ks are "bounded" iff there are Ls such that the Ks are injectable into the Ls but not conversely. Then we have

(R5') if but only if the Fs and Gs are bounded, pretend they have numbers #(F) and #(G), subject to the conditions that #(F) <#(G) iff F «G, and #(F)=#(G) iff F≈G.
(R5') gives us a nicely numerical way of conceptualizing the relation between the sets and the proper classes. Recall that the Fs form a finite class iff there is a natural number n such that n = #(F). What we can say now is that the Fs form a set-sized class – a set – iff there is a number N of any sort – finite or infinite -- such that N = #(F). Another way to put it is that a class is a set iff it is not as big as the entire universe.

Thirteenth Day, autonomy. Our ancestors realize that the issue of the Fs' number does not really depend on "F"'s beinga predicate we understand, or even on the Fs forming a "nice" plurality. This together with our earlier linkage of infinity and sets leads us to think of the the sets and classes as in some sense "complete." Just as the rationals (representable as m/n) are only some of the reals (which outrun our representations), the "nice" classes (definable by "nice" predicates) are only some of the classes. A theory of classes as (in the game) autonomous objects rather than predicate-images is needed. This is invented by Zermelo and refined by others, especially von Neumann, to become our present-day axiomatic theories of sets and classes, in particular ZFC and NBG.

Fourteenth Day, plenitude. Our ancestors discover the Lowenheim-Skolem theorem and come to realize that ZFC and NBG, being first-order theories, cannot enforce their desire that the universe of sets should be inclusive as it can possibly be. They remember that Zermelo originally conceived set theory in second order terms. They begin tinkering around and realize that second-order replacement gets you maximal width (Zermelo 1930), while an axiom to the effect that the non-sets form a set gets you maximal height (McGee 1998).

Much, much later, forgetting. These mathematical metaphors prove so extraordinarily useful that they are employed on a more or less full time basis. As generation follows upon generation, the knowledge of how the mathematical enterprise had been launched begins to die out and is eventually lost altogether. People begin thinking of mathematical objects as genuinely there. Some, ironically enough, take the indispensability of these objects to science as a proof that they are there -- ironically, since it was that that very indispensability that led to their being concocted in the first place. [17]

XIX. Case Study

I'd like now to explore how the "gameskeeping" conception of mathematics might be applied to an episode, or trend, in the history of mathematics. Penelope Maddy has noted something paradoxical in Quine's picture of how mathematics works. Quine sees mathematics as continuous with "total science" both in its subject matter and in its methods. This somehow leads to conclusions dramatically at odds with those reached by working mathematical scientists. Quine's idea in a nutshell is that in set theory, as elsewhere in science, we should keep our ontology as small as practically possible. He is prepared to

recognize indenumerable infinites only because they are forced on me by the simplest known systematizations of more welcome matters. Magnitudes in excess of such demands, e.g., beth-omega or inacessible numbers, I look upon only as mathematical recreation and without ontological rights. Sets that are compatible with [Godel's axiom of constructibility V= L] afford a convenient cut-off... [18].
Quine even proposes that we opt for the "minimal natural model" of ZFC, a model in which all sets are constructible and the tower of sets is chopped off at the earliest possible point. Such an approach is "valued as inactivat[ing] the more gratuitous flights of higher set theory..."

Valued by whom? Actual set-theorists, one would assume. But no. To them, cardinals the size of beth-omega are not even slightly controversial. They are guaranteedby an axiom introduced already in the 1920s (Replacement) and accepted by everyone. Inaccessibles are considered innocent too, except by the lonely few who suspect that ZF may be inconsistent. (Its consistency is deducible in ZFC from the existence of an inaccessible.) As for Godel's axiom of constructibility, it has been widely criticized – including by Godel himself – as entirely too restrictive. Here is Moschovakis, quoted by Maddy:

The key argument against accepting V=L...is that the axiom of constructibility appears to restrict unduly the notion of an arbitrary set of integers (1980, 610).
Set-theorists have wanted to avoid axioms that would "count sets out" just on grounds of arbitrariness. They have wanted, in fact, to run as far as possible in the other direction, seeking as fully packed a set-theoretic universe as the iterative conception of a set permits. All this is reviewed in fascinating detail in Maddy's book; see in particular her discussion of the rise and fall of Definabilism, first in analysis and then in the theory of sets.

If Quine's picture of set theory as something like abstract physics cannot make sense of the field's plenitudinarian tendencies, can any other picture do better? Well, clearly one is not going to be worried about multiplying entities if the entities are not assumed to really exist. But we can say more. The likeliest approach if the set-theoretic universe is make-believe would be (A) to articulate the clearest intuitive conception possible, and then, (B) subject to that constraint, let all heck break loose.

Regarding (A), some sort of constraint is needed or the clarity of our intuitive vision will suffer. This is the justification usually offered for the axiom of foundation, which serves no real mathematical purpose – there is not a single theorem of mainstream mathematics that relies on it -- but just forces sets into the familiar and comprehensible tower structure. Without foundation there would be no possibility of "taking in" the universe of sets in one intellectual glance.

Regarding (B), it helps to remember that sets "originally" came in to improve our descriptions of non-sets. E.g., there are infinitely many Zs iff the set of Zs has a proper subset Y that maps onto it one-one, and uncountably many Zs iff it has an infinite proper subset Y that cannot be mapped onto it one-one. Since these notions of infinitely and uncountably many are topic neutral -- the Zs do not have to meet a "niceness" condition for it to make sense to ask how many of them there – it would be counterproductive to have "niceness" constraints on when the Zs are going to count as bundleable together into a set. [19] It would be still more counterproductive to impose "niceness" constraints on the 1-1 functions; when it comes to infinitude, one way of pairing the Zs off 1-1 with just some of the Zs seems as good as another.

So: if we think of sets as "originally" being brought in to help us deal more effectively with non-mathematical objects, a restriction to "nice" sets would have been unmotivated and counterproductive. It would not be surprising, though, if the anything-goes attitude at work in these original applications were to reverberate upward to contexts where the topic is sets themselves. Just as we don't want to tie our hands unnecessarily in applying set-theoretic methods to the matter of whether there are uncountably many space-time points, we don't want to tie our hands either in considering whether there are infinitely many natural numbers, or uncountably many sets of such numbers.

A case can thus be made for (imagining there to be) a plenitude of sets of numbers; and a "full" power set gathering all these sets together; and a plenitude of 1-1 functions from the power set to its proper subsets to ensure that if the power set isn't countable, there will be a function on hand to witness the fact. This I hope gives some of the flavor of why the preference for a "full" universe is not terribly surprising on a gameskeeping conception of the theory of sets.

APPENDIX on Hilbert and Field

It is widely accepted that mathematical instrumentalism — versions of which have been advanced by David Hilbert and Hartry Field -- runs into problems with Godel's Incompleteness Theorems. Let's review how this is supposed to work.

An instrumentalist is someone who distinguishes two sorts of sentence: the "contentual" and the "ideal." Contentual statements we accept just to the extent that they seem (likely to be) true. Ideal ones are "accepted" not because they're true, but for the help they give in reaching true contentual results from contentual premises. For instance, they help us to "see" which contentual results follow from our contentual premises; they help us to prove those results in fewer steps; they allow us to prove lots of different results using similar methods. The ideal sentences are valued, in short, for reasons of convenience.

Now, if all the ideal sentences are doing is speeding up proofs and the like, it will be hard to defend our reliance on them unless a case can be made they're not going to lead us astray. If we can't feel as confident about ideally proven results as about those with contentual proofs, then reliance on the ideal sentences would seem irresponsible. Engineers may be willing to give up quality control for speed (etc.), but philosophers, and philosophically-minded scientists and mathematicians, are not.

Hilbert had an answer to this. He thought he could prove in a purely contentual way that the use of ideal sentences never leads us astray. Since the contentual lines up for him with the finite, we can put the goal like this: he thought he could prove using finitary mathematics (FM, often identified with PRA = primitive recursive arithmetic) that infinitary mathematics (IM) is real-sound: any finitary results provable in IM are in fact true.

This proposal was, the story goes, scotched by the Second Incompleteness Theorem. FM is being asked to prove that 'if finitary sentence S is IM-provable, then S.' But we know from the Second Incompleteness Theorem (or rather, the closely related Lob's Theorem) that FM can't even prove all instances of the weaker claim that 'if S is FM-provable, then S.'

One might wonder, though, whether a finitary proof of real-soundness should really have been the goal. Hilbert took the reliability of FM for granted; he had no ambition of proving real-soundness for it. But then why should we attribute to him the more ambitious goal of proving real-soundness for IM? His concern should really have been to show that in FM that IM is no more unreliable than FM is. One way to show it would be to establish that any FM-sentence provable in IM is provable already in FM -- that is, that IM is conservative over FM.

What Godel's Second Theorem certainly does do is undermine an attractive strategy for establishing conservativeness. The strategy is to prove in FM that IM is consistent. If we assume with Hilbert that FM is complete with respect to finitary statements -- it either proves or refutes every one -- then from the consistency of IM it follows that IM is conservative (with respect to finitary statements) over FM. The argument is simple: for any finitary statement S that IM proves, FM proves either S or its negation. If it were the negation then IM (which we assume to contain FM) would be inconsistent. So it must be that FM already proves S.

The problem with this otherwise attractive strategy is that, by Godel, FM can't even prove its own consistency; so it certainly can't prove the consistency of the more powerful IM.

The possibility remains though of proving conservativeness by some less demanding method. After all, one might say, for FM to "know" that IM is conservative over it, there is no need for FM to "know" that IM is consistent or even that it itself is consistent. There is no need for FM to know that it is consistent because conservativeness could still hold even if it were not consistent; indeed it would hold, because if FM proves everything then it certainly proves all finitary sentences provable in IM. (See Simpson, "Partial Realizations of Hilbert's Program," for a discussion of what might still be possible in the aftermath of Godel.)

Field is concerned to address the reliability issue as well. He starts by offering a platonistic (and so ideal) proof of the conservativeness of set theory (ST) over nominalized science (NS): any nominalistic result entailed by NS+ST is entailed already by NS. Critics pointed out that if "entails" is understood proof-theoretically, then unless NS is first-order — and as Field initially formulates it, it is not -- the result is not even true. The added set theory enables us to tease out semantic consequences of NS that NS is not in a position to prove.

Another question Field faced was this: how can we trust your platonistic proof before we get the assurances of reliability (of platonistic methods) that the proof itself was supposed to provide?

These complaints led Field to offer a nominalistic proof of conservativeness in which entailment is understood not semantically or proof-theoretically but as a modal primitive. But the proof only works for first-order languages, and there's a question of whether Field's nominalized science can be formulated in a wholly first-order way.

It might be replied that conservativeness becomes less important when the underlying logic lacks a complete proof procedure, as it will on the non-first-order proposals under discussion. Why should it be a problem if adding ST enables us to extract genuine but previously undeducible consequences of NS?

This is quite right as far as it goes, but it begs the question of whether ST doesn't perhaps enable more than this, viz. the deduction of falsehoods. It's one thing to say that failures of conservativeness don't automatically make for failures of soundness when you lack a complete proof procedure. It's another to explain what, in the absence of conservativeness, is supposed to convince us that soundness is indeed preserved when ST is added on.

Now, the above is predicated on the idea, shared by Hilbert and Field, that all the ideal sentences are there for is convenience, most notably the shortening of proofs. How are things changed if we say (as in the main body of this paper) that the appeal to ideal elements is representationally essential?

The first point is that while calculational convenience may not be worth enough to justify risk-taking on the soundness front, added risks might be justified if there were more at stake, such as the possibility of representing important facts about the material world. The pressure to establish a conservativeness result is greatly reduced if we move beyond "calculational" instrumentalism to "expressive" instrumentalism.

The second point is that the idea of conservativeness gets less of a grip if ideal elements due to their representational role are ubiquitous. Physics as actually practiced doesn't contain a whole lot of math-free sentences, so it's hard to feel inspired by the goal of preserving deductive relations among them.

The third point is that the expressive advantages that ideal elements afford — e.g. the ability to represent the notion of "finitely many" -- are just the kind to kill any possibility of a complete proof procedure. Ironically then, the objects whose use was supposed to be legitimated by a conservativeness result have the effect of making conservativeness less desirable, since in the absence of a complete proof procedure you can have real-soundness without it.

The fourth point is that for us, "ideal vs. contentual" is a misnomer, since math-infused sentences have concrete truth-conditions just as much as nominalistic sentences, provided they're understood metaphorically. That ideal objects buy us access to new concrete truth-conditions was indeed the reason for bringing them on board. (Note that the metaphorical truth-conditions of a pure-mathematical sentence, e.g., "there are infinitely many primes," will be be necessarily satisfied or necessarily frustrated. Such a sentence doesn't talk about concrete objects at all, so its acceptability in the game does not depend on how concrete objects may or may not behave. This may help to explain the feeling that mathematical claims are either necessary or impossible.)

Fifth, given that ideal claims have (metaphorical) truth-conditions, the idea of a truth-preserving inference makes sense not just for "contentual" sentences but all sentences. An inference is truth-preserving iff when you assign the sentences involved the appropriate sorts of content —- metaphorical in the case of ideal sentences, metaphorical = literal in the case of contentual —- the conclusion can't be false unless some of the premises are false.

The sixth point is more or less immediate given the fifth: our real concern should be not in making sure that the deductions math enables between contentual, that is, nominalistic, sentences, are truth preserving, but in making sure that inferences as among arbitrary sentences are truth-preserving. The question is: if I am playing the math game as a way of gaining greater descriptive purchase on the material world, and I draw logical consequences from sentences that are true in the game — and so really true as far as their metaphorical contents are concerned — will these consequences will also be true in the game? Is truth-in-the-game closed under logical consequence?

Now, it has to be conceded that there are (or can be) games where truth-in-the-game isn't closed under logical consequence. Consider the game in which we make as if moods are containers which people occupy. "He's in a good mood," "I'm coming out my mood," and so on. Suppose a player of the game were to reason that since "X is in an FG" logically entails "X is an a G," from "he is in a good mood" one ought to be able to infer "he's in a mood."

This inference is truth-preserving as far as literal content is concerned; how can you be in a good mood without being in some mood or other? But that is not to say that it preserves truth on the intended figural reading So understood, the inference takes us from (what may well be) the truth that he is feeling upbeat to (what may well be) the untruth that he is, well, feeling moody. It follows that that there can be no general guarantee that reasoning, in a game, in accordance with logic will never lead you astray — never lead you from a true (metaphorical) content to a false one

The question for us is whether this can occur in the particular game or games that we play with mathematical objects.

I see no good way of ruling the possibility out; I can't even rule it out that our mathematical theories are inconsistent. This would bother me a great deal if the game were one that we played for reasons of convenience. It bothers me less if it's a game we have got to play on pain of losing our representational grip on reality.





[1] "....if [arithmetic statements] were viewed as having a sense, the rules could not be arbitrarily stipulated; they would hafe tobe so chosen that from formulas expressing true propositions could be derived only formulas likewise expressing true propositions...Such matters, however, lie entirely oputside formal arithmetic and only arise when applications are to be made" (section 91).
[2] If we were willing to be this obscurantist, we should have just have shrugged off the original trilemma...
[3] I am indebted here to unpublished work of Ana Carolina Sartorio.
[4] This worry becomes especially acute if we are plenitudinous platonists, as to escape epistemological worries we might have to be.
[5] Have some notion of representational conservativeness; the same concrete configurations are ruled in and out?
[6] Actually it proves some deductive assistance in regular science too. Explain. But not enough.
[7] Nature is arithmetical even if arithmetic isn't true.
[8] Better, such and such is part of the game's content if "it is to be imagined .... should the question arise , it being understood that often the question shouldn't arise" (Walton 1990, 40). Subject to the usual qualifications, the ideas about make-believe and metaphor in the next few paragraphs are all due to Walton (1990, 1993).
[9] Walton 1993, 40-1.
[10] Ibid., 46. I should say that Walton does not take himself to be offering a general theory of metaphor.
[11] The theory might be a collection of axioms; it might be that plus some informal depiction of the kind of object the axioms attempt to characterize; or it might be an informal depiction pure and simple.
[12] As opposed to true-according-to-such-and-such-a-background-theory.
[13] Talk about the unclarity among mathematicians about whether a proposed new axiom is "true" or "fitting." This is what you'd expect on the story to come (?).
[14] "Laughing all the Way to the Bank" (about worshipping by the river Ganges).
[15] Thanks here to Hartry Field.
[16] Assume disjointness throughout.
[17] Even after all this time, there lingered on a few features of the practice that hinted at the mathematical objects' true provenance. Impatience. Indeterminacy, insubstantiality, expressiveness, silliness, unavailability, etc. ("A Paradox of Existence.") A different sort of evidence is given in the following....
[18] Quine 1986
[19] Except to the extent that such constraints are needed to maintain consistency.