Módulos de los SBCs

1 ‘‘N O M I N A L I S M’’ ^{A N D} ‘‘R E A L I S M’’¹

Nominalism is a large subject. In our book (Burgess and Rosen 1997) my colleague Gideon Rosen and I distinguished a negative or destructive side of nominalism, which tells us not to believe what mathematics appears to say, from a positive or reconstructive side, which aims to give us something else to believe instead. We noted that there were a few nominalists who contented themselves with the negative side, conceding that mathematics is useful, insisting that what it appears to say is not true, and letting it go at that, without attempting any reconstrual or reconstruction of mathe-matics. We expressed some surprise that there were not more such destruc-tive nominalists, since as compared with reconstrucdestruc-tive nominalism, destructive nominalism has what Russell in another context called ‘‘the advantages of theft over honest toil’’; and if nothing else was clear from the work of Hartry Field, Charles Chihara, Geoffrey Hellman, and other reconstructive nominalists whose work we surveyed, it was clear that the amount of honest toil that would be required for a nominalistic reconstrual or reconstruction of mathematics would be quite considerable.

Today, a couple of years after publication, it is beginning to seem that the main achievement of our book will have been to provide a decent burial for the hard-working, laborious variety of nominalism. For almost every-thing that has come forth since from the nominalist camp has represented a light-fingered, larcenous variety, which helps itself to the utility of math-ematics, while refusing to pay the price either of acknowledging that what mathematics appears to say is true, or of providing any reconstrual or reconstruction that would make it true. The usual label for this variety of

1 I have decided to keep this paper in the form in which it was originally written for oral delivery, adding footnotes to supply citations of the literature, and for clarification at a few points where experience has shown misunderstanding of my intended sense may be likely.

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nominalism is ‘‘[mathematical] fictionalism.’’² Not only has fictionalism become the most widely pursued form of nominalism, but even some former anti-nominalists have been wavering or drifting in its direction, including even the author of one of the most eloquent early criticisms,

‘‘Mathematics and Oliver Twist.’’ I will take the liberty, therefore, of substituting ‘‘fictionalism’’ for the ‘‘nominalism’’ label in the official title of our symposium.

While I am at it, I had better say something about the ‘‘realism’’ as well.

It is an even larger subject, since there is hardly any bit of philosophical terminology more diversely used and overused and misused than the R-word. There seems to be a systematic difference between the way in which the word is understood by many of those who describe themselves as

‘‘realists’’ and the way in which it is understood by most of those who describe themselves as ‘‘anti-realists.’’ For many professed ‘‘realists,’’ realism amounts to little more than a willingness to repeat in one’s philosophical moments what one says in one’s scientific moments, not taking it back, explaining it away, or otherwise apologizing for it: what we say in our scientific moments is all right, though no claim is made that it is uniquely right, or that other intelligent beings who conceptualized the world differ-ently from us would necessarily be getting something wrong. For many professed ‘‘anti-realists,’’ realism seems rather to amount to a claim that what one says to oneself in scientific moments when one tries to under-stand the universe corresponds to Ultimate Metaphysical Reality; that it is, so to speak, a repetition of just what God was saying to Himself when He was creating the universe.

The weaker position might be called anti-anti-realism, and the stronger position capital-R Realism. Quine says somewhere that his ‘‘realism’’ and his

‘‘pragmatism’’ are reconciled by his ‘‘naturalism.’’ This is a hard saying, but I think it can be explained along the following lines. First, ‘‘realism’’ here means anti-anti-realism: the refusal to apologize while doing philosophy for what is said while doing mathematics or science. Second, ‘‘pragmatism’’

2 In earlier publications I called this stance ‘‘instrumentalism,’’ while Rosen called it ‘‘constructive nominalism,’’ by analogy with van Fraassen’s ‘‘constructive empiricism.’’ (Incidentally, though van Fraassen is best known for his fictionalism about unobservable physical entities, he is also a fictionalist about abstract mathematical entities.) However, the analogous positions in other areas of philosophy (e.g. modality) are today generally called ‘‘fictionalist,’’ and on this ground Rosen adopted the

‘‘fictionalist’’ label, and I now follow him. Unfortunately, there has been another use of ‘‘fictionalist’’

in philosophy of mathematics, by Hartry Field and his students, for whom it includes all rejectionist views, all views that hold that standard mathematics is false, whether fictionalist in our sense or reconstructive. This usage seems to be out of alignment with the usage of ‘‘fictionalism’’ in other areas of philosophy, and for that reason to be avoided.

means rejection of capital-R Realism, rejection as unjustified of any claim that mathematics and science give us a God’s-eye view of capital-R Reality.

Third, ‘‘naturalism’’ means adherence to a conception of epistemology as an inquiry conducted by citizens of the scientific community examining science from the inside, rather than an inquisition conducted by philosophers foreign to science judging science from the outside.

And the reconciliation? Naturalism teaches us to look at our scientific, philosophical, and other forms of intellectual endeavor as activities of bio-logical organisms with cognitive capacities that, though extensive, stop well short of omniscience. As such, none of these endeavors can succeed in achieving a God’s-eye view of Reality. And therefore there is no reason to apologize for one of them, science, failing to achieve such a view, and every reason not to suppose that another of them, philosophy, could do better.

Should I, therefore, replace ‘‘realism’’ by ‘‘naturalism’’ in the title of our symposium? No, for unfortunately ‘‘naturalism’’ too has been used and overused and misused in multiple senses. In particular, while in Quine’s sense naturalism abstains from imposing philosophical constraints on science, there is another equally widespread but diametrically opposed conception in which ‘‘naturalism’’ consists precisely in imposing such a philosophical constraint, namely, the constraint that no entities are to be assumed that do not stand in natural cause-and-effect relations with us. So I will just replace ‘‘realism’’ by ‘‘anti-fictionalism,’’ making the title of our symposium come down to this: ‘‘fictionalism in philosophy of mathe-matics: pro and con.’’

2 L I T E R A R Y G E N R E S

To begin with the pro side, it is impossible to quarrel with the proposition that mathematics is in some respects like fiction. For indeed, anything is like anything else, in some respect. I even think the comparison may be illumi-nating, as to the nature of fiction, or rather, as to some of the questions philosophers raise about fiction, and in particular about the status of fictional characters. The hard-headed view is that they just do not exist.

Another view is that they are abstract entities of some sort or other. Yet another view is that they are mental entities. Now the view that mathe-matical entities are mental entities, though a perennial favorite of amateur philosophers, has been in disrepute among professional philosophers since Frege’s trenchant critique of psychologistic philosophies of mathematics more than a hundred years ago. I think that fiction is enough like mathe-matics to suggest that the view that fictional entities are mental entities is

equally dubious. For example, one problem for mentalistic theories of mathematical entities is that there are too many such entities for minds to have created each one; and a little reflection shows that the situation is similar with fiction: a novelist may write of the doings of a vast army with thousands of officers and myriads of soldiers, while in fact describing only a couple of dozen of them individually. But I am straying from our topic, which was supposed to be what, if anything, comparison with fiction can show us about mathematics, and not the reverse.

Reverting to that topic, then, I have said on the one hand that it can hardly be denied that mathematics is like fiction in some respects, most obviously in consisting of large bodies of manuscript and printed and now electronic writing. On the other hand, it must be said that there is a sense in which mathematics is clearly non-fiction: there is a well-established prac-tice of classifying writing as ‘‘fiction’’ and ‘‘non-fiction,’’ and setting aside attempted theoretical definitions and analyses, attending rather to the criteria by which in actual practice such classifications are made, clearly mathematics counts as non-fiction. The compilers of the New York Times best-seller list will never put any mathematical work, however wonderful, at the top of the fiction column, and not just because nothing even by Andrew Wiles will ever sell like Stephen King. Nor will any librarian catalogue, say, the Proceedings of the Cabal Seminar as an ‘‘anthology of short stories based on the characters created by Georg Cantor.’’ Now of course misclassifications are sometimes made, and mischievous hoaxes and pious frauds sometimes succeed, but these represent mistakes in applying general criteria to specific cases. I do not think it even makes sense to suggest that the general criteria are themselves mistaken, and it seems unquestionable that by those criteria mathematical writing is not literally fictional writing: mathematics is not in all respects the same as fiction.

So the question is: in which respects is mathematics like, and in which respects is it unlike, fiction? That in part depends on the species of the genus fiction one considers. The first species one thinks of will probably be the novel, and comparison with novels is in fact common among professed fictionalists and their critics, as with Oliver Twist, mentioned earlier. There has been, however, a minority – including especially the late Leslie Tharp³– who have preferred a comparison with mythology. In a slightly different vein, Steve Yablo, in an interesting paper,⁴has suggested a comparison with metaphor. Metaphor of course is not a genre of fiction but a figure of

3 Rescued from oblivion in Chihara (1989).

4 The reference was to an on-line version, but the paper has since appeared in print as Yablo (2000).

speech, and I think that to speak as Yablo does of a metaphor running on for volumes and volumes and volumes is to stretch the concept of ‘‘meta-phor’’ well beyond breaking point. But perhaps much of the content of Yablo’s suggestion could be preserved if we took the comparison to be between mathematics and parables or fables.

Be that as it may, I believe the comparison with fables is the most apt of the candidates I have considered, and comparison with novels the least so.

Novels almost always are attributable to identifiable individual authors:

Proust or Flaubert, Trollope or Dickens. Some fables are attributable to such authors, Lafontaine for instance; others are traditional. Mathematics also consists of both traditional elements and elements with identifiable authors. Novels are almost always unique. Fables tend to be retold over and over in variant versions by different writers, so that we have Aesop’s version, Lafontaine’s version, and many latter-day retellings of the fox and the crow, for instance. Mathematics likewise gets retold by textbook writer after textbook writer. The characters in one novel seldom reappear in another, and even those who do reappear, like Swann or Palliser, do so only in comparatively few stories, all by the same author. This is so with some characters of fable, but many, like the clever fox, reappear in whole cycles of tales. The same mathematicalia, p and e, the sine and cosine functions, 0 and 1 and 2 and so on, reappear through whole libraries of mathematical works. Again, characters encountered in novels are generally of the same species as those encountered in daily life, while those in fables are, as one dictionary definition reminds us, beings of a different order: ‘‘animals that talk and behave like human beings.’’ Mathematics, too, has objects even more unlike those of any other subject, and it is for precisely that reason that there is thought to be a philosophical problem about them.

Yet more important is the matter of application, which in literature typically takes the form of a ‘‘message.’’ The fable typically though not invariably has a ‘‘moral,’’ while to demand one of the novel is virtually the definition of Philistinism. I am reminded in this connection of what Nabokov says in his posthumously published lectures about Bleak House and its supposed concern with reform of the Court of Chancery.

At first blush it might seem that Bleak House is a satire. Let us see. If a satire is of little aesthetic value, it does not attain its object, however worthy that object may be. On the other hand, if a satire is permeated by artistic genius, then its object is of little importance and vanishes with its times while the dazzling satire remains, for all time, as a work of art. So why speak of satire at all? . . . Such cases as Jarndyce did occur now and then in the middle of the last century although, as legal historians have shown, the bulk of our author’s information on legal matters

goes back to the 1820s and 1830s so that many of his targets had ceased to exist by the time Bleak House was written. But if the target is gone, let us enjoy the carved beauty of the weapon. (Nabokov 1980, p. 64)

The question of applications is crucial in the case of mathematics, because though it would be a kind of Philistinism to demand that every piece of mathematics have one, many do; and it is precisely because many do that many philosophers have opposed nominalism, this being the least com-mon denominator of all ‘‘indispensability arguments.’’

Still more important, however, is a feature common to all genres of fiction. The most important single respect in which fictionalists hold mathematics to be like novels or fables or whatever is in being a body of falsehoods. In particular the existence theorems of mathematics are sup-posed to be untrue: these say there exist, for instance, prime numbers greater than 10¹⁰¹⁰, whereas according to mathematical fictionalists, and indeed all nominalists, there are no such things as numbers at all.⁵

3 _{N O N}-L I T E R A L L A N G U A G E

Nominalism, to repeat, is a large subject. In our book, Rosen and I distinguished two varieties of reconstructive nominalists. Those of one variety, the hermeneutic, insist that their reconstruals of mathematics reveal what, contrary to superficial appearances, deep down mathematical lan-guage has meant all along: mathematical theorems are true, but while what mathematical theorems appear to mean implies the existence of mathe-matical entities, what they really mean does not. This position might be summed up in the formula, ‘‘There are no numbers, and some of them are primes greater than 10¹⁰¹⁰.’’ Reconstructive nominalists of the other variety, the revolutionary, concede that their reconstructions of mathematics are not analyses of current mathematics, but amendments to it; not exegeses, but emendations.⁶

5 I have been concerned here with comparison between mathematics and fiction less for its own sake than for its bearing on the issue of nominalism. The comparison deserves a much fuller examination.

For an extended discussion of analogies and disanalogies, containing extensive references to the further literature, see Thomas (2000, 2002).

6 This is not the place to discuss reviews of Burgess and Rosen (1997) at any length, but it may be mentioned that many adherents of and sympathizers with reconstructive nominalism have wished to claim that there is some third alternative. For instance, it is sometimes said that a nominalist interpretation represents ‘‘the best way to make sense of ’’ what mathematicians say. I see in this formulation not a third alternative, but simply an equivocation, between ‘‘the empirical hypothesis about what mathematicians mean that best agrees with the evidence’’ (hermeneutic) and ‘‘the construction that could be put on mathematicians’ words that would best reconcile them with certain philosophical principles or prejudices’’ (revolutionary).

We did not in the book discuss a division between hermeneuticists and revolutionaries among the fictionalists, but such a distinction can be made.⁷ The hermeneutic fictionalist maintains that the mathematicians’

own understanding of their talk of mathematical entities is that it is a form of fiction, or akin to fiction: mathematics is like novels, fables, and so on in being a body of falsehoods not intended to be taken for true. According to the hermeneutic fictionalist, the anti-nominalist philosopher is being more royalist than the queen (of the sciences); is being a kind of fundamentalist, taking literally what was never so meant.

Consider, for instance, the question when the ‘‘reification’’ of numbers first occurred historically, in the main line of development leading up to modern mathematics. What I take to be a fairly conventional view dates the reification of natural numbers to about the time of Archytas of Tarentum and other Pythagoreans, among whom we begin to see the transition from the use of numerals as adjectives, as in ‘‘six oxen were sacrificed to Zeus,’’ to their use as nouns, as in ‘‘six is a perfect number’’; and it dates the reification of real numbers from about the time of Omar Khayyam and other medieval Islamic and Hindu mathematicians, who in contrast to Greek mathematicians treated ratios of geometric magnitudes as numbers, as things that could be added and multiplied. According to Chihara (1990), however, these dates are all wrong. The reification of number actually occurred not circa 500B C Eor 1000C E, among mathematicians, but some time in the 1960s and 1970s, and among philosophers: it is a rash and recent innovation of Quine and other ‘‘literalist’’ philosophers, myself and my fellow symposiast included.

Chihara’s argument for this claim is essentially that the mathematicians he has questioned have either expressed puzzlement when asked about their ontological commitments, or have repudiated suggestions that they are committed to any ontology at all. Yablo’s argument is on the face of it different, but I believe at bottom similar. He begins by calling attention to a very interesting phenomenon, the ‘‘paradox’’ of his title. Let me attempt my own presentation of it.

According to an old story, when Lindemann settled the ancient problem of squaring the circle, his colleague Kronecker reduced him to tears by

7 The term ‘‘hermeneutic fictionalism’’ was taken as the title for a large-scale study by Jason Stanley (2001), which pursues at length a number of the points to be made below, and many more also, through several different areas of philosophy where fictionalism has become fashionable. In an alternative terminology, hermeneutic reconstructive nominalism and hermeneutic non-reconstructive nominalism are called content hermeneuticism and attitude hermeneuticism respectively. This

7 The term ‘‘hermeneutic fictionalism’’ was taken as the title for a large-scale study by Jason Stanley (2001), which pursues at length a number of the points to be made below, and many more also, through several different areas of philosophy where fictionalism has become fashionable. In an alternative terminology, hermeneutic reconstructive nominalism and hermeneutic non-reconstructive nominalism are called content hermeneuticism and attitude hermeneuticism respectively. This