Fictionalism, on the other hand, is the view that (a) our mathematical sentences and theories do purport to be about abstract mathematical objects, as platonism suggests, but (b) there are no such things as abstract objects, and so (c) our mathematical theories are not true.There's much going for fictionalism. For example, it seems it makes no difference, as far as mathematical arguments are concerned, whether or not mathematical objects "really" exist or not. That is, in mathematics we never reason "Since mathematical objects are just figments of our imagination, it follows that..." or "Since mathematical objects exist in some independent Platonic realm, we have that...". As far mathematics is concerned, it seems there is no observable difference between, say, naturals really existing and our pretending they exist. Of course, mathematics is not a hermetic realm. We make observations such as
(*) Even if Goldbach's conjecture is true, it might be that there is no proof of Goldbach's conjecture using mathematical principles we accept, or could come to accept.all the time. Similarly, we draw conclusions about the behaviour of computer programs from mathematical statements. And so on. In (*) "Goldbach's conjecture is true" -- the use of "true" is inessential, since we may just replace the phrase with the statement of the conjecture -- is used factually, so to speak. Obviously, on any usual understanding, the observation does not mean
(*') Even if it is inherent in the stories we tell about naturals that Goldbach's conjecture is true, it might be that there is no proof of Goldbach's conjecture using mathematical principles we accept, or could come to accept.(Indeed, it is not even clear whether (*') makes any sense at all.) This is, essentially, "the problem of objectivity in mathematics". As Kreisel famously noted -- though where he noted this is obscure, as is the case with many an observation usually attributed to Kreisel; apparently it's from a review of Wittgenstein's notes on philosophy of mathematics -- in the question of objectivity in mathematics the real issue is the objectivity of mathematical statements, not existence of mathematicsl objects. But let's set this question aside for a moment.
Fictionalism is so named because in many forms it derives its appeal from an analogy with fiction. That is, we may make sense of the claim that Captain Planet is formed by combining the powers of the rings of the Planeteers, and agree that he is not instead formed by combining the powers of Harry Potter and Tristram Shandy, while also agreeing that, in reality, Captain Planet does not exist at all. Similarly, we may be tempted to say that mathematical objects (and possibly other abstracta as well) don't really exist. This is problematic, however. In case of Captain Planet what we deny when we deny that there is any such entity is entirely clear, just as it is clear what we deny in denying that Sherlock Holmes, Harry Potter, and so on, really exist. There simply are, in the real world, no such people. What of mathematical objects? In denying that they exist, just what it is we are denying? The suggestion that mathematical objects might exist but it simply happens they don't is on the face of it rather baffling. We are owed some account of what their existing, if they did exist, would amount to. (And, in absence of such an account, the sometimes suggested solution to the problem of objectivity, that mathematical statements are objective in the sense that it is determined as a matter of fact whether they would be true if mathematical objects existed and were such that the stories we tell about them would be true, is meaningless as well.)
A natural answer suggests itself: mathematical objects (and possibly other purely abstract things) are a degenerate case of fiction. That is, for such objects, there is simply no existence they could have in addition to being parts of our mathematical and abstract stories. If we take this view we naturally reject the claim (b) in the definition of fictionalism quoted above. That is, rather than denying that mathematical objects exist, we instead gleefully accept that they exist -- after all, presumably we accept many mathematical statements that trivially imply the existence of mathematical objects, e.g. "There is a prime smaller than 10" -- but that this existence is not at all analogous to the existence of physical objects in any sense. (We might here recall Kreisel's caution against thinking of large infinite sets as akin to humongous physical objects.) We may of course also accept usual statements expressing the Platonist position as metaphorical or poetic expressions of our natural attitude towards mathematical statements, something like Hardy's tautology "317 is prime because it is".
So far so good, but we're still left with the problem of objectivity. If mathematical objects exist only as parts of our mathematical stories, indeed can only exist as parts of such stories, how do we account for our natural inclination to regard truth of such as statements as "n is prime" as a factual matter. The answer, that truth of such statements amounts to nothing more than their being in some sense inherent in the stories is obviously unpalatable. It is after not true even in the stories themselves, and in asking whether Goldbach's conjecture holds, or whether there is a measurable cardinal we are not, on the face of it, asking anything about any stories at all, but rather about naturals and sets.
I don't have any good answer to the problem of objectivity, but will instead offer some general reflections, in my characteristically frustrating style. Kreisel in his hilariously rambling Second Thoughts Around Some of Gödel's Writings writes:
Subjectively speaking, the most striking general object lesson I believe I have learnt from logic is really little more than a confirmation of common sense, the distinction betweenIt is not at all given that either the observation that mathematical stories seem to be a degenerate case of fiction, or that our natural attitude is to take at least some mathematical statements as factual (and this observation applies just as well to intuitionism), are of the latter sort rather than the former, that is, that they are more than interesting observations, insightful remarks about mathematics. We must milk them in some interesting way, in various practical contexts, before it becomes apparent whether they have any theoretical import in the philosophy of mathematics (understood as the philosophical study of actual mathematics, and the concepts implicit in actual mathematics, instead of study of classical ontological, epistemological, metaphysical questions when formulated about mathematical entities and mathematical statements). It is sterile to put forth philosophical theories and systems, arguments against and for such systems, counter-arguments to such arguments, enumerating all logically possible combinations of these or those stances, and so on, at this point. (It is not sterile in the sense of producing an endless stream of disseratations, papers, talks, and such like, of course.)
bright ideas and germs of theories;
in the sense that the former function best as remarks (constatations in French, Konstatierungen in German), and simply do not lend themselves to much theoretical elaboration, while the latter do. (NB. The latter need not be more useful than the former, by any realistic measure of usefulness.)