Logic put to good use

It is very good to see wild fantasies about superintelligent machines, consciousness and what not supported by bandying about random Gödelian formalities. Very impressive stuff indeed.

Fictionalism and existence

The Stanford Encyplopedia of Philosophy article on Fictionalism in the Philosophy of Mathematics gives the following definition of fictionalism

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 between

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.)

It 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.)

Dummett on strict finitism

Recently, I've been going through (some of) the writings of Michael Dummett. Dummett is indubitably a master of philosophical argumentation, but often one feels he's "insufficiently profound", and his masterful weaving of different threads of reasoning amounts merely to the philosophical equivalent of "logic chopping". Nevertheless, there is much food for thought in his writings, and, of the modern philosophers of mathematics who might actually taken to have a project -- and not only primarily a technical project -- in the old grandiose sense of the term, of basing all of philosophy on the theory of meaning, and related considerations, he's possibly the most worthy of serious consideration. I'll later have something to say about all that, but in this post I wish to address just a single point.

In the 1970 paper Wang's Paradox Dummett takes on the task of defending intuitionistic anti-realism of his kind from the counter-argument that meaning-theoretic considerations (of the sort he adduces) should lead one to adopt rather the stance of "strict finitism". In contrast to Dummett's anti-realistic intuitionism, in strict finitism we demand that the meaning of mathematical language be explained in terms of our actual abilities. Dummett for example happily accepts that (n)(n is a prime \/ n is not a prime), since we may, "if we so choose", for any given n go through every m < sqrt(n) and check whether it divides n. We are, in Dummettian terminology, in possession of an effective method of deciding whether a given natural is a prime or not. Of course, this is a theoretical ability -- in reality, if given a sufficiently large natural I'm not at all capable of deciding its primality. Thus we may say that Dummettian anti-realism is a theoretical sort of anti-realism, explaining the meaning of mathematical language not in terms of our actual practice and abilities but rather in terms of a theoretical and idealised account, a conceptual picture, of that practice and those abilities. The question, then, is how, on the Dummettian account of meaning, we can learn to understand statements the meaning of which hinges on such a theoretical account? Shouldn't we rather take seriously the idea that meaning is grounded in our practice and abilities, investigating the real limitations of these abilities?

In Wang's paradox Dummett writes

But, even if no one were disposed to accept arguments in favour of the strict finitist position, it would remain of greatest interest , not least for the question whether constructivism, as traditionally understood, is tenable position. It can so only if, despite surface similarity, there is a disanalogy between the arguments which the strict finitist uses against the constructivist and those which constructivist uses against the platonist. If strict finitism were to prove internally coherent, then either such disanalogy exists or the argument for traditional constructivism is unsound, even in absence of any parallel incoherence in the constructivist position.

Dummett then goes on to consider what is involved in the strict finitist account of the naturals -- this is by far the most interesting part of the paper -- and finally concludes that strict finitism is incoherent, on basis of an argument ("Wang's paradox") to the effect that reasoning about vague predicates in general is literally inconsistent or incoherent.

Let's set aside the question of whether Dummett's argument is convincing (it is not). The curious thing is that after presenting this conclusion Dummett seems content. However, the argument against Dummettian anti-realist intuitionism is not addressed at all! That is, it might well be that strict finitism is incoherent. It doesn't at all follow that Dummettian anti-realist intuitinism is coherent -- for the counter-argument, arguing that strict finitism rather than anti-realist intuitionism in fact follows on meaning theoretic grounds might well be valid even if strict finitism is incoherent.

Now, Dummett does not formulate the argument against anti-realist intuitionism quite as I did in the above. Rather, as he sees things, it is the analogue of his argument against "the Platonist". (At that time Dummett apparently took it for granted that the only reason someone would hold that the law of excluded middle is true for mathematical statements is belief in "Platonism", existence of mathematical objects in some sense analogous to existence of physical objects.) Still, it remains baffling that while he notes that in order for anti-realist intuitionism to be coherent there must be some disanalogy he does not in fact locate any such disanalogy, seemingly happy to merely establish the incoherence of strict finitism to his satisfaction. It is of course possible that the purpose of the paper was merely to report some philosophical arguments against vague predicates, some seemingly unpalatable observations about phenomenal properties and observational preciates, and so on. Still, one would have expected at least some comment on the seemingly pressing question of the validity of the meaning-theoretic argument for anti-realist intuitionism...