[FOM] Fictionalism About Mathematics

Alan Weir Alan.Weir at glasgow.ac.uk
Sun Mar 11 11:37:37 EDT 2012

Richard Heck writes: 'many ... would agree with Lewis's famously funny rebuttal of fictionalism and its  kin in *Parts of Classes*.'

Funny but I think also a little sad that such a great philosopher should run down his profession in this way. I worry it might encourage the attitude that if you can't do mathematics you teach it, and if you can't even teach it you become a philosopher of maths. There is sometimes discernible an attitude that mathematics is a proper intellectual discipline, philosophy, including philosophy of  maths, a much shabbier enterprise populated by folk who couldn't make the cut in maths and had to settle for something less; maybe even some philosophers of maths share this and are afflicted by such low self-esteem. Not Lewis for sure, nor Richard nor other top rate philosophers of maths sympathetic to his line but it would be a worry if that attitude percolated through to the younger generation.

Not much evidence of that as far as I can see right enough, but to turn to the substantive issue, I'd contrast intuitionism with fictionalism here. Take not Dummett, whose position was nuanced and a little ambiguous perhaps, but someone who takes from Dummettian critiques the view that most of the mathematics taught in maths departments (and mathematical physics departments) is intellectually incoherent. Therefore, she boldly concludes, it should be abandoned and energy devoted to trying to explain how it managed to be instrumentalistically successful. I don't think we should rebut such a position by simply saying 1 million or whatever mathematicians can't be wrong. True, there's only so much we can read, but if someone one knew independently was a good philosopher said this, one should look at the arguments.

What one can say in this case, though, in contrast with fictionalism, is that the consensus in pure maths is relevant. Even if I can't spot a flaw in my intuitionist's arguments, they won't be conclusive and it is reasonable to say to her-  you have to be really, really sure of your case, and also know the discipline really well, before taking the incredibly bold step of writing off such an apparently successful discipline. (And since  the intuitionist arguments, both Brouwer inspired and Dummett inspired, are in my view pretty hopeless, I'd have no problem saying that.)

But fictionalism is different, more irenic, at any rate as I (as a non-fictionalist) understand it. The fictionalist will usually say that it is correct to affirm the existence of twin primes, incorrect to deny that. So it is not obvious that the fictionalist clashes with the practice of the jobbing mathematician (and that's how I think lots of fictionalists want it.)

Of course that raises lots of philosophical questions: it's correct but untrue? It's true in some 'light sense' but not in any weightier one? If correctness = 'true in the story' as in Harry's gloss, that can't mean, 'has been published and will never be challenged' for we have good reason to believe false 'theorems' will have been published and remain undetected, and anyway most mathematical theses will be such that neither they nor their negations will ever be concretely inscribed. Does it mean 'is provable from axioms which constitute 'the story''? Then we seem to be moving towards formalism.

But these are philosophical questions to be resolved by philosophers, I can't see that the views of pure mathematicians will be relevant here to a view like this, as opposed to intuitionism. Applications are different, witness the debates about whether Hartry Field's programme, even if it worked for Newtonian gravitation, could be applied to quantum mechanics. It might then be that a fictionalist would be well-advised to defer to a mathematical physicist (or philosopher of physics with a deep insider knowledge of some particular area) who says to her, your programme might work in these simpler cases, but not in this research specialism of mine.



Professor Alan Weir
Roinn na Feallsanachd/Philosophy
Sgoil nan Daonnachdan/School of Humanities
Oilthigh Ghlaschu/University of Glasgow
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