# [FOM] Lewis on Fictionalism About Mathematics

Staffan Angere Staffan.Angere at fil.lu.se
Tue Mar 13 08:10:58 EDT 2012

There is one version of the quote in his "Mathematics in Megethology", Philosophia Mathematica, vol. 1, 1993, pp. 3-23. That paper was included in some edition of Parts of Classes. Apolpgies for posting it in (almost) full, but I think that it definitely has affected a lot of philosophers to not take fictionalism and other forms of anti-realism about mathematics seriously, in a misguided attempt to be "naturalistic":

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"If there are no classes, there are no homeomorphisms, there are no complemented lattices, there are no probability distributions, ... For all these things are standardly defined as one or another sort of class. If there are no classes, then our mathematics textbooks are works of fiction, full of false 'theorems'. Renouncing classes means rejecting mathematics. That will not do. Mathematics is an established, going concern. Philosophy is as shaky as can be. To reject mathematics for philosophical reasons would be absurd. If we philosophers are sorely puzzled by the casses that constitute mathematical reality, that's our problem. We shouldn't expect mathematics to go away to make our life easier. Even if we reject mathematics gently-explaining how it can be a most useful fiction, 'good without being true'-we still reject it, and that's still absurd. Even if we hold on to some mutilated fragments of mathematics that can be reconstructed without classes, if we reject the bulk of mathematics that's still absurd.

That's no argument, I know. But I laugh to think how \emph{presumptious} it would be to reject mathematics for philosophical reasons. How would \emph{you} like to go and tell mathematicians that they must change their ways, and abjure countless errors, now that \emph{philosophy} has discovered that there are no classes? Will you tell them, with a straight face, to follow philosophical argument where it leads? If they challenge your credentials, will you boast of philosophy's other great discoveries: that motion is impossible, that a being than which no greater can be conceived cannot be conceived not to exist, that it is unthinkable that anything exists outside the mind, that time is unreal, that no theory has ever been made at all probable by evidence (but on the other hand that an empirically ideal theory can't possibly be false), that it is a wide-open scientific question whether anyone has ever believed anything, and so on, \emph{ad nauseam}? Not me!" (pp. 14-15)

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I think, personally, that Lewis is misunderstanding what it means to be a naturalist here: mathematicians are definitely more reliable than philosophers when it comes to questions of mathematics. However, "external" questions such as the existence of sets (as apart from the existence of certain specific sets in, say, ZFC) are not obviously mathematical questions at all, so there may be no reason to believe mathematicians to have a better understanding of them than anyone else. A different way to put it is that even if a philosopher is in no position to question what a mathematician says, she may very well discuss its interpretation.

Staffan Angere
University of Bristol

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Från: fom-bounces at cs.nyu.edu [fom-bounces at cs.nyu.edu] för W.Taylor at math.canterbury.ac.nz [W.Taylor at math.canterbury.ac.nz]
Skickat: den 12 mars 2012 07:51
Till: fom at cs.nyu.edu
Ämne: Re: [FOM] Fictionalism About Mathematics

Quoting Alan Weir <Alan.Weir at glasgow.ac.uk>:

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

Maybe I missed the link; but where does one come across this Lewis essay?

wfct

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