[FOM] Fictionalism About Mathematics

[Aatu Koskensilta]

Quoting T.Forster at dpmms.cam.ac.uk:

> It is true that there are people in Philosophy departments who  
> espouse fictionalism, but i have yet to meet a working mathematician  
> who does. The problem i have with this kind of talk is this:  
> ficticious objects have real counterparts. People in novels, plays  
> etc, are fictions. We (sort-of) know how to deal with this beco's  
> there are *real* people as well. But if mathematical entities are to  
> be fictions, what real things are they fictions of? And if they  
> aren't, why call them fictions?

   It is in this context perhaps a pertinent observation that there  
are in fact fictional mathematical objects in a perfectly  
straightforward sense. Greg Egan, for instance, has written a story  
that has the discovery of an inconsistency in PA as a plot point (if  
my memory does not play me false). A natural number coding a proof of  
a contradiction in PA in Egan's story is an example of a fictional  
mathematical object. It's fictional in just the same way Sherlock  
Holmes and Harry Potter are, in that there is in fact no such natural.

   Much of the appeal of fictionalism no doubt has to do with the  
observation that in our everyday mathematical reasoning it seems to  
make no difference whether we think naturals (sets, function spaces,  
what have you) figments of our collective mathematical imagination or  
as inhabitants of some independent mathematical reality. We don't,  
after all, find proofs that start with

   Since sets are only figments of our imagination, we have that...

or

   It follows from Lemma 2, given that sets inhabit an independent reality
   a bit like physical objects do, that Theorem 3 can't be generalized to
   cover...

or anything of the sort. Similarly, when I explain that the axiom of  
choice -- or impredicative comprehension or what not -- is for me an  
evident principle, most pleasing to the intellect, allowing me to make  
mathematical use of the "informal" idea sets are arbitrary extensional  
collections to be conceived "quasi-combinatorially" as Bernays put it,  
philosophical doctrines about the ontological status of mathematical  
objects simply don't seem to enter into it in any apparent way. And so  
on. But of course it does not follow there's no difference between  
imaginary or fictional sets or naturals and actual sets or naturals,  
in the everyday sense we use when discussing, say, science fiction  
stories involving mathematical imaginings.

   To connect traditional philosophical discussions and debates, over  
nominalism, Platonism, fictionalism, with the mathematical experience  
of the working mathematician, we'd need to find some point of real  
mathematical interest that in some meaningful way turns on such  
matters. Otherwise there's no reason to expect mathematicians to have  
any opinion here -- except perhaps out of some misguided sense of  
intellectual duty inspiring them to occasionally mumble incoherently  
about formal theories, reality, etc, just as scientist when cornered  
resort to mouthing naive and ill-digested bits from the philosophy of  
science that have no real force or meaning for them in their everyday  
business of successful sciencing.

-- 
Aatu Koskensilta (aatu.koskensilta at uta.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus