[FOM] Fictionalism About Mathematics
Aatu Koskensilta
Aatu.Koskensilta at uta.fi
Mon Mar 12 14:20:31 EDT 2012
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
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