[FOM] Remedial mathematics?
Timothy Y. Chow
tchow at alum.mit.edu
Tue May 24 10:29:10 EDT 2011
On Mon, 23 May 2011, Jeremy Shipley wrote:
> Existentialism (Frege): Existence (ie, intuition of logical and
> geometric objects) precedes essence (ie, consistent axiomatic
> Essentialism (Hilbert): Essence precedes existence.
> it strikes me as odd that fom practitioners should appeal to
> existentialism, even if it is accepted in ordinary mathematical
> reasoning. I would have thought that even if ordinary mathematics is
> existentialist, most fom practitioners would be essentialist. Otherwise
> why care about consistency proofs in the first place, which find their
> relevance, or at least their historical motivation, within an
> essentialist program?
Part of the problem here is that you're anachronistically lifting an old
debate into the modern world. In the old days, it wasn't so clear that
every mathematical argument could be easily formalized in a variety of
ways, nor was it clear that we couldn't hope for a finitary proof of the
consistency of infinitary set theory. F.o.m. practitioners may in the
past have looked to consistency proofs as providing a road from deep
skepticism about infinitary reasoning to absolute certainty in its
legitimacy, but thanks to Goedel they now recognize that as a chimera.
They are therefore frustrated when mathematicians who ought to know better
continue to cling to the old way of thinking, insisting that the only way
to "know for sure" that some system is consistent is to give a finitary
proof of it, while simultaneously failing to practice what they preach
(since mathematicians continue to claim to "know for sure" lots of things
without any finitary proof).
Getting back to your question, I do not think that f.o.m. practitioners
are "appealing to existentialism." They simply recognize that the
"trivial proof" of the consistency of PA that I sketched can be formalized
in ZF (for example), and don't worry about it further. ZF is stronger
than PA, but so what? Again, that should bother you only if you think
there's something illegitimate about infinitary reasoning and retain a
craving for a finitary proof of the consistency of PA. Neither the
existence of a finitary proof of the consistency of PA, nor the
consistency of PA itself, is considered an interesting mathematical
problem any more, because the issues are well understood.
*Relative* consistency of various formal systems is still interesting from
a f.o.m. perspective because they help us paint a picture of the landscape
of formal systems and where they stand relative to each other. But f.o.m.
practitioners no longer regard consistency proofs as functioning as
definitive refutations of skeptical doubts.
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