FOM: structuralists; quasi-empiricists
Stephen G Simpson
simpson at math.psu.edu
Tue Jul 21 18:16:13 EDT 1998
This is a response to the last part of Jeremy Avigad's posting of 15
Jul 1998 16:05:39.
About structuralism:
> Many category theorists ... may feel that proof theorists are
> missing the point entirely. ... Taken alone, neither the
> ontological nor the structural account tells the whole story; but
> taken together, they complement each other nicely.
My problem with the category theorists, Bourbakians, and other
structuralists is precisely that they fail to recognize this. They
delude themselves into thinking that structure alone is important.
This has resulted in further fragmentation and isolation of
mathematics from the rest of human knowledge. Jeremy, you are trying
to make friends with the structuralists, but is it worth it?
About "quasi-empiricism":
> traditional forms of foundationalism have come under attack by a
> number of philosophers and mathematicians. People like Lakatos,
> Kitcher, Putnam, Tymoczko, Barwise, Putnam, Hersch, Manders, and
> many others have suggested that the axiomatic approach is a
> misguided search for absolutes that neglects some of the most
> interesting aspects of mathematical activity.
I reviewed this anti-foundational trend in my FOM posting
FOM: Tymoczko's book; "quasi-empiricism"; the gold standard
of 1 Feb 1998 22:17:36. Hersh seems to have gone off on a strange
political tangent. What I find truly bizarre about the self-styled
"quasi-empiricists" is that they refuse to acknowledge the current
standards of rigor in mathematics. Jeremy, do you have any idea what
is eating these people? Why are they so bitterly and irrationally
opposed to f.o.m.? Are they angry at f.o.m. for disparate reasons, or
is there a common denominator?
> one can, at the same time, recognize the great strides made in
> mathematical logic over the last century, while still maintaining
> that there are additional aspects of mathematics that are worthy of
> sustained inquiry. I, for one, would welcome an account of
> mathematical proof that explains why some expositions are more
> explanatory than others; an account of meaning that shows how
> mathematical concepts and notation can evolve, and yet still denote
> the same structures; and account of rationality that explains how
> we have come to accept the axioms and forms of argument that we do;
> an account of what makes a certain classical theory "different"
> from a constructive one, even though in principle we can interpret
> one in the other; an account of what it means for a
> number-theoretic proof to "use" analytic methods, even though we
> know that ultimately the whole thing can be couched in the language
> of pure set theory; a deeper analysis of the way the structural and
> ontological components of a mathematical theory interact; an
> account of mathematical reasoning that includes heuristic aspects.
Yes, I agree. All ot these could be terrific research projects.
However, I don't think the quasi-empiricists will ever contribute
anything of value to any of them. They are too consumed with blind
hostility to f.o.m.
-- Steve
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