FOM: Friedman,fom,& Sheep's.Shop.
Robert Tragesser
RTragesser at compuserve.com
Fri Feb 20 11:38:18 EST 1998
Dear Steve Simpson & Harvey Friedman et al.:
The fault is surely mine, but I'm starting to feel
like a customer in a Simpson-Friedman owned Sheep Shop:
as in the sheep's shop in _Through the Looking Glass-, the shelves
full of so many wonderful goods and goodies, but when Alice
reaches for one, it vanishes:
I'm hoping for a very concise, direct, no nonsense, no
rhetoric, concrete, scientific and professional (re-)statement
(so that we all have it in one place without having to use Star
Fleet search engines to gather all the bits and decide for ourselves
among the variety of formulations) of:
[1] The EXACT aim(s) of f.o.m.(on S-F's view).
[2] The EXACT role(s) that set theory is to play
in reaching those ends.
[3] The EXACT reasons for the (absolute?) superiority of
set theory in reaching those ends.
[4] The unqualified successes [goodies which
won't disappear] of set-centered f.o.m..
[5] The qualified successes.
[6] Reasonable prospects.
[7] Visionary prospects.
In addition: there is a serious problem which needs to
be addressed. Ken Manders and others (e.g., philosophical historians of
mathematics, like Mancosu) have been drawing attention
to the conceptual contents and styles of parts of mathematics and
of mathematical methods. Felix Klein (almost) observed that most
of mathematics can be coded in set theory (its concepts "defined"
therein) only after the real mathematics has already been done by
other than set-theoretical tools:
The point is that most mathematics operates within conceptual
frames which are phenmenologically quite distinct from set-constructs:
Harvey Friedman wrote,
[[[[["The use of algebra - groups, rings, modules, and fields, etcetera, in
myriad contexts is a much more impressive set of tools that "make it
natural to neglect much and focus on a little, in a way that actually works
on things like a 300 year old question in Diophantine equations," and much,
much more. Yet the algebraists do not attempt to elevate these notions to
the status of foundations of mathematics in any accepted sense of the word.
Similar remarks apply to general topology and graph theory.
Again, these algebraic notions *could* play a meaningful role in genuine
f.o.m. if one could appropriately carry out a project like this: Give a
general theory of what kinds of algebraic structures are "important" and
classify them. E.g., why groups? as opposed to other things."]]]]]]
Those [algebraic constructs] are not only an impressive set of
tools,
but in practice one can't do without them; & they are
materially/conceptually
distinct from the set theoretic organon (tools).
This distinction is more than just psychological.
If the set-based f.o.m.er is truly scientific and professional,
mustn't they account for this profound difference and radically justify
its suppression?
It is perhaps an unfortunate accident, but the serious mathematics
I learned from others in seminars happened to be presented by
mathematicians
who held set theory in considerable contempt (if taken as more than a
useful language); one speaking of it as "the theory of the empty set" and
another (referring to the initials of one axiomatization of set theory):
"NBG - No Bloody Good"). They were not constructivists, either.
In both cases they were not just being vicariously nasty minded.
Their message was pedagogical: if you find yourself using set-theoretic
reasoning,
you are going to be in trouble. (A graduate number theory course I sat in
on, the instructor said something like this about mathematical induction --
if you find yourself trying to use mathematical induction, think harder!).
Robert Tragesser
Professor in History and Philosophy of Science and Mathematics
Connecticut College
Ph: 1-860-399-6305/6328
e-mail: RTragesser at compuserve.com
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