FOM: "structual" foundations

Harvey Friedman friedman at
Thu Mar 5 11:55:00 EST 1998

Response to McLarty 5:23PM 3/5/98.

Your posting again shows a fundamental misunderstanding of foundations of
mathematics, as well as foundations of subjects, which tends to make your
approach incapable of yielding any significant contribution to foundations
of mathematics. As I have repeatedly said, there is the possibility of a
genuine autonomous, coherent, and comprehensive structural foundation of
mathematics. Your understanding of the usual set theoretic f.o.m. seems to
be insufficient for you to realize what the true difficulties are. It is
not essential or urgent that one have this kind of f.o.m., given the
flexible and satisfactory nature of set theoretic foundations.
Nevertheless, it is a good problem in f.o.m., and an interesting and
imaginative solution would be far more important for f.o.m. than the sum of
all work done in category theory to date is for f.o.m.

I am reluctant to let you have the last word defending this kind of pseudo
foundations since over 300 people are listening, and some of them might
benefit from being warned about this. This may help people see what kind of
work has to be done to make structual f.o.m. a reality.

I have decided not to respond to your posting about what you quite wrongly
think are internal inconsistencies in Feferman's written point of view on
the matter, in the expectation that someone else would correct you.

One new way I thought of that might help clarify the situation is the
following question. What are the major open problems in "autonomous,
coherent, and comprehensive" f.o.m. from your point of view? And what has
been acheived in "autonomous, coherent, and comprehensive" f.o.m. from your
point of view that is comparable to the work of Aristotle, Frege, Cantor,
Cauchy, Zermelo, Hilbert, Russell, Godel, and Cohen? The answer "copying
them" is not satisfactory.

I leave you with the following pun. There is a categorical rejection of
category theory as a comprehensive, autonomous, coherent foundation of
mathematics in the mathematical community. People rightly view categories
as naturally resting on sets, when presented with the usual way in which
category theory is developed. And this doesn't prevent them from using
categories to good effect any more than it prevents them from using groups,
rings, fields, graphs, etcetera.

Response to Pratt 8:58AM 3/5/98

Pratt quoted my quotes from Silver and Feferman:

From: Silver
>What I'm looking for is a development of category theory, or
>perhaps topos theory, that is based on some underlying *conception*.
From: Feferman
>... the notion of topos is a relatively sophisticated mathematical
>notion which assumes understanding of the notion of category and that in
>turn assumes understanding of notions of collection and function. ...
>Thus there is both a logical and psychological
>priority for the latter notions to the former...

Pratt then quoted me:

From: Friedman
>You could directly address these two quotes [Silver and Friedman].

Pratt writes:

>You may be forgetting that I already did, in FOM mail dated respectively
>Thu, 22 Jan 1998 16:12:05 -0800 and Thu, 20 Nov 1997 10:16:03 -0800.

Apparently not satisfactorily, since neither Silver nor Feferman discussed
your (Pratt's) statements.

More information about the FOM mailing list