[FOM] Voevodsky's views

[Michael Blackmon]

On Fri, 2011-05-20 at 12:59 -0700, Walt Read wrote:
> This discussion has been enlightening in many ways and I thank
> Friedman for initiating it, especially for his willingness to take
> bold positions that have stimulated strong responses. At this point it
> seems to be running in two parallel threads. On the one hand we have
> much technical discussion of FOM issues from what I take to be the
> view of most people on this list, elaborating on the major
> developments of the last 100 or so years. As often happens most of
> these developments are inward-looking, foundations of FOM more than
> foundations of math, to the point where Friedman, e.g., argues for
> their value in illuminating mathematical practice rather than as
> foundational. This may be a valuable contribution of FOM, and one I'm
> personally sympathetic to, but it's certainly not the original intent.
> Math makes a peculiar epistemological claim that requires a different
> kind of foundation than, say, physics. If FOM is really about
> foundations of math, then the mathematicians have to be the judges of
> its success. As is frequently noted here, mathematicians seem at best
> unimpressed by the technical developments. And so here on the other
> hand we have substantial and serious mathematicians, personified by
> Voevodsky, who apparently feel that FOM has failed to accomplish even
> something as basic as foundations of elementary arithmetic. For me
> this raises a couple questions.
> 
> 1) Is the foundational goal defunct? Do we feel that FOM has
> found/created, or soon will, a foundation for math adequate to address
> mathematicians concerns? Or is FOM now just a branch of math
> comprising roughly logic and set theory, of interest mostly to
> specialists?

That is a rather depressing possibility, which I think is entirely
untrue. Consider all the applications of foundational techniques to
fields like general topology, algebra, etc. Moreover, the field of
set-theoretic topology is still very much alive (at least from my
perspective.) The foundational technology developed within fom is not
just used to settle foundational issues. The two best and most
accessible examples I can think of would be a) The Borel Conjecture
(Strong measure zero \implies countable) and b) The product of C.C.C.
topological spaces remaining C.C.C.

In summary the current approach to foundations seems well suited to deal
with the bulk of mathematics. Moreover, it is only in areas like
Category Theory where fom as a whole, seems to be lacking real solid
presence (and not for lack of trying.) The structure of Category Theory
and the mindset of some of its practitioners (for example Voevodsky)
tend to discourage the classical methods of fom.

> 
> 2) Is the lack of confidence from mathematicians due to poor
> communication or perhaps to simple ignorance or arrogance on their
> part or are their concerns legitimate? If mathematicians reasonably
> feel that they need foundations and FOM as it exists isn't working for
> them, where would they go?

I would argue that this is definitely the case. However, not in such
stark terms as "arrogance" or "ignorance." Consider the impact that
certain independence results have had on entire fields of mathematics
(the best example would be Algebra.) 


> 
> -Walt
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