[FOM] Voevodsky's views

[Walt Read]

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?

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?

-Walt