[FOM] Alternative foundations?

[Victor Marek]

In (New York Times) Book Review of February 16, 2014, there is a piece by
Professor Edward Frenkel of Math/UC Berkeley, entitled "Ad Infinitum." This
is a review of a book by Max Tegmark, a physicist from M.I.T. The book in
question is entitled "Our Mathematical Universe" and appears to be one of many
recent books presenting (an absurd in my opinion, and certainly untestable)
hypothesis of existence of multiverses (other texts on this hypothesis include
books by David Deutsch and by Brian Greene).

Regardless of the merits of the multiverses and other antics by physicists,
a fragment of the review caught my attention, as it pertains to FoM.

Here it is verbatim:

"I tried to process this information, but didn't feel much. Let's go back to
the notion of `mathematical structure.' We read in the book that it is a `set
of abstract elements with relations between them,' like the set of whole
numbers with operations of addition and multiplication. However there is a lot
more to math than such mathematical structures. Objects other than sets are
necessary and they now become widespread. Moreover, there is an effort underway
to overhaul the foundations of math in which set theory is no longer central.
So mathematical structures constitute but a small island of modern mathematics.
Why would someone who believes that math is at the core of reality try to
reduce all of reality to this island? Where would the rest of math then reside?
Unfortunately these questions are not addressed."

So, I wonder, what  is this "effort underway to overhaul the foundations of
math in which set theory is no longer central."

Of course, with the logic education from 1960ies, it must be me who is behind
times, not Professor Frenkel. Still, maybe we should try to see what are these
efforts to overhaul Foundations of Mathematics. Specifically, what are these
objects that are not (representable as) sets?

I believe Professor Frenkel opinions bear on the business of FoM, and maybe we
can see what is going on in the communities beyond FoM.


Victor W. Marek                                 Department of Computer Science
marek at cs.uky.edu                                        University of Kentucky