[FOM] Alternative foundations?

[Kreinovich, Vladik]

I think Eduard Frenkel refers to Homotopy Type Theory http://en.wikipedia.org/wiki/Homotopy_type_theory, a recent Institute of Advanced Studies project which resulted in a book http://homotopytypetheory.org/book/ which explicitly emphasizes that its category-based foundations are an alternative to the usual set-theoretic one, this was a joint effort of high-quality mathematicians and computer scientists who are interested in automating formal proofs in mathematics. The book is available both for sale and for a free downloading.

I agree with Dr. Marek that this mailing list is very appropriate for discussing this and similar approaches. 

-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of Victor Marek
Sent: Tuesday, February 18, 2014 7:59 AM
To: fom at cs.nyu.edu
Subject: [FOM] Alternative foundations?

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.