I’ve been folloiwing a discussion on the Foundations of Mathematics List (FOM) regarding these two videos (1, 2) on what noted Field’s Medalist Vladimir Voevodsky thinks regarding the consistency of Peano Arithmetic and the foundations of mathematics. His comments have caused quite a stir on the list. Below is a message to Voevodsky by Harvey Friedman and Voevodsky’s reply.
“Dear Professor Voevodsky,
I have become aware of your online videos at http://video.ias.edu/voevodsky-80th and http://video.ias.edu/univalent/voevodsky. I was particularly struck by your discussion of the “possible inconsistency of Peano Arithmetic”. This has created a lot of attention on the FOM email list. As a subscriber to that list, I would very much like you to send us an account of how you view the consistency of Peano Arithmetic. In particular, how you view the usual mathematical proof that Peano Arithmetic is consistent, and to what extent and in what sense is “the consistency of Peano Arithmetic” a genuine open problem in mathematics. It would also be of interest to hear your conception of what foundations of mathematics is, or should be, or could be, as it appears to be very different from traditional conceptions of the foundations of mathematics.
Respectfully yours,
Harvey M. Friedman
Ohio State University
Distinguished University Professor
Mathematics, Philosophy, Computer Science”
And Voevodsky’s reply:
“Dear Harvey,
such a comment will take some time to write …
“To put it very shortly I think that in-consistency of Peano arithmetic as well as in-consistency of ZFC are open and very interesting problems in mathematics. Consistency on the other hand is not an interesting problem since it has been shown by Goedel to be impossible to proof [sic].
Vladimir.”
I think that this reply still leaves a lot of room for interpretation. It could simply mean that Hilbert’s program is no longer interesting or it could be an call for finitism in foundations or it could indicate the possibility of fruitful mathematics by means of searching for an inconsistency proof of these well established systems. “Curiouser and curiouser!” Cried Alice …
