Last week, the highly distinguished Princeton Professor Ed Nelson announced a proof that the Peano axioms for arithmetic are inconsistent — and hence so is arithmetic itself. If true, this would be much bigger news than faster-than-light neutrinos. It would be bigger news than a discovery that the South had won the American Civil War. It would be far, far bigger news than a discovery that all life on Earth was intelligently designed.
There are, after all, multiple proofs that Peano Arithmetic (that is, the fragment of arithmetic described by the Peano axioms) is consistent. Among those, the simplest and most convincing (to the overwhelming majority of mathematicians) is this: The axioms of Peano Arithmetic, and therefore the theorems of Peano Arithmetic, are all true statements about the natural numbers — and a set of true statements cannot contradict itself.
Ed Nelson rejects that argument because (exempting himself from that overwhelming majority) he doesn’t believe in the set of natural numbers — or perhaps even in individual numbers when those numbers are very large. (How do you know that 810000 exists? Have you ever counted to it?)
Needless to say, this announcement — and the announcement of a forthcoming book providing details — generated more than a flurry of excitement on the math blogs — including one of my very favorite blogs, the n-Category Cafe. After Fields Medalist Terry Tao raised a specific technical objection to Nelson’s argument, Nelson showed up in the comments section to defend himself — and then Tao showed up to expand on his objections. Nelson responded, Tao re-responded, and then Nelson posted:
You are quite right, and my original response was wrong. Thank you for spotting my error.
I withdraw my claim.
Just to be clear, here: That’s Ed Nelson cheerfully acknowledging that the book-length argument he’s been painstakingly constructing for (probably) years, and which was intended to shake the mathematical world to its foundations, doesn’t work. This says so many good things about the culture of mathematics, and so many good things about the Internet, and so many good things about the way they interact (see here and here for more examples), and it says those things so eloquently, that I see no further need for comment.
(On the other hand, if you’re hungry for additional comments, the philosopher Catarina Dutilh Novaes provides some good ones here.)
The Internet’s impact on mathematics is a huge huge thing. Not quite as huge as an inconsistency in Peano Arithmetic, but huge enough to count as a marvel.