The discussion about Goedel’s theorem made me wonder about this question: when was the last time a widely quoted mathematical result turned out to be false? I’ve seen preprints that had proofs I didn’t believe, and I know that journals occasionally print results that are wrong, but does anyone know of a result that was once widely accepted, but then turned out to be wrong?
I’m sorry if my last post misled anyone. Goedel’s theorem is true. Lots of people have checked the proof. I’ve checked the proof. There are multiple proofs (the usual proof is Rosser’s, not Goedel’s original proof), as well as vast generalizations.
Here’s a two line proof of the theorem (some details are left out):
At the time, Goedel’s result was very surprising, but by modern standards it’s almost obvious. The set of provable theorems of arithmetic is recursively enumerable. It’s not that surprising that first-order arithmetic would be at least as expressive as Turing machines, which implies that the set of provable theorems is not recursive. Goedel’s theorem follows.
I intended the post to be tongue-in-cheek. My real point was that a certain number of papers on Arxiv are apparently written by cranks.
According to a new paper on Arxiv, Goedel’s theorem is false. There you have it.
This paper, On the Euler angles for SU(N), gives a definition/derivation of Euler angles for SU(n). The paper claims that the construction first appeared in 2002. Does anyone know if that’s true? I thought for sure that I’d seen Euler angles defined for SU(n) (and SO(n)) sometime in the 90s.
Week 222 of This Week’s Finds in Mathematical Physics is up.
I was websurfing a football site, Football Outsiders, when I came across a link to an essay called, of all things, Football’s Hilbert Problems. This is not an isolated phenomenon. It’s modeled after a essay from 2000, Baseball’s Hilbert Problems.
The (sort-of) Nobel Prize in Economics has been announced. The contributions of one of the two, Robert Aumann, are almost purely mathematical work in game theory.
His most interesting idea is that of correlated equilibria. The usual definition of equilibrium in a non-cooperative game, Nash equilibrium, rules out certain kinds of cooperation, even when that cooperation is in the self-interest of each player. Correlated equilibria allow randomized strategies which rely on a random event that is known to both players. Some details about correlated equilibria can be found
here.
(As an aside, the Wikipedia entry for Aumann is unusually bad, so it’s a good candidate for updating, if anyone’s interested. There’s also no entry for correlated equilibrium.)
The November issue of the Notices of the AMS is available.
Some highlights:
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Sigfpe has been posting on his blog about one of my minor obsessions, semirings. A semiring is a ring without subtraction. There are lots of semirings that arise in both pure and applied mathematics. Here are some examples that show just how common they are:
The last example has an interesting extra property: you can take infinite “sums“ by taking the infimum of an infinite set of elements. You can set up an extended analogy between the reals with usual arithmetic operations and the min-plus semiring. In this analogy, integration becomes optimization. You can even extend the analogy as far as the Fourier transform. The min-plus analogue of the Fourier transform is the Legendre transform that arises in classical mechanics. Sigfpe explains the analogy here.
He has also posted about an application of semirings to understanding the game Tetris.
Here’s a bizarre theorem I’d never heard of before: Blumberg’s theorem. It states that for any real function, there is a dense subset of the reals on which the function becomes continuous.