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Comment by ncmncm | Hacker News Reader

Well, this is correct as far as it goes.

But it raises the question of what, in modern parlance, is “canon”.

As far as I can tell, mathematics becomes canonical if enough mathematicians think it’s beautiful. There is plenty of seriously presented mathematics that never achieves canon status. The practical effect of being canon is that other mathematicians care to do proofs with it, or about it.

Riemann’s hypothesis is canon even though it hasn’t been proven, because so many proofs use it as an axiom.

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AFAIK, Riemann's hypothesis is not quite canon but many mathematicians posit that it is likely to be true due to a variety of "evidence" they feel leans in that direction.

I think the point is that a conjecture or open question can be canon. Hilbert's problems are an example: https://en.wikipedia.org/wiki/Hilbert%27s_problems

The P=NP question would be an example of a canonical unsolved problem in CS.

I have seen a paper that says it's undecidable, and then seems to choose a continuation in which it is. I am not equipped to evaluate it.

http://phys.lsu.edu/~fmoxley/bbm.pdf

I'm also not equipped to understand that paper, but my find tool is equipped to say the words "decidable", "undecidable", and "decidability" are not present.

Oops, wrong link.

http://phys.lsu.edu/~fmoxley/r1.pdf

I tried to think of any mathematics that used to be canon but isn't anymore. All that comes to mind is the Principia Mathematica, demolished by Gödel. Is there mathematics that has just become unfashionable, not because anybody found anything wrong with it, but just because it turned out not to lead anywhere interesting?

It would be hard to claim that such mathematics existed coeval with (non-avian) dinosaurs.

Euclid's five axioms.

For one thing we learned to understand "axioms" as something less than self-evidently necessary truths. For another we learned that the original axioms didn't really prove what they were thought to prove -- at least up to modern notions of rigour.

We would now understand the original Euclidean proofs to have been "sneaking in" other, implicit, axioms disguised as obvious deductions, and so modern formulations of the geometry need more elaborate axioms.

Did axiomatic geometry become umfashionable? Or did it just get prettied up?

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