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Results tagged with ho.history-overview
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user 3106
History and philosophy of mathematics, biographies of mathematicians, mathematics education, recreational mathematics, communication of mathematics.
102
votes
Conway's lesser-known results
Although it is well known that Conway was able to quickly calculate the day of the week of any given date, it is less well known that one part of the algorithm is easy to remember and useful in practi …
96
votes
What if Current Foundations of Mathematics are Inconsistent?
Contrary to popular opinion, there is no single foundation for mathematics. Probably you're referring to ZF or ZFC, but most mathematics can be developed on the basis of axioms that are logically muc …
46
votes
Bourbaki's definition of the number 1
These calculations have been carried out by José Grimm; see [1] as well as [2].
According to one version of the formalism in the original Bourbaki, Grimm gets
$$16420314314806459564661629306079999627 …
- 82.7k
46
votes
Logic in mathematics and philosophy
I agree with the commentators that the question is rather too broad, but here's an attempt to answer it anyway.
Readers of MO will likely have less familiarity with non-mathematical logic, so it might …
- 82.7k
45
votes
Proofs of theorems that proved more or deeper results than what was first supposed or stated...
The example given by Wojowu in the comments seems worth posting as an answer.
In the NOVA special The Proof, Ken Ribet says the following.
I saw Barry Mazur on the campus, and I said, "Let's go for a …
45
votes
Accepted
What were the main ideas and gaps in Yoichi Miyaoka's attempted proof (1988) of Fermat's Las...
Here's some information from Barry Cipra's June 1988 article "Fermat's Theorem remains unproved" in Science magazine.
Parshin showed that the arithmetical version of a certain inequality involving ge …
- 82.7k
45
votes
Proofs that require fundamentally new ways of thinking
Technically, the following are not proofs, or even theorems, but I think they count as insights that have the quality that it's hard to imagine computers coming up with them. First, there's:
Math …
45
votes
Endless controversy about the correctness of significant papers
Stanley Yao Xiao's comment has been upvoted so highly that it seems worth posting as an answer.
There is a currently unresolved controversy over Shinichi Mochizuki's claimed proof of the abc conjectur …
41
votes
Widely accepted mathematical results that were later shown to be wrong?
I'm surprised that this one has not already been mentioned. Voevodsky wrote an article explaining that one of the main motivations for his interest in homotopy type theory and univalent foundations w …
41
votes
What are some examples of theorem requiring highly subtle hypothesis?
The convergence conditions for the Fourier series of a function $f:S^1 \to \mathbb{R}$ are a good example. The investigation of convergence conditions for Fourier series was a major motivation for Can …
40
votes
Widely accepted mathematical results that were later shown to be wrong?
R. B. Kershner's paper "On Paving the Plane," Amer. Math. Monthly 75 (1968), 839–844, announced the classification of all convex pentagons that tile the plane. Kershner said that "The proof...is extr …
34
votes
Why is it still common to not motivate results in publications?
Others have given excellent answers. In general I do think that the pendulum has swung too far in the rabbit-out-of-a-hat direction, and it would help the mathematical community if there were more sh …
34
votes
What are some examples of theorem requiring highly subtle hypothesis?
This example has been mentioned elsewhere on MO but seems worth reproducing here. The abstract of Amnon Neeman's paper A counterexample to a 1961 “theorem” in homological algebra says:
In 1961, J …
33
votes
Fields of mathematics that were dormant for a long time until someone revitalized them
Back in 2008, I was trying to find a publisher to bring Dominic Welsh's classic book Matroid Theory back into print. I approached the American Mathematical Society, who solicited opinions from four ex …
33
votes
Shing-Tung Yau's doubts about Perelman's proof
First, let me make some preliminary remarks. We sometimes like to think that "being proved" is a black-and-white property, but in fact there are shades of gray. At one extreme are things like the in …