120
votes
Axiom of choice, Banach-Tarski and reality
There are two ingredients in the Banach-Tarski decomposition theorem:
The notion of space, together with derived notions of part and decomposition.
The axiom of choice.
Most discussion about the ...
- 47.9k
113
votes
The enigmatic complexity of number theory
I'm not a number theorist, but FWIW: I would talk, not so much about Gödel's Theorem itself, but about the wider phenomenon that Gödel's Theorem was pointing to, although the terminology ...
106
votes
Small ideas that became big
In a letter to Frobenius, Dedekind made the following curious observation: if we see the multiplication table of a finite group $G$ as a matrix (considering each element of the group as an abstract ...
80
votes
Small ideas that became big
The problem of the seven bridges of Königsberg is surely one of the best-known examples of this. Euler apparently didn't even consider this problem to be mathematical when he solved it, but in doing ...
74
votes
Accepted
How should a "working mathematician" think about sets? (ZFC, category theory, urelements)
Set theory provides a foundation for mathematics in roughly the same way that Turing machines provide a foundation for computer science. A computer program written in Java or assembly language isn't ...
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67
votes
Small ideas that became big
Cantor's monumental investigation of the infinity started very innocently as a method to understand the uniqueness of the representation of a function by trigonometric series.
63
votes
Accepted
Are there any fields of academic mathematics whose epistemic status as math is controversial within the academic community?
There are some speculative mathematical concepts that come to mind, such as the field of one element or motives, though perhaps these are more classifiable as "potential future mathematics" ...
59
votes
Pressure to defend the relevance of one's area of mathematics
Overall, people in academia in general and mathematicians in particular are very lucky in being free to study (and being able to make a good living) according to the standards of their discipline, ...
55
votes
Accepted
Why is integer factoring hard while determining whether an integer is prime easy?
What I think you're asking for are examples of search problems that seem to be hard, while a corresponding decision problem is solvable in polynomial time (but not totally trivial). It is true that ...
- 80.3k
54
votes
Taking a theorem as a definition and proving the original definition as a theorem
From a conversation I had with Gian-Carlo Rota when I was undergraduate, I know that one simple but important example that he specifically had in mind was the calculus of vector fields (whether ...
52
votes
Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false
Girolamo Saccheri in his Euclides Vindicatus (1733) essentially discovered Hyperbolic Geometry, by building around the hypothesis that the angles of a triangle add up less than 180°. This was widely ...
51
votes
Request for examples: verifying vs understanding proofs
Don Zagier has a well-known paper, A one-sentence proof that every prime $p\equiv 1\pmod 4$ is a sum of two squares. An undergraduate mathematics major should be able to verify that this proof is ...
49
votes
Accepted
In what respect are univalent foundations "better" than set theory?
I like your analogy with programming languages. If we think of ST as a low-level programming language and UF as a high-level one, then one advantage of UF is obvious: it is more convenient to write ...
49
votes
On critical reviews of Hawking's lecture "Gödel and the end of the universe"
Alon Amit: "There are some things that break my heart more thoroughly than reading nonsensical conclusions from Gödel's Theorems to the limitations of physics published by eminent scientists, but they ...
- 183k
49
votes
Why not adopt the constructibility axiom $V=L$?
Let me add to the existing answers a point which may seem "vulgar" at first, but I think is actually important:
V=L is complicated.
And whether or not this ought to be a reason to not raise it to ...
49
votes
Accepted
Swimming against the tide in the past century: remarkable achievements that arose in contrast to the general view of mathematicians
After mathematicians had been been taught for decades that a consistent theory of the calculus based on infinitesimals was impossible, Abraham Robinson was certainly swimming against the tide when he ...
48
votes
Contemporary philosophy of mathematics
Let me mention a few current issues on which I have been involved in the philosophy of
mathematics. Of course there are also many other issues on which people are working.
Debate on pluralism. First, ...
47
votes
Accepted
Hilbert's alleged proof of the Continuum Hypothesis in "On the Infinite"
The article "Hilbert and Set Theory" by Dreben and Kanamori devotes Section 7 to this argument and an analysis of its flaws. Dreben and Kanamori use the translation provided by van Heijenoort, so that ...
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46
votes
Are there mistakes in the proof of FLT?
No there are not any mistakes in these papers of any interest. In the 1990s there were a bazillion study groups and seminars across the world devoted to these papers; I personally read all three of ...
- 41k
45
votes
In what respect are univalent foundations "better" than set theory?
This is a question that has been discussed a lot on the Foundations of Mathematics mailing list (unfortunately with more polemics than necessary IMO—though I confess that I may have been guilty ...
45
votes
Proofs of theorems that proved more or deeper results than what was first supposed or stated as the corresponding theorem
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 ...
43
votes
The enigmatic complexity of number theory
What references, especially books, have been devoted to specifically addressing the source of the deep roots of the diversity and complexity of number theory?
To a first approximation, I would say ...
43
votes
Small ideas that became big
Integration by parts would seem like a good example. Whoever first used it to integrate a function such as $x\exp(x)$ could certainly not have anticipated the fundamental role it would once play in ...
43
votes
Proofs of theorems that proved more or deeper results than what was first supposed or stated as the corresponding theorem
In theoretical computer science, an extractor is an algorithm that takes a weak source of randomness (i.e. a distribution that may be far from the uniform distribution) and produces a much stronger ...
43
votes
Is pure mathematics useful outside of mathematics itself?
Adding beauty and joy to the world, contributing to humanity’s understanding: these are direct and immediate benefits from pure mathematics, even if they are not fiscal.
43
votes
Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false
Computational complexity theory involves investigating illusory worlds, since so many of the results depend on unanswered questions. A vivid example is given by Russell Impagliazzo's paper "A ...
42
votes
Axiom of choice, Banach-Tarski and reality
It's notable that most of the "bread and butter" mathematical consequences of the axiom of choice are actually consequences of countable choice. (Every infinite set contains a countable subset, a ...
- 42.4k
42
votes
Accepted
Is pure mathematics useful outside of mathematics itself?
This is not really an answer to the question as asked, but I believe it's important and relevant to your problem, and too long for a comment.
I will not here express any opinion about the validity or ...
41
votes
Pressure to defend the relevance of one's area of mathematics
These questions can be an opportunity for a bit of soul-searching; that's a not a bad thing once in a while.
You can say that you want to work on whatever you find interesting and no further ...
Only top scored, non community-wiki answers of a minimum length are eligible