118
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 ...

111
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 ...

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105
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 ...

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79
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 ...

72
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 ...

68
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.

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61
votes

Accepted

### Why should we believe in the axiom of regularity?

Regularity (aka Foundation) can be seen philosophically as an axiom of restriction. It is not necessarily saying “all the things you consider as sets must be well-founded”. It can be read saying “...

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, ...

59
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" ...

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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 ...

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53
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 ...

52
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 ...

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51
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 ...

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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, ...

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48
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 ...

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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 ...

47
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 ...

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47
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 ...

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 ...

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 ...

Community wiki

44
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 ...

Community wiki

43
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 ...

Community wiki

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 ...

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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 ...

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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.

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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
votes

Accepted

### Why aren't functions used predominantly as a model for mathematics instead of set theory etc.?

Let me explain one sense in which using functions or sets provides
exactly equivalent foundations of mathematics, in a way that is
connected with some deep ideas in set theory. There is a
translation ...

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 ...

Community wiki

42
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 ...

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