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1500 questions
1052
votes
292 answers
Examples of common false beliefs in mathematics
The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while they are learning mathematics, but quickly abandon…
gowers
- 28,729
523
votes
3 answers
Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?
Is there any polynomial $f(x,y)\in{\mathbb Q}[x,y]{}$ such that $f\colon\mathbb{Q}\times\mathbb{Q} \rightarrow\mathbb{Q}$ is a bijection?
Z.H.
- 5,273
426
votes
16 answers
Why do roots of polynomials tend to have absolute value close to 1?
While playing around with Mathematica I noticed that most polynomials with real coefficients seem to have most complex zeroes very near the unit circle. For instance, if we plot all the roots of a polynomial of degree 300 with coefficients chosen…
Andrej Bauer
- 47,834
422
votes
91 answers
Video lectures of mathematics courses available online for free
It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some universities do that (albeit to a very limited extent),…
alex
- 966
401
votes
84 answers
Proofs without words
Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results?
(One could ask if this is of interest to mathematicians, and I would say yes, in so far as the kind of little gems…
Mariano Suárez-Álvarez
- 46,795
395
votes
23 answers
Thinking and Explaining
How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words differ, or describe how they are connected for…
Bill Thurston
- 24,832
382
votes
53 answers
Widely accepted mathematical results that were later shown to be wrong?
Are there any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significant amount of time later, possibly even after being used in proofs of other…
Roman Starkov
- 131
382
votes
115 answers
Not especially famous, long-open problems which anyone can understand
Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please.
Motivation: I plan to use this list in my teaching, to motivate general education…
David Feldman
- 17,466
356
votes
30 answers
Geometric interpretation of trace
This afternoon I was speaking with some graduate students in the department and we came to the following quandary;
Is there a geometric interpretation of the trace of a matrix?
This question should make fair sense because trace is coordinate…
B. Bischof
- 4,782
331
votes
16 answers
What's a mathematician to do?
I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of question and since it is worrying me increasingly…
muad
- 1,402
325
votes
34 answers
Why is a topology made up of 'open' sets?
I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of many examples, but it's never been obvious to me…
Minhyong Kim
- 13,471
317
votes
22 answers
Why do so many textbooks have so much technical detail and so little enlightenment?
I think/hope this is okay for MO.
I often find that textbooks provide very little in the way of motivation or context. As a simple example, consider group theory. Every textbook I have seen that talks about groups (including some very basic…
Michael Benfield
- 435
293
votes
8 answers
Philosophy behind Mochizuki's work on the ABC conjecture
Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy behind his work and comment on why it might be…
James D. Taylor
- 6,178
291
votes
34 answers
What are some reasonable-sounding statements that are independent of ZFC?
Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC."
For example, suppose $A$ is an abelian group such that every short exact sequence of abelian groups…
Anton Geraschenko
- 23,718
290
votes
125 answers
What are some examples of colorful language in serious mathematics papers?
The popular MO question "Famous mathematical quotes" has turned
up many examples of witty, insightful, and humorous writing by
mathematicians. Yet, with a few exceptions such as Weyl's "angel of
topology," the language used in these quotes gets the…
John Stillwell
- 12,258