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145
votes
38answers
24k views

Demonstrating that rigour is important

Any pure mathematician will from time to time discuss, or think about, the question of why we care about proofs, or to put the question in a more precise form, why we seem to be so much happier with ...
116
votes
39answers
34k views

Most interesting mathematics mistake?

Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincaré's 3d sphere characterization or the search to prove ...
116
votes
16answers
16k views

When should a supervisor be a co-author?

What are people's views on this? To be specific: suppose a PhD student has produced a piece of original mathematical research. Suppose that student's supervisor suggested the problem, and gave a few ...
94
votes
94answers
12k views

What would you want to see at the Museum of Mathematics?

EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no ...
91
votes
20answers
13k views

Mathematical habits of thought and action which would be of use to non-mathematicians

Once again I come to MO for help with something I'm writing for the public. Which habits of mathematicians -- aspects of the way we approach problems, the way we argue, the way we function as a ...
63
votes
10answers
7k views

Have you solved problems in your sleep?

I have hit upon major (for me—relative to my trivial accomplishments) insights in my research in various sleep-deprived altered states of consciousness, e.g., long solo car-drives extending ...
63
votes
6answers
2k views

Good ways to engage in mathematics outreach?

Greetings all, I have often heard that it would be good if we as a community did more in the way of mathematics outreach: more to explain what it is we do to the community at large, more to expose ...
58
votes
15answers
6k views

Sophisticated treatments of topics in school mathematics

Sophisticated mathematical concepts typically shed light on sophisticated mathematics. But in a few cases they also apply to elementary mathematics in an interesting way. I find such examples ...
56
votes
26answers
5k views

Proof synopsis collection

I hate to keep going with the big lists, but the question about one-sentence summaries of topics/areas spurred this question...and I just can't help myself! Definition (Fraleigh): A proof synopsis ...
56
votes
6answers
7k views

Still Difficult After All These Years

I think we all secretly hope that in the long run mathematics becomes easier, in that with advances of perspective, today's difficult results will seem easier to future mathematicians. If I were ...
49
votes
5answers
6k views

Changing field of study post-PhD

I am doing my PhD in algebraic graph theory, for not much more reason than that was what was available. However, I love deep structure and theory in mathematics, and I do not particularly want to be ...
47
votes
17answers
9k views

Is rigour just a ritual that most mathematicians wish to get rid of if they could?

"No". That was my answer till this afternoon! "Mathematics without proofs isn't really mathematics at all" probably was my longer answer. Yet, I am a mathematics educator who was one of the panelists ...
47
votes
6answers
5k views

Does a referee have to check carefully the proof ?

I have always checked very carefully the papers I was refereeing when I wanted to suggest "accept". Actually I spend almost as much time checking the maths of a paper I referee than checking the maths ...
47
votes
7answers
4k views

Has anyone thought about creating a formal proof wiki with verifier?

Mathematics has undergone some rather nice developments recently with the adoption of new techologies, things like on-line journals, the arXiv, this website, etc. I imagine there must be many further ...
45
votes
19answers
6k views

Are there proofs that you feel you did not “understand” for a long time?

Perhaps the "proofs" of ABC conjecture or newly released weak version of twin prime conjecture or alike readily come to your mind. These are not the proofs I am looking for. Indeed my question was ...
42
votes
8answers
3k views

Published results: when to take them for granted?

Two kinds of papers. There are two kinds of papers: self-contained ones, and those relying on published results (which I believe are the vast majority). Checking the result. Of course, one should ...
40
votes
9answers
6k views

How does a mathematician choose on which problem to work?

Main question: How does a mathematician choose on which problem to work? An example approach to framing one's answer: What is a mathematical problem - big or small - that you solved or are ...
37
votes
14answers
3k views

A set for which it is hard to determine whether or not it is countable.

I got thinking recently, while trying to come up with a problem, that I did not know of any sets which were reasonable to define but for which it was very difficult to determine whether or not they ...
36
votes
25answers
7k views

Where can square roots come from when they are not distances?

In a recent survey "Supergeometry in Mathematics and Physics", Kapranov points out cases in which observable quantities of immediate interest are represented as bilinear combinations of more ...
35
votes
33answers
4k views

Structures that turn out to exhibit a symmetry even though their definition doesn't

Sometimes (often?) a structure depending on several parameters turns out to be symmetric w.r.t. interchanging two of the parameters, even though the definition gives a priori no clue of that symmetry. ...
34
votes
4answers
3k views

Famous vacuously true statements

I am interested to know other examples vacuously true statements that are non-trivial. My starting example is Turan's result in regards to the Riemann hypothesis, which states Suppose that for each ...
34
votes
9answers
4k views

Dimensional Analysis in Mathematics

Is there a sensible and useful definition of units in mathematics? In other words, is there a theory of dimensional analysis for mathematics? In physics, an extremely useful tool is the Buckingham Pi ...
32
votes
6answers
3k views

Negative impact of wrong or non-rigorous proofs

The recent talks of Voevodsky (for example, http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/2014_IAS.pdf), which describe subtle errors in proofs by him as well as others, as well ...
31
votes
6answers
4k views

Why is the Gaussian so pervasive in mathematics?

This is a heuristic question that I think was once asked by Serge Lang. The gaussian: $e^{-x^2}$ appears as the fixed point to the Fourier transform, in the punchline to the central limit theorem, as ...
30
votes
2answers
1k views

Applications of Lawvere's fixed point theorem

Lawvere's fixed point theorem states that in a cartesian closed category, if there is a morphism $A \to X^A$ which is point-surjective (meaning that $\hom(1,A) \to \hom(1,X^A)$ is surjective), then ...
28
votes
15answers
2k views

Objects which can't be defined without making choices but which end up independent of the choice

It happens a lot of times that when one defines a new object (ring, module, space, group, algebra, morphism, whatever) out of given data one first chooses some additional structure. And sometimes ...
25
votes
5answers
2k views

Collaboration or acknowledgment?

This post is a sequel of: When should a supervisor be a co-author? This time the topic is about the interaction between two professional mathematicians (in particular junior-senior, but not ...
24
votes
4answers
3k views

Overview of the interplay of Harmonic Analysis and Number Theory

I'm kind of disappointed that the question here was never sharpened. The Laplacian $\Delta$ on the upper half-plane is $-y^{2}(\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}))$. Suppose $D$ ...
23
votes
5answers
3k views

Where do surreal numbers come from and what do they mean?

I know about Conway's original discovery of the surreal numbers by way of games, as well as Kruskal's way of viewing surreal numbers in terms of asymptotic behavior of real-valued functions, leading ...
23
votes
3answers
2k views

An elementary problem in Euclidean geometry [closed]

This problem was first put to me by Luke Pebody (who did not know the answer at the time) and after some work I am yet to find a proof or counterexample. I would be grateful of any insights. Call a ...
22
votes
2answers
1k views

What might extraterrestrial mathematics look like? [closed]

In an extensive anthropological joint research project concerning the necessities in the development of life and civilisation my group is concerned with mathematics. This forum seems to be extremely ...
22
votes
4answers
1k views

Expert, Intuitive, Organizing Analogies

In learning a new area it is very helpful to have high-level intuitive analogies that keep track of the various parts of an important argument or strategy. Experts have a store of such things, and ...
20
votes
2answers
1k views

Center of mass from the abstract point of view, or could the ancient Greeks invent modern analysis?

This is a very open-ended question, which may or may not have a perfect answer, and for which I have a few ideas but nothing like a clear picture. However, I guess it won't hurt to ask to see if ...
19
votes
3answers
1k views

Which way for reading the proofs?

I am a master student in mathematics. For me a large part of doing mathematics is thinking about, reading and verifying the proof of theorems that I find them in my field of study. I can do this ...
18
votes
3answers
2k views

What would remain of current mathematics without axiom of power set? [closed]

The power set of every infinite set is uncountable. An infinite set (as an element of the power set) cannot be defined by writing the infinite sequence of its elements but only by a finite formula. By ...
18
votes
4answers
1k views

What is the term for combining functions $f_1,f_2,\dots,f_n$ into a tuple $(f_1,\dots,f_n)$?

This is an embarrassingly simple question, but I was not able to find a definitive answer from literature search. Suppose one has some collection of functions $f_1: X \to Y_1, \dots, f_n: X \to Y_n$ ...
18
votes
3answers
1k views

Results that are easy to prove with a computer, but hard to prove by hand [closed]

Consider the assertion: There is no completely multiplicative function $f:\mathbb{N}\rightarrow \{\pm 1\}$ with $\left|\sum_{n\leq x}f(n)\right|\leq 2$ for all $x\geq 0$. One can write a very short ...
18
votes
5answers
757 views

Online high quality colloquium talks

In my department we're thinking about showing online lectures one day per week at lunch, as sort of a virtual colloquium appropriate to mathematics undergraduates as well as faculty. To start with ...
17
votes
1answer
2k views

Why do people use “formal calculation” to describe informal calculations?

Many times, I see the word formal being used to describe a calculation that is not rigorous. I would think that such calculations should rather be termed informal than formal. What is the explanation ...
16
votes
36answers
3k views

Basic results with three or more hypotheses

Consider the following statement of the Arzela-Ascoli theorem. Theorem. Let K be a compact topological space and let S be a subset of C(K). Then S is relatively compact if and only if S is uniformly ...
16
votes
14answers
3k views

What are some examples of “chimeras” in mathematics?

The best example I can think of at the moment is Conway's surreal number system, which combines 2-adic behavior in-the-small with $\infty$-adic behavior in the large. The surreally simplest element ...
16
votes
7answers
2k views

Unexpected applications of the fact that nth degree polynomials are determined by n+1 points

I had a funny idea for proving an identity in Euclidean geometry. While it didn't end up being a very nice proof strategy in my case, I would still like to collect nice examples of where the proof ...
16
votes
5answers
640 views

Mathematics of privacy?

I wonder to which extent the current public debate on privacy issues (not only by state sniffing, but e.g. by microtargetting ads too an issue) offers interesting questions in mathematics? Can we ...
16
votes
2answers
1k views

What are the applications of operator algebras to other areas?

Question: What are the applications of operator algebras to other areas? More precisely, I would like to know the results in mathematical areas outside of operator algebras which were proved by ...
15
votes
7answers
2k views

Is this a rational function?

Is $$\sum_{n=1}^{\infty} \frac{z^n}{2^n-1} \in \mathbb{C}(z)\ ?$$ In a slightly different vein, given a sequence of real numbers $\{a_n\}_{n=0}^\infty$, what are some necessary and sufficient ...
15
votes
1answer
993 views

Relation between math and piano music

What, if any, is the relation between Cantor's function and Ligeti studio: Devil's Staircase?
15
votes
3answers
788 views

Thom's Principle: rich structures are more numerous in low dimension

Marcel Berger states Thom's Principle as: "rich structures are more numerous in low dimension, and poor structures are more numerous in high dimension." This is in Geometry II ...
14
votes
12answers
3k views

Why semigroups could be important?

There is known a lot about the use of groups -- they just really appear a lot, and appear naturally. Is there any known nice use of semigroups in Maths to sort of prove they are indeed important in ...
14
votes
3answers
1k views

Card game / options pricing / Brownian bridge question

We play a game. I shuffle a deck of cards and start dealing them face up. After any card you can say "stop", at which point I pay you 1 dollar for every red card dealt and you pay me 1 for every black ...
13
votes
2answers
848 views

Is there a name for sets for which it is easier to test membership than to find members---and vice versa?

This is a question my son Bob asked me. For some sets it is relatively easy to test for membership but a lot more difficult to find members, and for others the reverse is true. Here is an elementary ...