122
votes
Every mathematician has only a few tricks
$$
\sum_{i=1}^m\sum_{j=1}^n a_{i,j}=\sum_{j=1}^n\sum_{i=1}^m a_{i,j}
$$
(and its variants for other measure spaces).
I still get misty-eyed whenever I read something that capitalizes on this trick in ...
Community wiki
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 ...
Community wiki
105
votes
Accepted
Conceptual reason why the sign of a permutation is well-defined?
(This is a variant of Cartier's argument mentioned by Dan Ramras.)
Let $X$ be a finite set of size at least $2$. Let $E$ be the set of edges of the complete graph on $X$. The set $D$ of ways of ...
88
votes
Every mathematician has only a few tricks
A very useful generic trick:
If you can't prove it, make it simpler and prove that instead.
An even more useful generic trick:
If you can't prove it, make it more complicated and prove that instead!
Community wiki
79
votes
Endless controversy about the correctness of significant papers
(Also mentioned in Oliver Nash's comment)
From a February 2017 article in Quanta Magazine called "A fight to fix geometry's foundations" (the original has relevant links in the text):
...in 2012, ...
Community wiki
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 ...
76
votes
Every mathematician has only a few tricks
In combinatorics: shove it into OEIS, and see what's up.
Also, add more parameters!
Note: the Macdonald polynomials were introduced by adding more parameters to the Jack and the Hall-Littlewood ...
Community wiki
76
votes
Every mathematician has only a few tricks
Dennis Sullivan used to joke that Mikhail Gromov only knows one thing, the triangle inequality. I would argue that many mathematicians know the triangle inequality but not many are Gromov.
Community wiki
76
votes
Conceptual reason why the sign of a permutation is well-defined?
This is obviously a subjective question, but I teach this in two phases
(1) I need to know this fact very early, so I give the quickest definition I know:
$$\mathtt{sign}(\sigma) = \frac{\prod_{i<j}...
72
votes
17 camels trick
Slack variables in linear programming. Quote from the link:
In an optimization problem, a slack variable is a variable that is added to an inequality constraint to transform it to an equality. ...
Community wiki
68
votes
Small ideas that became big
Cantor's monumental investigation of the infinity started very innocently as a method to understand the uniqueness of the representation of a function by trigonometric series.
Community wiki
64
votes
17 camels trick
Lagrange multipliers: to extremize $f(x,y)$ subject to constraint $g(x,y)=0$, add a variable $\lambda$ and
introduce an auxiliary function
$$
{\mathcal {L}}(x,y,\lambda )=f(x,y)-\lambda \cdot g(x,...
Community wiki
63
votes
17 camels trick
Maybe this example is too elementary for this site, but I'd say the proof that a closed subset of a compact set is compact itself looks like this. Given an open cover of a $A$ where $A$ is a closed ...
Community wiki
63
votes
Is data science mathematically interesting?
I will stay away from the academic politics of hiring "professors of data science", but if I interpret the question more specifically as "does data science offer problems of mathematical interest", I ...
Community wiki
61
votes
Endless controversy about the correctness of significant papers
The Jordan curve theorem asserts that a simple continuous closed curve separates the plane into two distinct connected open sets.
Whether Camille Jordan's original proof is correct or not seems to be ...
Community wiki
61
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" ...
Community wiki
60
votes
Accepted
Is spherical trigonometry a dead research area?
It is not. As a proof, I will mention three relatively recent papers where I am a co-author:
M. Bonk and A. Eremenko, Covering properties of meromorphic functions, negative curvature and spherical ...
59
votes
Is amateur research in mathematics viable?
This is possible. I have at least two friends who studied mathematics (in the graduate school), did not defend their PhD, and found some jobs not related to mathematics. Still they do research, and ...
Community wiki
57
votes
Each mathematician has only a few tricks
The question is worded in a way that seems to imply we might speak of other mathematician's tricks, but I'm not sure I know the tricks of even my closest collaborators, except by osmosis; so I hope it'...
55
votes
Every mathematician has only a few tricks
Integration by parts has allegedly earned some people big medals.
Community wiki
54
votes
17 camels trick
The AM-GM inequality (as well as other inequalities, for instance the discrete version of Jensen's inequality) can be proved with the following trick due to Cauchy, known as "forward-backward ...
Community wiki
53
votes
17 camels trick
The usual way to prove the Strong Nullstellensatz from the Weak Nullstellensatz is the so-called Rabinowitsch trick.
In short, adding an extra variable allows us to apply the "weak" version of the ...
Community wiki
53
votes
What mathematical problems can be attacked using DeepMind's recent mathematical breakthroughs?
This is a bit speculative, and perhaps too challenging for an undergraduate project, but I wonder if an AlphaGeometry type approach might be possible for the task of automatically upper bounding sums ...
51
votes
Fascinating moments: equivalent mathematical discoveries
Consider $n$ evenly spaced points on a circle representing $\mathbb{Z}^n$.
Two sets of points
with the same multiset of distances between them (measured by the shortest distance around
the circle) are ...
Community wiki
51
votes
Conceptual reason why the sign of a permutation is well-defined?
Draw a braid on $n$ strings sloppily, so that no three strings pass through the same point. Convince yourself that the parity of the number of crossings is the same in every other drawing of the ...
50
votes
Accepted
Why aren't proceedings from ICM 2014 on mathscinet?
We have had difficulty obtaining the requisite permissions from the publisher. The ICM2014 website has the Legal Disclaimer: "The Seoul ICM Organizing Committee, the legal copyright owner of the ...
50
votes
Changes forced by the pandemic
Online seminars
Research (and other) seminars have gone virtual. The obvious advantage is, that anyone can attend from basically all over the world. The page https://researchseminars.org/ compiles a ...
Community wiki
50
votes
Situations where “naturally occurring” mathematical objects behave very differently from “typical” ones
Most finite groups empirically are 2-groups (in the sense of being a p-group with $p=2$ not in the other sense of the word). There are a lot of them. Conjecturally almost all finite groups are 2-...
50
votes
What do you do when you're stuck?
Here is an answer which may be math-specific: If you are stuck in some proof of some claim that you believe is true:
Add the missing piece as assumption and continue as planned.
Community wiki
Only top scored, non community-wiki answers of a minimum length are eligible
Related Tags
gm.general-mathematics × 344soft-question × 78
big-list × 56
ho.history-overview × 23
reference-request × 19
mathematics-education × 18
nt.number-theory × 16
co.combinatorics × 14
lo.logic × 11
linear-algebra × 10
ca.classical-analysis-and-odes × 10
journals × 10
career × 10
pr.probability × 9
real-analysis × 9
mp.mathematical-physics × 9
big-picture × 9
mathematical-philosophy × 9
recreational-mathematics × 9
polynomials × 8
inequalities × 8
terminology × 8
intuition × 8
mathematical-writing × 8
closed-form-expressions × 8