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In Gian-Carlo Rota's "Ten lessons I wish I had been taught" he has a section, "Every mathematician has only a few tricks", where he asserts that even mathematicians like Hilbert have only a few tricks which they use over and over again.

Assuming Rota is correct, what are the few tricks that mathematicians use repeatedly?

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    $\begingroup$ A mathematician never reveals their tricks. $\endgroup$ Jun 15 '20 at 14:39
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    $\begingroup$ Going to MO, because that way all the tricks are pooled. $\endgroup$ Jun 15 '20 at 14:43
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    $\begingroup$ I once heard a Fields Medallist say that his research consisted of interchanging the order of summation and applying the Cauchy-Schwarz inequality. $\endgroup$ Jun 15 '20 at 15:38
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    $\begingroup$ I am not sure what to expect from answers to this question. I always thought that the point Rota was trying to make was that a mathematician has only a small set of "tricks" that the mathematician has personalized deeply enough to always reach for and use them. Certainly we all "know" a lot more mathematics than a few tricks, but we natively are all truly fluent in a much narrower range than one would naively expect. I thought the point was that the set of techniques was particular to each mathematician. What am I missing here? @MartinSleziak's suggestion for an answer format seems reasonable. $\endgroup$
    – Jon Bannon
    Jun 15 '20 at 22:41
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    $\begingroup$ I think this question misunderstands the quote; I'm surprised at so many upvotes. I think the point is that every mathematician has a few tricks that are her own. We don't necessarily know each other's tricks. That's why the observation has some real content.... i.e. it's not that everyone else is super clever and you are average because they too only have a few tricks, it's just that you do not understand their tricks, you only understand your tricks $\endgroup$
    – T_M
    Jun 16 '20 at 9:45

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Terence Tao wrote a paper, Exploring the toolkit of Jean Bourgain. The abstract reads:

Gian-Carlo Rota once asserted that "every mathematician only has a few tricks". The sheer breadth and ingenuity in the work of Jean Bourgain may at first glance appear to be a counterexample to this maxim. However, as we hope to illustrate in this article, even Bourgain relied frequently on a core set of tools, which formed the base from which problems in many disparate mathematical fields could then be attacked. We discuss a selected number of these tools here, and then perform a case study of how an argument in one of Bourgain's papers can be interpreted as a sequential application of several of these tools.

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I went through all the responses, and I'm surprised that the following trick has not already been posted, given its ubiquity: Use antisymmetry in a partially ordered set. That is, if $a\le b$ and $b\le a$ then $a=b.$

Examples:

  1. Real numbers: $a\le b$ and $b\le a$ implies $a=b$ (can be helpful when directly showing equality is difficult)
  2. Divisibility of positive integers: $a\mid b$ and $b\mid a$ implies $a=b$ (extremely common in number theory)
  3. Subsets of a set: $a\subseteq b$ and $b\subseteq a$ implies $a=b$ (for example, in locus proofs of classical geometry)
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From a physicist point of view I want to mention this trick and its generalization for operators:

      "Two commuting matrices are simultaneously diagonalizable"

(for physicists all matrices are diagonalizable). Of course the idea is that if you know the eigenvectors of one matrix/operator then diagonalizing the other one is much easier. Here are some applications.

1)The system is translation invariant : Because the eigenvectors of the translation operator are $e^{ik.x}$, then one should use the Fourier transform. It solves all the wave equations for light, acoustics, of free quantum electrons or the heat equation in homogeneous media.

2)The system has a discrete translation symmetry: The typical system is the atoms in a solid state that form a crystal. We have a discrete translation operator $T_a\phi(x)=\phi(x+a)$ with $a$ the size of the lattice and then we should try $\phi_k(x+a)=e^{ik.a}\phi_k(x)$ as it is an eigenvector of $T_a$. This gives the Bloch-Floquet theory where the spectrum is divided into band structure. It is one of the most famous model of condensed matter as it explains the different between conductors or insulators.

3)The system is rotational invariant: One should then use and diagonalize the rotation operator first. This will allow us to find the eigenvalue/eigenvectors of the Hydrogen atom. By the way we notice the eigenspace of the Hydrogen are stable by rotation and are therefore finite dimension representations of $SO(3)$. The irreducible representations of $SO(3)$ have dimension 1,3,5,... and they appears, considering also the spin of the electron, as the columns of the periodic table of the elements (2,6,10,14,...).

4)$SU(3)$ symmetry: Particle physics is extremely complicated. However physicists have discovered that there is an underlying $SU(3)$ symmetry. Then considering the representations of $SU(3)$ the zoology of particles seems much more organized (A, B).

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  • $\begingroup$ If "for physicists, all matrices are diagonalizable", what about the matrix $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$? $\endgroup$
    – ogogmad
    Nov 10 '20 at 9:23
  • $\begingroup$ Is it for physicists "almost all" $\approx$ "all"? $\endgroup$
    – ogogmad
    Nov 10 '20 at 9:25
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    $\begingroup$ @ogogmad For physicists all matrices are diagonalizable except for the matrices than are not. $\endgroup$
    – RaphaelB4
    Nov 10 '20 at 13:36
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This trick is called “stupid argument” by some of my collaborators and me.

Let’s say you have a property that is defined on testing using cubes on all scales. Now you have some regular set (say again a cube or a ball) in which the property holds, that is, if you test with a cube that is contained in this regular set then it holds. You might come in this situation for example by local transformation of sets with this property, like flattening the boundary in PDE. Now to get it for all cubes you make the following case distinction: if the concentric cube of half the size is contained in the regular set, you use your assumption. Otherwise, the original cube contains a cube of 1/4 side length compared to the original one that is completely outside the regular set and for this one you get the property usually trivially.

Since this is a trick, I kept it somehow mysterious. Applicability includes things like measure theoretic dimensions, metric properties like porousity and so on and the exact details why it’s trivial “outside” and why testing with comparably smaller objects is ok depends a bit on the specific property one aims for.

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The Fundamental Theorem of Calculus, that is $$\int_0^1 \frac{d}{dt} \psi_t dt =\psi_1 - \psi_0.$$ This "trick" is used throughout differential topology/geometry, for example in showing that the de-Rham cohomology is invariant or for a uniform bound on the period of a negative gradient flow line of the Rabinowitz action functional in constructing Rabinowitz--Floer homology. Actually, the trick consists of cleverly bringing the statement in question down to the form where one can apply the fundamnetal theorem of calculus. Also, in Floer theory in general, Ascoli's theorem (or Arzelà-Ascolis theorem) is used in an exceeding amount.

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    $\begingroup$ The fundamental theorem of calculus has served me well in my research. $\endgroup$
    – Deane Yang
    Jun 17 '20 at 20:48
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Find something that can be computed (a special case, a simplification, or just something of the same flavor as the real problem). Then stare at the data and look for patterns.

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There is but one major trick, and furthermore, many of the other answers are applications of it. Let's call it

T R A N S L A T I O N

The idea is very simple; you translate your problem to a language in which it is simple to solve, so you solve it, and then (if necessary) translate your solution back to the original language. Alternatively, you can think of this as finding the right angle of attack to solve your problem.

  1. Conjugation? Say $W$ is a sequence of Rubik moves that twists two corners. Find moves $V$ that puts the corners you want to twist in the correct position. The n simply apply $VWV^{-1}$.
  2. Change of variables? Translate your integral in $x$ to an integral in $u$ (with translated differential and bounds of course). Find the antiderivative as a function of $u$, and translate back to a function of $x$ (or evaluate if definite).
  3. Diagonalization? Find an eigenbase (essentially a convenient language to understand your matrix). Change basis by conjugation, and your matrix action suddenly looks much simpler.
  4. Analytic geometry? Translate your geometric problem to convenient algebraic manipulations.
  5. etc.
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  • $\begingroup$ This is usually some form of abstraction, and indeed it can be quite powerful. In some sense, this is even the definition of mathematics. Finding an abstraction that unifies and answers easily many apparently unrelated questions. $\endgroup$
    – Deane Yang
    Jun 17 '20 at 20:47
  • $\begingroup$ I also mention the Rubik's cube in my answer, so let me note that when I got my first Rubik's cube as a kid, I certainly invented conjugation immediately. But I did not invent commutators, and was not able to solve the cube. (I don't disagree with your answer though.) $\endgroup$
    – Ville Salo
    Jun 18 '20 at 12:48
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If a function with connected domain is locally constant, then it is constant.

Connectedness doesn't need to be understood topologically: one manifestation of the trick is that a sequence whose consecutive terms are equal must be constant. If the domain is nice enough, it extends to periodic functions as well (a locally periodic function has the same periodic pattern everywhere).

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Using some form of Yoneda's Lemma, see your particular structure as representable functor in an abstract category where your desired constructions are obvious.

This functorial point of view is very nice in algebraic geometry.

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John Allen Paulos has referred to the parable of the Texas sharpshooter a number of times. To paraphrase:

A man driving across the US notices that in a west Texas town, there are a lot of barns with targets painted on their side. Each of the targets have bulletholes directly within the bullseye. Impressed, the man inquires in a local diner about the ace shooter who lives in the town. The townsfolk inform him that a local resident just likes to shoot randomly at barns, and then later on will paint the target right where the bullets land!

Although parable is often referred to as a fallacy of logical reasoning, there appears to be more of a mathematical trick to the logic - namely, conditioning on a random event may be fruitful, even if the probability of the specific random event is low.

This Texas sharpshooter parable to me seems similar to, for example, the post-selection tricks of scatter-shot boson sampling and variants used in recent (2020) experiments. Therein one has limited control over the generation of individual photons, so one merely post-selects on the particular $M$ crystals that generated the photons.

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Two come to mind for me.

  1. "When in doubt, differentiate!" -Chern (or so I've heard). As a result, it's been useful for me to check the implications of $d^2=0$ on differential forms.

  2. I have not used it in research (I have moved away from analysis somewhat), but I love trying to use Jensen's inequality when I come across an analysis problem. If I recall correctly, I solved two problems on my analysis prelim exam using said inequality.

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    $\begingroup$ When working in coordinates, as with PDEs, checking what happens when you commute partial derivatives is indeed quite useful. Jensen’s inequality is really just the definition of convexity. So it encompasses all other inequalities based on convexity. Looking for convexity and then applying the direct consequence of its definition is an impressively useful and powerful “trick” $\endgroup$
    – Deane Yang
    Jun 17 '20 at 20:44
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Proof "tricks" that are routinely used by many:

  • Induction
  • Contradiction

Computational and proof "tricks":

  • Interchanging the order of summation/integration
  • Counting the same thing in multiple ways
  • Looking for patterns (compute special cases, etc.)

Less applicable to as many problems, but still applicable to a wide range of problems in fields like Computer Science, we have the Repertoire Method.

More specialized in mathematics, there are also various methods related to exponential sums, e.g., van der Corput's, Vinogradov's, etc.

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Geometrise!

It worked well for Newton in his Principia when he didn't think that mathematicians would swallow the results he had found for calculus.

For Lie, when he considered PDEs and their solutions.

It worked well for Minkowski when he geometrised Special realtivity and this was helpful for Einsteins work on GR; although he has said that he didn't think of his theory as geometrical, he thought of physically as a unification of inertia and gravity.

Also for Noether when left a note about Betti numbers were better understood as groups and also lectured on them.

It also worked well for Zariski and Grothendieck when they geometrised lots of number theory.

Also Mechanise!

It worked for Archimedes when calculating various volumes.

It also worked for Witten, when he given a problem by Atiyah when he asked Witten to discover a physical understanding of the HOMFLY knot invariant.

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Taylor expansion. Much of the classical theory of statistics (and some of its modern extensions) revolves around performing a second-order Taylor expansion of the likelihood.

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A closed discrete subset of a compact space is finite!

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Let $\mathcal{M}$ be the set of all mathematicians of all times. When you write:

Assuming Rota is correct, what are the few tricks that mathematicians use repeatedly,

it seems that you interpreted Rota's words as follows:

There is a set of tricks $\mathcal{T}$, with $|\mathcal{T}|\ll 10^{10}$, such that every $m\in\mathcal{M}$ uses only tricks from $\mathcal{T}$,

when in fact he meant:

For every $m\in\mathcal{M}$, there is a set of tricks $\mathcal{T}_m$, with $|\mathcal{T}_m|\ll 10^{10}$, such that $m$ uses only tricks from $\mathcal{T}_m$.

Therefore, you should first specify $m$ to get a description of $\mathcal{T}_m$. Most of the posted answers address indeed this kind of question after having selected a suitable subset $S\subset \mathcal{M}$.

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