What do named "tricks" share?

Question

There are a number of theorems or lemmas or mathematical ideas that come to be known as eponymous tricks, a term which in this context is in no sense derogatory. Here is a list of 11 such tricks (the last of which I learned at MO):

• the Whitney trick
• the DeTurck trick
• the Cayley trick
• the Rabinowitsch trick
• the Klee trick
• the Moser trick
• the Herglotz trick
• the Weyl trick
• the Karatsuba trick
• the Jouanolou trick
• Minty's trick

Edit: List augmented from the comments and answers:

• the Eilenberg–Mazur swindle
• the Parshin trick
• the Atiyah rotation trick
• the Higman trick
• Rosser's trick
• Scott's trick
• the Craig trick
• the Uhlenbeck trick
• the Alexander trick
• Grilliot's trick
• Zarhin's trick [For any abelian variety $A$, $(A \times A^{\vee})^4$ is principally polarizable.]
• Kirby's torus trick
• Trost's Discriminant trick
• The Brauer trick. Discussed in Gorenstein's Finite Simple Groups.

Further Edit. And although my original interest was in eponymous (=named-after-someone) tricks, several non-eponymous tricks have been mentioned, so I'll gather those here as well:

• the determinant trick
• the kernel trick
• the W-trick

Some of those listed above do not yet have Wikipedia pages (hint, hint—Thierry).

I (JOR) am not seeking to extend this list (although I would be incidentally interested to learn of prominent omissions), but rather I am wondering:

Is there some aspect or trait shared by the mathematical ideas or techniques that, over time, come to be named "tricks"?

Answer 1

How about the following (which I think applies to some of these tricks but not others): a trick is something whose usefulness is not fully captured by any particular set of hypotheses, so it would limit its usefulness to write it down as a lemma.

Answer 2

One well-known trick is a way to evaluate the Gaussian integral $G = \int_\mathbb{R} e^{-x^2}dx = \sqrt{\pi}$ by writing $$G^2 = \left(\int_\mathbb{R} e^{-x^2} \, dx\right)\left(\int_\mathbb{R} e^{-y^2} \, dy\right) = \int_{\mathbb{R}^2} e^{-(x^2+y^2)} \, dx \, dy$$ which when transformed to polar coordinates becomes $$G^2 = 2\pi \int_0^\infty e^{-r^2} r \, dr = \pi \int_0^\infty e^{-u} \, du = \pi$$ via the substitution $u=r^2$. It appears this idea is due to Poisson.

In a 2005 note in the American Mathematical MONTHLY, R. Dawson has observed that this is a trick that only works once; there are no other integrals that can be evaluated by this method. Specifically:

Theorem. Any Riemann-integrable function $f$ on $\mathbb{R}$, such that $f(x)f(y) = g(\sqrt{x^2+y^2})$ for some $g$, is of the form $f(x)=ke^{ax^2}$.

See: Dawson, Robert J. Mac G., On a “singular” integration technique of Poisson, Am. Math. Mon. 112, No. 3, 270-272 (2005). ZBL1088.26500.

So if a technique is a trick that works twice, this one is definitely still a trick.

Answer 3

I'll take a stab at this.

I think that the term "trick" is used to connote a technique that achieves something as if by magic. If I make a cake by combining flour, sugar, and eggs and baking, that is simply a standard technique, but if I make the cake by putting the ingredients into a top hat and waving a wand over it, that is a magic trick. The way that the Weyl unitary trick makes complex groups behave like compact ones seems like a magic trick. (For those of you trying to follow this half baked analogy, the cake is complete reducibility of representations, the oven is integration, and the hat is ... uhhh.... )

Answer 4

To my way of thinking, the other answers are missing an important element, a necessary feature for a mathematical tool or method to be called "trick."

Namely, in order to be called a "trick," a method or technique must involve artifice or misdirection of some kind. When we treat a mathematical object as something that it isn't really or when we pretend that something is other than it is in order to advance an argument (which is not to suggest that the mathematics is not correct), then we are using trickery. When we solve a problem by placing our focus on something else, in which we aren't actually interested as such and which may even be silly in some way—a kind of misdirection—but by doing so we become successful in the original problem, then we are using trickery. When we replace a robust concept, in which we are really interested, with a modified version of it, perhaps even an absurd version of it, but which makes the argument work, then we are using trickery.

For example, with Craig's trick, we replace a formula $\varphi$ with the conjunction with itself $\varphi\wedge\varphi\wedge\dots\wedge\varphi$ repeated many times over. The new assertion is just silly and we don't actually care about it as such, although of course it is logically equivalent to $\varphi$. How could it possibly help? The point is that we can use the new assertion to code some extra information into an axiomatization or presentation: the number of times it was repeated. By this artifice, we can deduce that every computably enumerable theory has a computable set of axioms. The same idea works in many other contexts. For example, every c.e. presentable group has a computable presentation, by sufficiently repeating relations suitably in the presentation.

With Scott's trick, the issue to be solved is that the equivalence class of an object forms a proper class, which can cause certain problems, and so we replace that equivalence class with the set of rank-minimal members of the class. If we think of this fake equivalence class as the real thing, then everything works great! This trick is surprisingly robust, and can be used to find small canonical sets of representing structures in almost any situation. For example, in ZFC there is a definable manner of choosing a set of groups from each group isomorphism class: the rank-minimal groups from that class. This is a trick, because we don't really care much about that particular collection as such.

With Rosser's trick, we replace the concept of a theory $T$ proving a sentence $\sigma$, with: $T$ proves $\sigma$ by a proof for which there is no shorter proof of $\neg\sigma$. When you think of "proof" using this concept, then Gödel's incompleteness theorem is improved to the Gödel-Rosser theorem, where one can drop Gödel's extra hypotheses about $\omega$-consistency. This is a trick, because we don't actually want to think about "proof" using Rosser's concept, except that it makes the argument work.

In many of the other tricks, we do something that seems a little absurd at first, misdirecting our attention from the original problem to this other thing, which may seem irrelevant at first, but when we follow it more fully it provides the answer we seek.

In each case, we replace the concepts or objects in which we are truly interested by concepts or objects that we don't actually care about as such and which in several cases are comical versions of the original, except that they make the argument work.

Answer 5

https://en.wikipedia.org/wiki/Rosser%27s_trick

"A technique is a trick that works twice"

Note that Grothendieck never published his proof of the Grothendieck-Riemann-Roch theorem because he felt that the proof depended on an "astuce" (trick) rather than flowing naturally.

Answer 6

Scott's Trick is called a "trick" because it is not actually necessary for the completion of the proof in which it is involved; however, without the trick the proof is massively more tedious. Although the other tricks may not have a widely-agreed-upon-reason for being a trick, I suspect that they may be called such for similar reasons.

Answer 7

There's always Feynman's trick. It's more common amongst physicists; Feynman, afterall was a physicist. Although quite standard now, back in Feynman's hayday he was able to solve many complicated integrals by adding another variable, using an auxiliary function, solving a basic integral involving the auxiliary function, and then differentiating said variable to get the correct integral. This is common now, but when Feynman would use it, people wouldn't understand how he'd solved such complicated integrals so fast. It's very diverse too, in that it applies in many scenarios. One example I always found interesting, off the top of my head, was using it to prove that the Gamma function interpolates the factorial in one line:

$$\int_0^\infty e^{-x}x^{n}\,dx = (-1)^{n}\frac{d^{n}}{dt^{n}}\Big{|}_{t=1}\int_0^\infty e^{-tx}\,dx = \frac{d^{n}}{dt^{n}}\Big{|}_{t=1} \frac{(-1)^{n}}{t} = n!$$

though is of no way limited to this single, obvious, instance. One can solve the $\text{sinc}$ integral using this. One can simplify the amount of work one needs by using this little trick.

It is a trick in the sense that it's very simple. It's a little magic, where persons usually go "where'd you think of using that extra variable and that second function and taking the derivative?" It's basic and easy to understand, isn't a theorem or a lemma really (putting down hypotheses or conditions blurs its simplicity). It's just a technique one tries when solving integrals; like partial fractions or trigonometric substitution; except it takes a while to get a hang of using, and at first seems extraneous. I think that's an important fact to something being a trick: it isn't obvious, but it's easy, and it takes a while to get in the hang of using it. As a kicker, Feynman's trick can go very far with little work; Feynman made a living off using this trick.

Answer 8

I've long known the adage that a "trick" works only once whereas a "method" works in multiple instances, or maybe is expected to work in yet unanticipated future instances.

But there's another POV: a trick is something whose efficacy cannot be anticipated, but only by hindsight is seen to work. All methods I've seen of finding $\int \sec x \ dx$ are "tricks". I've always leaned toward viewing unanticipatability as the essence of trickhood.

But I also like Qiaochu Yuan's answer.

Answer 9

While tricks have names because they wind up being associated with with some particular mathematician, tricks are tricks because something important goes on "behind the curtain."

For instance, to prove $$ (a_1 b_1 + \cdots + a_n b_n)^2 \leq ({a_1}^2 + \cdots + {a_n}^2) ({b_1}^2 + \cdots + {b_n}^2), $$ write \begin{align*} A &= ({a_1}^2 + \cdots + {a_n}^2)\\\ B &= (a_1 b_1 + \cdots + a_n b_n)\\\ C &= ({b_1}^2 + \cdots + {b_n}^2), \end{align*} then we must show $$ B^2 \leq AC. $$ Equality clearly holds when $A = 0$. Otherwise, since $\mathbb{R}$ has no negative squares, for all $x \in \mathbb{R}$, $$ 0 \leq (a_1 x - b_1)^2 + \cdots + (a_n x - b_n)^2. $$ Expanding the squares, $$ 0 \leq Ax^2 - 2Bx + C. $$

The quadratic expression vanishes whenever $$ x = \frac{B}{A} \pm > \sqrt{\left(\frac{B}{A}\right)^2 - > \frac{C}{A}}. $$

If $x = \dfrac{B}{A}$, then $$ 0 \leq A\left(\frac{B}{A}\right)^2 - 2 B\left(\frac{B}{A}\right) + C = \frac{B^2}{A} - 2 \frac{B^2}{A} + C = - \frac{B^2}{A} + C, $$ thus $$ B^2 \leq AC. $$

Answer 10

Would regarding a scalar as the trace of a $1\times1$ matrix be considered a "trick"?

Here's an occasion where that's useful:

https://en.wikipedia.org/wiki/Estimation_of_covariance_matrices#Maximum-likelihood_estimation_for_the_multivariate_normal_distribution

Answer 11

Looooong before I was old (then I was ancient, and now I am archeological), this is what I said:

        The difference between a method and a trick is understanding.

That's all.

Trick => method (evolution):   Observe, that once a "trick" is understood it becomes a method. Often, it is still called a trick for various reasons like tradition, recreational value, inertia, etc.

And just a distant association (poetry): once a kenning becomes common, it gets simplified to a metaphor--the metaphor is then like a hint (of that kenning). Perhaps this is one of the reasons why literary critics often are confused, they say that kenning is a kind of a metaphor (hmmm....). Thus we get another evolution: 1.trick => method, 2.kenning => metaphor. (Didn't I say distant?)

Answer 12

A trick is a mathematical life hack.

A life hack is a simple but unexpected solution to a somewhat frequently occurring problem. So it is with a trick; it provides a simple and unexpected solution to a problem that seems baffling to those who don't know the trick. It is important that the trick be unexpected to the uninitiated; if a problem is manifestly a "standard" problem that we expect to have a "standard" solution that we can look up in a "standard" reference, then the standard solution will not be considered a trick. It is also important that the trick have an "easy when you know how" feel to it; if the method is too laborious then we are unlikely to call it a trick.

Answer 13

I would like to mention an important trick in Vanishing theorem in algebraic geometry-The "cyclic cover trick", which is in the MO question: what is the cyclic cover trick?

Answer 14

A "trick" is something that can be clearly seen by hindsight to succeed, but could not be anticipated.

For example: $$ \int \sec x\, dx = \int \frac{(\sec x)(\sec x + \tan x)}{\sec x + \tan x} \, dx = \int \frac{du}u $$

Answer 15

A nice trick by Edmund Landau:

Let us suppose that we need to show the validity of Bertrand's postulate for every $n<4000$ (as in Erdős's famous 1932 paper). According to Erdős, in the case under consideration, Landau's teachings imply that one does not have to look for a prime number in all of the intervals

$$(1,2], (2,4], (3,6], (4,8], \ldots, (3999, 7998],$$

and that it suffices to ponder the following list of fourteen primes in which each of them is smaller than twice the other:

$$2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503, 4001.$$

Indeed, if $N \in [2,4000) \cap \mathbb{N}$, let us denote by $p_{N}$ the greatest prime in the list that is smaller than or equal to $N$; then, if $p_{N+1}$ is the prime in the list that comes right after $p_{N}$, it holds that $p_{N+1} \in (N,2N]$ and we are done.

If a trick is an idea which can be used only once, then the previous "Bemerkung" by Landau (as Erdős refers to it in the aforementioned paper) is definitely deserving of being declared as one, right?

Answer 16

Your question reminds me of Prof. Tao's description of the tensor power trick. Although sounding as an indirect answer, it might be useful to show the POV of a an exceptional mathematician (and profound thinker) about a topic he masterly treated. Verbatim from here: The amplification trick is a deceptively simple one, but it can become particularly powerful when one is arbitraging an unintuitive symmetry, such as symmetry under tensor powers. Indeed, the “tensor power trick”, which can eliminate constants and even logarithms in an almost magical manner, can lead to some interesting proofs of sharp inequalities, which are difficult to establish by more direct means.