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 10 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

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.]

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"?

I am aware this is a borderline question; feel free to close if it unduly distracts.

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

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 3

One well-known trick is a way to evaluate the Gaussian integral $G = \int_\mathbb{R} e^{-x^2}dx = \sqrt{\pi}$ by writing $$G^2 = \left(\int_\mathbb{R} e^{-x^2}dx\right)\left(\int_\mathbb{R} e^{-y^2}dy\right) = \int_{\mathbb{R}^2} e^{-(x^2+y^2)}dxdy$$ 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. MacG. On a "singular" integration technique of Poisson. American Mathematical Monthly 112 (2005), 270-272.

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

Answer 4

http://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 5

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 6

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 7

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

Here's an occasion where that's useful:

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

Answer 8

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 $$ B^2 \leq AC. $$