The n-Category Café

8.12.4 Clever tricks and gamesmanship

Two very different attitudes toward the technical workings of mathematics are found in the literature. In 1761, Leonhard Euler complained about isolated results which ‘are not based on a systematic method’ and therefore whose ‘inner grounds seem to be hidden’. Yet in the 20th century, writers as diverse in viewpoint as Feller and de Finetti are agreed in considering computation of a result by direct application of the systematic rules of probability theory as dull and unimaginative, and revel in the finding of some isolated clever trick by which one can see the answer to a problem without any calculation.

For example, Peter and Paul toss a coin alternately starting with Peter, and the one who first tosses ‘heads’ wins. What are the probabilities $p, p'$ for Peter or Paul to win? The direct, systematic computation would sum $(1/2)^n$ over the odd and even integers:

$$p = \sum_{n = 0}^\infty \frac{1}{2^{2n+1}} = \frac{2}{3}, \qquad p' = \sum_{n = 1}^\infty \frac{1}{2^{2n}} = \frac{1}{3}.$$

The clever trick notes instead that Paul will find himself in Peter’s shoes if Peter fails to win on the first toss: ergo, $p' = p/2$, so $p = 2/3, p' = 1/3$.

Feller’s perception was so keen that in virtually every problem he was able to see a clever trick; and then gave only the clever trick. So his readers get the impression that:

1. probability theory has no systematic methods; it is a collection of isolated, unrelated clever tricks, each of which works on one problem but not on the next one;

2. Feller was possessed of superhuman cleverness;

3. only a person with such cleverness can hope to find new useful results in probability theory.

Indeed, clever tricks do have an aesthetic quality that we all appreciate at once. But we doubt whether Feller, or anyone else, was able to see those tricks on first looking at the problem.

We solve a problem for the first time by that (perhaps dull to some) direct calculation applying our systematic rules. After seeing the solution, we may contemplate it and see a clever trick that would have led us to the answer much more quickly. Then, of course, we have the opportunity for gamesmanship by showing others only the clever trick, scorning to mention the base means by which we first found the answer. But while this may give a boost to our ego, it does not help anyone else.

Therefore we shall continue expounding the systematic calculation methods, because they are the only ones which are guaranteed to find the solution. Also, we try to emphasize general mathematical techniques which will work not only on our present problem, but on hundreds of others. We do this even if the current problem is so simple that it does not require those general techniques. Thus we develop the very powerful algorithms involving group invariance, partition functions, entropy, and Bayes’ theorem, that do not appear at all in Feller’s work. For us, as for Euler, these are the solid meat of the subject, which make it unnecessary to discover a different new clever trick for each new problem.

We learned this policy from the example of George Pólya. For a century, mathematicians had been, seemingly, doing their best to conceal the fact that they were finding their theorems first by the base methods of plausible conjecture, and only afterward finding the ‘clever trick’ of an effortless, rigorous proof. Pólya (1954) gave away the secret in his Mathematics and Plausible Reasoning, which was a major stimulus for the present work.

Clever tricks are always pleasant diversions, and useful in a temporary way, when we want only to convince someone as quickly as possible. Also, they can be valuable in understanding a result; having found a solution by tedious calculation, if we can then see a simple way of looking at it that would have led to the same result in a few lines, this is almost sure to give us a greater confidence in the correctness of the result, and an intuitive understanding of how to generalize it. We point this out many times in the present work. But the road to success in probability theory goes first through mastery of the general, systematic methods of permanent value. For a teacher, therefore, maturity is largely a matter of overcoming the urge to gamesmanship.