## October 19, 2019

### Jaynes on Clever Tricks

#### Posted by Tom Leinster

I’ve posted before about the eminently browsable, infuriating, provocative, inspiring, opinionated, visionary, bracing, occasionally funny, unfinished book Probability Theory: The Logic of Science that arch-Bayesian apostle of maximum entropy Edwin Jaynes was still writing when he died.

All I want to do now is to share section 8.12.4 of that book, Clever tricks and gamesmanship, which acts as balm for anyone who feels they’re not good at “tricks”. The rest of this post consists of that section, verbatim.

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.

Posted at October 19, 2019 5:52 PM UTC

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### Re: Jaynes on Clever Tricks

This is more of a drive-by comment than a properly considered thought, but I find Jaynes’s choice of example here a bizarre one to justify his point, because it is precisely not an isolated trick! Rather, it is a prototype or toy version for a whole body of conditioning-flavoured arguments in probability theory, bringing in ideas such as symmetry and information.

I would go as far as to say that Feller’s argument is the probabilists’ way of thinking and the arguments with series are the analysts’. Feller’s approach also seems, dare I say it, more categorical in exploiting invariance/symmetry?

Posted by: Yemon Choi on October 19, 2019 8:35 PM | Permalink | Reply to this

### Re: Jaynes on Clever Tricks

I see your point. Nevertheless, Jaynes positions himself as the champion of arguments such as symmetry, information and invariance:

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.

(Emphasis added.) It would be good to see a better-chosen example.

Posted by: Tom Leinster on October 19, 2019 9:08 PM | Permalink | Reply to this

### Re: Jaynes on Clever Tricks

I have the feeling that prototypes of general techniques being shorn of context and presented as “clever tricks” is a fairly general phenomenon.

The technical result that de Finetti is most closely associated with — i.e., the thing people named for him — is the de Finetti theorem, which is all about symmetry. (Consequently, so too are its quantum generalizations.)

Posted by: Blake Stacey on October 19, 2019 9:47 PM | Permalink | Reply to this

### Re: Jaynes on Clever Tricks

I have the feeling that prototypes of general techniques being shorn of context and presented as “clever tricks” is a fairly general phenomenon.

I guess one can separate out two different phenomena here.

The first involves arguments that look like clever tricks, but are in fact instances of an established general technique. Any such argument could be rewritten in terms of the general theory, thus lessening the impression that one has to be amazingly clever in order to do the subject.

The second phenomenon involves clever tricks that really are one-offs, at least as far as we know. There’s no general theory into which they embed. In that case, a more radical solution is needed if the subject is to be made accessible to the less ingenious.

Posted by: Tom Leinster on October 20, 2019 12:25 AM | Permalink | Reply to this

### Re: Jaynes on Clever Tricks

I guess Tom is too modest to quote himself, but here’s what he had to say about cleverness some 10 years ago:

It’s great for mathematics to be visionary, beautiful, etc., but if cleverness is the first quality that comes to mind then it suggests to me that something is insufficiently understood.

That seems to align with the idea that something may seem clever at first, but in fact turn out — when sufficiently understood — to be a “prototype or toy version” of a fundamental conceptual idea.

Posted by: Mike Shulman on October 20, 2019 4:13 AM | Permalink | Reply to this

### Re: Jaynes on Clever Tricks

This reminds me of an anecdote that Halmos told about von Neumann.

Then there is the famous fly puzzle. Two bicyclists start twenty miles apart and head toward each other, each going at a steady rate of 10 m.p.h. At the same time a fly that travels at a steady 15 m.p.h. starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner till he is crushed between the two front wheels. Question: what total distance did the fly cover?

The slow way to find the answer is to calculate what distance the fly covers on the first, northbound, leg of the trip, then on the second, southbound, leg, then on the third, etc., etc., and, finally, to sum the infinite series so obtained. The quick way is to observe that the bicycles meet exactly one hour after their start, so that the fly had just an hour for his travels; the answer must therefore be 15 miles.

When the question was put to von Neumann, he solved it in an instant, and thereby disappointed the questioner: “Oh, you must have heard the trick before!”

“What trick?” asked von Neumann; “all I did was sum the infinite series.”

Posted by: Oscar Cunningham on October 19, 2019 10:53 PM | Permalink | Reply to this

### Re: Jaynes on Clever Tricks

I’ve learned to be wary of anything in category theory that provokes the word clever. I don’t mean that I like my category theory stupid, of course. But if I ever catch myself thinking of something I’ve done as ‘my clever trick’ (and I have), then I know I’m in trouble. Each of those three words should ring alarm bells. It’s great for mathematics to be visionary, beautiful, etc., but if cleverness is the first quality that comes to mind then it suggests to me that something is insufficiently understood.

Posted by: Todd Trimble on October 20, 2019 1:24 AM | Permalink | Reply to this

### Re: Jaynes on Clever Tricks

Thank you, and Mike, for remembering! Yes, that popped into my head too. It struck me that although Jaynes was such an anti-Bourbakist, and so withering about the standard modern rigorous manner of writing, here he was making points that would strike a chord with many category theorists.

I mean, here’s what he said about the intrusion (as he saw it) of measure theory into probability theory:

We could convert many 19th century mathematical works to 20th century standards by making a rubber stamp containing this Proclamation, with perhaps another sentence using the terms ‘sigma-algebra, Borel field, Radon-Nikodym derivative’, and stamping it on the first page.

I suppose it’s not really a paradox. He simply wanted to be free of what he saw as extraneous clutter, whether it was the mental clutter of ad hoc clever tricks or the linguistic clutter of measure theory and sets and pedantic precision. I don’t say I agree — I probably don’t — but I think I understand his point.

Posted by: Tom Leinster on October 20, 2019 12:29 PM | Permalink | Reply to this

### Re: Jaynes on Clever Tricks

Seeing that quote again, I feel a certain kinship between Jaynes and Errett Bishop. I don’t have anything with me to quote right now, but I seem to remember the latter also having withering things to say about excessive generality and formality. I wonder whether he ever opined on cleverness.

Posted by: Mike Shulman on October 20, 2019 3:24 PM | Permalink | Reply to this

### Re: Jaynes on Clever Tricks

That’s interesting; I’d be interested in any details you should find in the lore on Bishop.

I cannot speak with any authority on the mathematics of Bishop as analyst/geometer, but from what I glean (especially from this article), much of it was in the interface between complex analytic and algebraic geometry and functional analysis. My rough impression based on this is that he was not averse to using appropriately abstract tools there, and I also think almost every mathematician is interested to some degree in what generality their theorems hold.

It must be added that by all accounts, including the linked article, he was considered an astonishingly powerful mathematician during an an era in which abstraction and generality were generally highly valued. Whatever he himself may have opined on “cleverness”, his contemporaries considered him enormously clever, a genius in fact.

An aversion to formalism on his part wouldn’t surprise me much, from what I understand (which, again, isn’t a great deal).

Posted by: Todd Trimble on October 21, 2019 12:47 AM | Permalink | Reply to this

### Re: Jaynes on Clever Tricks

Regarding generality, what I’m half-remembering is a quote about the edifice of point-set topology “collapsing down to size” when we restrict it constructively to metric spaces. But maybe I’m misremembering or misinterpreting it.

Posted by: Mike Shulman on October 21, 2019 11:41 AM | Permalink | Reply to this

### Re: Jaynes on Clever Tricks

In writing that passage, Jaynes was thinking in the context of physicist and engineers who apply the theory - for most of them, Fellers argument seems like a trick. For the mathematician, Fellers argument is more logical. Fellers work, pushing the frequency interpretation of probability, was clearly more couched in the language of math, but fortunately, all the good math in the world can not help you when you start out on the wrong philosophical footing. Namely, the idea that statistics was somehow more objective that “subjective probability’’ (conditioned on your background info.), was erroneous from the get go. It’s remarkable that Bayesian statistics is now the preferred method of analysis for most statisticians.

Posted by: kirk sturtz on October 20, 2019 2:58 AM | Permalink | Reply to this

### Re: Jaynes on Clever Tricks

I don’t think it’s true that Bayesian statistics is now the preferred method of analysis for most statistics, but it has certainly grown in popularity since the early/mid 20th century, mainly because computational advances have made it possible to actually carry out Bayesian analysis in many situations where it was not previously possible.

Posted by: Evan Patterson on October 20, 2019 3:37 AM | Permalink | Reply to this

### Re: Jaynes on Clever Tricks

In this example the formulas with series are important because they show you don’t have to be sneaky to answer the question: these series are pretty much just the definition of the probabilities for Peter or Paul to win. The series make sense even if you don’t know how to sum them — which is good, because in fancier problems such a closed-form formula may not exist.

The ‘trick’ is important because it illustrates a nice combination of symmetry and conditioning. If one has a powerful enough framework this won’t seem like a ‘trick’: it’ll seem like an example of some techniques.

I think it’s really important for people to invent tricks and then for people — possibly different people — to study these tricks, generalize them, and clearly explain the principles that underlie these tricks, so they are no longer ‘tricks’.

So I’m glad that some people love tricks and other people hate tricks.

Posted by: John Baez on October 20, 2019 3:07 AM | Permalink | Reply to this

### Re: Jaynes on Clever Tricks

So I’m glad that some people love tricks and other people hate tricks.

Jaynes’s book contains many, many mentions of Feller, who he seems to have known well. Almost every mention explains how Feller was wrong on some point or other. Nevertheless, he seems to have held him in grudging respect. A footnote on p.466 reads:

Since we disagree with Feller so often on conceptual issues, we are glad to be able to agree with him on nearly all technical ones. He was, after all, a very great contributor to the technical means for solving sampling theory problems, and practically everything he did is useful to us in our wider endeavors.

It’s a very back-handed compliment and if I’d been Feller I wouldn’t have appreciated it much (“oh, he’s technically able”). But I suppose he’s acknowledging your point: it’s important that we have a wide diversity of mathematical life-forms.

Posted by: Tom Leinster on October 20, 2019 12:42 PM | Permalink | Reply to this

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