6
$\begingroup$

It is now common to come across the term "left as exercise" in mathematics textbooks, and from there a comical usage of such terms was developed, typically by applying them to absurdly difficult propositions.

Who is the first one to use such phrase? When did this phrase gain its popularity? In general, what's the history of this term (and its close relatives, where the solution to a problem is omitted)?

Improve this question
$\endgroup$
3
  • 5
    $\begingroup$ The history is left as an exercise to the reader. $\endgroup$
    – Jon Custer
    Dec 19, 2022 at 22:29
  • 5
    $\begingroup$ The practice goes back at least three centuries. Edmund Wingate, Mr. Wingate's Arithmetick, 11th ed., London, J. Philips 1704, p. 79: " And therefore I shall conclude ... with the following Questions, whose Answers are annexed to them, and may be found out by the preceding Rules; but the operations are purposely omitted, and left as an exercise for the Learner." p. 544 "... may be resolved in another way, but I leave that as an Exercise to the Wit of the ingenious Reader." $\endgroup$
    – njuffa
    Dec 20, 2022 at 2:03
  • 1
    $\begingroup$ With all respect to those who have replied or commented, I suggest that some uses of 'left as an exercise' are, at least, more legitimate than 'proof by intimidation'. For example. some trigonometric developments and proofs can be very lengthy in expansions, rearrangements, &c, without involving any difficulty of principle beyond sheer volume. 'Left as an exercise' can save paper, ink and boggling the reader's eyesight. :) $\endgroup$
    – terry-s
    Dec 27, 2022 at 19:00

2 Answers 2

Reset to default
9
$\begingroup$

The oldest I can recall is Descartes in La Géométrie (1637):

But I shall not stop to explain this in more detail, because I should deprive you of the pleasure of mastering it yourself, as well as of the advantage of training your mind by working over it, which is in my opinion the principal benefit to be derived from this science. Because, I find nothing here so difficult that it cannot be worked out by any one at all familiar with ordinary geometry and with algebra, who will consider all that is set forth in this treatise.

Subsequent authors shortened the expression considerably. :)

Improve this answer
$\endgroup$
1
  • 1
    $\begingroup$ Original bottom of p. 301: "Mais ie ne m'areste point a expliquer cecy plus en detail, a cause que ie vous osterois le plaisir de l'apprendre de vous mesme, & l'utilité de cultiver vostre esprit en vous y exerceant, qui est a mon avis la principale, qu'on puisse [p. 320] tirer de cete science. Aussy que ie n'y remarque rien de si difficile, que ceux qui seront un peu versés en la Geometrie commune, & en l'Algebre, & qui prendont garde a tout ce qui .." $\endgroup$
    – njuffa
    Dec 25, 2022 at 11:17
5
$\begingroup$

While I can't state the origin of the use of the specific term left as exercise, and further research is needed, I think it can be framed in a broader context.

Left as exercise can be considered a special case of proof by intimidation.

Proofs by intimidation are important rhetorical devices, useful when you don’t know exactly what you are saying, but you want to persuade the audience of your argument. They are studied in rhetorical logic:

[…] rhetorical logic, a promising field of science, of great value to those writing research proposals. It provides new, and utterly convincing tools for closing embarrassing gaps in your reasoning, without resorting to brute-force methods.$^1$

The sharpest example of proof by intimidation is:

Proof: Trivial. $\Box$

Proofs by intimidation can be introduced by phrases as

"Clearly..."

"It is self-evident that..."

"It can be easily shown that..."

"Left as exercise".

Here a possible definition of proof by intimidation:

[...] “Proof by Intimidation.” The aim here is to make something sound terribly difficult, using as much jargon as possible, and then ending with “so obviously X holds.” Though the argument may be completely obscure, even totally incorrect, proof by intimidation is understood by everyone who is too vain to admit they don’t understand you.$^2$

Left as exercise and Trivial are the most efficient proofs by intimidation, as they make immediately, and without waste of words, the audience feel stupid, and accept the thesis in silence.

The mathematician Giancarlo Rota said, in his book Indiscrete Thoughts, that the expression proof by intimidation was introduced by Marc Kac after the lessons of William Feller:

He took umbrage when someone interrupted his lecturing by pointing out some glaring mistake. He became red in the face and raised his voice, often to full shouting range. It was reported that on occasion he had asked the objector to leave the classroom. The expression "proof by intimidation" was coined after Feller's lectures (by Mark Kac). During a Feller lecture, the hearer was made to feel privy to some wondrous secret, one that often vanished by magic as he walked out of the classroom at the end of the period. Like many great teachers, Feller was a bit of a con man. — Rota, Gian-Carlo, 1932–1999. (1997), Indiscrete Thoughts.$^3$

$^1$ https://www.es.ele.tue.nl/~tbasten/fun/rhetoric_logic.pdf

$^2$ ibid.

$^3$ https://en.wikipedia.org/wiki/Proof_by_intimidation

Improve this answer
$\endgroup$
2
  • 2
    $\begingroup$ Wow! I was noticing this arrogance in many mathematical papers (mainly applications in chemistry): It can be easily shown that...yes indeed. $\endgroup$
    – AChem
    Dec 20, 2022 at 6:12
  • 2
    $\begingroup$ When I was a student , in a course of mathyematical analysis, the frequent phrase of the professor 'it is very simple' before a proof terrorized the students. $\endgroup$
    – BakerStreet
    Dec 20, 2022 at 10:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.