# What is an instance of a mousetrap proof?

A part of the first chapter of the book The spirit and the uses of the mathematical sciences talks about the beauty of mathematics. The author quotes from a lecture of Hasse and introduces the notion of a mousetrap proof. I feel that the author gives a lucid exposition on what talked about, but regretfully without an example of a proof of such type.

Seeking after is at least one example of a proof of such type.

The original paragraph is in the following formulation, the chapter having which is entitled The characteristic features of mathematical thought and written by PROF. DR. RER. NAT. J. WEISSINGER:

Criteria of beauty at the second level, according to Hasse, are purposefulness and elegance. Purposefulness means that at every point of a proof we should know where we stand and should have the goal in view. The opposite of this is the so-called mousetrap proof, in which we are nudged forward conclusion by conclusion until suddenly the door snaps shut. We feel ambushed, extraordinarily stupid, and irritated by the esoteric ingenuity of the author, yet when we try to gnaw at the bars of the logical conclusions, we are compelled to admit that the proof is solid and without flaws.

Beauteous, is not it? Here the boldfaced words are those quotation-marked in the original text.

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For those without a copy of the book, it would help if you could quote (or summarize) the book's discussion of "mousetrap" proofs. – Nate Eldredge Jul 9 '14 at 13:42
Oh, thanks for your interest; I would love to. But since the book is for now not at hand, tomorrow will be fine. – Gudson Chou Jul 9 '14 at 13:49
I conjecture that "RER. NAT." is the end of "Dr. Rer. Nat.", not the beginning of J. Weissinger's name. – Andreas Blass Jul 10 '14 at 1:36
Thanks so much for your prompt comment. I should have written out all the title, lest there be any unnecessary confusion. – Gudson Chou Jul 10 '14 at 1:43

The following is an excerpts from here (an inaugural lecture of some sort at Heidelberg university):

"Auf der zweiten Stufe der Schönheitskriterien stehen nach Hasse die Forderungen nach Eleganz und nach Zielstrebigkeit. Die letzte bedeutet, daß man an jeder Stelle eines Beweises weiß, wo man steht, und das Ziel vor Augen sieht. Das Gegenteil eines solchen Beweises ist der sogenannte ,,Mausefallenbeweis''. Man tastet sich Schluß für Schluß vorwärts und mit einem Mal fällt die Klappe zu: Man kommt sich überrumpelt und außerordentlich dumm vor; aber so viel man auch an den Gitterstäben der logischen Schlüsse zu nagen sucht, man ist gefangen, der Beweis ist lückenlos richtig."

Euclid's proof of the Pythagorean theorem is not a "mousetrap proof" in this philosophical sense. When it is called by that name then because the movements of the quadrangles in Euclid's proof look like what's happening when a mousetrap fires.

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Much appreciated. I feel some sense of a mousetrap proof, if not a complete understanding. Maybe this notion is not teachable? – Gudson Chou Jul 9 '14 at 13:26

I find that notion (or rather its German translation Mausefallenbeweis) only in the context of the German philosopher Schopenhauer, who coins that in the context of Euclid's proof of the Pythagorean theorem: It logically forces the reader to believe the result without much insight (as opposed to some other proof that uses just a picture and one immediately understands).

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Thanks so much for it. I once guessed this point, but later I thought that if so then every proof by contradiction will become a mousetrap proof. But then why bothering to give a proof by contradiction a new name? – Gudson Chou Jul 9 '14 at 13:06