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"Is this really a proof?" is the exact question e-mailed to me today from an undergraduate mathematics student whom I know as a highly competent student. The one sentence question was accompanied with the following demo:

enter image description here

I am looking for a down-to-earth, non-authoritative answer who one may give to such a student. What would be your answer if you were faced with such a question?

Update after closure. Reading the comments you may realized that most of them answer the original title of the post: "Is this really a proof?" Of course, the answer to such a question is as clear as the daylight for MO users. And, such a question should be closed asap. But, the actual question was (is) in the body of the post, and it was (is) what your constructive answer would be to such a student if you were faced with such a question. Now, with the change of the title, the actual question is much more clearer, and I hope, worthy of MO attention.

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    $\begingroup$ No. It's not an actual proof. It's an excellent example, though. $\endgroup$
    – Asaf Karagila ♦
    Commented Aug 7, 2013 at 9:41
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    $\begingroup$ I'm with Asaf. A very pretty device. But you know the angles, lengths and volumes are not exact. And there are a lot of other shapes of right triangle. A proof has to cover all that. $\endgroup$
    – Colin McLarty
    Commented Aug 7, 2013 at 9:44
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    $\begingroup$ I want one of those. $\endgroup$
    – Paul Reynolds
    Commented Aug 7, 2013 at 10:24
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    $\begingroup$ No proof. A magician could use containers with different depths to 'prove' fake theorems. $\endgroup$
    – Uwe Stroinski
    Commented Aug 7, 2013 at 10:47
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    $\begingroup$ Well, it's a fun device, but aside from the obvious objections given before, it also doesn't give any insight into why the theorem is true. Isn't that really what a proof is good for? $\endgroup$
    – Todd Trimble
    Commented Aug 7, 2013 at 13:57

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Approximate equality in one or more examples is all that can be demonstrated by physical measurements and pictures. This certainly is a useful first step in proof of "exact" (ideal) equality, or in suggesting that there might be an underlying causal mechanism. But there are many geometric dissection problems/puzzles (which I cannot quickly locate, unfortunately) which seem, pictorially, to decompose a figure into pieces whose total area does not add up to the true total. Of course, the "catch" is a tiny imprecision in drawing.

Nevertheless, if a physical demonstration is not deliberately "rigged" to give a deceitful result, an accidental measured-equality is on the whole very convincing, in the same way that so-called Monte-Carlo testing is fairly convincing.

Methodologically, too, I suppose one routinely checks the plausibility of an assertion before allocating much effort to proving it, and physical demonstrations can be quick and effective. (Maybe construction of such an elaborate model as in the demo above wouldn't be usual!)

For that matter, the Euclidean picture-drawing rules-of-proof are themselves a fairly stylized game, as corroborated by Hilbert-et-al's eventual observation that there were some implicit assumptions. Not that the two-thousand-year-old conclusions were wrong, but only that some visual/physical assumptions were being used, in addition to a supposed axiomatic set-up.

Pictures and physical demos certainly capture a diffident audience's attention better than narrative.

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  • $\begingroup$ @DanFox, yes, I realize that the demo above was not a random-measurement situation, which is what I meant to contrast to. Thus, that demo, obviously contrived with some trouble, at best demonstrates plausibility. I guess I was unclear... $\endgroup$
    – paul garrett
    Commented Aug 7, 2013 at 13:38
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    $\begingroup$ One such dissection problem is here: math.stackexchange.com/a/348298/18398 $\endgroup$
    – JRN
    Commented Aug 7, 2013 at 13:50
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    $\begingroup$ @JoelReyesNoche... Excellent! :) Thanks! $\endgroup$
    – paul garrett
    Commented Aug 7, 2013 at 13:51
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Clearly it is not a proof. In the same way we could "demonstrate" the squaring of the circle (and it has been done more or less that way already by many amateurs). However, isn't it nice to "see" Pythagoras that way ?

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