FOM: Visual proofs -- two examples
Moshe' Machover
moshe.machover at kcl.ac.uk
Mon Mar 1 15:16:05 EST 1999
At 10:12 am -0500 1/3/99, Joe Shipman wrote:
>You're assuming something I'm not ready to assume yet. The reason I presented
>those two proofs was because I wanted FOMers to elucidate
>
>1) what it was about THOSE proofs that made them convincing
>2) why we are persuaded that the proofs could be "easily converted" into a
>rigorous sentential proof.
I can only speak for myself. I found these proofs not only visually
persuasive; but examining them step-by-step I could see how to turn them
into logically correct proofs.
>It is not obvious to me that 1) and 2) are the same. Can you give an example
>of a proof that is as "intuitively persuasive" as the two I gave that
>cannot be
>easily converted into a proof?
Clearly, if you can see how to turn a visual/intuitive argument into a
proper proof, then it is persuasive, but the converse is false.
There are many examples of arguments that seem visually persuasive but
cannot be turned into a proper proof *except by adding an axiom or
condition, or modifying one of the key definitions, or proving a lemma that
turns out to be quite hard to prove*.
Here are a few such examples (none of them terribly original...):
(i) Euclid's proof that an equilateral triangle can be constructed with a
given side (additional postulate needed for correct proof);
(ii) Euler's formula for the vertices, edges and faces of polyhedron
(famous case-study by Lakatos; definition of polyhedron had to be worked
out).
(iii) Any visual argument that takes for granted Jordan's curve theorem
(hard `lemma' needed; think of such an argument presented *before* Jordan
had proved his theorem!)
>There is a pragmatic assumption experienced mathematicians make, that a proof
>which is sufficiently convincing can be converted to a rigorous sentential
>proof. This assumption is justified by experience.
I think it is the other way around: they regard a purported proof as
convincing, hence a real proof, if they can see how to convert it into a
correct sentential proof (of course, I don't mean a *formalized* proof; it
takes an experienced logician to see how to formalize a correct informal
proof).
>It is similar to the
>common assumption (Church's thesis) that any function we can calculate by an
>effective procedure is recursive.
I don't think this is the correct analogue. Rather, the correct analogue is
an argument `by Church's Thesis' that a *particular* function defined in
intuitive verbal terms is recursive. (This kind of argument is often used
by Rogers in his book.) I would accept such an argument as convincing only
if I can see how the appeal to Church's Thesis can be eliminated (say by
using the Recursion Theorem etc.).
[...]
>I would like people to explain why these proofs, which are only persuasive
>because of the pictures they evoke, are rigorizable without begging the
>question by assuming that anything that is persuasive is rigorizable.
Personally, I don't think that proofs, which are *only* persuasive because
of the pictures they evoke are always rigorizable--except by adding an
axiom or a condition, or modifying a key definition or by a similar device.
But then what is ultimately proved is not the same proposition as the one
that the original `proof' purported to prove.
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