FOM: Geometric proofs
Joe Shipman
shipman at savera.com
Tue Feb 23 08:59:32 EST 1999
Hersh writes:
>I agree that Tarski can give a first-order proof of everything
>in Euclid's Elements.
>Nevertheless, mathematicians use geometric proofs, even
>if you claim you can prove they aren't proofs.
>There's "Visual Complex Analysis," I forget the author's name,
>it should be in your library.
>There's Arnold's books on ordinary differential equations.
>Mathematics magazine for years has published "proofs without words."
>Recently a collection of proofs without words was published
>as a book entitled "Proofs without words." If it's not in
>your library, just look at some issues of Math Magazine.
>Why do mathematicians do this?
>Because a geometric proof is often more convincing, more perspicuous,
>more interesting, more memorable.
Reuben,
Thanks very much for reminding us of these "geometric" proofs. In the
case of the "proofs without words", the diagrams represent a compact
visual representation of some formal information, and it is always easy
to translate the geometric proof into a formalized one which represents
the quantities depicted in the diagram and their relationships with
formal symbols. But you are correct that the wordless "proof by
diagram" is *primary* in the psychological sense. This does not
invalidate Steve's point, because in all of the wordless proofs I've
seen the translation is easy and straightforward, and for some
algebraically-minded individuals who are not good at visualization the
formal translation may actually be easier to understand. More
interesting would be a result that wordless proofs could be in some
*measurable* way superior, for example more compact or more "generic".
I haven't seen "Visual Complex Analysis", but from what I remember of
Arnold's books it is also straightforward to translate the proofs into a
diagramless form; however there may be an additional benefit here to the
visual proofs, namely that they allow one to *understand* the proof and
not just verify it (the "proofs without words" in Mathematics magazine
can also be said to improve one's understanding, but in those cases the
results are elementary enough to be easily "understandable" even without
a diagram). I would enjoy being proved wrong here -- can you give an
example from Arnold or from the "Visual Complex Analysis" book where
translating the proof is not obviously possible? Even if there is no
such example, the apparent necessity of visual proofs for full
understanding (and the related necessity of geometric or visual
reasoning for finding the proofs in the first place) has great
foundational significance because it suggests that when we make proofs
in a formal language *primary* we lose something important (though if a
visual-formal translation is always possible the loss doesn't affect our
metamathematical results but only our philosophical conclusions).
What I'm *really* interested in is if there is geometric reasoning which
is NOT obviously translatable into a formal language [or is only
translatable with the help of new geometric axioms independent of ZFC].
As I remarked earlier, I think low-dimensional topology is a good place
to look for this. Does anyone have any candidates?
-- Joe Shipman
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