FOM: Visual proofs response to Shipman
Moshe' Machover
moshe.machover at kcl.ac.uk
Tue Mar 2 12:37:13 EST 1999
I would like to respond *briefly* to Shipman's (Mon, 01 Mar 1999 16:47)
reply to my posting, without lengthy rehashing and multiply nested quotes.
1. It seems to me that *any* good (ie prima-facie persuasive) visual
`proof' can be rigorized--ie turned into a proof proper--*if* you allow
adding extra axioms or conditions or modifying a key definition. However,
if you do make such modifications then, in an important sense, the theorem
that is eventually proved (correctly) is not the same one that the original
visual argument puported to prove. For example, Euclid's purported proof of
the constructibility of an equilateral triangle with given side is a faulty
argument in support of a proposition about a not-necessarily-continuous
plane (continuity is not postulated). The rigorized version is a correct
proof of a theorem about what we call the Euclidean plane, which is
continuous (continuity is postulated).
It is of course historically true, as Shipman points out, that no-one had
any problem about adding continuity postulates. But this is beside the
point. I suspect that the fact that people did not make much fuss, and had
the impression that (despite the addition of crucial new postulates) the
"same" theorem is being proved, was largely due to some kind of intuition
about *physical* space--an intuition which is most probably faulty anyway.
[I am aware that the above view of Euclid is a travesty from the standpoint
of the history of mathematics. I am viewing the matter anachronistically,
from the standpoint of modern mathematics. I am not trying to see matters
from Euclid's own viewpoint.]
2. My main point is that in practising and teaching mathematics, one must
cultivate maximal use of visual/intuitive arguments as a means of
understanding and invention; and at the same time utmost suspicion of such
an argument as a means of validation, so long as it is not clear that it
can be rigorized. This suspiciousness is not a logician's luxury, but is
essential for doing modern mathematics.
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