FOM: geometrical reasoning

[Michael Zeleny]

>From: Charles Parsons <parsons2 at fas.harvard.edu>
>>Steve Simpson wrote:
>>>Jacques Dubucs writes:

>>>>there is an increasingly fashionable tendancy to consider again
>>>>geometric reasoning by means of diagrams or figures as somehow
>>>>irreducible to logical inference.

>>>I wasn't aware of this trend.  Could you please summarize some of the
>>>arguments here on FOM?

>>	Actually, this train of thoughts, which emphazises rather strong
>>precedence of geometry over logic, is more represented in "continental"
>>(i.e: european) areas than elsewhere. This trend, which could be qualified
>>(or which qualifies itself) as "neo-transcendantalism" or "hermeneutism"
>>for reasons which would be tedious to explain in a sketchly presentation,
>>is mainly inspired by the French mathemaician Rene Thom (winner of a Field
>>medal in 1958). As few texts from this school are still written or
>>translated in English, the best is probably to give some short excerpta
>>from an english paper by a good representant of it (L. Boi, "The
>>'revolution' in the geometrical vision of space in the XIXth century, and
>>the hermeneutical epistemology of mathematics", in D. Gillies (ed.),
>>"Revolutions in Mathematics", Oxford: Clarendon Press, pb 1995):

>That's helpful, but perhaps you could also add references to some of the
>principal expressions of this kind of view in French.
>
>As regards the history of geometry, serious work has been done by Kenneth
>Manders to try to describe accurately and to understand the practice of
>reasoning from diagrams. I don't know what of this is published. (Ken, are
>you listening?) That's independent of the question whether, given where we
>stand today, the view you describe has any merit.

I would be interested in seeing this last question addressed in a way
that contravenes Gorgias' third maxim.  For it would seem that putting
it in writing from left to right, and thence from the top downwards,
must either traduce the doctrine or fall short of expressing it.  One
way to approach the claim of geometrical reasoning's irreducibility
might lie through exploring the history of Euclid's "pons asinorum".
But what sort of communicable evidence is admissible in such cases?

Cordially -- Mikhail Zeleny at math.ucla.edu * MZ at ptyx.com ** www.ptyx.com
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