FOM: geometrical reasoning

[Stephen G Simpson]

Robert Tragesser's posting of 18 Feb 1999 20:41:38 suggests the
existence of scientifically significant distinctions between physical
and mathematical and computational methods of reasoning, even when
these methods lead to identical or at least compatible conclusions.
And I guess Tragesser is trying to hint that the same kinds of
distinctions may come into play within mathematics, in the sense that
a piece of non-logical geometrical or diagrammatic reasoning may be
sometimes be somehow more satisfying than a sequence of syllogisms
leading to identical or at least compatible conclusions.

In my opinion, Tragesser's point is correct as far as it goes, but it
misses the point of the remark of Dubucs in 12 Feb 1999 06:02:14 which
initiated this thread.  According to Dubucs:

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

Later, in 16 Feb 1999 09:48:52, Dubucs quoted from an essay by someone
named Boi in a book called `Revolutions in Mathematics':

  "Thom asserts that the "phantasmatic" solution which characterizes
  modern mathematics is that which consists in generating the
  continuous from the discrete. It is the paradigmatic solution
  proposed by those who have developed a conception of mathematics
  based on set theory according tho which our understanding of the
  continuum is reduced to and explained by the (discrete) model of the
  field of real numbers R (e.g. of the real straight line), and any
  purely geometric explanation of the continuum is dismissed in favour
  of a reduction to the arithmetical model" (p. 190) 

  "There are certain fundamental intuitions which inspire and guide
  mathematical discovery and development; they are alinguistic and
  cannot be completely formalized. An example is the spatial
  continuum, which cannot be reduced to any axiomatic construction"
  (p. 207)
  
In other words, according to Thom and Boi, the understanding achieved
by geometrical intuition is somehow totally incompatible with and
inaccessible to logical formalization.  According to them, the
standard set-theoretical or arithmetical explanation of the continuum
is worthless.  It is to be discarded in favor of an unspecified kind
of purely geometrical understanding.  Even the standard
definition-theorem-proof model of mathematical exposition is to be
discarded, in favor of -- what?  There is a certain mystical or
Brouwerian overtone here.  It is a call for rebellion against the
allegedly crippling bonds of logic.

I now have Boi's essay and I'll comment more on it later.

-- Steve