FOM: geometrical reasoning Reply-To: simpson@math.psu.edu

[dubucs]

	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):
	"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)

>
>Doesn't the work of Euclid, Hilbert, Szmielew, ... in foundations of
>geometry put this idea to rest once and for all, by showing that
>geometrical reasoning can be formalized within the predicate calculus?
>Do the naysayers have any serious arguments?
>
>-- Steve

	I've been unable to found precise arguments in the relevant
literature on this point (as you can see from the above quotations, the
disagreement with the main trend of foundations is much wider). Thus let me
play devil's advocate, for I'm not sure that your own formulation is very
apposite:
	1) As regards Euclid, the average reader of his "Elements" is
tempted, quite reversely, to susbscribe to the thesis of the irreducibility
of geometrical reasoning to logical one. For consider already the very
first proposition of Book I: "on a given line segment AB, it is possible to
construct an equilateral triangle". Euclid proves it along the familiar
way, by considering two circles of radius !AB! respectively centered on A
and B. The intersection of these circles is visible on the figure, but its
existence does not logically follow from the postulates ! (cf. e.g. H.
Eves, "A Survey of Geometry", 1963, Boston: Ally & Bacon, p. 373 sq.).
Whence the impression, as the correctness of the result seems indisputable,
that geometrical proof is something different from logical proof (this
impression is arguably one of the sources of Kant's doctrine that
geometrical reasoning is synthetic).
	2) Of course, Hilbert is well-known, inter alia, for having
explicited the assumptions (e.g. continuity) which remained hidden in
Euclid. Whence the conclusion that the only geometrical proofs that are
irreducible to logical ones are, after all, the BAD geometrical proofs !
But is it truly the correct lesson from the "Grundlagen der Geometrie" ? As
far as the first-order fragment of his system is only concerned, geometrical
reasoning can be indisputably reproduced in predicate calculus. But
Hilbert's Grundlagen (and specially his formulation of Euclid's "hidden
assumptions") extends far away from first-order language, in such a way
that the relevant notion of consequence is not axiomatisable ...

	Jacques Dubucs

	Institut d'Histoire et de Philosophie des Sciences et des Techniques
	Universite de Paris I Sorbonne Pantheon
	13, rue du Four, 75006 Paris