# FOM: logic, geometry, and intuition

dubucs dubucs at ext.jussieu.fr
Tue Feb 23 08:53:32 EST 1999

```	Dear Steven Simpson,

Here my ultimate word on the topic, to call on other participants
to evaluate and discuss my last claims on geometry and its relationship to
logic.

(I assume here that higher-order geometrical notions as continuity
are really required by our scientific representation of the world; but this
assumption could certainly be contested (cf my 21 Feb posting to FOM), and
I'd interested to have opinion of FOMers on this point).

1) The euclidean structure of space is described without intuitive
residue and with a maximal precision (i.e.: categorically) by Hilbert's
system H of the Grundlagen der Geometrie

2) However, we don't have, and we never will have, because we can't
have, any sound system S of formal recursive rules with the property that
any geometrical truth (i.e. sentence that is true in the euclidean
structure) were derivable from H by means of S.

I regard both 1) et 2) as established and significant facts from
which any serious discussion of geometry from a logical standpoint should
start.

2) means that we can't have formal access from H to all the
sentences that hold in the unique model of H. Maximal precision in the
description of geometrical structure, or completeness of a system of formal
rules for deriving geometrical truths, one has to choose.

Which branch of the alternative one chooses is at a large extent
delicate matter, mix of scientific opportunity and philosophical
inclination.

For my part, I suggest the following remark as an argument
favouring the first option. Henkin-style strategies to extend first-order
deductive completeness to the so-called "so-called "higher-order logics""
basically rest on the interpretation of "higher-order" as "many-sorted".
This train of thoughts is certainly of great merit in pure logic. But as
far as geometry is concerned, this interpretative decision implies that we
have to renounce to consider lines, polygons, polyedres, a.s.o. as sets of
points, and it leaves us with the archaic conception of them as heterogens
kinds of geometrical beings, each sui generis. A terribly long time has
been necessary to deliver us from such an aristotelician image of the
geometrical entities as divided in incommensurable genus. Much hard
mathematical work has been done at the end of the 19th century to overcome
this prejudice. To my knowledge, the strongst contest against this
mathematical splendid work is just contained in Bergson's writings. He
writes at lenght, is so many words, that a continuous segment can't be
built up out of points, for continuity is something absolutely basic. This
is precisely what neo-transcendantalists claim. And this is precisely what
I contest. Prof. Simpson, are you sure you were happy to live in such a
vicinity ?

Leaving FOM tribune for a moment, I'm confident that my arguments
will be weighted here in the light of reason, not ad hominem, ad nationem,

Jacques Dubucs
IHPST	CNRS Paris I
13, rue du Four, 75006 Paris
Tel: (33) 01 43 54 60 36
Fax: (33) 01 44 07 16 49

Jacques Dubucs
IHPST CNRS Paris I
13, rue du Four
75006 Paris
Tel (33) 01 43 54 60 36
(33) 01 43 54 94 60
Fax (33) 01 44 07 16 49

```