# FOM: geometrical reasoning: reply to Black

dubucs dubucs at ext.jussieu.fr
Fri Feb 19 22:42:15 EST 1999

```In his posting of Feb 19, Robert Black makes interesting claims which
deserve some further remarks:
>
>1.  Euclid's proofs make essential use of his diagrams, which (among other
>things) led Kant to think that geometrical, and more generally
>mathematical, proofs make essential use of intuition.

a) Yes, it's crucial to notice that the question in debate does not concern
the use of intuition to get "true axioms" (as e.g. in G"odel), but its use
in the course of geometrical proofs.
b) Some years ago, Hintikka (after Beth) gave an interesting answer to this
question: geometrical proofs actually use intuition, but "intuition" is to
understand here as "instanciation", thus as a purely logical ingredient
(roughly expressed: we need, in the course of geometrical proofs, to grasp
SINGULAR objects). However subtle it may be, this heroical attempt to
vindicate Kant does not fit very well with textual evidence in Kant's
writings. A reasonably argumented exegetical discussion on this point is
probably out the scope of FOM, but arguments could be produced to show that
Kant had quite different ideas in the mind, with effect that geometrical
reasoning is genuinely irreducible to logical one.
c) It's to be stressed that Kant's geometrical intuition is not empirical
intuition of the figure, and that its doctrine is not as childish as "Let
you see the truth of the theorem on the blackboard". The obvious flaw of
this last slogan is the lack of generality of the result of the "proof": we
get by this way no guarantee that the theorem applies to another figure, on
another blackboard (it's the famous "Locke-Berkeley problem"). Actually,
Kant alludes rather to "transcendantal" intuition (an expression which is
not easy to briefly explain, surely Charles Parsons could help !). Sketchly
expressed, it's the intuition we have of our own cognitive and perceptual
constitution. Therefore, when we use it in a proof, the result may have
general (while not strictly universal) value: it applies to any figure (of
the same kind, e.g. triangle) that human beings (or creatures with a
similar constitution) can have the experience of.
d) It could be of some interest to reflect on this train of thoughts
outside any reference to kantian or transcendantalist context, if only to
make this viewpoint commensurable with the main tradition of f.o.m. and to
make a discussion possible. Tentatively, I'd like to briefly suggest a
comparison with the so-called "preferential logics" that are advanced in
some artificial intelligence quarters. Roughly stated, B is a "preferential
consequence" of a set E of premisses if B is true in any model (of a
certain privilegied or "prefered" class) in which each member of E is true.
Mutatis mutandis, the "prefered" class of models would correspond to the
kantian "intuitional possibilities", opposed by Kant to the "conceptual
possibilities" which encompasse the whole class of all the models. In other
words, according to this suggestion, geometrical ("transcendantal")
intuition a priori selects a category of models, and it's left to the
logical reasoning to establish that the sentence to be proved is valid in
this category. The allegued impossibility (or inadequation) of SAYING (by
way of axiomatic characterization) what are the distinctive features of the
prefered class of models (and, therefore, of working as usual with
reference to ALL the models of the axioms) would be a very natural
counterpart of Kant's motto that we have only intuitive (NOT CONCEPTUAL)
knowledge of the a priori conditions of our possible experience.
>
>2.  Hilbert showed that by adding extra axioms the essential use of
>diagrams and spatial intuition in proofs in Euclidean geometry could be
>avoided.
>
>3.  However, you can only get a categorical axiomatization of the whole of
>geometry in second-order logic, which is of course not recursively
>axiomatizable.
>
Of course, at first sight, one can't disagree. However, let me provoke :
Surely, Hilbert has provided us with a fully logical characterization of
the euclidian space, in the sense that his axioms define it uniquely up to
isomorphism. But does this result establish that geometrical REASONING is
LOGICAL reasoning ? To get the conclusion, you have to assume that if B is
logical consequence of E, then there is a "logical reasoning" that leads
from E to B. In the context of the present discussion, "logical reasoning"
should mean "reasoning without appeal to intuition (of course, we are
speaking of reasoning not as psychich process (which is indisputably not
free of intuition, uncertain heuristics, luck, irrationality, a.s.o), but
as result of such a process, like a chain of syllogisms, or something
like). Now, the best explanation we have, for a chain of inferences, of
being intuition-free and without gap, is probably: whose correction does
not depend on the interpretation of the symbols that occur in it, but only
on their shape, and which can be effectively checked. In short, "logical
reasoning" in the relevant sense should be more or less "formal proof".
Admitting that, the probleme is that in our case, namely higher-order
geometry, B can be logical consequence of E without being conclusion of a
"logical reasoning" starting from E (of course, the objection does not
apply at all to first-order or "weak-second-order" geometry Steve Simpson
evoked in a recent posting).
Note of course that the "full" second order geometry with his assumptions
of continuity is of course taken by the neo-transcendantalists as the
"true" geometry
>
>Like Charles Parsons, I'd be grateful for references to where these claims
>are being made beyond Luciano Boi's paper.  (Boi mentions an article by
>Thom in an Italian encyclopedia which unfortunately isn't in our library.)

As I've said in my first posting, there is, to my knowledge, very few
references in English language. To tell the truth, I'm (delightfully)
suprised, viewing the composition of FOM, to be asked to give references in
my own idiom (let me take the opportunity of apologizing for my poor
English, I hope that the lines above make however sense to the FOMers).
Here some items:

Luciano Boi, Le probleme mathematique de l'espace, Springer 1995

L. Boi, D. Flament, J. Salanskis (eds.), 1830-1930, A Century of
Geometry, Springer, "Lectures Notes in Physics", 402-1992

R. Thom, Paraboles et catastrophes, Paris: Flammarion, 1983
R. Thom, Apologie du logos, Paris: Hachette, 1990
R. Thom, Stabilite structurelle et morphogenese, 2d ed., Paris:
Intereditions, 1977
>
>Also:  haven't we been here before?  A hundred years ago Poincare in his
>neoKantian attacks on logicism was saying something very like this.  And he
>too was sliding from a brilliant presentation of point (5) above (e.g. in
>'Les definitions mathematiques et l'Enseignement') to the wholly
>unjustified conclusion that mathematical proofs couldn't be captured in
>logicist systems.

Poincare is indisputably an heroe for this trend. But there is
other sources too, specially when one looks to their claim of the absurdity
of deriving continuum from discrete. To my view (but I suspect that members
of the school would fully disagree!), it's obviously also, in this respect,
a revival of Bergson's diatribes against the denaturation of the intuition
of the continuum by conceptual thought ...
>
Jacques Dubucs
IHPST CNRS Paris I
13, rue du Four
75006 Paris

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

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