FOM: geometrical reasoning

[Robert Black]

The following five claims seem to me to be obviously true:

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.

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.

4.  The Dedekind/Cantor analysis of the continuum is not the only possible
one, and there is a fruitful mathematical project of exploring alternatives.

5.  Mathematicians think informally in 'intuitive' (and often geometrically
pictorial) ways; without an ability to think in this way one would often
never get a proper understanding of what a formal proof is really about.
Much of learning mathematics concerns learning to think with the right
pictures (and learning their limitations).

However, none of this seems to me to come *anywhere near* supporting the
*extremely* radical claim that there are forms of geometric reasoning in
mathematics which are irreducible to logical inference.  It's far from
clear just what this claim amounts to, but I take it to be something like:
there are valid mathematical proofs using geometrical notions which cannot
be formalized.  Now of course there are Goedelian reasons for thinking that
no particular formal system captures the whole of geometry, but what is
being said here seems to be much more radical than that - it seems a
regression to a Kantian position according to which each individual proof
needs its own special appeal to intuition.  If this were really so, it
could surely be shown by example (as Kant attempted to do).  But what are
the examples?  If it really is true we would have to totally rethink our
attitudes to logic and foundations.

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.)

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.

Mic Detlefsen, incidentally, has argued that Poincare really only meant
that the logicist proofs, though valid, weren't really *mathematical*.  I
don't actually agree with him, but Mic, are you out there, and do you have
anything to say about this neopoincareism?

Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD

tel. 0115-951 5845