FOM: geometrical reasoning: reply to Black

[Robert Black]

Jacques Dubucs writes:

>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

But suppose someone were to use their geometrical intution to get a
geometrical theorem not provable in ZFC.  [Just to fix ideas, it might be a
proof/disproof of (a geometrical statement of) the continuum hypothesis,
for example.]  I don't think this is remotely likely, but suppose.  What
should we then say and do?

I think the first thing we would do is identify the intuitive principle
used and add it to our axioms of geometry.  We would then reexamine our
identification of Euclidean space with R3 and either see that the new
principle extended to a new axiom of set theory settling CH, or
alternatively that the new axiomatization gave us a conceptualization of
space for which the proof of isomorphism with R3 failed.  Either way we
would have captured the relevant reasoning in logical formalization.

So even such an unlikely discovery wouldn't be enough to establish the
claim being made about the indispensability of intuition in proof.  Rather
one would have to be talking about an intuitive method of proof of
uncodifiable potentially infinite variety.  And this stretches my credulity
too far all together.

>	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 ...

You may well be right, but if you want the trend to be taken seriously, I
wouldn't shout it from the housetops!

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

tel. 0115-951 5845