FOM: geometrical reasoning: reply to Black
dubucs
dubucs at ext.jussieu.fr
Sat Feb 20 21:50:23 EST 1999
Robert Black writes:
>
>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.
As far as I understand your first sentence, you admit therefore the
possibility of getting a proposition that were true in the unique model of
Hilbert's system for Euclidean geometry (I guess it's the sense of your
words "geometrical theorem"), without to obtain this proposition by means
of a formal proof from the axioms, nor by a reasoning able to be
immediately formalized by such a proof. It was just my point. Now, you are
leaving the firm ground of logic for the shifting area of likehood
("probably") and strategy ("what we should say and do in such a case"). Let
me follow you here.
a) Once again, I consider as established that the debate is about what I
termed "reasoning as result" (the piece of argument offered to
intersubjective evaluation), not about creative reasoning as subjective
individual process
b) Nor we are concerned by intuition in the minimal sense elicited by
Hilbert, namely perceptual intuition of symbols as spatio-temporal items
(incidentally, Hilbert was not absolutely correct on this point, for
mathematical symbols as intervening in proofs are not merely empirical or
concrete objects, rather equivalence classes modulo equiformity, but let it
go at that)
c) I can't grasp your distinction between "intuitive method of proof" and
"intuition in proof" (presumably, proof which makes room for intuition in
an acception that were different and stronger from Hilbert's one explained
above), except in the following way, which is obviously irrelevant to the
discussion: "intuitive method of proof" is "shortened or informal proof",
or "abstract of proof" in the sense picked out by Whitehead in Principia
Mathematica (Introduction, p. 3), in short "proof able to be less or more
tediously reducible to standard formal proof from axioms". Of course,
realistically conceived, mathematical activity deals with such "intuitive
methods of proofs", but that's not the issue.
d) Actually, my argument was directed against the idea, restated by Simpson
in a recent posting, that Hilbert's categorical characterization of the
euclidean structure had definitely settled, by the same way, the question
of geometrical reasoning. To tell my point more clearly, we have to meet
the following alternative:
(i) EITHER we take the first-order fragment (or some suitably
defined extension of it, a la Simpson) of geometry as sufficient for our
purposes. But in that case, we have to accept the following onus probandi:
to show that a discretized image of the world is intelligible and
defensible. This is a (very interesting) research programme, not a
self-evidence
(ii) OR we take seriously, as something in some way required by our
Weltbild, the higher-order assumptions of current geometry. But in that
case, we have to be prepared to accept as convincing evidence pieces of
reasoning that fall short of mechanical checkability, for there is no
possibility of deriving all the true propositions of this geometry from the
axioms by means of formal (i.e. recursive) inference rules.
It is worth noticing that, if one chooses the second branch, one is by no
way committed to the neo-transcendantalist claim made by Boi in the passage
I quoted formerly, namely that "the spatial continuum cannot be reduced to
any axiomatic construction". If "reduced to" is understood as "described
by" (and I can't grasp any other manageable sense), this claim is merely
extravagant, of course.
e) All these considerations do not undermine the significance of Hilbert's
achievements in the Grundlagen. The categorical characterization of the
Euclidean structure is clearly a great conquest of human reason. But it is
not less clear that Hilbert himself did not conceive his system as a
startpoint for mechanically deriving the truths of geometry. You will
easely convinced of this point by trying to use in a formal proof his
Vollst"andigkeitsaxiom, which says in so many words that the the model
described by the whole system is maximal among the models of the preceding
axioms !
f) I fully agree with your description of what would probably happen if we
had an "intuitive" proof of an unprovable (in formal sense of proof)
proposition, namely that "we would add the intuitive principle used to our
axioms". It's in accordance with our modern notion of proof that, if we are
convinced by a non-mechanically checkable argument ½ establishing that B is
consequence of a set E of hypotheseis, we try to extract the ingredient X
of ½ that blocks checkability to obtain a formal proof of B from E U {X}.
Another manner of telling the point is to say that we admit intuition as a
tool for grasping "true" axioms, but that we reject it in proofs from
axioms. If I'm right, the distinction between both roles is less strict
than generally supposed.
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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