FOM: intuition

[Alexander R. Pruss]

Hi!

[To introduce myself: I have a Ph.D. in mathematics and am completing a
Ph.D. in philosophy now.  My math research has been in complex analysis and
probability theory, while my philosophy thesis is on the nature of
possibility (and, as ~M~ is L, also of necessity :-) ).]

I rather think this issue has been discussed before here, but let me ask.
I've been bothered by this for years.  There are two species of
mathematician's intuition.
The weak species generates hunches, such as: "Given these partial results
and these intuitions, I conjecture that..." Since conjectures are often
disproved, the weak species might be as often wrong as right, and so it is
not all that interesting for my purposes.

The strong species of mathematician's intuition, however, makes
stronger assertions, such as: "It's now easy to show that p follows from
q", which mathematicians (I submit, based on personal experience, which I
think is not idiosyncratic)  make without actually having gone through the
proof that q entails p.  Obviously, outside of logic, almost no math paper
includes every step of the proof.  The standard when refereeing is that
the reader should be capable of filling in the missing steps.  But I
suspect that in practice in many published papers there are steps that
neither the referee nor the author has filled in.  Nonetheless, both the
referee and the author have a strong mathematician's intuition that these
steps can be filled in.  And given the extreme rarity of errata in
mathematical publications, they are generally right.

This suggests that the stronger form of mathematician's intuition
is extremely reliable.  Why is it reliable, though?  One could try to
point to the mathematician's experience, but when we're talking about
logical space, I'm not sure experience helps much.  After all, two
mathematical problems can be apparently very similar and have very
different answers. Experience is tied, I think, to inductive reasoning,
and hence inapplicable to deductive reasoning (unless one has actually
seen literally the same problem, perhaps in a different guise--but I am
not aware of evidence that this would be enough to account for all the
cases of mathematical intuition).  There is something really strange here.
We seem to have two reliable ways of epistemic access to mathematical
truth: one way is proof and the other is the strong form of mathematical
intuition.  While much has been said about the epistemology of proofs, the
intuitions seem much more mysterious.

Another way to highlight the problem is this.  I said errata are
rare.  However, as we sadly know, errors in math papers are perhaps not
all that rare.  But when a paper has been published after the usual
refereeing process, then any errors are likely to be confined to small
details of the proof, details that the reader can fix, and such errors do
not merit an erratum.  Why is it that small errors of detail in proofs
usually do not vitiate the main theorems?  After all, speaking completely
abstractly, as someone has noted, a mathematical "proof" with an error in
it is like a person who traced his genealogy to William the Conqueror with
only two gaps in the proof.  If so, then a mathematical "proof" with a gap
provides no evidence at all for the truth of the claim it attempts to
demonstrate.  And yet I assume that generally the main theorems are right,
even if some detail in the proof is wrong.  Why?  For instance, I
once published a very long paper giving among other things a
characterization of the subtrees of the regular tree of degree d which for
fixed size (i.e., fixed number of segments) minimize the first Dirichlet
eigenvalue of the Laplacian.  I would be very surprised if there were
no errors at all in the proofs--it's a monster paper and I found minor
errors in most proofreadings of it.  But I would also be quite surprised
if the characterization were false and if the errors were not easily
rectifiable.  Why is this confidence in the result justified, even though
I have good reason to think that there are errors in the proof?  Do we
have another epistemic route to mathematical truth besides proof?

(A related question: Is numerical experiment a route to
mathematical truth.  If I have checked the Andreev-Matheson Conjecture (an
inequality involving polynomials) numerically for 20 million
pseudorandomly chosen cases, this seems to give me confidence in thinking
the Conjecture is probably true.  But why is it any evidence at all?
Surely inductive reasoning does not work in mathematics?)

Alex Pruss

--
Dr. Alexander R. Pruss    ||  e-mail: pruss+ at pitt.edu
Graduate Student          ||  home page: http://www.pitt.edu/~pruss
Department of Philosophy  ||  alternate e-mail address: pruss at member.ams.org
University of Pittsburgh  ||  Erdos number: 4
Pittsburgh, PA 15260      ||
U.S.A.                    ||
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   "Philosophiam discimus non ut tantum sciamus, sed ut boni efficiamur."
       - Paul of Worczyn (1424)