FOM: indubitability (or "certainty")

[M. Randall Holmes]

There are several issues which Hersh and others confuse in the matter of
"indubitability".

1.  A correct mathematical proof compels assent to its conclusion, if
one accepts the axioms.

Without this, the whole concept of rigor makes no sense at all.  One
cannot be more or less rigorous if there is no standard of perfect rigor
to approximate.

We may _not_ doubt that the conclusion of a valid argument follows
from the premises.  We do have explicit standards, which we can spell
out, as to what constitutes a valid argument.  This is the precise
sense in which mathematics is indubitable.  No natural science is
indubitable in this sense.

This is not "19th century foundationalism"; it is a standard known
to Euclid (though he did not perfectly exemplify it).

2.  It is possible to doubt the correctness of something presented to
one as a completely formal mathematical proof because one cannot grasp
it all at once; one cannot tell whether it is such a proof or not.
This is not a challenge to the indubitability of mathematics; it is a
realistic problem with human capabilities.  Formally correct proofs
can be presented in ways which make it easier (or harder) to check
them.  One can use automatic proof checkers (one can also doubt their
capabilities -- and this is also not the same thing as doubting that
correct proofs compel assent).  There is some relevance here to
Brouwer's demand for "unitary intuition" (referring to another
thread).

3.  One can doubt the truth of axioms (such as Infinity or Choice).  This
doesn't affect the validity of mathematics -- one still must accept that
if Infinity is true, such and such conclusions follow.  An even more 
radical form of such doubt is found in constructive mathematics (re
excluded middle).

4.  In practice, completely formal mathematical proofs are seldom (but
not "never") presented.  The standards for what constitutes an
acceptable approximation to a formal proof cannot be formally spelled
out.  The actual process by which theorems are accepted is not a
formalizable one (though one hopes that it is a careful
approximation).  From time to time mistakes are found; a "proof" has
been accepted which we become convinced cannot be cashed in for a
valid argument.  The actual social practice of presenting and verifying
informal proofs is presumably what Hersh is interested in; but this practice
makes no sense in the absence of a standard of correct argument with the
property that a correct argument, once recognized as correct, compels assent.

5.  It is possible to do experimental or probabilistic work; there is
nothing wrong with this as long as it is carefully labelled as such, and
not as (even abbreviated) formal proof.

6.  One can also be slow-witted or deliberately obtuse (see .sig); one
can refuse assent even when one must assent.

Sincerely, M. Randall Holmes

holmes at math.idbsu.edu or mrh29 at dpmms.cam.ac.uk
http://diamond.idbsu.edu/~holmes

Boise State University and the University of Cambridge
must be held harmless for any silly thing I may say.

"And God posted an angel with a flaming sword at the gates 
of Cantor's paradise, that the slow-witted 
and the deliberately obtuse might not glimpse 
the wonders therein."  (Holmes, with apologies to Hilbert)