FOM: Consensus vs Indubitability

[Joe Shipman]

>In brief, there's
>no contradiction between your claim about the consensus and my claim
that
>indubitability is no longer considered an appropriate goal for the
>philosophy of mathematics.

Reuben, I agree with your response to Steve.  But it is invalid to
conclude that
     "rigorous foundations are unnecessary"
just because
     "a big motivation for developing rigorous foundations was to
restore indubitability".

It is characteristic of scientific subjects that "consensus" is
attainable, because of the reproducibility of results (either by
repeating experiments in the case of physical sciences, or by
recapitulating proofs in the case of mathematics).  This consensus does
NOT imply indubitability, just that everyone agrees that the result in
question meets the accepted professional standards.  As I said earlier,
a crisis arose late in the last century, which threatened the status of
mathematics as a scientific subject.  Mathematics survived the crisis
because the development of a completely rigorous foundation restored
consensus about what had and had not been proved.

When I criticized your sociological definition of mathematics a few
months ago, I said that it did not recognize the special role played by
rigorous proof in attaining the conditions (reproducibility and
consensus) of your definition.  It is possible to regard this special
role as only having been filled in the last couple of centuries
(pre-1800 proofs were much less rigorous, and the discovery of
non-Euclidean geometries and the standards of rigor set by Gauss mark a
turning point); but my claim is that rigorous proof (not necessarily in
ZFC but "in principle" reducible to a formal system) is still necessary
to maintain consensus and will be for the forseeable future.

I am willing to expand the notion of proof to include long humanly
unverifiable computer calculations and "probabilistic proofs" like
Rabin's primality test, because they are reproducible in the same way
physical experiments are so professional consensus is still attainable;
but both of these inherently involve "ordinary" rigorous proofs.  That
is, we accept the 4-color theorem not only because our computers verify
the calculations of Appel and Haken's computer, but because they
provided a perfectly standard rigorous proof that their algorithm was
correct; and we accept as prime a number n certified with
p>(1-(2^-1000)) by Rabin's algorithm because there is a real proof that
at most 1/4 of the potential "witnesses to compositeness" lie and we
believe we have selected 500 witnesses in {1,2,3,...,n-1} sufficiently
randomly.  (We recognize that there is a tiny chance we might be wrong,
but there is a consensus that "n is prime" which is similar to the
consensus for a more conventionally proved theorem; there is an explicit
lower bound of p=2^-1000 to the "dubitability", but in fact the other
sources of dubitability for difficult theorems are normally even larger
than that because mistakes do occasionally go uncaught for a long time!)

On the other hand, "the distribution of twin primes follows the
theoretically plausible guess up to 10^16"  (a computer-aided result of
Thomas Nicely), while consensually acceptable as an established result
because of the combination of Nicely's proof that his algorithm is
correct and the experimental reproducibility of his output, does NOT
allow us to reach a consensus that "there are infinitely many twin
primes" itself has been established!  What is missing is the rigorous
logical connection of the twin prime conjecture to the computer
calculation.  When mathematicians stop caring about the distinction
between these two situations, I will admit that you are right about the
dispensability of rigorous proof.

On the other hand, I hope that if another "crisis of consensus" occurs
which engages mathematicians outside of logic and f.o.m, and if it is
resolved by the general acceptance of new axioms, that you will admit
that I am right!  (It is of course possible that it will be resolved in
a different way, by mathematicians deciding the propositions in question
are meaningless for example, but if Harvey succeeds in his current
project that will not be an option.)

-- Joe Shipman