FOM: Hersh & consensus

[Martin Davis]

Reuben Hersh cannot bring himself to say that the assertions in mathematics
are *true* or *correct* in some objective sense. Instead he speaks of
"consensus".

> Mathematics can be about anything, as long as its
> subject matter is immaterial (not a natural science) and as long as 
> it attains an extremely high degree of reproducibility and consesus.

Let us consider some well established theorem of elementary number theory,
say, Lagrange's theorem that every positive integer is the sum of four
squares. Is it really just that professional mathematicians have all agreed
to accept this ("consensus") or is there in fact some objective fact of the
matter? 

Now, by "consensus", Reuben can only mean consensus among mathematicians. To
avoid the vacuous "mathematical truths are just those that mathematicians as
a class believe" as explanation, one would have to take Reuben's
characterization as implying that whenever the subject matter is
"immaterial" and the experts in that subject matter all agree, then they are
doing mathematics. So we find, for example that Catholic Theologians and
Orthodox Rabbis are mathematicians and that the dogmas of their faiths are
mathematical theorems. Reductio ad absurdum.

Suppose we are told that someone has found some very large integer which a
direct computation has shown not to be representable as the sum of four
squares. Do we keep an open mind or do we just laugh this out of court? Why?
A "consensus" can be overturned. There was once a consensus that no
continuous function could be nowhere differentiable. But that consensus was
clearly different than that for Lagrange's theorem. And when Weierstrass
produced his example, that instantly overrode the consensus, despite those
who "viewed with horror" such a monstrosity.

Let me say it. There is mathematical *truth*. It is genuine and objective.
It is when consensus arises *because* mathematical truth has been attained
that the subject advances. Some may believe the dogmas of a religion as
fervently as Lagrange's theorem. But when pressed it will be admited that it
is a matter of "faith". My belief in the truth of Lagrange's theorem is not
a matter of faith.

Martin