FOM: Hersh & consensus

[Charles Silver]

On Fri, 2 Jan 1998, Martin Davis wrote:

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".

	I too feel that "consensus" is inadequate, but I also am having
trouble figuring out how one can determine what is "beyond consensus." 
That is, when a consensus about some mathematical statement has been
reached, what is it (if there is an "it") that is *responsible* for the
consensus?

Reuben Hersh:
> > 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.

Martin Davis:
> 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? 

	Let us suppose you (Martin Davis) are correct that it is not
merely that professional mathematicians have reached some agreement, but
that there is *something* *else* responsible for that agreement.  What
could that something else be?  I think Barwise and Feferman have provided
partial answers to this.  I will cite one of Feferman's 10 theses below.


M.D.:
> 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.

	Yes, I think some reference to subject matter has to be provided
to avoid vacuousness.  Perhaps something like this: mathematics is *about*
underlying patterns or structures of the "imagination". <== I am calling
attention here to Solomon Feferman's use of 'imagination' in a recent
post, and I agree that 'imagination' seems awfully mysterious, but I don't
know what should replace it. The following is Feferman's first thesis (on
what mathematics is):

S. Feferman:
1. Mathematics consists in reasoning about more or less clearly and
coherently [sic] groups of objects which exist only in our imagination.


M.D.:
> 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.

	Let's try this as one possible way to sharpen Feferman's "groups
of objects" in 1 above. Suppose we say that mathematical theorems are not
true of some particular kind of thing, but true of anything having certain
agreed-upon structural properties.  (I think Barwise said this better in
an earlier post.) One could then think that mathematical theorems are true
of any representative of a class of isomorphic structures. (This is
somewhat like a computer program implementing a single concrete data
structure that exemplifies the desirable characteristics of an abstract
data structure.)  Then, would this make mathematics not *about* a single
thing, but about any isomorphic copy of structures that we hold in our
"imagination"?  So, is sameness of imagined structures the fundamental
reality underlying mathematics? 


Charlie Silver
Distinguished Husband of Professor Susan Silver &
(Undistinguished) Lecturer in Logic at Smith College