FOM: Comment on Hersh/Shipman exchange

[Moshe' Machover]

In my view, what Hersh needs to explain is not only what distinguishes
mathematical `consensus' from other kinds of consensus. Rather, assuming
(at least for the sake of argument) that mathematical ideas are social
constructs, he needs to explain what distinguishes mathematical rigour from
epistemological standards in other domains of socially constructed ideas.

This is illustrated by Shipman's statement about chess:

> White wins in chess if you remove Black's Queen from the initial position.

I think Hersh is right in asserting that this is a mathematical *statement*
(conjecture); although it is not (yet?) a *theorem*. Sufficiently good
chess players, having no mathematically rigorous grounds for believing this
statement, believe it on other grounds. OK.

But how do we know that the statement itself (not the present grounds for
believing it!) is mathematical? Clearly, we recognize that *in principle*
it is capable of mathematical proof or disproof. We understand what it
would mean to prove (or disprove it).

More generally, we recognize that a statement is mathematical by
recognizing that it is capable of *rigorous treatment* (proof, disproof,
proof of independence from stated postulates...). Hilbert's problems were
all clearly recognized as mathematical problems before any of them had been
solved.

Hersh says:

> I say that there is human mental activity, concerned
> with human mental constructs (ideas) which has the science-like
> property of reproducibility, or consistency if you prefer, and which
> consequently achieves a high consensus comparable to that achieved
> by experimental sciences like physics or chemistry.

Wrong. The consensus/consistency achievable by mathematical proof is
incomparably greater (and qualitatively different) from anything achievable
in the empirical sciences. No empirical science is rigorous in anything
like the sense in which mathematics is.

What needs to be expalined is what distinguishes mathematical statements
from statements on other human mental constructs that makes the former, but
not the latter, capable of this very special kind of consensus/consistency,
of rigour.

I think that the social-constructivist view of mathematics (which Hersh
seems to support) must deny the absolutely unique quality of mathematical
rigour.


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