FOM: Internal/External Characterizations. Response to Silver

Moshe' Machover moshe.machover at
Sat Jan 24 16:55:27 EST 1998

Silver says:

>	In a very general sense, I don't see that Hersh is in such deep
>water.  In terms of a characterization I favored earlier, it seems to me
>there are two sides to his view about mathematics.  The two sides are:
>Agreement and Aboutness.  "Agreement" is all about social consensus;
>"Aboutness" is what mathematics is about.  I think they correspond to your
>internal/external distinction. ...
>... The novelty in his approach, I think, is that he thinks the nature
>of mathematical consensus is a major factor in determining what
>mathematics *is*.  Since what mathematics *is* would already have been
>explained internally--according to this suggestion--he'd have to establish
>a kind of bridge between the two, a kind of "if and only if" between the
>two sides that would forge them together.

This is just the point! The external part of Hersh's characterization, the
special quality of consensus that is uniquely possible in mathematics (and,
I insist, is qualitatively different from anything possible in the
empirical sciences, let alone the humanities) would then have to be
*explained* by the internal `aboutness' of mathematics. Thus the latter
only is fundamental, and the former merely derivative. It is only the
`aboutness' of mathematics that is really determinative of what mathematics

As I argued in previous messages, this is borne out by the fact that we can
tell that a proposition is mathematical (in a given context, which fixes
the meaning of the terms mentioned in the proposition) even without knowing
its truth value, let alone the degree of consensus about it. (Example `P =
NP'. No-one knows whether it is true, and I have read in this list some
animated differing views about whether a majority of experts *think* it is
true. Still, we all know it is a mathematical proposition. I predict with
certainty that if and when it is proved or refuted, an overwhelming
consensus about its truth value will rapidly be achieved among experts. My
prediction is sociological, but it derives from the 'aboutness' of the

Therefore Hersh's sociological observations about consensus among
mathematicians--whether or not they are correct--cannot in principle be
used to characterize mathematics.

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