FOM: Reply from Hersh on truth in mathematics

[Joe Shipman]

   Date:         Wed, 23 Dec 1998 11:58:34 -0700 (MST)
   From:        Reuben Hersh <rhersh at math.unm.edu>
     To:         Vladimir Sazonov <sazonov at logic.botik.ru>, Joe Shipman
<shipman at savera.com>,
        Charles Silver <csilver at sophia.smith.edu>


My starting point is the question of Shipman and Silver,
"What happens to truth in my humanist philosophy of math?"

        The thought seems to be that placing something in
the social realm casts
doubt on the use of "true" or "false."  But just consider:

        I an a citizen of the U.S.
        I am a professor emeritus of UNM.
        Next semester I will be teaching a student named Robert Money a
        reading course in geometry.
        I will have to pay my 1977 U.S. income tax, and a penalty
        and interest for being late.
        Bill Clinton is president of the U.S.
        He is not a member of congress.

        etc  etc  etc.

        You might say statements 3 and 4 are not quite true, only highly

probable.  I think most anyone would agree that the other statements
are true.  I think anyone would admit that they are not about physical
objects or about inner mind states, but about situations in the social
world.

        So it is commonplace to use true and false for statements
about social objects or situations.

        Then why should putting math into the social world create any
difficulty about truth?

        I think the difficulty is about kinds of truth.  If I am crazy,
let's say paranoid, then what seems clearly true to me (I am Napoleon
or Adolf Hitler) could be false.  But if I were paranoid, the
multiplication table could also seem false.  Don't you get the
crank mail that I get?

        What's different about math isn't that it has truth values, but
the way those truth values are verified.

        E.g., verifying the statements above would be a matter of
talking
to people who are supposed to know, or looking at archives or
records that are considered reliable.

        The truth is, in math we also do those things--but not for
publication!  For publication, we are supposed to make a "proof" or
else cite some other publication where a proof is made.

        These facts in  no way contradict the social status of math
objects.  The essential point is that "proof" is a procedure which
humans have evolved, in interaction, over time.  There's not sufficient
reason
to claim it has reached its final state and will evolve no further.

        Proof is something that people do (forgive my overlooking
extraterrestrials.)  It's something they do together in groups or
 even individually as part of a given mathematical culture (see
Lobatchewsky in yesterday's.)

        I think the resistance to acknowledging this plain fact is
the belief that our proof is something infallible, transcendental,
should I say God-given?  Yet it's something we do, and whether we do
it right is checked by  fellows and females  (not gods or demigods.)
it's part of our whole superstructure of universities, publications,
granting agencies etc.

        Yet we (not I) think it transcends all that, and Lagrange's 4
squares theorem (for instance) "obviously" was true before the creation
of the solar system.

Please understand that by Proof I mean what real mathematicians really
do
and call "proof," not what logicians study apart from  real mathematical

practise  Proof is  a human construct, like universal grammar, musical
harmony,
chess, checkers and what not.

         Youur objection to humanist philosophy
of mathematics is not because of a deficiency of truth, but rather
because it denies  a kind of religious  attitude toward mathematical
proof.

        So what is mathematical truth?  It's a statement that is
corrctly
proved according to contemporary standards of proof, or it's a
conjecture
that can eventually be so proved.  For the conjecture, it is true or
false without our knowing which.  For the theorem, we take it as true,
on
the hypothesis that its proof is correct.

        If a big theorem of today is declared false next year, then
we would say that we thought it was true but we were mistaken; or
possibly that it was true and today's critics of it are mistaken.
An issue that would be decided mathematically, not philosophically.

There's nothing surprising in all this, if we recognize that proved
theorems are human constructions, and may contain errors.  We want to
think of our mathematical knowledge as absolute, infallible, not
conditional.  The difficulty in Theorem X being accepted today and
rejected tomorrow is not for the humanist but for the absolutist.  He
wants to think accepted proofs are indubitable.  If theorem X is
accepted
today and rejected tomorrow, he'll be embarrassed, not I.

Reuben Hersh