FOM: Re: your mail

[Joe Shipman]

Reuben Hersh wrote:

> Joe, I am going to answer head on your question, what if mathematicians
> unanimously agree that A is true, and later agree that it is false.
> Do I claim that the truth value of a mathematical statement switches
> back and forth in such an eventuality?
>
> By the way, this is not a mere fictitious supposition.  For around 2,000
> years all mathematicians agreed that the statement "Euclid's theorems
> follow logically from his axioms"  was true.  Moritz Pasch in 1880 or so
> demonstrated that this mathematical statement is false.
>
> Of course I, like any other mathematician, would say that the statement
> was false all along; the unanimous mathematicians were mistaken.
>
> In previous letters, I gave examples of statements about social realities
> which were unequivocally true.  (I know you can make up counterexamples
> involving fraud or masquerade or mistaken identity.  The truth here
> is not absolute, indubitable truth, but practical, un-ignorable truth.)
>
> My point was that mathematical truth, being a special kind of social
> truth, also can be decisive practically, without being indubitable
> abslutely.
>
> Just as my draft notice may have been made out before I know about
> it, (even before I ever know about it if I happen to walk in front of a
> moving truck before receiving it), so mathematical statements are true
> or false according as how they are implied or negated by the existing
> body of mathematical concepts and knowledge.  True or false in the
> first place; I try to find out which, in the second place.
>
> The unanimity of mathematicians was cited by me as a distinctive
> charateristic of mathematics, not as the definition of mathematical
> truth.
> Mathematical concepts are social in their mode of existence, and they are
> nonetheless real, as real as mental states or physical mountains and
> oceans.  Being real, they do or don't have various properties, which we as
> mathematicaisn try to discover.
>
> This is something I said in my book over and over again.  Mathematical
> objects are the subject of objective truths; our knowledge of them
> is a consequence of their prior objective validity.
>
> This may sound to you like back-door Platonism.  But I always argued
> that the strength, the attractiveness, of Platonism is just that it
> accepts and recognizes the objective truths of mathematics.  Its
> fallacy is in attributing these objective properties to "abstract
> objects" which are nothing more than transcendental wisps.  Attaching
> mathematical truths to the social realities they actually adhere to
> keeps objective truth without attaching it to ghosts and spirits.
>
> The unanimous opinions of mathematicians is a characterisitic feature
> of mathematics; it also is in practise the main way mathematicians
> decide what to believe.  In view of examples like Pasch, I would
> like to add, "in the long run."  If some disputed point ultimately becomes
> decided by nearly all mathematicians, that ultimate opinion as such would
> outmode or outdate earlier ones.
>
> But of course this is still not absolutely certain knowledge.  You
> could imagine a social degeneration, say following major nuclear
> holocaust, where standards of mathematical argument could deteriorate,
> and  earlier knowledge be lost.  Like Archimedes was lost to the
> Dark Ages.
>
>  I hope I have  given a clear-cut answer to your question.
>
> Mathematical statements, like many other social statements, can be,
> often are, objectively true or false.  Mathematical research is the
> effort to learn the truth of a few of the more interesting mathematical
> statements.  The near unanimity of mathematical belief is testimonhy
> to the validity of the methods of mathematical research.  These methods
> include formal deduction, but are by no means limited to such.
>
> Reuben Hersh

Reuben,

Thanks for a thoughtful response.

You are right about Pasch, but for an instructive contrast consider the
superficially similar case of the parallel postulate.

First of all, remember that until Godel's Completeness theorem, entailment and
logical implication were distinct.  It was generally thought that Euclid's
other axioms *entailed* the parallel postulate (in no possible worlds was the
parallel postulate false and the other axioms true), but it was not *as*
generally thought that the other axioms *implied* the parallel postulate
(implication being a stronger notion, equivalent to "provable entailment"),
and neither the entailment nor the implication of the parallel postulate by
the other axioms was regarded as "proven".  The entailment was just a
generally believed conjecture, and the implication was a somewhat less
generally believed conjecture.  Therefore the parallel postulate was
indispensable, geometry could not be done rigorously without it, but
mathematicians for 2000 years were bothered by this and tried to actually
settle the conjecture by one of three methods:
 1) Actually proving entailment (which is the same as showing implication)
 2) Disproving implication (which is a proof-theoretical investigation
distinct from and weaker than disproving entailment)
 3) Disproving entailment (a model-theoretic investigation showing a possible
world in which Euclid's other axioms were true but the parallel postulate was
false)

As we know, eventually Lobachevsky, Bolyai, and Gauss independently succeeded
in 3) -- long before Pasch in 1880. One reason it took so long was that most
previous mathematicians had tried to accomplish 1) which was doomed to fail.

But although the conjecture may have been "wrong" the mathematicians were
*not* "mistaken" because they KNEW that it had not been PROVEN that the
parallel axiom followed from the others -- that's why they continued to
include it as an axiom when teaching geometry!  Mathematicians today would
agree that the twin prime conjecture is true with at least as much unanimity
as the ancients had about the parallel postulate being entailed by the other
axioms -- if by some miracle the twin prime conjecture is disproven that will
be a shock but  there will be no question of a "mistake" because nobody is
claiming to have a proof of the infinitude of twin primes.

On the other hand, Pasch's discovery was in a certain sense trivial -- he
found something which everyone else had overlooked because it was "too
obvious", which was indeed an important discovery, but one which was simple
and immediately assented to universally.  It is not fair to claim that
mathematicians were simply mistaken when they said "Euclid's theorems follow
from his axioms", because this statement was a metamathematical rather than a
mathematical statement until proof theory had been developed in the late 19th
century.  Pasch's discovery coincided with large advances in the formalization
of mathematical proof.  Of course they WERE wrong to say "Euclid's theorems
follow from his axioms", but they were not "mathematically wrong" until that
statement became clearly mathematical.

One feature of your account of mathematics, with its de-emphasis of
formalization, is that the distinction between mathematics and metamathematics
is more difficult to uphold, so you may still be able to get away with saying
that prior to Pasch mathematicians were "mathematically wrong" about the
sufficiency of Euclid's axiomatization.  The reason I don't want to do this is
I think that mathematicians are not going to be "mathematically wrong" for
2000 years.

This discussion doesn't affect our main issue, because we both recognize that
mathematicians can be unanimously wrong about whether a mathematical statement
has been proven [a much stronger assertion than that they can be unanimously
wrong in their opinion of the truth of some conjecture].  I'm glad to see you
add "in the long run"; this avoids the problems I raised about the viability
of the notion of truth in your account of mathematics, though it raises two
others:
 1) We can not be certain of mathematical knowledge because we don't ever know
if it has been "long enough"; furthermore the "long run" may never arrive if
civilization collapses
 2) There is no identification of the overturning of a unanimous consensus
with the discovery of a "mistake", an identifiable error in reasoning, even
though that seems to be a feature that is always present.

-- Joe Shipman