FOM: Re: foundationalism (fwd)
Martin Davis
davism at cs.nyu.edu
Sat Sep 26 14:09:07 EDT 1998
On Fri, 25 Sep 1998, Reuben Hersh wrote:
>
> THANKS, MARTIN, FOR YOUR THOUGHTFUL CRITIQUE.
>
> i THINK IT'S ONLY FAIR FOR ME TO CRITICIZE YOUR CRITIQUE. I WILL HAVE TO
> MAKE REFERENCE TO "THE MATHEMATICAL EXPERIENCE" AND "WHAT IS MATHEMATICS,
> REALLY?" I DON'T KNOW IF YOU HAVE THE FIRST, BUT I KNOW FOR SURE YOU
> HAVE THE SECOND, AND HAVE EVEN READ IT.
>
> IN BETWEEN QUOTES FROM MY LETTER ABD YOUR CRITICISM, I WILL PLACE MY
> CRITICISM OF YOUR CRITIQUE IN ALL CAPS, FOR CLARITY.
>
> On Thu, 24 Sep 1998, Reuben Hersh wrote:
>
> Here in foundations the story was different. There were three
> schools, logicism, formalism, and intuitionism. All sought to
> repair the foundations of mathematics, after the damage they had suffered
> froom the Antinomies. None of them succeeded in their mission. In the
> course of an unsuccessful philosophic quest, they all created some
> original and important mathematics.
>
> YOU CRITICIZED:
> " Bad history. Logicism was begun by Frege; his work was completed and
> his magnum opus at the printers when he learned of the Russell paradox.
> The effort to avoid methods though to be illegitimate because of
> non-constructivity or use of impredicative definitions goes back at least
> to Kronecker. The Weierstrass-Cantor-Dedekind foundation for analysis
> involved modes of thought troubling to many mathemticians including
> Poincare, Borel, and Weyl and leading to rich philosophical
> discussions.
>
> THIS IS INFORMATIVE, BUT I DON'T SEE HOW IT RELATES TO WHAT I
> WROTE.
Sorry I was unclear. My paragraph criticized your sentence beginning "All
sought ..."
> YOU WROTE:
> "There is no reason to think that Brouwer cared especially
> about the antinomies: he drew the line well below the level at which they
> became issues.
>
> I AGREE, THAT WAS A SLIP-UP.
>
> YOU WROTE:
>
>
> Hilbert certainly was concerned with the antinomies. But he
> was at least at much bothered by the attack on the core of modern analysis
> by Brouwer and Weyl.
>
> I WROTE:> I was bothered not so much by the impasse in which
> foundationalist philosophy of mathematics found itself.
> I was much more disturbed by the obvious (to me) fact that
> all three were incredible. The pictures of mathematics they
> offered did not at all resemble the mathematics I knew as student,
> teacher, and researcher.
>
> Platonism (including its special case, logicism) invoked
> a mathematical world eternal, unchanging, and independent of human
> actions. How this ghost world related to the material world wasn't
> even recognized as a problem! (Later I learned about Benacerraf
> throwing the same problem at his fellow philosophers of math.)
>
> YOU CRITICIZED:
>
> "Logicism as a movement was mainly concerned to provide a seamless
> development of mathematics beginning with purely logical notions and
> eventually leading up to the totality of mathematical discourse. As such
> it can be followed without any necessary ontological commitments.
>
> YOU SAY "LOGICISM CAN BE FOLLOWED WITHOUT ANY NECESSARY
> ONTOLOGICAL COMMITMENTS". I SAY "LOGICISM PRODUCED FRUITFUL
> ADVANCES IN LOGIC AND MATHEMATICS, BUT ITS PHILOSOPHICAL PROGRAM
> WAS UNSUCCESSFUL." IT LOOKS TO ME LIKE WE'RE SAYING THE SAME THING,
> WITH DIFFERENCT EMPHASIS. RATHER THAN SAY THE PHILOSOPHICAL PROGRAM WAS
> UNSUCCESSFUL, YOU SAY THE ONTOLOGICAL COMMITTMENTS ARE UNNECESSARY.
> HAD THEY BEEEN SUCCESSFUL, YOU WOULDN'T NEED TO PUSH THEM ASIDE.
Again I was evidently unclear. I was criticizing your assertion that
logicism is a special case of Platonism.
> If (as Dedekind seems to have) one thinks of "set" as a logical notion, the
> logicist program has been a great success, tacitly followed by
> mathematicians from Halmos to Bourbaki.
>
> SOME AUTHORS (E.G. THE KNEALES, AS CITED IN W.I.M.R.), THINK
> THAT THE INTRODUCTION OF THE AXIOM OF INFINITY MEANT THE FAILURE
> OF LOGICISM. IT'S BELIEVED THAT AS AN AXIOM OF LOGIC THE AXIOM
> OF INFINITY IS NOT COMPELLING.
>
> AS I SAID MORE THAN ONCE IN THE MATHEMATICAL EXPERIENCE AND IN
> W.I.M.R., IN QUOTATIONS PROVIDED TO YOU, LOGICISM AND
> FORMALISM WERE SUCCESSFUL AS CONTRIBUTIONS TO MATHEMATICS OR LOGIC, BUT
> UNSUCCESSFUL AS PHILOSOPHIC PROGRAMS ATTEMPTING TO GIVE AN ADEQUATE
> ACCOUNT OF THE NATURE OF MATHEMATICS. YOU MAY REMEMBER THE QUOTE FROM
> RUSSELL, IN BOTH BOOKS,WHERE HE SAYS THAT HE WAS SEEKING A FIRM BELIEF TO
> REPLACE HIS LOST BELIEF IN CHRISTIANITY. HE SAYS HE THOUGHT HE COULD FIND
> CERTAINTY IN MATHEMATICS. WHEN HE FOUND MATHEMATICS WANTING, HE ATTEMPTED
> TO GIVE IT A SOLID FOUNDATION, AS HE SAYS, BY SETTING IT FIRST ON AN
> ELEPHANT, THEN ON A TORTOISE, ETC. UNTIL "AFTER MANY YEARS OF ARDUOUS
> TOIL" HE GAVE UP.
>
> DOESN'T SOUND LIKE HE THOUGHT LOGICISM WAS A BIG SUCCESS!
>
> WHAT ABOUT FREGE? IN HIS LATER YEARS HE DECIDED THAT HIS
> ATTEMPT TO FOUND MATHEMATICS ON LOGIC WAS NOT ONLY A FAILURE, BUT
> FUNDAMENTALLY MISTAKEN. HE RETURNED TO KANT'S IDEA OF INDUBITABLE
> INTUITIONS INSTEAD OF LOGIC.
>
> TOO BAD HE DIDN'T NOTICE WHAT A HUGE SUCCESS LOGICISM WAS.
>
> I THINK THIS DISAGREEMENT IS A MATTER OF EMPHASIS. YOU
> HAVE WRITTEN THAT YOU PREFER TO LEAVE MATTERS OF ONTOLOGY TO
> THE PHILOSOPHERS. THAT SUGGESTS THAT THE PHILOSOPHICAL
> PROGRAM OF LOGICISM DOESN'T INTEREST YOU. SO IT'S SUCCESS
> OR FAILURE ARE OF NO ACCOUNT. WHAT MATTERS IS ITS CONTRIBUTION
> TO LOGIC AND MATHEMATICS.
>
> I, ON THE OTHER HAND, AM STRONGLY INTERESTED IN PHILOSOPHICAL
> ACCOUNTS OF THE NATURE OF MATHEMATICS.
>
> SO I SAY LOGICISM WAS A FAILURE IN ITS PHILOSOPHICAL
> PROGRAM. THEN YOU TELL ME THAT IT'S REALLY A GREAT SUCCESS, PROVIDED YOU
> IGNORE IT'S PHILOSOPHICAL SIDE. JUST SAYING THE SAME
> THING, WITH DIFFERENT EMPHASIS.
>
> YOU CALL MY LETTER "BAD HISTORY." IS IT GOOD HISTORY TO
> CONCENTRATE ON THE ASPECT OF HISTORY CONGENIAL TO
> YOUR INTEREST, AND MINIMIZE THE LESS CONGENIAL ASPECT, WITHOUT
> REGARD TO THE ACTUAL VIEWS OF THE HISTORICAL FIGURES IN QUESTION?
>
> MAYBE YOU DO THINK THAT'S GOOD.
Alonzo Church has characterized logicism as the view that logic and
mathematics are related as the elementary and advanced part of the same
subject. So whether you regard the program as a success depends on where
you draw the line. If the assumption (which Dedekind and Frege thought
they could prove) that there is a set containing infinitely many elements
is not taken to be part of logic, then logicism failed. Logicism was
primarily a scientific program and has to be judged by its accomplishments
and failures, not by what various folks have said about it at different
times.
The sense in which logicism succeeded is that there is a seamless
development of all of mathematics from a simple foundation involving the
the purest of abstractions. This is not just a contribution to logic. It
is a satisfactory durable largely accepted F.O.M.
> AS TO FORMALISM, YOU SEEM TO IDENTIFY THE FORMALIST POSITION
> ON THE NATURE OF MATHEMATICS WITH THE VIEWS OF ONE GREAT FORMALIST,
> DAVID HILBERT. BUT HILBERT'S BRAND OF FORMALISM IS NO LONGER AROUND.
> FORMALIST THINKING IS AROUND, AND I WAS
> TALKING ABOUT FORMALISM AS YOU MAY HEAR IT EXPOUNDED NOW.
If formalism is not the philosophical views and mathematical program of
Hilbert, then I have no idea what you might be talking about. Is it gossip
in math common rooms you're trying to attack? Which thinkers? And what
coherent did they have to offer as a program for foundations?
> I APPRECIATE YOUR TELLING ME THAT HILBERT NEVER SAID MATHEMATICS WAS
> MEANINGLESS. I KNOW THAT. (LAST SENTENCE OF MARTIN'S LETTER, SEE BELOW.)
> REFER TO PAGE 336 OF THE MATHEMATICAL EXPERIENCE:
> "HILBERT'S WRITINGS AND CONVERSATION DISPLAY FULL CONVICTION THAT
> MATHEMATICAL PROBLEMS ARE QUESTIONS ABOUT REAL OBJECTS, AND HAVE
> MEANINGFUL ANSWERS WHICH ARE TRUE IN THE SAME SENSE THAT ANY STATEMENT
> ABOUT REALITY IS TRUE. IF HE WAS PREPARED TO ADVOCATE A FORMALIST
> INTERPRETATION OF MATHEMATICS, THS WAS THE PRICE HE CONSIDERED NECESSARY
> FOR THE SAKE OF OBTAINING CERTAINTY."
>
The point is just that he was not advocating "a formalist interpretation
of mathematics". He was proposing to *use* a formalization of mathematics
as a *tool* for proving consistency. It is as though you explain an
applied mathematician's using PDEs to study fluid flow by saying "X is
prepared to countenance a PDE interpretation of fluid flow as the price for
obtaining useful results."
As to the rest: I was replying to what I presumed was a careful considered
exposition of your developing views. All of your "see - I knew that,
because look what I wrote" is beside the point. You say P. I criticize P.
You can say:
1. Oh, I guess you're right, or
2. No, you didn't understand, let me be clearer, by P I meant P1, P2, and
P3, or
3. In criticizing my statement P, you said Q, and Q is ridiculous for the
follwoing reasons ..., or even
4. Your criticism shows such a combination of malice and ignorance that
it's useless for me to continue this.
What makes no sense is for you to say: "How can you think I need to be
told Q, look what I wrote, It proves that I knew Q all along." To which I
of course retort: "If you knew Q how could you have said P?"
Be well,
Martin
More information about the FOM
mailing list