FOM: Re: foundationalism (fwd)
rhersh at math.math.unm.edu
Sat Sep 26 01:00:09 EDT 1998
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.
" 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
THIS IS INFORMATIVE, BUT I DON'T SEE HOW IT RELATES TO WHAT I
"There is no reason to think that Brouwer cared especially
about the antinomies: he drew the line well below the level at which they
I AGREE, THAT WAS A SLIP-UP.
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.)
"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.
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.
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.
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."
THESE SENTENCES ESSENTIALLY ARE
REPEATED ON P 160, 3D PARAGRAPH OF W.I.M.R., FOLLOWED BY A
QUOTE FROM HILBERT ABOUT THE NEED FOR CERTAINTY.
SEE ALSO P. 237 AND P. 247 OF W.I.M.R.
HILBERT'S MOTIVATION RELATIVE TO WEYL'S FLIRTATION WITH BROUWER, WHICH
YOU BRING UP, IS MENTIONED ON PAGE 159 OF W.I.M.R. AND PAGE 335 OF THE
IF YOU READ MY 6-PAGE EXPOSITION OF FORMALISM IN W.I.M.R.
YOU MAY PRONOUNCE IT "A WEAK PARODY' OR "BAD HISTORY TAKEN FROM
UNDERGRADUATE TEXTBOOKS". I DON'T COMPETE
WITH YOUR EXPERTISE ABOUT THE HISTORY OF LOGIC. BUT YOU MIGHT ADMIT THAT
A ONE-SENTENCE REFERENCE AS PART OF AN AUTOBIOGRAPHICAL LETTER NEED NOT
PACK ALL THE DETAILS AND QUALIFICATIONS OF A 6-PAGE ARTICLE.
> > Formalism, removing or denying the content, the meaning, from
> > mathematics, was equally indigestible, for in my experience the
> > meaning of mathematical sentences, words, and ideas was exactly what
> > made them interesting, what made it worth the trouble to analyze them.
> > Trying to explain math by making it meaningless was to me just as
> > absurd as trying to understand literature by making it meaningless.
This shows the problem of trying to understand these things from
> undergraduate textbooks rather than by going back to the real sources.
> This is a weak parody of what Hilbert was trying to accomplish. By proving
> the consistency of a formal system in which ordinary mathematics was
> embedded, by methods that such as Brouwer and Weyl would find
> unobjectionable, Hilbert hoped to refute their criticisms. For this
> purpose it was crucial to represent things in a purely formal symbolic
THIS IS ON PAGE 159--164 OF W.I.M.R.
ALSO 335--338 OF THE MATHEMATICAL EXPERIENCE
> For Hilbert this was an instance of the tried-and-true method of ideal
> elements in which the expansion of the mathematical universe is
> accomplished by a consistency proof. Since with arithmetic, he was down to
> mathematical bedrock, a formalist reconstruction of mathematics was
But Hilbert never "denied" the content of mathematics.
SEE MY REPLY ABOVE.
> IN CONCLUSION, I LOOK FORWARD TO YOUR CRITIQUE OF THIS CRITIQUE
OF YOUR CRITIQUE. I FEEL SURE THAT WHATEVER MOCKING
CHARACTERIZATIONS YOU CHOOSE TO THROW NEXT TIME, YOU WON'T REPEAT THE ONES
YOU USED SO EFFECTIVELY THIS TIME.
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