FOM: Hersh's incoherent attack on formalism and logicism

Thu Oct 1 18:14:20 EDT 1998

DEAR STEVE,

On Thu, 1 Oct 1998, Stephen G Simpson wrote:

> Reply to Hersh 30 Sep 1998 16:19:42.
> 
> 1. Frege and modern logic
> 
>   Hersh writes:  
>    > You say "you must dismiss Frege's work as a failure."  Not at all.
>    > I wrote, page 141 of W.I.M.R.:  "Frege's introduction of quantifiers is
>    > consdered the birth of modern logic."
>   
>   Do *you* consider Frege's work to be the birth of modern logic?
>   
>   Do *you* think modern logic is of value for philosophy of mathematics?
>   
>   You don't seem to think so.

	I'M SORRY, I SHOULDN'T HAVE USED THE ABBREVIATION W.I.M.R.
	IT STANDS FOR THE TITLE OF MY BOOK WHICH FOR ALL I KNOW
	YOU MAY STILL HAVE A COPY OF.  "WHAT IS MATHEMATICS, REALLY?"
	WHEN I QUOTE MY OWN PRINTED WORDS, OF COURSE I AM EXPRESSING
	MY OWN OPINION.  THE POINT IS, IF YOU BOTHERED TO READ WHAT
	I SEND YOU, EITHER BOOKS OR MESSAGES0 YOU WOULDN'T WASTE TIME
	WITH SUCH AN ABSURD QUESTION.  THE ANSWER IS YES.  YOUR LAST
	SENTENCE ABOVE IS, AS THEY SAY, "OFF THE WALL."
>   
> 2. The axiom of infinity: Hilbert's program and formalism
> 
>   > You research program doesn't respond to my remark.
>   
>   Which remark, and which research program?  
	STOP PLAYING GAMES.  YOU'RE BRIGHT, YOU KNOW WHICH REMARK AND
WHICH RESEARCH PROGRAM.> 
>   One of your remarks (12 Sep 1998 18:06:45) was as follows:
>   

>    > One famous difficulty is [the] axiom of infinity.  You can't do
>    > modern math without it.
>   
>   One of "my" research programs (actually it involves a number of
>   people) responds directly to your remark.  It does so by examining
>   the extent to which modern mathematics is reducible to finitism.
>   This is in the context of Hilbert's program, which you have
>   dismissed as a failure.
>   
	I NEVER DISMISSED IT.  I SAID IT DIDN'T ACHIEVE ITS
	ORIGINAL GOAL.  SEE THE ESSAY ON THIS SUBJECT BY JOHN
	VON NEUMANN IN W.I.M.R.?

>   Are you willing to rescind your dismissal?  THERE IS NO DISMISSAL.

Are you willing toexamine evidence against your remark?
	SURE, OF COURSE.>   
>    > You say, "If you are unwilling to study the role of infinity in
>    > mathematics, then how can you expect anyone to take your comments
>    > on it seriously?"
>    > 
>    > My comment, that the axiom of infinity is not inituitively
>    > plausible as an axiom of logic, is not mine.  It has been made by
>    > others ....
>   
>   Even if the comment has been made by others, you repeated it, so you
>   must take some responsibility for it.  You can't hide behind others.
> 	RIGHT.  I'M NOT HIDING.  I QUOTED OTHERS TO BACK UP OR
	REINFORCE OR FORTIFY MY OWN OPINION.
>   However, I wasn't referring to that particular comment.  (More on
>   that comment below, in connection with logicism.)  Rather, I was
>   referring to another of your comments concerning the axiom of
>   infinity:
>   
>    > One famous difficulty is [the] axiom of infinity.  You can't do
>    > modern math without it.
>   
>   I say again: How can you expect anyone to take this comment
>   seriously, if you are not willing to examine evidence for and
>   against it?
>   
	I DON'T EXPECT ANYONE TO  TAKE IT SERIOUSLY.  WHETHER
	ANYONE TAKES IT SERIOUSLY IS ENTIRELY UP TO THEM.
	MY IMPRESSION IS THAT RUSSELL INTRODUCED THE AXIOM
	OF INFINITY BECAUSE HE COULDN'T DO WITHOUT IT.  IF
	MODERN RESEARCH HAS SHOWN THAT RUSSELL WAS WRONG, THEN
	MY REMARK WOULD ALSO BE WRONG.

	I AM PUZZLED, THOUGH, BY THE FACT THAT PEOPLE CONTINUE
	TO REFER TO ZF AS THE FOUNDATIONAL AXIOMS OF SET THEORY.
	IF YOU ARE THROWING OUT THE AXIOM OF INFINITY, SHOULDN'T IT
	BE ZFS (ZERMELO-FRAENKL-SIMPSON)?

>    > You say, "You present a a caricature of Hilbert's work, then
>    > attack the caricature."  No.  I used the word "formalism in the
>    > common, colloquial sense, not in Hilbert's sense.  There is no
>    > caricature and no attack.
>   
>   Here you seem to be evading the fact that Hilbert is generally
>   regarded as the originator of formalism.  Do you dispute this
>   conventional view of the history of formalism?
>   
>   But, all right, let's take you at your word and assume that you
>   never attacked Hilbert's formalism.  Let's assume that you were
>   attacking somebody else's formalism.
>   
>   Who are these hitherto unnamed formalists?  
	CURRY, HENLE

Do you recognize a
>   difference between their views and those of Hilbert?  YES.

Or are you
>   merely attacking coffee-room chatter, as Martin Davis suggested?
	MY MAIN CONCERN IN WIMR IS THE PHILSOPHY OF MATHEMATICS
	IN THE SENSE OF THE PHILSOPHICAL VIEW OF MATHEAMTAICS
HELD BY MATHEMATICIANS.  THE COFFEE ROOM CHATTER YOU SNEER AT IS
IMPORTANT, BECAUSE THE VIEW OF MATHEMATICS HELD BY MATHEMATICAINS IS
IMPORTANT. OF COURSE, YOU ARE FREE TO SNEER AT IT AND DISMISS IT IF YOU
CH;OOSE TO DO THAT.

>   >    > You say, "You were arguing that it's OK to
dismiss Hilbert's
>    > views without a hearing."  As I keep trying to explain, I never
>    > referred to Hilbert's views at all.  The word formalism has more
>    > than one meaning.  I can't believe you're unaware of that.
>   
>   I'm *not* aware of that.  I accept the conventional view that
>   Hilbert is the originator of formalism.  If you have some other kind
>   of formalism in mind, please tell me who originated it and how it
>   differs from Hilbert's formalism.
>   
	FORMALISM IN COMMON SPEECH SAYS THAT MATHEMATICS IS JUST FORMULAS
AND CALCULATIONS.  MEANING IF ANY IS EXTRAMATHEMATICAL.  I DON'T KNOW WHO
ORIGINATED IT.  I BELIEVE FORMALISM WAS ONE OF THE BELIEFS THAT FREGE
ATTACKED.  SORRRY I CAN'T GIVE YOU THE PAGE REFERENCE.  HILBERT
ORIGINATED HIS "PROGARAM" BUT FORJALISM WAS KNOWN BEFORE THAT.

>   Here are the real questions:
>   
>   Do *you* think Hilbert's program is of any actual or potential value
>   for philosophy of mathematics?
>   	YES.
>   Do *you* think the research of your other formalists (who are they?)
>   CURRY AND HENLE

has any actual or potential value for philosophy of
mathematics

		?YES
>   
> 3. The axiom of infinity: set theory and logicism
> 
>   Hersh writes:  
>    > You seem incapable of dealing with this well known fact.
>   
>   Here you are referring to the well known fact that the axiom of
>   infinity is not generally regarded as a logical axiom.  I accept
>   that fact, and I understand the reasons for it, at least in the
>   context of Russell's type theory and ZF set theory.  In this sense,
>   one could say that these theories do not represent a *total*
>   vindication of the logicist program.  But it's going too far to say,
>   as you do, that the logicist program as a whole is a mistake or a
>   failure.
	I SAID IT FAILED TO ACHIEVE ITS ORIGINAL GOAL. THAT
	DOESN'T MEAN IT WAS A FAILURE AS A WHOLE OR A MISTAKE.
	I HAVE TOLD YOU OVER AND OVER THAT I RECOGNIZE THE ACHIEVEMENTS
	OF LOGICISM.  LIKE, LOOK BACK AT THE VERY FIRST QUESTION ON Y;OUR 
	LIST!!!>   
>   By the way, there is an alternative set theory known as New
>   Foundations (= NF), going back to Quine.  I don't know too much
>   about it, but my impression is that it attempts to carry out the
>   logicist program by deriving the axiom of infinity and others from
>   some logical principles.  Naturally there are costs to this.  As I
>   say, I am not an expert on this.  The FOM subscriber list includes
>   some experts on NF: Thomas Forster, Randall Holmes.
>   
>   Also, there is some recent work of Harvey Friedman about motivating
>   the axioms of set theory in a more logical way, as an outcome of a
>   theory of mathematical predication.
>   
>   Do you regard this kind of f.o.m. research as legitimate?  Do you
>   regard it as having potential interest for philosophy of
>   mathematics?
>   YES
> 4. Demonization
> 
>   Hersh writes:
>    > what do you mean, "demonize"?  When you attribute such motives to
>    > me, it's I who am being demonized.  To criticize or even reject
>    > foundationalism isn't demonizing anything.  It's what people do
>    > in the course of finding their philosophical beliefs.
>   
>   

You have gone beyond what I regard as legitimate philosophical
>   criticism.  

You have attacked foundationalism as anti-"humanistic",
>   anti-life in a sense, 

	IF YOU CAN CITE WHEN AND WHERE I DID SUCH A THING, 
	I WILL GLADLYL RETRACT AND APOLOGIZE.
~~
	MY GUESS IS THAT YOU ARE DAZED BY THE NAME I CHOSE FOR
	MY OWN IDEAS--HUMANISM.  THEN, BY A MISUSE OF LOGIC, YOU
	CONCLUDE THAT I AM CALLING YOU AND YOUR FRIENDS "ANATI-HUMANISTIC'
	WHICH COULD ONLY MEAN DISAGREEING WITH HUMANISM, AS YOU
	CERTAINLY DO.  BUT ON THE WAY YOU DROP A FEW LETTERS AND
	IMAGINE I CALLED YOU AND YOUR FRIENDS ANATI-HU;MAN. BIG
	MISTAKE.  NEXT TIME PAY BETTER ATTENTION

and you have tried to artificially link
>   foundationalism with religion and with authoritarian or totalitarian
>   politics.  I don't think "demonize" is too strong a term to describe
>   your behavior.

	I PRESENTED HISTORICAALLY THE CONNECTION BETWEEN RELIGION
	ABD PHILOSOPHY OF MATHEMATICS, FROM PLATO TO LEIBNIZ..I WOULD
	BE INTERESTED IF YOU BOTHERED TO READ THAT, AND TELL ME
	IF THERE ARE ANY MISTAKES OR FALSIFICATIONS THERE.  IN ANY
	CASE ESTABLISHING A LINK WITH RELIGION IS NOT DEMONIZING!
	AS FAR AS  POLITICS, THIS DISCUSSION
	WAS A FEW PAGES OF CHAPTER 11.  THERE I TABULATEAD
	THE PHILOSOPHICAL VS. POLITICAL VIEWS OF 23 PHILOSOPHERS.
	IT TURNED OUT THAT WHAT I CALL THE "MAINSTREAM"
 PHILOSOPHERS WERE MOSTLY RIGHT WING, AND THE OTHERS WHOM I CALL
"MAVERICK" WERE MOSTLY LEFT WIHG.  THAT'S IT. NOTHING ABOUT
AUTHORITARIAN
OR TOTALITARIAN.  IT WOULD BE NICE IF YOU LOOKED OVER THOSE FEW PAGES
AND TELL ME IF I MADE ANY MISTAKES. IF NOT, THEN I HAVE TO CONCLUDE
THAT YOU THINK IT'S OFFENSEIVE EVEN TO RAISE THE QUESTION OF
CORREELATION
BETWEEN PHILOSOPHY AND POLITICS.  FINE, IF THAT'S WHAT YOU THINK, SAY SO.
BUT WHERE IS THE DOMINIZING? >

	IT WAS DUMMETT WHO READ FREGE'S JOURNAL AND FOUND HE WAS A NAZI.
WAS DUMMETT DEMONIZING FREGE, OR DID FREGE DEMONIZE HIMSELF?
	BROUWER WAS TRIED BY HIS OWN UNIVERSITY AND CONVICTED OF
COLLABORATION WITH THE NAZI OCCUPATION.  THIS IS IN HIS BIOGRAPHY.
DID I DEMONIZE BROUWER? DID HIS BIOGRAPHER?  OR DID BROUWER DEMONIZE
HIMSELF?  MAYBE YOU THINK FACTS LIKE THESE SHOULD BE KEPT FROM
THE PUBLIC.  BUT IT'S TOO LATE.  IT'S IN YOUR LIBRARY BACK THERE IN
PENNSYLVANIA.

>    > It's weird to tell me I regard the pursuit of certainty as "evil
>    > incarnate."
>   
>   It's not at all weird to tell you this, in light of your attempt to
>   demonize foundationalism, on the explicit grounds that the
>   foundationalists (Frege, Brouwer, Hilbert, ...) were motivated by a
>   quest for certainty.
>  
 SAYING SOMEBLDY WAS MOTIVATED BY A QUEST FOR CERTAINTY IS NOT NOT NOT
DEMONIZING ANYBODY!  LOOK UP THE WORD IN YOUR DICTIONARY WHICH YOU SAID
YOU OWNED~!!! >   

 > to be fair, you'll have to accuse Sol of demonizing,
attacking, >    > and being "so hostile" to fom!!
>   
>   Not at all.  Sol Feferman has never attempted to demonize f.o.m. by
>   saying that it is anti-humanistic and linking it to totalitarian
>   politics, as you routinely do.
> 
> 5. Misinterpretation
> 
>   Reuben Hersh writes:
>    > I asked why you consistently persist in misinterpreting me.
>    > You didn't answer, of course.
>   
>   OK, I'll answer.  The answer is that I don't accept the premise of
>   your question.  The premise of your question is that I am
>   misinterpreting you.  I don't accept that premise.  I don't think I am
>   misinterpreting you.  I think my interpretation of you is correct.  To
>   put it colloquially, I think I've "got your number".
> 	GREAT!  LET'S HEAR IT!  WHAT IS THE NUMBER OF ME THAT YOU'VE GOT?
	I'M EAGER TO HEAR.

        REUBEN HERSH

> -- Steve
> 
> 