FOM: Hersh's incoherent attack on formalism and logicism

[Stephen G Simpson]

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.
  
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?  

  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.
  
  Are you willing to rescind your dismissal?  Are you willing to
  examine evidence against your remark?
  
   > 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.

  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?
  
   > 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?  Do you recognize a
  difference between their views and those of Hilbert?  Or are you
  merely attacking coffee-room chatter, as Martin Davis suggested?
  
   > 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.
  
  Here are the real questions:
  
  Do *you* think Hilbert's program is of any actual or potential value
  for philosophy of mathematics?
  
  Do *you* think the research of your other formalists (who are they?)
  has any actual or potential value for philosophy of mathematics?
  
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.
  
  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?
  
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, 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.
  
   > 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.
  
   > 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".

-- Steve