FOM: Hersh's pointless attack on logicism and formalism

[Stephen G Simpson]

Hersh Thu, 24 Sep 1998 14:34:06 writes:
 > I offered to teach a course ... on Foundations of Math.  ...
 > 	I had expected naively that this would be like teaching
 > number theory or differential geometry. ....

That expectation was indeed very naive, if not downright arrogant.

Evidently Hersh found out that teaching a course in f.o.m. is *not*
like teaching a course in number theory, differential geometry, etc.
Many people, f.o.m. professionals and laymen alike, were already aware
of this.

I would interpret Hersh's discovery as more evidence for something
that I have been saying all along: f.o.m. is not just another branch
of mathematics.  Its goals and subject matter are very different.  I
would formulate the difference as follows: F.o.m. focuses on the most
basic concepts of mathematics, with an eye to the unity of human
knowledge.  Thus f.o.m. necessarily involves a lot of philosophical
overtones and motivation which are not essential in core mathematical
subjects.

Not everyone would agree with the views expressed in the previous
paragraph.  For example, Shoenfield has said that he regards f.o.m. as
just another branch of mathematics.  I would imagine that this view is
reflected to some extent in Shoenfield's lecture style: very dry, very
precise, very little in the way of philosophical motivation.  Perhaps
Shoenfield does not even regard philosophy of mathematics as a
legitimate subject.

Returning to Hersh, I still don't know what Hersh means by
"foundationalism" and why he is so hostile to it.  Hersh says that
foundationalism is the pursuit of indubitability, but as Martin Davis
pointed out, indubitability is a straw man.  Hersh identifies Frege,
Russell, Hilbert, ... as the enemy, but he admits that their research
was of great scientific value.  I now suspect that Hersh's views in
opposition to "foundationalism" are completely incoherent.  We may
never get to the bottom of Hersh's real intent.

 > If my words against logicism, formalism and intuitionism haved been
 > hurtful, I'm sorry.

Hersh's words are not hurtful, only pointless and incoherent.  He
displays a lot of hostility toward "foundationalism", but he doesn't
present what I would regard as a serious scientific or philosophical
argument against the work of Frege, Russell, Hilbert, G"odel, et al.
Nor does he present any serious philosophical or scientific ideas of
his own.  

Let's look at Hersh's views on two specific f.o.m. programs, logicism
and formalism.

1. Logicism

Hersh formulates the goal of logicism as:

   to make mathematics indubitable by reducing it to logic.

Hersh then attacks logicism on the grounds that this supposed goal was
allegedly not achieved.  Such a loaded approach is obviously not a
good way to evaluate anything.  A better approach would be to
formulate the goal of logicism somewhat differently:

   to investigate the extent to which mathematics is reducible to
   logic, and along the way to develop technical tools which can be
   used to investigate other related philosophical or foundational
   questions concerning the nature and basic concepts of mathematics.

This is *much* more fruitful.  With this formulation, one can talk
about the achievements which flowed from the logicist program:
identification of the predicate calculus as an explication of
mathematical and non-mathematical reasoning; identification of the
axioms of set theory; set-theoretic foundations; a far-reaching
explication of mathematical rigor in terms of provability in ZFC; the
independence phenomenon; etc etc.  Some of Hersh's pronouncements make
me wonder whether Hersh is even aware of these developments, let alone
whether he has thought seriously about them.

2. Formalism

Hersh states the goal of formalism as:

   to make mathematics indubitable by reducing it to the manipulation
   of meaningless formal symbols.

This caricature of Hilbert's program is worse than useless.  In his
posting of 25 Sep 1998 23:00:09, Hersh explicitly dismisses Hilbert's
actual views, on the grounds that "Hilbert is no longer around".  This
makes me question whether Hersh is really interested in this subject,
as he claims to be.

A much more accurate and fruitful formulation of Hilbert's program is
in terms of finitistic reductionism:

   to investigate the extent to which mathematics can dispense with
   actual infinities; to investigate the extent to which mathematics
   is reducible to finitism; to develop technical tools which can be
   used to investigate these and related philosophical or foundational
   questions concerning the nature and basic concepts of mathematics.

With this formulation of Hilbert's program, one can discuss the
achievements which flowed from it: conservation results, consistency
proofs, primitive recursive functions, G"odel's first and second
incompleteness theorems, Gentzen-style ordinal analysis, etc etc.
Concerning the specific issue of finitistic reductionism, considerable
progress has been made, as detailed in my paper "Partial Realizations
of Hilbert's Program", JSL 53, 349-363,
http://www.math.psu.edu/simpson/papers/hilbert/.  That paper contains
the following estimate:

  at least 85 percent of existing mathematics can be formalized within
  WKL_0 or WKL_0^+ or stronger systems which are conservative over PRA
  with respect to Pi^0_2 sentences.

Since PRA (= primitive recursive arithmetic) is finitistic, this
estimate would if true represent a significant vindication of
Hilbert's program of finitistic reductionism.

To Hersh, all of this is meaningless.  I raised these issues pointedly
in my posting of 17 Sep 1998 18:16:23 entitled "Hersh on the axiom of
infinity", but I got no response at all from Hersh.  Hersh 12 Sep 1998
18:06:45 loves to use the axiom of infinity to attack
"foundationalism", but he is adamantly unwilling to examine any and
all evidence concerning the actual role of the axiom of infinity in
mathematics.  Specifically, Hersh is uninterested in knowing the
outcome of Hilbert's program of finitistic reductionism.

In sum, I am beginning to wonder whether the views of Hersh and his
"humanist" followers concerning f.o.m. deserve to be taken seriously.

-- Steve