FOM: Hersh's unfruitful attack on logicism and formalism

[Stephen G Simpson]

Reuben Hersh writes:
 > I see that you can restate the goals of formalism and logicism
 > .... why not say that your are updating the goals?

I'm *not* updating the goals, at least not intentionally.  My
statement of the goals was intended to be reasonably accurate.

Let's take Frege. You say that Frege's goal was to reduce mathematics
to logic.  I say that Frege's goal was to investigate *the extent to
which* mathematics is reducible to logic.  From the scientific point
of view, these are two ways of saying the same thing.  But my way is
better, because it's more fruitful.  You must dismiss Frege's work as
a failure, while I can build on Frege's genuine and remarkable
achievements.

One of Frege's remarkable achievements is the invention of the
predicate calculus.  Don't you agree that this is a remarkable
achievement?  The predicate calculus is used to study all sorts of
interesting issues in f.o.m., especially issues concerning the logical
structure of mathematics.  Don't you agree that this approach is more
fruitful than yours?

Similarly for Hilbert.  You present a caricature of Hilbert's ideas,
then attack the caricature.  This may make you feel good, but it's not
scientifically fruitful.  It's much more fruitful to examine the
extent to which Hilbert's actual goals can be achieved.

 > [foundationalism] is a term coined by Imre Lakatos, to express his
 > insight that formalism. logicism, and intuitionism shared a common
 > goal of restsoring certainty to mathematics, ...

I'm glad you are now saying "certainty" rather than resorting to your
earlier slippery term, "indubitability".  

 > Foundationalism is a grab baG word that means logicism, formalilsm,
 > or ilntuitionism, based on the insight that they shared a comomon
 > goal.

OK, now at last we know what you are talking about.  When you demonize
"foundationalism", you are really demonizing logicism, formalism and
intuitionism, based on Lakatos's alleged insight that these three
programs share a common goal, viz. certainty.

I have a question.  Isn't certainty the goal of all science?  If so,
and if you regard the pursuit of certainty as evil incarnate
(anti-"humanistic" and so on), then doesn't it follow that you are
opposed to all science?

 > You challenge me to study how the axiom of infinithy is used.  You
 > are indignant that I don't immediately volunteer for this
 > assignment.

Of course I'm indignant.  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?

 > You seem to think that no one but an fom'er is allowed to be
 > interested in the nature of mathematical existence and knowledge.

I never said that.  People can be interested in all kinds of things.
Prior knowledge of a subject is not a prerequisite for being
interested in a subject.

 > I write that indubitabililty is no longer regarded as a viable
 > goal.  You say it's a straw man.  It seems your'e agreeing with me
 > while thinking you're attacking me.

I'm not agreeing with you.  Indubitability was never an issue.  It was
always a straw man.  As I said before, nothing is or can be
indubitable, because any yokel can doubt anything at any time.  The
real issue is scientific certainty, not indubitability.

 > You misquoted me seriously when you said I said Hilbert isn't
 > around.

I misquoted you, but not seriously.  You were arguing that it's OK to
dismiss Hilbert's views without a hearing.  That's unacceptable, no
matter what reasons you try to adduce.

 > to my knowledge, nobody is pushing Hilbert's ...

Are you familiar with the literature in this area?  Are you interested
in becoming familiar with it?  For instance, the 1988 Journal of
Symbolic Logic contains a symposium on Hilbert's program.  Have you
read it?

 > Joining the list did not mean becoming an fom'er or an uncritical
 > booster of fom.  The idea was that people with different viewpoints
 > may try to understand each, and possibly learn from each other.

Yes.  For instance, I'm trying to learn from you why you and other
quasi-empiricists are so hostile to f.o.m.

-- Steve