[FOM] Nelson's ultraformalism --to Slater

Mirco Mannucci mmannucc at cs.gmu.edu
Sun Nov 5 12:31:52 EST 2006

---- Hartley Slater <slaterbh at cyllene.uwa.edu.au> wrote:

> I have read this paper ('Syntax and Semantics'), and can readily see
> why it should be thought important.

Actually, I ought to  apologize, I am guilty of pointing everyone to
the WRONG paper: the one I had in mind is called "Mathematics and Faith"


and the amazing "confession" I was referring to is at page 7,
beginning with the sentence:

--I must relate how I lost my faith in pythagorean numbers--.

Anyway, the one you are discussing is along the same lines: Nelson, if
I understand him at all, essentially proposes a rigorous ultra-formalism.

  --->   Ultra-formalism = everything (in math) is syntax, syntax and
         nothing but syntax

I think that this is a FOM challenge for everybody, and I take it very
seriously (to the point of taking Nelson's logic itself to task, when he
proposes his own brand of finitistic arithmetics and finitistic
set theory).

I also  think that there are a few interesting topics to ponder here.

For instance:

1) can we keep a formalist approach in a coherent way, not just
at the mathematical level, but also at the meta-mathematical one? (stopping
finally the meta-delusion that there is a meta-level "above"  math)

2) can we give a positive  interpretation of the incompleteness phenomena in
the ultra-formalist approach? In other words, can we account for Godel's
incompleteness as saying something ABOUT our syntactical games, as opposed
to  referring to some "intended" pre-built platonic structures?
(the negative interpretation, saying--they are great theorems, but
they mean nothing---, is the cheap way out)

3) how does the math world looks to a ultra-formalist that does not allow
himself/herself the (expensive!) luxury of naively using infinitary
arguments of any type and shape?

I have my own view (which I shall present in due time), but I would like
to hear from others on 1)--> 3)

Best Wishes

Mirco A.  Mannucci

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