[FOM] Nelson's ultraformalism --to Slater

[Eray Ozkural]

On 11/5/06, Mirco Mannucci <mmannucc at cs.gmu.edu> wrote:
> 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)

[In the below, I drop "I think that" in some places, but all of the below
must be read as my ideas.]

I think that a computationalist approach may fare better at
this ambitious goal. Here is why. Abstract concepts help us predict
physical events better. However, all abstract concepts are computational.
There is no use for any abstract concept that does not lend to
computation. Therefore, we imagine mathematics to form by self-reflective
processes which have started finding out about facts of computation.
Proving 2+3=5 is a shortcut, short of counting, which is a computation.
In AI community, shortcuts have sometimes been called "chunking".
However, those approaches are unfortunately not capable of inventing
the ordinary kind of arithmetic (PA). Computational basis for such capability
on the other hand might arise from the application of inductive inference
(Reference  available upon request). It is unknown at the present if this
is the case.

Note also the relation to the halting problem, which seems to
correspond to the "golden standard" for computational
problems. The theorem 2+3=5 is an easy instance of the halting
problem, we can show a program that halts iff that is the case. On
the other hand, a little longer program can turn out to be very hard
(like twin primes conjecture).

However, I think the (ultra) formalist position does not necessitate getting
rid of potential infinity, because in my view the concept of potential
infinity has a crystal clear formalization which is a non-terminating
program, that everybody can objectively examine. I see no magic
with that, nor any assumption of Pythagorean philosophy (as well as
Platonism). It is false that numbers existed in some realm before
we thought them up. However, it is true that they exist in a
real sense of the word, as representations in our heads. This is a
formalist position, but it does not strip numbers of their intended
meaning (which is explained somewhat satisfactorily by the above
use-theory of meaning). Thus, potential infinity seems to remain as an
idealization that is often encountered in physical theory. Moreover, there are
no gaps in our reasoning when we make this idealization.

In particular, if we return to the twin primes conjecture, it seems
innocent enough to introduce an open ended loop to our program.
Why should we limit ourselves to only certain kinds of programs? They
are all the same to a computer. And furthermore, how are we going to
know which instance of the halting problem is admissible and which
is not? (This may be a stronger objection than it seems, since instances
of the halting problem do not necessarily refer to such abstract concepts
as integers. They are simply strings of symbols to a computer.)
Also, how are we going to know these upper bounds?

Thus, what I mean is, we need not look at what lies above, but what
lies beneath, in our heads. On the other hand, ultimately, all
meta-theory must be formalized accordingly. I believe some formal
meta-mathematical approaches comply with such restriction.

I am leaving the matter of actual infinity untouched, because that is
a topic that attracts too much debate and too few conclusions.

Finally:

> 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)

Is not this already achieved by Kolmogorov complexity theory? I know
that there has been widespread underrating of the theory, in favor of
saving the day for Platonic forms, which do not exist at any rate. In
particular,
some criticisms contain seriously flawed assumptions about computation. More
explanation available upon request, as that subject is beyond the scope of
the present post. However, I will just state that there are
incompleteness theorems
that are strictly about computational facts. I believe that counts as
syntactical
enough.

What do you have in your mind when you say that? Do you want to
state Godel's theorem in another way? Or do you want to prove a new
kind of incompleteness theorem?

Regards,

PS: Also, what do you think of Godel's statements when he said that
the second incompleteness theorem is also valid for finite systems. This
seems to weaken the finitist position a little, if we listen to Godel.

PS2: I am hoping that this will finally make some sense to the moderators.
I had expressed that it is best to understand set theory in terms
of computations. I can and will explain in much finer detail any of the
above points which hope to reconcile formalism and psychologism (the
latter termn has been used by Godel).

PS3: I apologize in advance if my ideas offend proponents of the
Pythagorean view.

-- 
Eray Ozkural, PhD candidate.  Comp. Sci. Dept., Bilkent University, Ankara
ai-philosophy: http://groups.yahoo.com/group/ai-philosophy