Re: article on schools of foundations of mathematics

examachine_at_gmail.com
Date: 01/19/05 

Date: 19 Jan 2005 05:05:17 -0800

I find it unfortunate that some sci.math audience is keen on the
weakest philosophical foundations mentioned AFAICT: platonism
(chapman?) and logicism (torkel?).

Why do people shy away from intuitionism, constructionism, formalism,
psychologism, physicalism, computationalism and other tenable positions
while they should try to understand them?

In particular, the logicists here seem to be of an especially strange
variety, who seem to think that all foundational matters have been
settled. Some of them have been so vulgar as to try to undermine
significant theorems like in Chaitin's information theory work, of
course they are simply following the footsteps of their luminaries who
have an equally inadequate understanding of computer science.

There is more to this strange logicism. They seem to think that logic
is the true foundation of mathematics, and that it is made up of
meaningless tautologies. I think anybody who did real mathematics would
know that mathematics has genuine and meaningful content. And anybody
who did logical reasoning would know that formal logic models certain
reliable forms of reasoning, which are also mirrored (to some extent)
by certain linguistic constructs.

If mathematics were meaningless, I'm afraid nothing would be.

Instead, there is always an irreducible essence to any significant
mathematical work, be it in your brain, in the computer, or on the
paper. There is also an spparent way in which mathematics is *used*,
especially in the scientific context. (The artistic aspects do not
interest me much when I speak of foundations)

Imagine that *no* mathematical theorem had any application whatsoever
(be it physical, psychological, etc.). If we could not use it to talk
and *think* about the world in any way, then it would be truly
meaningless. But it is not like that. On the contrary, it is chock-full
of meaning, even for some of the most abstract theories like
information theory or category theory! Likewise for logic. Logic is not
"good" because it is glorified. It is good, because it is the first and
most widely used artificial intelligence, e.g. a technique to mechanize
intelligence, to be conscious of certain modes of thinking in a very
very useful way. Logic started the day some ancestor of ours observed
his way of thinking and could distinguish logical thoughts from
illogical thoughts. (This reflection may have required a new kind of a
language)

The classical logicism which assumes a clear-cut boundary between
analytic and synthetic propositions must be abandoned. There can be
nothing in the world that is free of physics. Our philosophy of logic
can evolve.

These "classical logicists" must try to read and understand Godel's
philosophical work which shows the implausibility of radical
reductionism.

In addition, there seem to be certain realist elemenents in the
logicists here, which confuses things all the worse. It's extremely
difficult to tell if the logicists here favor Godel-like realism of
mathematical objects.

Sometimes, they seem to assume that certain logical facts or rules of
logic preside over everything else. Perhaps they think our universe
consists in logical sentences that float about in nothingness. I have
never been able to understand what a classical logicist thinks
a) what the world is
b) what the mind is

I would be glad if one of the resident logicism proponents could tell
this to me. What is the "mind" according to the school of logicism?
Regards, 

--
Eray

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