Existence of mathematical entities (Re: Successor Axiom: on what grounds TF?)

examachine_at_gmail.com
Date: 02/08/05 

Date: 7 Feb 2005 19:37:38 -0800

[I'm cross-posting a little, because this has gotten immensely
interesting.]

Jeffrey Ketland wrote:
> <examachine@gmail.com> wrote in message
> news:1107806519.664395.170850@c13g2000cwb.googlegroups.com...
> >
> > Jeffrey Ketland wrote:
> >> >Would you say, well, read this set
> >> > theory/topology textbook, it tells all about it?
> >>
> >> Actually, yes. I'd say, "If you want to learn about the geometry
of
> > physics,
> >> here's a nice textbook: Theodore Frankel 1997, _The Geometry of
> > Physics_
> >> (CUP)".
> >> Just checking, I see no discussion of ultra-finitist or nominalist
> >> philosophy in this textbook. Seems like the physicists are not
much
> >> interested in sceptical philosophical fairy tales. Rightly so.
> >> They're trying to understand the world, and will happily use any
> >> mathematics that
> >> happens to have been developed (even if it contradicts
> >> finitist/nominalist/constructivist scruples).
> >
> > No, it's just that they don't give real numbers the philosophical
> > significance you assign.
>
> I assign no "philosophical significance" to real numbers. Similarly,
> I assign no "philosophical significance" to atoms, quarks, genes or
> galaxies. Every mathematicized theory in science assumes the
> existence of mathematical entities.

I really wonder the opinion of physicists, theorists and engineers who
might be reading this.

Do any of the theories you use assume the "existence of mathematical
entities", e.g. Godel's mathematical realism?

Are not there any worthwhile competing points of view, for instance
such as physicalism, computationalism, functionalism, instrumentalism,
etc.?

Why do you spare the word "existence" so easily? I always thought of it
as a word to use very carefully, and indicate in which sense when I use
it. Surely, few people believe that mathematical objects exist in the
same fashion an apple exists.

And, obviously, to a philosopher of mind, what you say might sound
infinitely strange. I don't know how you would make that compatible
with physicalism, in particular with substance monism.

I am a computer scientist. We use integer formulas to calculate the
efficiency of our programs. I say a particular algorithm takes O(n^2)
time. Does the f(n)=n^2 function "exist"? Where except my head and on
paper? :) Surely, you must be joking Mr. Ketland, but I am finally
getting the hang of it. That was a really good one. :)

Are you aware that you just said all scientists must believe in
mathematical realism, or they are not scientists? That every
philosophical view about mathematics except mathematical realism is
counter-scientific?

I personally find mathematical realism to be a mysticist and
counter-scientific point of view, that's why perhaps it's relieving for
otherwise bored souls of some mathematicians. I know that
mathematicians and smart people in general have a tendency for
mysticism, including Godel. Well, one of my favorite mathematicians,
Gregory Chaitin wrote on Kabbalah in a paper, so what can I say? But
he's expressed skepticism of the reality of real numbers in another
paper. And he is more or less the successor to Godel even if you don't
like him, so make what you will out of this. Who'd know mathematical
realism better? Perhaps mathematics too evolves. :)

> This is trivial:
> it is what is meant by "mathematicized theory":
> i.e., one that quantifies over numbers, sets, functions, etc.
> What, for example, is a (scalar) physical quantity Q? It is a
*function*
> from a domain of physical entities to the *real line* R. So, e.g., if
x is a
> physical object, then Mass_kg(x) is a real number. (Different scales
are
> related by admissible transfomations, as explained in work on
representation
> theorems. E.g., Krantz et al. 1971 _Foundations of Measurement_.) All
of
> this is assumed without further comment. And this logically
contradicts
> nominalism, constructivism, finitism, etc.

All of this is assumed without further comment by the holy Roman
Church?

But you contradict yourself. You just said F=ma is true in one post,
and then in another post you said it was false. Which is it?

May I suggest you may be overlooking what the "real" philosophy behind
these "postmodernist guys" is? It is computationalism or in general
"machinism". I personally would not be content with anything less than
a mechanical theory of nature.

Also, you've just made one major hand-waving move by dismissing
constructivism in a single sentence. :) I have a lot of respect for
some of the smart people who pursued those movements.

Some physicists like Penrose, by assuming the things you say are
assumed without further comment, reached fairly absurd conclusions, for
instance about philosophy of psychology (aka mind).

If you believe in this non-mechanical, soul-like posits, then of
course, every kind of falsehood follows, including quantum souls or
mathematical gods. Let alone entire universes beneath the Planck scale.
It's a fantasy mine for many people, and that's what we are skeptical
of, nobody is skeptical of the physical world or brains or intellect.

Regards, 

--
Eray Ozkural

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