Science And Math Was: Python's Lisp heritage

[Gonçalo Rodrigues]

On Mon, 22 Apr 2002 18:10:03 GMT, Tim Daneliuk <tundra at tundraware.com>
wrote:

>"Gonçalo Rodrigues" wrote:
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>
>> 
>> You should understand that this is a philosophical viewpoint. The
>> mathematical field is roughly divided in the Platonist field, those who
>> believe that mathematical objects exist somewhere out there in space (to
>> quote Sun Ra) in a Wonderful World of Platonic Ideas, and the formalist
>> field of those who believe mathematics is essentialy a game we play on
>> paper with symbols with no "real" meaning attached to it besides the
>> convenience in describing natural phenomena.
>
>Yup, I'm aware of this - I didn't want to digress more into philosophy
>than absolutely necessary.  IIRC, the Enlightenment thinkers also viewed
>mathematics as a branch of natural science and they were (mostly) 
>Aristoteleans, right?
>
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>
>> You should be careful when you say this. There is *virtually no*
>> mathematical field that has not been applied.
>
>True enough, but that is not a sufficient condition to argue that mathematics
>is correspondent one-for-one to the natural world.  All I'm arguing here is that
>mathematics embraces a problem set *larger* than just the natural world and
>this cannot simply be a natural science.

Ahh, OK. Then I agree completely -- Mathematics *is* larger than its
would-be applications to the natural sciences. I think it was P. Erdos
who remarked once that all natural science is conceivably finite - e.g.
there would be a time when there would be nothing more to learn about
the universe - but mathematics is literally infinite.

>
>
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>
>> This is plainly not true. String theory models live in 10 and 24
>> dimensions. You would not want to disparage the biggest fad in current
>> theoretical physics, would you?
>
>
>Well, OK, I'll say it more formally:  Let n denote the hightest dimension
>used by any of the natural sciences.  Mathematicians routinely operate
>in k dimensions where k >> n.   Acceptable?

In quantum mechanics the state spaces are Hilbert spaces of infinite
dimensions (wether analytically or algebraically speaking). Believe me,
theoretical physics (or more correctly, mathematical physics)
uses/invents the same arsenal as mathematics. There is really no clean
separation.

>
>> 
>> Even so let me reiterate, that almost everything that mathematicians
>> have concocted is currently used in theoretical physics. To give you
>> just an example, my own, I use homotopy and higher order category
>> theory.
>> 
>> Even more when mathematics advances by theoretical physics promptings. I
>> could give you numerous examples, but just remind you that quantum
>> mechanics contributed a LOT (the celebrated Von Neumann, for example) to
>> the growth of abstract mathematics (functional analysis, Hilbert spaces,
>> operator algebras, etc.). Some of the more active work in mathematics
>> today is also prompted by theoretical physics.
>> 
>> Somehow (I do not intend to make this more precise now) new mathematical
>> formalisms and the discover of new laws in the Universe go hand in hand.
>
>
>Again, my claim was not that mathematics cannot be/is not appliable, merely
>that it is bigger than just its applications.
>
>> >
>> >I'm not choosing sides here on the math vs. science debate - both are important
>> >artifacts of the human intellect.  I merely take umbrage with the notion that
>> >mathematics is innately wired into the universe somehow.
>> 
>> See above to understand that what you are saying is really not that
>> obvious.
>
>
>Well almost nothing about epistemology is obvious.  As I said, I am willing
>to be persuaded otherwise, but it sure looks to me like mathematics is the
>construct I claim, not innate to the natural world...

The problem is what is exactly "innate to the world". A Platonist would
argue that there is a universe out there where the mathematical objects
exist independently. In short, If the universe ended right now, these
objects would continue to exist.

All the best,
Gonçalo Rodrigues