Science And Math Was: Python's Lisp heritage

[Gonçalo Rodrigues]

Just some clarifications.

On Mon, 22 Apr 2002 06:10:02 GMT, Tim Daneliuk <tundra at tundraware.com>
wrote:

>Carl Banks wrote:
>> 
><SNIP>
>
>> 
>> Sorry, but I don't agree that algorithms, or any mathematical concept,
>> is an invention of the human mind.  The algorithms have existed since
>> the beginning of time.  The quicksort had the property of requiring ~
>> n^2 comparisons in the worst case before it had ever been used or
>> applied.  We merely discovered this algorithm; we didn't create it.
>> 
>> So I say mathematics, including the study of algorithms, is a natural
>> science.  And, for now, that's all I have to say about it.
>
>I disagree strongly (but am willing to be convinced otherwise).  All of
>mathematics is a formal construct of the human mind created with the
>intent of removing ambiguity and enhancing our ability to describe what
>we *think* and what we *observe*.  We can (and do) create all kinds of new
>calculi that are problem- or domain-specific.  That is, we *invent*
>mathematics to suit our needs - mathematics is not one-for-one
>correspondent to the natural universe.

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.

>
>e.g., Calling the upper-bound of Quicksort N^2 is merely one way of looking at 
>the time complexity of this "idea".  There is not some hardwired constant
>in the universe that declares heapsort to be O*(N^2) - this is merely an
>artifact of the calculus used for Big-O analysis.
>
>The reason I take this position is pretty simple.  We are able to construct
>mathematical systems which describe intellectual abstractions for which
>there is no natural analog.   That is, mathematics can embrace way more than
>just the natural world and its workings.  (Theoretical mathematicians are famous
>for their contempt of "mere" Applied mathematicians whom they view as operating in a
>very limited, mechanical intellectual domain.)  For instance, even the theoretical

You should be careful when you say this. There is *virtually no*
mathematical field that has not been applied.

>physicists only need about 5 dimensions to describe Life, The Universe, and
>Everything, but mathematicians routinely work in arbitrary "n-space" for which
>there is not physical/natural correspondent.  Similarly, a mathematician
>can introduce ideas like an "Incompleteness Theorem" which is a meta-mathematical
>commentary on mathematics itself - something far beyond the domain and range of
>mere natural science.

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?

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.

>
>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.

>
>
>------------------------------------------------------------------------------
>Tim Daneliuk
>tundra at tundraware.com

best regards,
Gonçalo Rodrigues