NP Alabama Pi

[desert search for techno allah]

As with the architecture discussion, it seems not many are interested
in the philosophy of mathematics. But I will gladly continue this
discussion with anyone via email.

Paul Mackin writes:
>On Mon, 3 Jul 2000, desert search for techno allah wrote:
>
>> I think usually it's not thought that math is "in" nature somehow
>> in the sense I think you're meaning. That said, the relationship
>> is obviously there.
>
>Yes, I was thinking of in the extreme sense though it's hard to visualize
>what this might mean. Thanks for the reply.

The extreme sense is actually not all that odd a sense, in some
cases.

In the physical sciences often theorists attempt to come up with
"representation theorems" which show that they can take a physical
system and some representation of it, convert it into some more
tractable representation, do things with that, and then convert
back to the physical system. The picture to keep in mind here
is the analogy between temperature and the real number line -
the representation is conveniently depicted in a line of mercury
or alcohol in a thermometer (pointing the way to another representation-
identification - between the height of the mercury and the real
number line). These things get pretty complicated, but they're
similar in spirit.

There's a growing body of study, by scientists, that could loosely
be characterized as "formal systems study." All kinds of people
do this kind of research, often from more traditional fields -
mathematicians, computer scientists, physicists, biologists,
social scientists, etc. An odd thing about this research is
that it seems a lot more certain, in many respects, than traditional
science. At times, even as certain as mathematics. This gives
people pause because formal systems sciences give results about
_real world systems_ often based on _ideal systems_. For example,
take the Konigsberg bridge problem. In Konigsberg, there are
bridges arranged in a certain way over the river. It was once
a long-standing puzzle, whether or not one could make a complete
circuit of the bridge system, ending where one started, without
crossing paths - people would often try to solve the problem,
physically, for fun. The mathematician L. Euler proved that
it was impossible. Thing is, he proved it for a _graph_ which
represented the bridge system, with a series of nodes and
connections between them. But in a sense this is far more
certain than, say, a prediction about the bridges' stability
by a civil engineer.

The reason I bring this up is that there are philosophers of
science who want to argue that these formal systems which are
identified with the real ones are "in" the systems - they
are formal properties of them in much the same way their
structural integrities, e.g., are physical properties. This
is an odd but compelling argument; I don't think I believe it
but it's a tough one. A lot of the problems revolve around how
much identification is being done, as above in the more traditional
systems.

I can probably dig up a couple references if anyone's interested.


Josh

-- 
josh blog: http://www.public.iastate.edu/~kortbein/blog/
      tdr: http://www.public.iastate.edu/~kortbein/tdr/