Mountain Man's Global News Archive Is Mathematics a Science? Mati Meron (sci.physics)
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The Processes of Mathematics and Science

Date: Fri Jul 03 08:43:13 1998
From: meron@cars3.uchicago.edu
Newsgroups: sci.math,sci.physics,sci.logic,alt.philosophy.debate
Subject: Is Mathematics a Science?

Deduction is a clear cut process.  From A follows B, from B follows C.
You start with some facts or statments taken as true and proceed from
there.  The process may be long and tedious but is uniquely defined.
You cannot start with Euclid's axioms and reach a theorem stating that
in a right triangle a^2 + b^ < c^2.

What you do in physics is the opposite.  A priori you don't have the
basic laws.  You've observation, but only finite number of those and
they're always performed to a finite accuracy.  This is not sufficient
to uniquely identify the underlying physical laws.  In other words,
there is no process where by purely logical analysis of the data you
can arrive at a unique set of physical laws that could've give rise to
it.

So, what do you do?  You guess.  Not randomly of course, you make the
most educated guess you're capable of.  You utilize not only the data
at your disposition but also general ideas based on past experience as
well as (gasp!, horror!) beliefs.  Such as the belief that physical
laws should be inherently simple.  But even with all this stuff added
you're still not in a position where there is only a single possible
set left.  No, there is always an infinity of possible sets and you
pick one of them because it is "simple", because "it makes good sense"
etc.  This is the "law identification" part and it is inductive, not
deductive.

Now, that you've done it, you go into the deductive stage.  You
assume (temporarily) that the laws you've arrived at are indeed true
and you try to find out what can be deduced from them.  First of all,
of course, it must be possible, starting with these laws, to arrive at
results coinciding with the data you already have, else they are DOA.
But then you go further and generate predictions regarding things you
didn't measure yet.  And then you measure these things to see whether
your predictions come true and with each additional prediction that
pans out your confidence in your theory grows.  But it never becomes
an absolute confidence since it is always based only on a finite set
of information.  There is always a possibility that you'll reach a
region of the parameter space where the theory starts deviating form
the data (or even that more precise measurements in the region you've
already covered will indicate deviations).  And when this happens, the
whole process is repated over again.  Guess basic laws, work out
predictions, etc.  It is this eternal circle of
inductive-deductive-empirical-... that's at the core of science.

So, no, it is not a deductive process, though there are some parts of
it which are deductive.  In a deductive process the result is either
fully determined by the axioms and the logic, or fully indeterminate.

For a mathematician to, start with Euclid's axioms and derive a result
contradictin Pythagoras Theorem, is something which could occur only
through an error of logic.  But is wasn't an error of logic which
prevented Newton from arriving at relativity, since there was no
single theory following logically from the information at his
disposition.  There was a potential infinity of choices and he came
with the best guess that could have been formulated based on what was
known to him.  Couple hundred years later, when more was known, a
better guess could've been formulated and it is quite probable that
better guesses yet will be formulated in the future.

Now, am I making any sense here or am I just wasting my time?  I've
other stuff to do, you know.

Mati Meron                      | "When you argue with a fool,
meron@cars.uchicago.edu         |  chances are he is doing just the same"

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