The Difference between Science and Mathematics

Date: 09/18/2006 at 21:58:46
From: Carlee
Subject: How is a math project different from a science fair project?

I need to do a science fair project (I'm in 11th grade, pre-calculus)
and I was thinking about doing a math project, but my friend told me
that math projects involve a "completely different process" from the
normal scientific process.  I asked her what she meant, but she wasn't
really sure, so I was wondering if you could answer me that?

Date: 09/19/2006 at 10:25:49
From: Doctor Ian
Subject: Re: How is a math project different from a science fair project?

Hi Carlee,

She's right.  In math, we start with some definitions, and we see what
follows from those definitions.  For example, we define numbers, and
operations, and then we find that we can define a category of numbers
called "prime" numbers, and then we can ask questions about them and
try to answer the questions.  

What's important here is that we KNOW what the "rules" are, because we
CHOSE them.  If we want to use different rules, we're free to do that.

For example, if we start with the rules of Euclidean geometry, we find
out that in any triangle we construct, the sum of the interior angles
is 180 degrees.  That's a consequence of choosing those rules.  If we
choose a different set of rules, e.g., the rules of spherical 
geometry, we find that this consequence is no longer true, e.g., we 
can construct triangles where the interior angles add up to 196 
degrees, or 258 degrees or a lot of other values that aren't reachable
in Euclidean geometry. 

It's very important to note that none of that has anything to do with
what's going on in the world!  It's just making up games and playing
them.  So the concepts of "true" and "false" become matters of 
internal consistency.  Can the angles of a triangle add up to more
than 180 degrees?  The answer can be "yes" or "no", depending on which
game you're playing.  But it's all "math". 

In science, it works in the other direction.  We don't know what the
rules are, so we try to set up situations that can help us guess what
they are.  Or, more precisely, we guess what the rules are, expressing
our guesses in terms of mathematics.  Then we use math to generate
some predictions.  And we test those predictions against the world. 

So a mathematical question would be something like:

  Given [some set of definitions], 
  and noticing that [some pattern that seems to hold],
  can we prove that the pattern must hold,
  or that it does not hold for at least one case? 

A specific example of this would be 

  Given the rules of standard arithmetic 
  and the standard definition of "prime number",
  and noticing that it seems to be possible to 
  express every even integer greater than 2
  as the sum of two primes,
  can we prove that this must always be the case,
  or find a counter-example? 

(This particular example is called Goldbach's Conjecture, and it's an
open question, i.e., no one has been able to prove it either way.)

A scientific question, on the other hand, would be something like:

  If [some theory is true],
  then in [some situation],
  we should see [some result],
  and if we see a different result, 
  then the theory isn't true. 
  So, do we see that result?

A specific example of this would be

  If Einstein's theory of general relativity is true,
  then as light from a distant star passes close to the sun
  on its way to earth, 
  the apparent distance of the star should be shifted
  by such-and-such an amount.
  So, does that happen?

To be even more precise, when asking scientific questions, we're
_really_ trying to differentiate between two competing theories.  So
that would look like

  [Theory A] predicts that in [some situation],
  we will see [Result A].
  [Theory B] predicts that in the same situation,
  we will see [Result B].
  Which prediction is closer to what we actually observe?

and

  Einstein's theory of general relativity predicts that 
  as light from a distant star passes close to the sun
  on its way to earth, 
  the apparent distance of the star should be shifted
  by such-and-such an amount.
  Newton's theory of gravitation predicts that there will
  be no shift.
  Which prediction is closer to what we actually observe?

Note that in the case of science, it may be the case that NEITHER
theory actually captures the rules that govern what's going on.  We're
not asking which theory is TRUE, but which theory is MORE ACCURATE.  

But that's the best we can do--keep guessing, and checking our 
guesses.  Because even if we get to the point where our best guess
seems to predict the answer to every question we can think up, that
might just mean we haven't been smart enough to think of the questions
that would show us that the theory is wrong.  

Does this make sense?

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/