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### Is Math a Science?

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Date: 03/18/2001 at 20:03:10
From: James Rouzier
Subject: Is math considered a science?

On a discussion board I mentioned that math is a science. Someone
said that math was not a science. Then I thought I usually do not hear
math mentioned as a science. Is it considered a science?

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Date: 03/18/2001 at 22:10:09
From: Doctor Jeremiah
Subject: Re: Is math considered a science?

Hi James,

I think that math can be considered a science if you look at it from
the right perspective.

Let's say you have a hypothesis (imagine you are Fermat or
Pythagoras). How would you prove that you were right?  You would do
an experiment (the proof) and arrive at a conclusion. This is the
scientific method, and it does fit how mathematics is done.
Sometimes it takes a while to do enough experiments to prove your
theory. Look at Fermat's Last Theorem for an example.

However, my opinion might not be the same as other people's, so I am
leaving this question and maybe another "Math Doctor" will give you
his or her opinion as well.

- Doctor Jeremiah, The Math Forum
http://mathforum.org/dr.math/
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Date: 03/19/2001 at 12:46:41
From: Doctor Jordi
Subject: Re: Is math considered a science?

Hi James. I would love to give my two cents here.

Mathematics... Science?  Philosophy? Art?  I think it's a wonderful
combination of all.

For one, I cannot think of mathematics as entirely a science; the two
are fundamentally different in a very important aspect: in science  we
have to look at reality and then give explanations, usually enlisting
the aid of mathematics as a coherent language in which to frame our
explanations... but mathematics is done in many other situations
beyond science. Pure mathematicians are sometimes proud to claim how
useless their discoveries are. For an example of this, I recommend
G.H. Hardy's succulent essay, "A Mathematician's Apology," which can
be found in a popular compilation entitled _World of Mathematics_.

In science we experiment. We go into the "real world," observe
phenomena, go back to the drawing table, and try to explain these
phenomena. Then we go back out to the world, see if we can predict a
new phenomenon before it happens (when we can do that we usually say
that we have discovered "a fundamental law of nature"), and either
smugly rest for the day, or crawl back to the drawing table, slightly
disappointed if our hypothesis did not work as we intended.

This, in general, is what we call the "scientific method."

Mathematics is different. Although I agree with Dr. Jeremiah that
mathematics is becoming an experimental discipline, particularly with
the recent introduction of powerful calculating machines, it does not
rely on these experiments in order to claim "Eureka! I have discovered
a new truth!" Mathematics requires proof, and it's very picky about
what it considers proof to be. For a scientist, ten experiments with
consistent results might constitute proof, "within experimental
error."  For a mathematician, a googleplex of successful experiments
is not enough proof. Instead, we rely on logic, and this thing we call
"common sense," fundamental logical rules we believe no one will
dispute, very basic rules.

Mathematics is very often inspired by nature, but it is a purely
intellectual pursuit. It is just a bunch of ideas in our heads, like
philosophy. Unlike most of philosophy, there is some "glue" to it all,
some fundamental unity, something we call logic, reason, order.  Pure
abstract reasoning.

That's why I sometimes like to say that mathematics is applied
philosophy. Philosophy under the influence of very specific rules.

Then there's the aesthetics of it. The capacity of mathematics to be
an art. This is one of my favourite interpretations. The sheer
simplistic beauty, the awe one can feel when one reads an entire proof
and understands every aspect of it, when a surprising truth is found
by unsurprising means... this is a very personal experience, I think.
You really have to feel it in the flesh to understand it. That flash
of understanding when a complex problem has been solved. That simple
marvel of seeing many unrelated ideas congegrate under a single roof
of logic and order. This is what spurs the most romantic of
mathematicians to keep on trying to prove that ancient conjecture.

At least that's why _I_ do it.

I love these open-ended questions!  I like to think, just as everyone
else likes to think.  What do you think?

- Doctor Jordi, The Math Forum
http://mathforum.org/dr.math/
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Date: 03/19/2001 at 13:06:18
From: Doctor Ian
Subject: Re: Is math considered a science?

Hi James,

from three different math doctors!  Nicely done.

Math is most definitely _not_ a science. In science, (we assume that)
there is a set of rules (the fundamental laws of nature) in operation,
and the task is to figure out what the rules are by observing the
results that occur when the rules are followed. Basically, it's an
attempt to reverse-engineer the machinery of the universe.

In math, it's the other way around - we get to choose the rules, and
the task is to discover the results of choosing any particular set of
rules.

There is a superficial similarity, which leads some people to confuse
the two pursuits. In science, the way you test a theory is to codify
it as a set of rules, and then explore the consequences of those rules
- in effect, to predict what would happen if those rules were true.
You do the same thing in math - and in fact, the way it's done in math
serves as a model for the way it's done in science.

But here is the big difference: In science, as soon as your
predictions conflict with experimental data, you're done. You know
that your rules are wrong, and you need to start putting together a
new set.

In math, this kind of conflict can't arise, because there is no
necessary connection between any mathematical theory and the world.
The way you 'test' a set of rules in math is see whether the results
they produce are interesting enough to induce mathematicians to keep
playing with them.

We might summarize the situation this way: Science is the pursuit of
_the_ correct description of _this_ particular world; whereas math is
the pursuit of interesting descriptions of possible worlds. Whereas
scientific theories are right or wrong, mathematical 'theories' are
merely interesting or uninteresting.

more, or if you have any other questions.

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