Is Math a Science?

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?

Thank you for your time.

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/   

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/   

Date: 03/19/2001 at 13:06:18
From: Doctor Ian
Subject: Re: Is math considered a science?

Hi James,

Your question is interesting enough that you've managed to get answers 
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. 

I hope this helps.  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

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