What is Math?

Date: 09/14/2001 at 15:56:36
From: Rob Gleeson
Subject: What is math

What is math? Just how do you define it? Is it a study of patterns or 
whatever?

Date: 09/14/2001 at 23:35:33
From: Doctor Peterson
Subject: Re: What is math

Hi, Rob.

If you asked a dozen mathematicians (or lexicographers, even) what 
math is, you would probably get a dozen answers. You might want to 
gather different ideas by looking it up in various dictionaries or 
encyclopedias, and reading the introductory chapters of several 
popular books on math, which may tell what the authors think math is 
all about. There is no one "correct" answer.

You can also search our archives by going to

    http://mathforum.org/mathgrepform.html   

and typing in 

   what is math

selecting "that exact phrase." You will find a few opinions we have 
given.

My own favorite definition is that math is the study of abstractions. 
That is, we isolate one or a few features of some kind of object for 
study, and see what we can learn about the behavior of those features 
while ignoring everything else about them: features like number, 
shape, or direction. For example, when we work with numbers we are 
taking the concept of counting away from all other details about the 
things we are counting, such as color or name, and just thinking about 
how many there are. We learn to work with numbers as an abstract 
entity, so that we can add two numbers without having to think of them 
as representing two apples and three apples. When we finish our 
calculations with numbers, we can come back to the real world and know 
just how many apples we have.

It is often found that a concept that is first encountered in one part 
of our experience turns out to be useful in other areas as well. 
Having solved a problem in one context, we don't have to solve it 
again, because we solved it abstractly. For example, a common problem 
is to find the number of sides and diagonals of a polygon. It turns 
out that the same solution applies also to a question about the number 
of handshakes that occur if everyone in a room shakes hands with 
everyone else, and also to a problem about the number of different 
ways a student could choose two classes to take. They all look the 
same when you think of them abstractly. If I know the solution to one 
of these problems, I can transform a new problem into the known 
problem, and quickly find the answer. That kind of thinking is central 
to what math is.

As you go more deeply into math, you find that we end up studying 
abstractions of abstractions, such as systems of objects that behave 
like numbers, but don't follow all the same rules. This turns out to 
be surprisingly useful; for example, rotations of an object in space, 
thought of as if they were a sort of number, can be handled very 
neatly, even though they work very differently from numbers in some 
respects.

As you can see, math goes far beyond arithmetic, or even algebra and 
geometry. All sorts of logical thinking fit this description. And math 
is a very creative field, involving exploration of the unknown, not 
just learning rules we are told to follow. Mathematicians invent 
abstract worlds, and discover all the surprises in them that are never 
noticed by those who don't look for the abstractions behind the 
reality.

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