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### What is Math?

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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?
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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

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/
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