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/ |
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