Why study math ...

Because mathematics is intriguing. As Bertrand Russell put it, "mathematics is the subject in which we know neither what we are talking about nor whether what we say is true." So, perhaps, first and foremost, we should study it to find out what it is. We owe it to ourselves to at least know what mathematics is before we claim that it isn’t really important in our daily lives or before we close off career opportunities which depend on it.

Mathematics is about relationships, or put more "mathematically" , it is about structure. Even at its most basic level, relationships are the name of the game. First you learn to count, and the important relationship is "order". Then you learn to add and multiply; you start to pair off numbers to produce new numbers, according to what seem to be arbitrary and mysterious "laws". But you soon learn, much to your dismay, that although they might be mysterious, these laws are not arbitrary. If you don’t get them exactly right, the teacher puts a big red check next to your answer. And later, if you don’t get them right, you’re late and miss your friends or you don’t have enough money because you didn’t figure in the sales tax.

And soon, these relationships aren’t between numbers anymore; they’re between letters that merely represent numbers or letters that represent points in a plane. And you must learn to combine the letters according to the laws which govern the relationships among whatever the letters represent. And this is just the beginning. Next you learn "algorithms", relationships between relationships. For example, if you want to know your average electric bill, first you need to add, then to count, and finally to divide.

Eventually, if you survive all of this, which too few do (especially in the U.S.), you may get to the point that those arbitrary and unforgiving numbers disappear and what’s left is just the relationships. That’s what Russell meant what he said that mathematics was the subject in which we "do not know what we were talking about", because in the end we aren’t talking about what the numbers represent anymore or even about the numbers themselves; we are talking only about relationships and structure. At this point, mathematics is more like philosophy or art than it is like physics or economics. Mathematics is the study of form and structure. It is pure logic; it is a mind game, as challenging a mind game as any you will find.

Unfortunately, not too many Americans get to the point that mathematics is brain-teasing fun. They get lost somewhere along the way -- back where everything seemed arbitrary and unforgiving. They lose their way among the symbols and relationships. Even if they survive to and through the "letters representing numbers" leap of faith, they soon get lost in all the letters. What does "x + 2y = 1" mean? Is it "algebra" or is it "geometry"? Is it an "equation" or is it a "line"? Do the x and y represent "numbers" or "points"? And, which ever, what does it have to do with everyday life? After all, whenever you solve for the x’s and y’s in math books, they end up being 3 or -5; but whoever measured a "-5" or even a "3" for that matter; life is made up of messy, uneven numbers, not exact little integers. Math in school is at once too simple and too complicated.

Well, there may still be hope. And it’s an exciting time to be a mathematician. Mathematics currently has an opportunity to "get itself back together". Most mathematics problems can actually be approached from multiple perspectives. We can choose the structure we wish to work in. For a variety of reasons, we have over many centuries avoided certain structures in favor of others -- and we have suffered for it. For example, we have solved problems algebraically (the least intuitive approach) because the geometric solution involved drawing graphs which we did not want to do and the numerical solution involved making long lists and tables of numbers which we could not or would not do. In addition, even in the algebraic solution, we oversimplified the problems in order to make the solutions come out "nice". This is no longer necessary. We now have the tools to enable us to approach many problems from all three perspectives. There is mathematical software out there that can help us to solve problems in the structure best suited to them. Or we can solve them multiple ways, gaining insight and information from the relationships between the solutions. As a result, we can, in important new ways, concentrate more on the problem solving itself, deferring the implementation until later. We can in fact get back to the heart of most problems, discovering the relationships and formulating the algorithms. And once we have the algorithms, we are free to work with more realistic numbers. The potential is there to both recapture some of our lost intuition and experiment with real world problems. The move is on to revitalize mathematics and we can be a part of it.