What's so fascinating about Mathematics?

This page was last revised on Sept. 8, 2004

One thing that really fascinates me about mathematics is its very real permanence. It is essentially immortal...Once a theorem always a theorem would summarize this viewpoint adequately. Few other areas of human endeavour can boast this sort of inflexibility...Mathematicians become a part of history through their active role as researchers; I am referring here to the role of a mathematican as a discoverer/creator of new mathematics rather than as an educator. Pythagoras (ca. 585-500 B.C.) is remembered mostly for his result on the relationship between the sides of a right angled plane triangle, remember Pythagoras' theorem (?), something which is as true today, in 1996, as it was 2,500 years ago when he first discovered it, and which will be true forever more...

I always like to think about the role that mathematics, both elementary and advanced, plays in everyday life. What's really cool about science in general is that it uses the language of mathematics in order to make predictions about the state of the phenomena it is studying. For example, long after there is any thought left emerging from this planet of ours, the sun will continue in its orbit around the milky way, the moon will keep its revolutions around the earth, almost like clockwork, and the mathematics describing these motions will still be valid even though there is no one here to interpret them! Amazing isn't it? How can something be so permanent as to defy human reason?

I love the world of mathematics...I love the act of discovery; you push these symbols around according to some laws that everyone accepts in the field and then...voila'...out comes this result where the symbols now interact with one another and there is new meaning to what you initially put in. You've discovered, some would say uncovered, something new and pretty and interesting about the mathematical world you interact with, something which may or may not have any relevance whatsoever with the real world but, we believe, that someday all this new abstract mathematical stuff will, indeed, be useful to someone, somewhere, sometime in our future. How could Giovanni Ricci-Curbastro and Tullio Levi-Civita have known, in 1900, that their basic theory of tensors, a very abstract theory for their time, would become the cornerstone of 20th. century physics... that someone called Albert Einstein would come along and in 1916 would use their theory to produce a new theory of gravitation, or general relativity, as it is called these days, a theory that would change philosophy and launch our new era? They couldn't have known...and, in fact, they may not have even cared about so called applications at the time of publication. It's part of the duties of a mathematician to discover new things, to present new approaches to old problems and to create new mathematics regardless of its readership.

One of the most fascinating and unsolved problems of classical differential equations (a branch of mathematics that uses a whole lot of calculus) is called the problem of n bodies. In other words, assume Newton's laws hold in the universe we are studying, take, say 4 or 5 bodies (planets, asteroids, moons, stars, ...) and put them in some position in space and even initially at rest. The question is, what will be the relative position of these bodies, say, 10 million years in the future? Unless you do some very serious number-crunching using very powerful computers you won't know...well, what I mean is, there isn't any formula known to us that will tell you this in a few minutes. This problem dates back to Newton himself, and it remains essentially unsolved to this day. Notwithstanding this, the Galileo space probe still found its way to Jupiter because of the serious number-crunching experiments referred to above... This is remarkable and I even find it mystifying! Not only that but the "most remote object ever made by man" is now more than 8 billion miles away! What is it?