Beauty of Mathematics Part One--No Right Way to Think About Mathematics

I call this Part One because it will probably be a recurrant theme in my posts for a VERY long time. The rest of the title should be self-explanitory.

Saying there is a right way to think about mathematics is a little like saying there is a right way to think about sculpture or any other form of art. It just doesn't work. Film (to pick a relatively ubiquitous medium for the modern human) speaks uniquely to all who view it. The Godfather Trilogy, for instance, is to me a very sublime commentary on life and a moving tragedy in three parts. To another it may just be another crime film. Math is the same way. For many of my aspiring engineer friends, math is a set of tools for getting a job done. To me it is, amoungst other things, art.

Needless to say, it takes a particular breed to be passionate about what to many is a collection of formulas or a set of rules for balancing your checkbook.

But mathematics transcends this simple formulation as sculpture transcends the aftermath of attacking a rock with hammer and chisel. Think of something classical, like the Pythagorian Theorem. In its simplest form, it speaks of the length of the hypoteneus of a right triangle. In a broader sense, it is the shortest distance between two points in Euclidean space with the usual Euclidean distance. One geometric proof derives it from properties not of triangles, but circles. It has intimate connections to sine and cosine--the length of the opposite side of over the square of the length of the hypoteneuse is by definition sine. In calculus, it is critical in solving certain classes of integrals. It's the simplest non-trivial form of the dot product from vector analysis of a vector with itself (which is just the square of its usual Euclidean length). Each context worries about a version of what could be called a Pythagorean Theorem, which is simply a special case of more general ideas about metric or distance and each is equally valid.