The romance of mathematics

A couple of weeks ago, I found myself describing to someone how badly I had wished to be a mathematician in college.  Mathematics, I said (much to her surprise!), is the most romantic subject in the world.  As scientists, we explore and describe the nature of reality.  Mathematicians are bound by no such limit.  Like the greatest authors, they get to create entire worlds, realms that simply spring forth from their minds.  But math goes so far beyond even the most imaginative fiction - a few basic ideas can open up a window into limitless thought.  

 This fragment of the  Elements  is nearly 2000 years old.  (Image credit: University of Pennsylvania)

This fragment of the Elements is nearly 2000 years old.  (Image credit: University of Pennsylvania)

Sitting next to me on my desk is a copy of the Elements, history's greatest textbook.  Written more than 2000 years ago by the Greek mathematician Euclid, it chronicles the discovery of geometry.  In its modern printing, it runs nearly 500 pages in length.  But there in the front, on the first two pages, lies the really important stuff.  On these two pages, Euclid laid out all the rules he would be following.  For example, on the bottom of page two, he lists five "common notions":

  1. Things which are equal to the same thing are also equal to each other.
  2. If equals be added to equals, the wholes are equal.
  3. If equals be subtracted from equals, the remainders are equal.
  4. Things which coincide with one another are equal to one another.
  5. The whole is greater than the part.

Seems pretty obvious, right?  That's the real beauty shining through.  Notions like "the whole is greater than the part" are so obvious that most of us have probably never even thought to stop and realize that.  But, by combining that idea with definitions of things like pointline, and boundary, Euclid would go on to fill hundreds of pages.

 A page from Newton's  Principia Mathematica  showing proofs using Euclid's geometry.

A page from Newton's Principia Mathematica showing proofs using Euclid's geometry.

The English language today contains a couple hundred thousand words, yet Shakespeare needed fewer than a sixth of them to write every comedy, tragedy, and history he would ever pen.  What about Euclid?  He defined just 22 words on which to build all of modern math and science.  That's right: the Elements, unfolding from these two brief pages, is not solely a work of fantasy.  More than 1600 years later, Isaac Newton would discover physics and birth modern science in his Principia Mathematica.  Peer inside, and you'll discover that nearly every idea about how the physical world works is proven through Euclid's geometry.

This fact - that mathematics applies to the physical world - is the most remarkable in existence.  The notions of Euclid - that a line is both infinitely long and infinitely narrow or that a point has no size in any direction - have no physical manifestation.  Somehow, despite this, the implications of points, lines, surfaces, and boundaries define everything about the universe in which we live.

So now I ask you, how can mathematics not be the most romantic endeavor imaginable?  It exists at the intersection of reality and fantasy, imagination and deduction, thought and observation.  To practice math is to experience, if only for the most fleeting of moments, true wonder at even the simplest of things.  Its no surprise I felt such longing to be on the forefront of such a remarkable journey.