The Quadrivium: The Road to Truth

Plutarch tells a story about Archimedes, the great Greek mathematician who lived in the third century B.C., which illustrates the profound difference between ancient and modern attitudes towards mathematics.  A Roman army under Marcellus had besieged the city of Syracuse by land and sea, confident of taking it quickly.  But his forces were completely routed by an amazing array of catapults, grappling beams, projectile apparatus, and other machines.  Archimedes, famous for his mathematical studies of leverage and hydraulics, had developed all these machines at the request of the king of Syracuse to impress the common, pragmatic mind with his studies.  And that they did: his grappling beams applied his mathematical principles so well that they lifted entire Roman ships up into the air, shook the sailors into the sea, and then smashed the ships against the rocks!

Yet all this success meant little to Archimedes, who was enamored only of the beauty of the mathematical ideas themselves.  As Plutarch relates:

…[Archimedes] placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life; studies, the superiority of which to all others is unquestioned, and in which the only doubt can be whether the beauty and grandeur of the subjects examined, or the precision and cogency of the methods and means of proof, most deserve our admiration.

Imagine that!  He didn’t think of all the money he could make from his inventions, the great career he could have with the king, or the power he could win for himself or his country.  The truths he learned were beautiful and grand, the proofs he offered were precise and powerful, and that was all he desired.  [Read the entire amazing story of Plutarch at: http://www.math.nyu.edu/~crorres/Archimedes/Siege/Plutarch.html]

Archimedes’ attitude was shared widely by the ancient Greeks.  Euclid, whose work, The Elements, was the geometry textbook from his time through the eighteenth century, gave birth to a little story indicating his attitude to the subject.

Someone who had begun to read geometry with Euclid, when he had learnt the first theorem, asked Euclid, "What shall I get by learning these things?" Euclid called his slave and said, "Give him threepence, since he must make gain out of what he learns."

The Greeks realized that mathematics offered the young mind its first immediate encounter with Truth.  Early education taught basic reading and calculating skills, and developed a taste for beautiful and noble actions and ideas.  Through these efforts, the youth were made receptive to their culture.  But geometry, the study of shapes through measurement, and arithmetic, the study of fascinating kinds of numbers, contained truths that transcended any human society.  Right triangles and ellipses belong to no man; they are instances of a beautiful, ordered reality that is eternal and divine.  Students learned that such truth exists, and that their minds could see it. 

For them, this was sufficient justification for mathematical study.  But it had a higher purpose—to prepare the mind and heart for the pursuit of wisdom, philosophy itself.  "Let no one come to our school who has not first learnt the elements of Euclid," reputedly read a sign on the door of Plato’s Academy.  Centuries later Boethius developed the name, quadrivium, to indicate that arithmetic, geometry, astronomy and music are the “four-fold road” to philosophy.

Today, mathematics is viewed in a very different light.  Many signs point to a problem with our schools in teaching mathematics, but the interpretation of these signs shows that math is seen exclusively as a tool for engineering, science, computers, and careers.

A presidential panel declared math education in the United States "broken" yesterday and called on schools to focus on ensuring that children master fundamental skills that provide the underpinnings for success in higher math and, ultimately, in high-tech jobs.  Washington Post, March 14, 2008