The Pythagorean Secret.

I’ve never been especially good at math. Numbers get lost in my head sometimes, and I’ve always had trouble grasping much beyond the most simple of formulas. Calculus was always a mystery, and I never felt any particular interest in exploring mathematical language and concepts in any length. That is, until I learned about the Pythagorean secret.
St. John’s College, where I spent my freshman and sophomore years, requires a full mathematics course every year. I feared it, but I was surprised to find that because of the way that math was taught, I was actually able to grasp concepts more deeply then before. Because the college used no textbooks, we would read original texts, say Euclid or Ptolemy, and work over and argue over the proofs at the blackboard. It was a method of learning that eschewed rote formulas and valued placing everything that we learned in the larger context of knowledge.
It was in the course of this liberal arts approach to math that I came across the idea of irrational ratios. The Pythagorean school of philosophy was famous throughout ancient Greece for the skill of its mathematicians. They believed in, one might almost say worshiped, numbers, holding that the mysteries of life could be explained by understanding the ratios that existed between whole numbers. But then something happened to them when one of their own, Hippasus of Metapontum,. stumbled across a devastating secret. He realized that there was simply no ratio to √2. Or if you will, take a square and divide it diagonally:



The Pythagoreans believed that it would normally be possible to come up with a ratio that identified the relation of the diagonal to the side of the square. But in fact, there is no such ratio. The ratio between the diagonal and the side will always change according to the length of the side of the square, thus creating an “irrational ratio”, an infinite number like π that has no stable end. It goes on forever.
This shook the Pythagorean worldview. It also shook mine. My first semester at St. John’s I had read Plato and been impressed by his idea of the perfect form, the belief that Socrates articulates that for everything in the material world there exists somewhere the perfect form, or εἶδος, and that all material manifestations are but copies of this form. As a consequence, decisions in life are simple. How do we most closely manifest the perfect form in what we do?
But contemplating an irrational ratio sent a shiver through me. I had always assumed math to be absolute. Like the Pythagoreans, I trusted in the idea that all math could be resolved into a perfect form. But if it wasn’t so, what consequences did such an idea have? I thought of the square, and how we measure it. We say the side is one inch, but if there is no perfect inch, then how can we know for sure what an inch is? We might eventually round off, we might come to a general agreement and design a model inch at some government bureau, and such estimations may be useful in building bridges, but in the end, the inch was an arbitrary idea. In the end, the solid foundation I had once assumed wasn’t there, just what we collectively agreed to believe based on the evidence we had.
It was an epiphany that fundamentally changed my understanding of ideas of science, evidence and proof. I understood for the first time the provisional nature of all proof, how explanations can evolve. I felt a new admiration for the scientific method. Instead of pursuing some perfect form, science claimed only to have a theory, supported by observed information. Sure, there were theories, like the idea that 2 plus 2 equals 4, for which there was a great deal of evidence in favor and little in opposition. But a true scientist would always accept the possibility that, with new data, 2 plus 2 might indeed someday be proven to be something other then 4.
That is, if the scientist really understood and believed in the limits of his knowledge, and was truly dedicated to always advancing those limits. The Pythagoreans did not. They reacted to their discovery out of fear, and in one of the famous episodes of scientific history, they murdered Hippasus of Metapontum for discovering the irrational ratio.
The question then came to me this way. How would I react when faced with evidence that contradicted something I held to be an absolute truth? Would I respond with a defensive reactionary posture, or with an open mind? One of the great misunderstandings about the nature of science is that imperfections in a particular scientific theory somehow devalue science. They don’t. Point out a flaw in a theory to a true scientist and he will react not with dismay, but with the excitement of new data, new knowledge to expand his understanding. The dogmatist might cry “your theory has gaps!”. The scientist will respond by saying “I know. Let’s fill them in.”
I am not a scientist in the traditional sense. But I have come to believe in the skeptical and rational evaluation that lies behind the scientific method. I try to apply this skepticism to whatever new information comes my way, to weigh and understand it in the context of all the evidence. This is what studying math in a liberal arts program taught me. My calculus isn’t any stronger yet, but my mind is more open then before.