Chapter 1

The Pythagorean Answer

While the ancient Greek philosopher Pythagoras is doubtless best known for his eponymous geometric theorem, few schoolchildren who regularly use it are aware that Pythagoras founded and led one of the most fascinating religious cults in antiquity. His followers, the Pythagoreans, settled in Southern Italy in the early 6th century BCE and established one of the earliest known vegetarian societies. In addition to abstaining from meat, the Pythagoreans avoided drinking wine and eating beans for reasons that remain unclear. Members of the cult were also expected to never pick up objects that had fallen, never eat from a whole loaf of bread, never walk on highways, and never use a mirror beside a light. Women in Pythagorean communities were considered equal to men, and all property, including intellectual property, was communal. Cult initiates were selected for their merit and discipline, and had to spend five years listening to Pythagorean teachings in complete silence before taking a test to officially join the cult.

Over time, the Pythagoreans became considerably influential in aristocratic social circles and turned their attention to politics, perhaps attempting to impose their ways on others. This led to various deadly attacks on Pythagorean communities, culminating in many Pythagoreans being driven out of Italy. While individual cult members may have continued to practice their unique lifestyle, no organized communities appear to have been established elsewhere. The Pythagoreans seem to have died out in the 4th century BCE, before making a comeback with Neopythagoreanism in the 1st century BCE.

But what does any of this have to do with right triangles? The answer is that Pythagoras and his followers were, at their core, number-worshipers. When Pythagoreans observed the world around them, they reached the conclusion that the objects inhabiting it could be differentiated by various qualities. For example, different objects are characterized by different sizes, weights, textures, colors, shapes, smells, etc. But none of these qualities are universal! This book, for example, might smell moldy if you store it in a damp place, but many things lack any discernable smell, which means they couldn’t possibly be defined by smell. This led the Pythagoreans to realize that the only quality necessarily defined for all things is that they can be counted. In other words, while the world could at least theoretically exist without smells, moldy or otherwise, the world couldn’t possibly exist without numbers. Consequently, numbers should be seen as the very essence of reality! For the Pythagoreans, this meant that mathematical knowledge is fundamentally different from and superior to all other forms of knowledge. Mathematics is therefore more than just a bad childhood memory; it was a way for humans to interact with the transcendent.

The Pythagoreans drew upon the importance of numbers to develop a number-based theory to explain the universe. 1 represented reason, 2 represented opinion, 3 represented harmony, 4 represented justice, 5 represented marriage, 6 represented creation, 7 represented wisdom, 8 represented love, and so on. Odd numbers were considered masculine, while even numbers were considered feminine. The Pythagoreans’ extreme devotion to mathematics led to genuine scientific breakthroughs: they were the first to adopt the spherical Earth model, and one of the last Pythagoreans is credited with formulating the heliocentric theory – roughly 1,700 years before Copernicus! Needless to say, the Pythagorean creation narrative was delightfully on-brand. It was said that the universe was once unstructured and chaotic. A point then appeared, and when it moved, it begat lines, which moved to beget surfaces, which moved to beget all of the three-dimensional bodies in the universe. This elegant theory linking the creation of the world to mathematical dimensions was undoubtedly intuitive for the Pythagoreans, for whom mathematics was the ultimate truth.