He preached, for instance, the transmigration of souls—that people’s spirits could be reincarnated in the bodies of animals. To avoid harming their ancestors, therefore, Pythagoreans wouldn’t eat meat or beat a disobedient dog. Like the Zoroastrians in Persia, they believed that the world is locked in a contest between light and darkness, good and evil. Pythagorean communities were the monasteries of the ancient world, holding property in common and living by a rigid code. Several centuries after Pythagoras, Iamblichus of Chalcis wrote, “The aim of all the Pythagorean precision about what should and should not be done is association with the divine. This is their starting-point, and their way of life has been wholly organized with a view to following God.”³ To them, the evil in the world was the imprecise and the uncertain, which is why math was so important. Again, Iamblichus: “The Pythagoreans devoted themselves to mathematics and admired the accuracy of its reasonings, because it alone among human activities knows of proofs.”⁴

      Most mathematical proofs in those days took form in pictures of abstract shapes; the algebraic notation we use now wasn’t invented yet. The Pythagoreans considered these pictures sacred, combining geometry and mystery cult in a single scientific-religious mélange. The correspondence between mathematical ratios and musical scales especially fascinated Pythagoras. He believed that the movements of the stars and planets make a beautiful sound, playing always, which we don’t notice only because we’ve been hearing it our entire lives.

      The idea of mixing mathematics with mythology seems odd to us today. We memorize formulas, use them to do problem sets, and forget most of it when the test is over. But that’s not the way mathematics was, and continues to be, created. It’s a foray into the unknown that borders on mysticism. Polls suggest that among scientists mathematicians are most likely to believe in a God.⁵ Maybe spending one’s life immersed in abstractions makes a divine mind seem more plausible. But I wouldn’t really be one to know.

      Math didn’t come easily to me growing up. My father did his best to help. For a time in middle school, I would wake up each morning and find waiting for me a paper with a set of problems in his delicate, hieroglyphic handwriting. But math problems were the last thing I wanted to do in my first waking minutes, and downstairs I would hand them to him with dashed-off answers, if any.

      Dad was a real estate agent—of heroic ability, as his clients have always told me. Work kept him out late, and he would eventually come in the front door with a pile of papers in his arms, topped by a clunky old HP-12C calculator. His mind seemed like a calculator too. Given a date, he could instantly say how old somebody was at the time; given a price, he would produce the tax or interest as if by reflex. He always counted steps as he walked up and down them, automatically and insuppressibly. If it weren’t for him telling me that each flight in our house had eight steps, on dark nights I would have been content to feel each with my sock to tell if it was the last.

      A hobby of Dad’s was to make family trees of English royalty with his computer. They became dot-matrix murals that covered the long wall of his study. He had traveled to castles and cathedrals, and he would tell wonderful stories about them over dinner if something got him started. We had no blood of Stuarts or Hanovers or Windsors ourselves, of course, but they became a sort of extended family.

      As I was getting ready for bed, I might hear him singing part of a Verdi aria, accompanying himself on the piano, and after he went silent, if I crept out of bed, I would see him lying on the couch with headphones on. Maybe this time it was Wagner, or Puccini. On other nights he would ramble around the backyard in the dark, planning the next part of his garden.

      In high school geometry, math finally started to become something I could wrap my mind around; we started learning about proofs. We learned how to construct arguments from basic principles. All of a sudden math class was not simply a matter of calculating, but of discovering, and my attitude about it changed entirely. I took calculus my senior year and felt the exhilaration of late-night group study sessions, when the solution to a problem would finally come loose for us after hours of tugging at it from every direction. We learned to derive important theorems, masterpieces that had been composed in great minds of centuries past, and then used those theorems to derive more. In physics class, we used the math to predict the motion of tangible objects. I worked my way twice through a floppy paperback of Einstein’s Relativity. Like the correspondence Pythagoreans found between mathematics and musical scales, Einstein’s equations declared that the universe is not what it seems.

      Around the time Pythagoras died, at the beginning of the fifth century B.C.E., another Greek thinker of strange notions and lasting influence was born, also in southern Italy. Like Pythagoras, Parmenides of Elea treaded in the brackish region between religion and philosophy, myth and logos, and politics. The laws he established in Elea survived him by five hundred years. He wrote his treatise in the form of a poem, though its verses strain to accommodate their meticulousness. They seem ready to burst into prose at any moment. As did the epic poets, he attributed his inspiration to the whispers of a goddess who opened his eyes and moved his pen. But like the philosophers he strove to make no claim without reasons.

      The goddess guided him to divide his poem into two parts: the Way of Truth and the Way of Appearance. The second is a compendium of what he takes to be false opinions that people have about the world; it resembles Pythagoreanism. The first is an all-out attack on common sense, in the name of banishing logical contradiction. Nothing, Parmenides claims, cannot exist. Everything conceivable exists, unchangeably, eternally, in perfect unity, and in the shape of a sphere. What we see in the world that appears to change, to cease to exist, and to differ from other things is all illusion. That anything could not exist is a contradiction; if you conceive of a thing not existing, it then exists in the very conception. In this weird way, the goddess gets him to trust his mind before his eyes. Thoughts are the reality of the world, and logic is its native language.

      Among his fellow Greek sages, such totalizing notions were commonplace. Thales claimed that everything is really water, Anaximenes said everything is air, and Heraclitus answered that everything is like an ever-burning fire. Parmenides was different; a century and a half later, a commentator named Eudemus wrote, “Parmenides would not agree with anything unless it seemed necessary, whereas his predecessors used to come up with unsubstantiated assertions.”⁶

      Human minds make imperfect looms for pure reason. Even when such careful thinking doesn’t turn into an affront to the obvious, as it did for him, it gets tangled up often enough. Philosophy, for me, started to be of interest after forcing myself to go cold turkey on a years-long obsession with Star Trek and the show’s vision of a future made better by human reason. Yet the words of Spock to a precocious younger Vulcan in Star Trek VI seem