What Lies Behind Pythagoras’ Theorem

By Neel Burton | February 1, 2014

Pythagoras (570-500 BC) was born on the island of Samos in what is now Greece. On the advice of Thales of Miletus, he travelled to Memphis in Egypt where he came into contact with priests renowned for their wisdom. At the age of 40, he fled the tyranny of Polycrates to Croton in Southern Italy, where he established a philosophical and religious community. Those who entered the community’s inner circle were governed by a strict set of ascetic and ethical rules, forsaking personal possessions, assuming a mainly vegetarian diet, and—since words are so often careless and misrepresentative—observing the strictest silence. Some of the community’s more eccentric rules, such as ‘do not break bread’ or ‘do not poke the fire with a sword’ may have been riddles or allegories that required interpreting. Pythagoras’ brotherly community has been hailed as a prototype for later philosophical institutions such as Plato’s Academy, Aristotle’s Lyceum, and Epicurus’ Garden, and, indeed, for the monastic life and associated early universities.

Music played an important role in Pythagoras’ community. Pythagoreans recited poetry, sang hymns to Apollo, and played on the lyre to cure illnesses of both body and soul. It is said that, one day, Pythagoras passed by some blacksmiths at work, and found that their hammering on anvils produced especially harmonious sounds. He then found that the anvils were simple ratios of one another, one being half the size of the first, another two thirds of the size, and so on. This discovery of a relationship between numerical ratios and musical intervals led Pythagoras to believe that the study of mathematics was the key to understanding the structure and order of the universe. According to his ‘harmony of the spheres’, the heavenly bodies move according to mathematical equations that correspond to musical notes and form part of a grand cosmic symphony.

Pythagoras never separated religion from science and philosophy, which, even in his day, left him open to accusations of mysticism. No doubt under the influence of Orphism, an Ancient Greek mystery religion that arose from pre-Hellenic beliefs and the Thracian cult of Zagreus, he believed in the transmigration of the soul; that is, in the reincarnation of the soul over time into the bodies of human beings, animals, or plants (metempsychosis) until such a time as it became moral. He claimed to have lived four lives and to remember them all in great detail, and once recognized the cry of his dead friend in the yelping of a puppy. After his death, the Pythagoreans deified him, and attributed him with a golden thigh and the gift of bilocation (being in two places at once). But in his own lifetime Pythagoras had always been a paragon of modesty, declining to be called a ‘wise man’ or ‘sophos’, and preferring instead to be called ‘a lover of wisdom’ or ‘philosophos’—thereby coining the term ‘philosopher’.

It is said that Numa Pompilius, the second king of Rome, had been schooled by Pythagoras, whence his great wisdom and piety. This story is referred to and discredited by Plutarch and Livy, not least because the dates do not tally, with Pythagoras having lived from about 570 to 500 BC, and so considerably later than the semi-legendary King Numa. Even though Pythagoras and Numa never met, Pythagoras and the Pythagoreans exerted a strong influence on the Roman mind. In the Tusculan Disputations, Cicero indicates that Pythagoras rose to fame in southern Italy at just the same time that Brutus brought an end to the monarchy, and that many Roman usages derived from the Pythagoreans. Unfortunately, he does not elaborate on the nature of these usages. In his Natural History, Pliny the Elder tells us that, in 343 BC, during the war with the Samnites, the god Apollo ordered the Romans to erect one statue to the wisest and another to the bravest of all Greeks, with their choices falling upon Pythagoras for the former and Alcibiades for the latter. Pliny expresses surprise that they picked Pythagoras over Socrates, whom Apollo himself had called the wisest of all men. But the fact is that the Romans liked to think of the Greek-Italian Pythagoras as their very own philosopher, and spun all sorts of stories, such as the one about Numa, to better appropriate him.

Apart from this, Pythagoras also exerted a strong indirect influence on Roman thinking, and indeed on all philosophy and theology, through the teachings of Plato, the principal architect of the western mind. Aristotle, who was Plato’s pupil of twenty years, claimed that his master’s teachings owed much to those of Pythagoras; so much, in fact, that, in his History of Western Philosophy, Bertrand Russell upheld not Plato but Pythagoras as the most influential of all Western philosophers. Pythagoras’ influence is especially evident in Plato’s mystical approach to the soul and in his emphasis on mathematics and, more generally, abstract thinking, as a secure basis for the practice of philosophy.

Book Review: AC Grayling’s Friendship

By Neel Burton | November 14, 2013

It is striking that the great thinkers, from Aristotle to Augustine and Mencius to Montaigne, devoted so much of their time and thought to friendship, but almost none of either to marriage. Grayling’s timely treatise reacquaints us with a great but forgotten good that promises to fulfil so many of our practical, intellectual, emotional, and metaphysical needs. The book principally consists of a history of the philosophy of friendship capped by an account of canonical, often homosexual or homosocial friendships such as that of Achilles and Patroclus and Jonathan and David, who, in the Bible, describes the love of Jonathan as “better even than that of women”. Throughout, Grayling seeks to define friendship and, in so doing, explores its many forms, facets, charms, and consolations.

Perhaps in a desire to be modern, relevant, or politic, Grayling seems to reject the classical notion that, at its best and most meaningful, friendship is a highly elitist good. For the greats, only virtuous men can be ideal friends. Aristotle famously says that, while there are many ways for men to be bad, there is only one way for them to be good, and it is precisely in this sense that an ideal friend is ‘another self’—a historically important notion that Grayling severally dismisses. Because they are all one and the same, virtuous men are predictable, reliable, and therefore worthy of one another’s friendship. In contrast, bad people are in some way unlike themselves, and just as likely to hate other bad people as anyone else.

In my opinion, Plato, whom Grayling underrates, advances by far the most subtle and sophisticated of all theories of friendship, one far superior even to that of Aristotle. Despite the extravagant praise that he lavishes upon friendship, Aristotle is quite clear that the best and happiest life is not that spent in friendship, but in the contemplation of those things that are most true and therefore most beautiful and most dependable. There is a contradiction here: if the best life is a life of contemplation, then friendship is either superfluous or inimical to the best life, and therefore undeserving of the high praise that Aristotle lavishes upon it. It may be, as Aristotle tentatively suggests, that friendship is needed because it promotes contemplation, or that contemplation is only possible some of the time and friendship is needed the rest of the time, or even that a life of friendship is just as good as a life of contemplation. So much for Aristotle, one might say.

Plato’s Lysis may seem to fail in its task of defining friendship, but one should never take Plato or his mouthpiece Socrates at face value. There is far more to the Lysis than a couple of interesting but misguided thoughts on friendship. By discussing friendship with Lysis and Menexenus as he does, Socrates is not only discussing friendship, but also demonstrating to the youths that, even though they count each other as close friends, they do not really know what friendship is, and that, whatever friendship is, it is something far deeper and far more meaningful than the puerile ‘friendship’ that they share. In contrast to the youths, Socrates knows perfectly well what friendship is, and is only feigning ignorance so as to teach the youths: ‘…and I, an old boy, who would fain be one of you…’ More than that, by discussing friendship with Lysis and Menexenus as he does, Socrates is himself in the process of befriending them. He befriends them not with pleasant banter or gossipy chitchat, as most people ‘befriend’ one another, but with the kind of philosophical conversation that is the hallmark of the deepest and most meaningful of friendships. In the course of this philosophical conversation, he tells the youths that he should ‘greatly prefer a real friend to all the gold of Darius’, thereby signifying not only that he places friendship on the same high pedestal as philosophy, to which he has devoted (and will sacrifice) his life, but also that the kind of friendship that he has in mind is so rare and uncommon that even he does not possess it. If friendship ultimately escapes definition, then this is because, like philosophy, friendship is not so much a thing-in-itself as it is a process for becoming. True friends seek together to live truer, fuller lives by relating to each other authentically and by teaching each other about the limitations of their beliefs and the defects in their character, which are a far greater source of error than mere rational confusion. For Plato, friendship and philosophy are aspects of one and the same impulse, one and the same love: the love that seeks to know.

Just as philosophy leads to friendship, so friendship leads to philosophy. In the Phaedrus, Plato’s most important work on friendship (although not generally recognized as such—Grayling fails to mention it), Socrates and Phaedrus go out into the idyllic countryside just outside Athens and have a long conversation about the anatomy of the soul, the nature of true love, the art of persuasion, and the merits of the spoken over the written word. At the end of this conversation, Socrates offers a prayer to the local deities. This is the famous Socratic Prayer, which is notable both in itself and for the response that it elicits from Phaedrus.

Socrates: Beloved Pan, and all ye other gods who haunt this place, give me beauty in the inward soul; and may the outward and inward man be at one. May I reckon the wise to be the wealthy, and may I have such a quantity of gold as a temperate man and he only can bear and carry. —Anything more? The prayer, I think, is enough for me.

Phaedrus: Ask the same for me, for friends should have all things in common.

Plato may fail to define friendship in the Lysis, but in the Phaedrus he gives us its living embodiment. Socrates and Phaedrus spend their time together enjoying the beautiful Attic countryside while engaging in honest and open philosophical conversation. By exercising and building upon reason, they are not only furthering each other’s understanding, but also transforming a life of friendship into a life of joint contemplation of those things that are most true and hence most beautiful and most dependable. If only on the basis of his response to the Socratic Prayer, it is obvious that Phaedrus is another self to Socrates, since he makes the same choices as Socrates and even justifies making those choices on the grounds that their friendship requires it. Whereas Aristotle and Grayling try to tell us what friendship is, Plato lets us feel it in all its allure and transformative power.