It is hard to let go of Pythagoras. He has meant so much to so many for so long. I can with confidence say to readers of this essay: most of what you believe, or think you know, about Pythagoras is fiction, much of it deliberately contrived. Did he discover the geometrical theorem that bears his name? No. Did he ponder the harmony of the spheres? Certainly not: celestial spheres were first excogitated decades or more after Pythagoras’ death. Does he even deserve credit for his most famous accomplishment, analysing the mathematical ratios that structure musical concordances? Possibly, but there is little reason to believe the stories about his being the first to discover them, and compelling reason not to believe the oft-told story about how he did it. Allegedly, as he was passing a smithy, he heard that the sounds made by the hammers exemplified the intervals of fourth, fifth and octave, so he measured their weights and found their ratios to be respectively 4:3, 3:2, 2:1. Unfortunately for this anecdote, recently rehashed in the article on Pythagoras in Grove Music Online, the sounds made by a blow do not vary proportionately with the weight of the instrument used.
My problem is that to convince you of such deflationary truths I have to give an account which inevitably is less exciting than, for example, the following extract from Bertrand Russell’s well-known History of Western Philosophy (1946):
Pythagoras . . . was intellectually one of the most important men that ever lived, both when he was wise and when he was unwise. Mathematics, in the sense of demonstrative deductive argument, begins with him, and in him is intimately connected with a peculiar form of mysticism. The influence of mathematics on philosophy, partly owing to him, has, ever since his time, been both profound and unfortunate.
Or this from Roger Penrose in The Road to Reality: A Complete Guide to the Laws of the Universe (2005):
Although mathematical truths of various kinds had been surmised since ancient Egyptian and Babylonian times, it was not until the great Greek philosophers Thales of Miletus (c.625-547 BC) and Pythagoras of Samos (c.572-497 BC) began to introduce the notion of mathematical proof that the first firm foundation stone of mathematical understanding – and therefore of science itself – was laid. Thales may have been the first to introduce this notion of proof, but it seems to have been the Pythagoreans who first made important use of it to establish things that were not otherwise obvious. Pythagoras also appeared to have a strong vision of the importance of number, and of arithmetical concepts, in governing the actions of the physical world.
Both writers are wildly wrong, but Russell had a good excuse. He was voicing the received scholarly opinion of his time, mediated to him through the writings of John Burnet and F.M. Cornford, Russell’s one-time colleague at Trinity College, Cambridge. And that scholarly opinion was itself the codification, with properly footnoted sources, of a millennia-long tradition about Pythagoras and mathematics.
What happened between Russell and Penrose was the publication in 1962 of a very great work of scholarship, Walter Burkert’s Weisheit und Wissenschaft: Studien zu Pythagoras, Philolaus und Platon (revised version translated into English as Lore and Science in Ancient Pythagoreanism, 1972). The effect of Burkert’s book was to destroy for ever the alluring picture of Pythagoras as a mystical mathematician, a picture which has been endlessly recycled from antiquity to the Renaissance and beyond. Mystic, yes – or at least the leader of a religious cabal which believed in transmigration of the soul and was disciplined enough to take political power in several cities of southern Italy. But mathematician, no. Not at any rate if, with Russell and Penrose, we think of a mathematics based on deductive proof, as opposed to the fanciful numerology recorded in Aristotle’s On the Beliefs of the Pythagoreans, of which the following is an example:
Marriage, they said, is five, because it is the union of male and female, and according to them the odd is male and the even female, and five is the first number to be generated from the union of the first even number, two, and the first odd number, three; for the odd is for them (as I said) male and the even female.
Burkert is listed in Penrose’s bibliography, but in Penrose’s text Pythagoras still leads us along the road to reality with mathematical proof as his guide to understanding. The Burkert revolution has had zero effect on his impressive book.
I sympathise with Penrose. The problem is not just that beloved historical traditions die hard. Lore and Science is as dense a work of classical scholarship as you could fear to meet. To deconstruct the Pythagoras tradition, Burkert has to unravel so many obscure sources that his pages groan with footnotes citing ancient authors whom even specialists may not have heard of. The going is tough, the effect unremittingly negative. I have a vivid memory of the week, way back in 1978, when I struggled through my first reading of the book, so gripped that I stayed in bed, scribbling notes, all day every day. I had been brought up on a strong version of the Cambridge interpretation deriving from Cornford. Thanks to Burkert, I could no longer accept a word of that. But I hardly knew what to believe instead. The following week I was due to give my first ever lecture on Pythagoreanism. Had the two books under review been in the shops back then, I would have rushed to devour them. For both are written to answer the question: what now remains, in the wake of Burkert, to be said about Pythagoras and his followers?
The first thing to notice is how short both books are: each fewer than two hundred pages to compare with Burkert’s 535. There’s much less to say about Pythagoras now than there was when I was young. The textbook we studied then was The Presocratic Philosophers: A Critical History with a Selection of Texts by G.S. Kirk and J.E. Raven (1957), in which Raven (my undergraduate tutor) devotes 40 pages to Pythagoras and his early followers. In the revised second edition of this now standard work (Kirk, Raven and Schofield, 1983), Raven’s account of early Pythagoreanism is replaced by a mere 24 pages of entirely new material, written by Malcolm Schofield and much indebted to ‘Burkert’s . . . masterpiece of postwar classical scholarship’. I recommend it to anyone who wants to see the scraps of evidence, in Greek plus translation, from which must derive any up-to-date picture of Pythagoras and the ideas of the movement he founded.
Christopher Riedweg’s book is dedicated to Burkert, while Charles Kahn thanks Burkert for ‘superlative’ comments on the manuscript he sent to the publisher. Whenever Pythagoreanism comes up for scholarly study, the Burkert revelation is now everywhere, the anxiety of his influence omnipresent – but with different effects on different writers. Riedweg seems confused by it, both affirming and denying the break with tradition. Kahn, like Schofield, remains cool and collected. The difference shows in the heading of Kahn’s fifth chapter, ‘The New Pythagorean Philosophy in the Early Academy’.
That little word ‘new’ testifies that Kahn has managed to let go, for it accepts from Burkert that the origins of the traditional picture of Pythagoras are to be sought, not during the sixth century BC, when he lived and fought his political battles, not during the fifth century, when democratic forces ousted his followers from power in various cities of southern Italy, but late in the fourth century. That was when Speusippus and Xenocrates, the dominant figures in Plato’s Academy, sought to devise ancient authority for certain aspects of their late master’s philosophy. Theirs was a conscious construction whereby Pythagoras became the apostle of mathematics and a highly mathematising philosophy, full of anticipations of Platonic metaphysics. But instead of denigrating or dismissing it for the fictive construction it was, Kahn hails it as a new Pythagorean philosophy, a way of thinking that deserves to be tracked through the centuries from Ptolemy’s Harmonics through Copernicus to Kepler.
Riedweg half-agrees with this, and has a parallel section in his book entitled ‘Pythagoras as an Idea in the Middle Ages and Modernity – A Prospect’, beginning:
Had Pythagoras and his teachings not been since the early Academy overwritten with Plato’s philosophy, and had this ‘palimpsest’ not in the course of the Roman Empire achieved unchallenged authority among Platonists, it would be scarcely conceivable that scholars from the Middle Ages and modernity down to the present would have found the Presocratic charismatic from Samos so fascinating. In fact, as a rule it was the image of Pythagoras elaborated by Neopythagoreans and Neoplatonists that determined the idea of what was Pythagorean over the centuries (my italics).
Fine, but a reader who asks Riedweg, ‘What, then, did Pythagoras really stand for earlier, before the Academy set to work?’, gets a muddled, muddling answer. Legends are retold. Pythagoras’ golden thigh is put on display once more, alongside his gift of bilocation (he was seen simultaneously in two different cities). During a visit to the temple of Hera in Argos where, ages before, the Greeks had dedicated the booty they brought home from their victory over Troy, Pythagoras recognised among the exhibits the shield he had carried when, in a previous incarnation as the warrior Euphorbus, he was killed by Menelaus. After drinking at a well in Metapontum, he correctly predicted that an earthquake would occur in three days’ time.
Not that Riedweg buys into all this, but he does encourage his readers to marvel at a man around whom such legends grew. And in his anxiety not to let go he will defend the indefensible: for example, that Pythagoras invented the word ‘philosophy’ and was the first to make ‘cosmos’ mean ‘world-order’.
More important, in answer to the question I began from, ‘What did Pythagoras himself contribute to mathematics?’, Riedweg refers us to this passage from the opening book of Aristotle’s Metaphysics:
Contemporaneously with these philosophers [the Atomists Leucippus and Democritus] and before them, the Pythagoreans devoted themselves to mathematics; they were the first to advance these studies, and having been brought up in them, they supposed their principles to be the principles of all things.
‘They were the first to advance these studies’: it sounds conclusive, and has been endlessly cited as proof that the Pythagoreans (if not Pythagoras himself) were the founders of ancient Greek mathematics. But it is no such thing.
First, a mundane point of translation. Aristotle has set out to survey the contributions of earlier thinkers who discussed the question, ‘What are the fundamental principles of reality?’ He began with Thales, who said that all is water, then he went on to others who proposed other material principles, climaxing with the theory that all is atoms and the void. Now comes the sentence just quoted, with the key verb proa’gein translated as ‘advance’. This, the rendering that has prevailed in vernacular translations since the Renaissance (a time of enthusiastic Neopythagoreanism and Neoplatonism), seems to credit the Pythagoreans, if not with founding Greek mathematics, at least with being the first to raise standards to a high level.
But proa’gein simply means ‘bring forward’ – bring forward in any way one can bring something forward, which might include bringing forward a witness to testify in court. The medieval translators of Aristotle use verbs like producere or adducere, ‘to bring forward’ or ‘adduce’. The meaning then is that the Pythagoreans were the first to make mathematics bear witness in the metaphysical debate, or the first to adduce the principles of mathematics as the principles of all things. The point of saying they were the first is that in the next chapter Aristotle will discuss Plato’s contribution to metaphysics as a second, and somewhat different, mathematising account of the fundamental principles of reality. Exactly that contrast between a first (Pythagorean) and a later (Platonic) version of the thesis that the principles of mathematics are the principles of all things is what Aquinas provides in his commentary on the Metaphysics (c.1270-72). On this medieval, pre-Renaissance understanding of the passage, absolutely nothing is said about the history of mathematics itself. It is about mathematical, or pseudo-mathematical, contributions to the history of metaphysics, at least some of it in the style of the stuff about marriage quoted above.
The next question is: which Pythagoreans does Aristotle have in view when he introduces their contribution to the metaphysical debate? And how would he know what they thought? We are informed that the first Pythagorean to write and publish a book ‘On Nature’ was Philolaus (of Croton or Tarentum), born c.470 BC, which implies publication some time in the second half of the fifth century, fifty years or more after Pythagoras’ death. One of Burkert’s key achievements was to match up Aristotle’s reports on Pythagorean cosmology with the solid evidence of Philolaus’ book, of which a fair number of fragments remain for us too to study.
There were some enthralling ideas in this book. One was a revolutionary proposal to move the Earth. Not indeed to move it around the Sun, but Philolaus’ hypothesis of a central Fire around which circle Earth, a Counter-Earth we can never see, the Sun, Moon, the five known planets, and finally the outermost circle of the fixed stars, was a radical innovation on the standard geocentric scheme. Cicero’s and Plutarch’s reports of it excited Copernicus.
Another of Philolaus’ proposals, equally innovatory at the time, was to locate thought and reason in the brain instead of the heart, as was commonly believed; according to him, the heart is rather the seat of life and sensation. The idea that thought goes on in the brain was accepted by Plato, but long resisted by Aristotle, the Epicureans and Stoics. The crucial importance of the brain was only established beyond dispute in the third century BC, by Hellenistic doctors whose vivisections ranged from pigs to human prisoners in the jails of Ptolemaic Egypt. (Naturally, all that pain was inflicted for the sake of future human welfare.)
But there was also numerological fancy in Philolaus: ‘He called the number seven “motherless”,’ says a late source, ‘for it alone has neither the nature to generate nor the nature to be generated.’ This is confirmed and explained by Aristotle, though he does not expressly name Philolaus:
Since seven neither generates any of the numbers in the decad [the numbers one to ten] nor is generated by any of them, they [the Pythagoreans] called it Athena. For two generates four, and three generates nine and six, and four generates eight, and five generates ten, while four and six and eight and nine and ten are generated, but seven neither generates any of them nor is generated from any. Just this is the character of Athena, who is motherless and always virgin.
Philolaus intrigues because of his ability to combine innovative contributions to Presocratic physics with traditional Pythagorean number symbolism. So far as we can tell, the combination is unique, without parallel or predecessor. Certainly, none of his innovative ideas in physics can be traced back to the founding father of the movement, Pythagoras himself. And when it comes to mathematics properly so called, while Philolaus wrote about the ratios involved in dividing a musical scale, there is no sign that his conclusions were backed by mathematical proof.
Our information about ancient Greek achievements in mathematics begins, as Penrose rightly says, with Thales of Miletus, well before Pythagoras. Thales is credited with the discovery of several elementary geometrical theorems; one source expressly comments on the archaic vocabulary in which he announced that the angles at the base of an isosceles triangle are equal to one another. The story gathers pace in the second half of the fifth century, when Hippocrates of Chios (not to be confused with the famous doctor Hippocrates of Cos) showed how to square a lune, i.e. how to determine the area of a curvilinear figure shaped like a crescent moon. Hippocrates’ ‘quadrature of lunes’ is the earliest extant deductive proof in Greek mathematics, immediately recognisable as the ‘real thing’. He was also the first to compose an Elements: that is, a deductive treatise such as Euclid produced two centuries later in which theorems are inferred from definitions and other types of first principle laid down at the start. Oenopides of Chios was known for mathematical work on the ecliptic and may have been the first to require that only ruler and compass be used in the solution of simple problems. Theodorus of Cyrene was the first to prove, case by individual case, the irrationality of the square roots of the prime numbers from 3 to 17, while his pupil Theaetetus of Athens early in the fourth century produced the first general theory of irrationality and the first general account of the construction of the five regular solids (cube, tetrahedron, octahedron, dodecahedron, icosahedron).
This is powerful, mainstream mathematics, a far cry from the numerology of marriage. Yet not one of the names just mentioned is that of a Pythagorean, not one comes from southern Italy. Still, there is one name that prompts a question. Why would Theodorus begin his proofs with the irrationality of √3 if not because the irrationality of √2 was already known? Who, then, discovered this, the first and most elementary case of irrationality?
The simple answer is that no one knows. Numerous books (Penrose’s included) will tell you that the discovery was felt by the Pythagoreans as a great shock, for it threatened their attempt to explain the world in terms of whole-number ratios on the model of the musical concords. Ancient testimony to this claim is non-existent. All there is is a late story, found in the Neoplatonist Iamblichus’ Life of Pythagoras (fourth century ad), that divinity drowned at sea the Pythagorean who made the discovery public, in breach of the ban (itself of dubious historicity) on divulging to outsiders any detail of what took place within the school.
Enter now the first Pythagorean to be credited with a significant mathematical discovery, Hippasus of Metapontum in southern Italy. Date uncertain, the best estimate being that he was active around 450 BC in the generation before Theodorus. Now, according to the same late compilation by Iamblichus, Hippasus was the first to show how to construct a dodecahedron and to publish his discovery – in punishment for which he was drowned at sea. For all that sea travel in antiquity was a hazardous undertaking, with shipwreck a common occurrence, some scholars unite the two drowning stories and suppose that Hippasus’ punishment was for revealing both the fact of irrationality and the construction of the dodecahedron; it has even been suggested that he discovered irrationality in the course of working on the dodecahedron. Readers who prefer history to supernatural drama may be comforted to learn (on the not entirely reputable authority of Aristoxenus of Tarentum, a pupil of Aristotle and the leading music theorist of the fourth century BC) that Hippasus performed experiments with free-swinging metal discs of equal diameter and varying thickness which could validly verify the ratios of fourth, fifth and octave.
Be that as it may, the next candidate for a Pythagorean mathematician is Archytas of Tarentum in southern Italy. The founder of mathematical mechanics (later advanced by Archimedes), and of mathematical optics (later advanced by Euclid, Archimedes and Ptolemy), he also contributed to mathematical harmonics. A formal deductive proof has come down to us beginning, as do proofs in Euclid later, with a statement of the theorem to be proved: ‘A superparticular ratio cannot be divided into equal parts by a mean proportional placed between them.’ This shows that the tone, which has the superparticular ratio 9:8, cannot be divided equally, and hence that there is no true ‘semitone’. Last, but very far from least, in geometry he devised an amazing solution (drawing on earlier work by Hippocrates of Chios) to the problem of how to duplicate a cube. This was truly a giant.
But Archytas is a contemporary of Plato, whom in 361 BC he was able to rescue from virtual imprisonment by Dionysius II, tyrant of Syracuse. (As a leading politician in democratic Tarentum, seven times elected general, he could command both a ship to go to the rescue and the international clout to induce Dionysius to let Plato go.) Splendid as Archytas’ mathematical achievements are, they tell us nothing at all about Pythagoras two centuries earlier.
Not only is Archytas the first clearly attested important Pythagorean mathematician. He is also the last. By his time most of the Pythagorean communities had been broken by their political opponents. The death toll was high. The survivors, including Philolaus, fled to mainland Greece. Philolaus settled in Thebes, where he taught Simmias and Cebes, the two characters with whom in Plato’s Phaedo Socrates discusses immortality and transmigration of the soul:
Once, they say, he was passing by when a puppy was being whipped. He took pity and said: ‘Stop, do not beat it; for it is the soul of a friend I recognised when I heard it give tongue.’
‘He’ is Pythagoras, as described by a contemporary philosopher-poet, Xenophanes of Colophon. This is evidence as near the original as one could hope to find. Even if it is evidence only about what ‘they say’ (what more than second-hand stories could one expect from a man who set nothing down in writing?), there is independent early confirmation that Himself, as Pythagoras was called by the faithful, did teach that both before our birth and after our death our soul has other lives to live in a variety of animal bodies. Here at last we see through the mists of fiction to something that approximates historical fact.
Now, however many readers of this essay believe that their soul will survive death, rather few, I imagine, believe that it also pre-existed their birth. The religions that have shaped Western culture are so inhospitable to the idea of pre-existence that you probably reject the thought out of hand, for no good reason. Be patient. There are more exotica to come:
Abstain from beans. Eat only the flesh of animals that may be sacrificed. Do not step over the beam of a balance. On rising, straighten the bedclothes and smooth out the place where you lay. Spit on your hair clippings and nail parings. Destroy the marks of a pot in the ashes. Do not piss towards the sun. Do not use a pine-torch to wipe a chair clean. Do not look in a mirror by lamplight. On a journey do not turn around at the border, for the Furies are following you. Do not make a detour on your way to the temple, for the god should not come second. Do not help a person to unload, only to load up. Do not dip your hand into holy water. Do not kill a louse in the temple. Do not stir the fire with a knife. One should not have children by a woman who wears gold jewellery. One should put on the right shoe first, but when washing do the left foot first. One should not pass by where an ass is lying.
The list could be continued, on and on. Item one, ‘Abstain from beans,’ is the best known, its rationale much disputed in antiquity; one suggestion was that it is through bean blossoms that souls return to earth for their reincarnation. Item two puts paid to the widespread idea that the Pythagoreans were always strict vegetarians. Collectively, these injunctions were known as a ’kou’smata, ‘things heard’, implying that they were transmitted by word of mouth. A number of the prescriptions have parallels in ancient cult practice. But the important thing to my mind is the sheer quantity of the rules that constrain a Pythagorean life, and the minute scrupulosity they enforce.
Other a ’kou’smata were cast in indicative rather than imperative mood:
What are the isles of the blest? Sun and Moon. Pythagoras is the Hyperborean Apollo. An earthquake is a mass meeting of the dead. The purpose of thunder is to threaten those in Tartarus, so that they will be afraid. The sea is the tears of Cronus. The Pleiades are the lyre of the Muses, and the planets are Persephone’s dogs. The ring of bronze when it is struck is the voice of a daemon trapped within it.
Add these indicatives to those imperatives and one realises that the world the followers of Pythagoras inhabit is a world full of taboos and threatening forces. All the more reason to try to escape the cycle of reincarnation, with the aid of the Hyperborean Apollo, and reach the isles of the blest.
But meanwhile, there is the politics of our present life: ‘Three hundred of the young men, bound to each other by oath like a brotherhood, lived segregated from the rest of the citizens, as if to form a secret band of conspirators, and brought the city under their control.’ That is how, according to a Roman historian of the first century BC drawing on earlier historiographical sources, Pythagoras in the sixth century got to dominate the city of Croton, which soon came, with the aid of comparable cabals in other cities, to dominate much of southern Italy. Read in today’s world, his account may well make us shiver.
The story becomes the more chilling when one reflects that there might be a connection between the discipline required for successful conspiracy and the apparently arbitrary discipline imposed by a ’kou’smata. (This is a topic on which Riedweg, drawing from modern sociological studies of charisma and sectarian religion, has useful things to say in a chapter ominously entitled ‘The Pythagorean Secret Society’.) The more arbitrary the discipline, the more it works to reinforce belief in the cause. For only the truth of the belief and the righteousness of the cause could justify the hardship of submission. It is no accident that organisations like the Church of Scientology often insist that newly recruited acolytes cut themselves off from all contact with their families. The cost of ‘disconnection’, as this is called, is so terrible that membership of the church had better be a gain of unmatchable value.
All those years ago, when as an undergraduate I was studying the Cambridge interpretation of Pythagoreanism with John Raven, there came a knock on my door. Three young men of about my age came in to speak about the work of the Plymouth Brethren. In the course of our conversation, one of them said, in his quiet-spoken way, that his favourite pastime was bird-watching, but he had been persuaded to sell his binoculars to help finance the work of the Brethren. He was telling me how much the cause meant to him. I heard only the cruelty of a sect out to bind him by making him give up his most precious possession. For the more he sacrificed, the more he would need, psychologically, to believe in the cause.
I do not mean that the Plymouth Brethren are insincere, or that Pythagoras did not believe in his cause as whole-heartedly as his followers were disciplined to do. Let it be the case that Pythagoras sincerely believed, and got his followers to believe, that he was the Hyperborean Apollo and that, as Euphorbus, he had fought Menelaus during the Trojan War. That only makes it all the more clear that he belongs to the history of politically intrusive religious movements, not to the history of philosophy or science. Even less does he deserve his traditional place in the history of mathematics.