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Kirk and J. Raven , in which Raven my undergraduate tutor devotes 40 pages to Pythagoras and his early followers. 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. 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. Theirs was a conscious construction whereby Pythagoras became the apostle of mathematics and a highly mathematising philosophy, full of anticipations of Platonic metaphysics.

Return to Mathematical Circles: A Fifth Collection of Mathematical Stories and Anecdotes

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. Legends are retold. 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.

Not that Riedweg buys into all this, but he does encourage his readers to marvel at a man around whom such legends grew. 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.

LINCOLN’S YARNS AND STORIES

But it is no such thing. First, a mundane point of translation. 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.

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. 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.

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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?

There were some enthralling ideas in this book. One was a revolutionary proposal to move the Earth.


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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. 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. 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.

Teaching with Original Historical Sources in Mathematics

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. Who, then, discovered this, the first and most elementary case of irrationality? The simple answer is that no one knows. Ancient testimony to this claim is non-existent.


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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 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.

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. 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. 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. 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. Once, they say, he was passing by when a puppy was being whipped.

This is evidence as near the original as one could hope to find.

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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.

My (Portable) Math Book Collection [Math Books]

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:.

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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. It may be strange to watch mathematicians, who at other times pride themselves upon their insistence on preciseness, repeat without hesitation apocryphal anecdotes without bothering one bit about their authenticity. However, if we realize that these are to be regarded as anecdotes rather than as history, and if we pay more attention to their value as a catalyst, then it presents no more problem than when we make use of a heuristic argument to explain a theorem.

Besides, though many anecdotes have been embroidered over the years, many of them are based on some kind of real occurrence. Of course, an ideal situation is an authentic as well as amusing or instructive anecdote. Failing that we still find it helpful to have a good anecdote which carries a message. There are plenty of examples of anecdotes which serve to achieve the aims set out in Eves preface.

I will give only two examples.