Zhmud calls these cosmological acusmata into question a, — , noting that some only appear in Porphyry, but Porphyry explicitly identifies Aristotle as his source and we have no reason to doubt him VP The best analogy for the type of account of the cosmos which Pythagoras gave might be some of the myths which appear at the end of Platonic dialogues such as the Phaedo , Gorgias or Republic , where cosmology has a primarily moral purpose.
Should the doctrine of the harmony of the spheres be assigned to Pythagoras? Certainly the acusma which talks of the sirens singing in the harmony represented by the tetraktys suggests that there might have been a cosmic music and that Pythagoras may well have thought that the heavenly bodies, which we see move across the sky at night, made music by their motions.
The first such cosmic model in the Pythagorean tradition is that of Philolaus in the second half of the fifth century, a model which still shows traces of the connection to the moral cosmos of Pythagoras in its account of the counter-earth and the central fire see Philolaus.
If Pythagoras was primarily a figure of religious and ethical significance, who left behind an influential way of life and for whom number and cosmology primarily had significance in this religious and moral context, how are we to explain the prominence of rigorous mathematics and mathematical cosmology in later Pythagoreans such as Philolaus and Archytas?
It is important to note that this is not just a question asked by modern scholars but was already a central question in the fourth century BCE. What is the connection between Pythagoras and fifth-century Pythagoreans? The tradition of a split between two groups of Pythagoreans in the fifth century, the mathematici and the acusmatici , points to the same puzzlement.
The evidence for this split is quite confused in the later tradition, but Burkert a, ff. The acusmatici , who are clearly connected by their name to the acusmata , are recognized by the other group, the mathematici , as genuine Pythagoreans, but the acusmatici do not regard the philosophy of the mathematici as deriving from Pythagoras but rather from Hippasus.
The mathematici appear to have argued that, while the acusmatici were indeed Pythagoreans, it was the mathematici who were the true Pythagoreans; Pythagoras gave the acusmata to those who did not have the time to study the mathematical sciences, so that they would at least have moral guidance, while to those who had the time to fully devote themselves to Pythagoreanism he gave training in the mathematical sciences, which explained the reasons for this guidance.
This tradition thus shows that all agreed that the acusmata represented the teaching of Pythagoras, but that some regarded the mathematical work associated with the mathematici as not deriving from Pythagoras himself, but rather from Hippasus on the controversy about the evidence for this split into two groups of Pythagoreans see the fifth paragraph of section 4. For fourth-century Greeks as for modern scholars, the question is whether the mathematical and scientific side of later Pythagoreanism derived from Pythagoras or not.
The picture of Pythagoras presented above is inevitably based on crucial decisions about sources and has been recently challenged in a searching critique Zhmud a. In many cases, he argues, the evidence suggests that early Pythagoreanism was more scientific and that religious and mythic elements only gained in importance in the later tradition.
One of the central pieces of evidence for this view is that the tetraktys does not first appear until late in the tradition, in Aetius in the first century CE DK 1. Zhmud himself agrees that sections 82—86 of On the Pythagorean Life as a whole go back to Aristotle but suggests that the acusma about the tetraktys was a post-Aristotelian addition a, — Once again source criticism is crucial.
If the acusma in question goes back to Aristotle then there is good evidence for the tetraktys in early Pythagoreanism. If we regard it as a later insertion into Aristotelian material, the early Pythagorean credentials of the tetraktys are less clear. Although there is no explicit evidence, Pythagoras is the most likely candidate to fill these gaps. Thus between Thales, whom Eudemus identifies as the first geometer, and Hippocrates of Chios, who produced the first Elements , someone turned geometry into a deductive science Zhmud a, In each case Zhmud suggests that Pythagoras is that someone.
Such speculations have some plausibility but they highlight even more the puzzle as to why, if Pythagoras played this central role in early Greek mathematics, no early source explicitly ascribes it to him. Of course, some scholars argue that the majority have overlooked key passages that do assign mathematical achievements to Pythagoras. In order to gain a rounded view of the Pythagorean question it is thus appropriate to look at the most controversial of these passages. Some scholars who regard Pythagoras as a mathematician and rational cosmologist, such as Guthrie, admit that the earliest evidence does not support this view Lloyd , 25 , but maintain that the prominence of Pythagoras the mathematician in the late tradition must be based on something early.
Others maintain that there is evidence in the sixth- and fifth-century BCE for Pythagoras as a mathematician and cosmologist. Thus the description of Pythagoras as a wise man who practiced inquiry is simply too general to aid in deciding what sort of figure Herodotus and Heraclitus saw him as being. There is more controversy about the fourth-century evidence. Zhmud argues that Isocrates regards Pythagoras as a philosopher and mathematician a, However, it is hard to see how the passage in question Busiris 28—29 supports this view.
Nowhere in it does Isocrates ascribe mathematical work or a rational cosmology to Pythagoras. What Isocrates emphasizes about Pythagoras is what the rest of the early tradition emphasizes, his interest in religious rites. The same situation arises with Fr.
If the words in question were by Aristotle they would be his sole statement that Pythagoras was a natural philosopher. The case of Fr. The further problem with Fr. This awkward repetition of the same story about two different people immediately suggests that only one story was in the original and the other was added in the later tradition. This suggestion is strikingly confirmed by the fact that Aristotle does tell this story about Anaxagoras in his extant works Eudemian Ethics a11—16 but not about Pythagoras.
Aristotle only knows Pythagoras as a wonder working sage and teacher of a way of life Fr. What about the pupils of Plato and Aristotle? As discussed in the second paragraph of section 5 above, Eudemus, who wrote a series of histories of mathematics never mentions Pythagoras by name.
Arguments from silence are perilous but, when the most well-informed source of the fourth-century fails to mention Pythagoras in works explicitly directed towards the history of mathematics, the silence means something.
There are only two passages in which Pythagoras is explicitly associated with anything mathematical or scientific by pupils of Plato and Aristotle. Moreover, Aristoxenus explains what he means in the final participial phrase. This is consistent with the moralized cosmos of Pythagoras sketched above in which numbers have symbolic significance.
Xenocrates is being quoted here in a fragment of a work by a Heraclides Barker , — , perhaps Heraclides of Pontus. There is controversy whether the quotation of Xenocrates is limited just to what has been quoted in the previous sentence or whether the whole fragment of Heraclides is a quotation of Xenocrates.
If the second sentence is accepted then Xenocrates clearly presents Pythagoras as an acoustic scientist. It seems most reasonable, however, to accept only the first sentence as belonging to Xenocrates.
If the quotation from Xenocrates does not break off at that point, there is no other obvious breaking point in the fragment and the whole two pages of text must be ascribed to Xenocrates. The problem with ascribing it all to Xenocrates is that Porphyry introduces the passage as a quotation from Heraclides, which would be strange if everything quoted, in fact, belongs to Xenocrates.
If just the first sentence comes from Xenocrates, then all he is ascribing to Pythagoras is the recognition that the concordant intervals are connected to numbers. In such a context Xenocrates would not be making the point that Pythagoras discovered the whole number ratios but rather that he found out that concords arose in accordance with whole number ratios, perhaps from musicians who discovered them first not being the issue , and used this fact as another illustration of how things are like numbers.
Thus, the fragments of Aristoxenus and Xeoncrates show that Pythagoras likened things to numbers and took the concordant musical intervals as a central example, but do not suggest that he founded arithmetic as a rigorous mathematical discipline or carried out a program of scientific research in harmonics.
It should now be clear that decisions about sources are crucial in addressing the question of whether Pythagoras was a mathematician and scientist. The Pythagorean Question 2. Sources 2. Life and Works 4. The Philosophy of Pythagoras 4. Was Pythagoras a Mathematician or Cosmologist?
The Pythagorean Question What were the beliefs and practices of the historical Pythagoras? Apollonius of Tyana died ca. It is possible that this work is by another otherwise unknown Apollonius.
Life and Works References to Pythagoras by Xenophanes ca. The Philosophy of Pythagoras One of the manifestations of the attempt to glorify Pythagoras in the later tradition is the report that he, in fact, invented the word philosophy.
Referred to as DK. Wilson tr. Athenaeus, , The Deipnosophists , 6 Vols. Gulick tr. Barker, A. Becker, O. Bremmer, J. Burkert, W. Minar tr. Cornelli, G. Delatte, A. Diels, H. Hicks tr. Rolfe tr. Gemelli Marciano, L. Laks and C. Louguet eds. Granger, H.
Guthrie, W. Hahn, R. Heath, T. Heinze, R. Renger ed. Huffman, C. II: Contextes , M. Dixsaut ed. Long ed. Preus ed. Frede and B. In it, the principles of what is now called Euclidean Geometry were deduced from a small set of axioms.
When Euclid wrote his Elements around BCE , he gave two proofs of the Pythagorean Theorem: The first, Proposition 47 of Book I, relies entirely on the area relations and is quite sophisticated; the second, Proposition 31 of Book VI, is based on the concept of proportion and is much simpler.
He may have used Book VI Proposition 31, but, if so, his proof was deficient, because the complete theory of Proportions was only developed by Eudoxus, who lived almost two centuries after Pythagoras. Euclid's Elements furnishes the first and, later, the standard reference in geometry. It is a mathematical and geometric treatise consisting of 13 books. It comprises a collection of definitions, postulates axioms , propositions theorems and constructions and mathematical proofs of the propositions.
Euclid provided two very different proofs, stated below, of the Pythagorean Theorem. This is probably the most famous of all the proofs of the Pythagorean proposition. In right-angled triangles the figure on the side opposite the right angle equals the sum of the similar and similarly described figures on the sides containing the right angle.
In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle. Euclid I 47 is often called the Pythagorean Theorem , called so by Proclus — a Greek philosopher who became head of Plato's Academy and is important mathematically for his commentaries on the work of other mathematicians — and others centuries after Pythagoras and even centuries after Euclid.
Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making them easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics twenty-three centuries later.
Although best known for its geometric results, Elements also includes number theory. It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers and the Euclidean algorithm for finding the greatest common divisor of two numbers. The geometrical system described in the Elements was long known simply as geometry , and was considered to be the only geometry possible.
Today, however, this system is often referred to as Euclidean Geometry to distinguish it from other so-called Non-Euclidean geometries that mathematicians discovered in the nineteenth century. At this point in my plotting of the year-old story of Pythagoras, I feel it is fitting to present one proof of the famous theorem. For me, the simplest proof among the dozens of proofs that I read in preparing this article is that shown in Figure Start with four copies of the same triangle.
See upper part of Figure See lower part of Figure In the seventeenth century, Pierre de Fermat — Figure 14 investigated the following problem: for which values of n are there integer solutions to the equation. Fermat conjectured that there were no non-zero integer solutions for x and y and z when n was greater than 2.
He did not leave a proof, though. Instead, in the margin of a textbook, he wrote that he knew that this relationship was not possible, but he did not have enough room on the page to write it down. His conjecture became known as Fermat's Last Theorem. This may appear to be a simple problem on the surface, but it was not until when Andrew Wiles of Princeton University finally proved the year-old marginalized theorem, which appeared on the front page of the New York Times.
Today, Fermat is thought of as a number theorist, in fact perhaps the most famous number theorist who ever lived. It is therefore surprising to find that Fermat was a lawyer , and only an amateur mathematician.
Also surprising is the fact that he published only one mathematical paper in his life, and that was an anonymous paper written as an appendix to a colleague's book. Because Fermat refused to publish his work, his friends feared that it would soon be forgotten unless something was done about it. His son Samuel undertook the task of collecting Fermat's letters and other mathematical papers, comments written in books and so on with the goal of publishing his father's mathematical ideas.
Samuel found the marginal note the proof could not fit on the page in his father's copy of Diophantus's Arithmetica. In this way the famous Last Theorem came to be published. His graduate research was guided by John Coates beginning in the summer of Together they worked on the arithmetic of elliptic curves with complex multiplication using the methods of Iwasawa theory. He further worked with Barry Mazur on the main conjecture of Iwasawa theory over Q and soon afterwards generalized this result to totally real fields.
Taking approximately 7 years to complete the work, Wiles was the first person to prove Fermat's Last Theorem, earning him a place in history. Wiles was introduced to Fermat's Last Theorem at the age of He tried to prove the theorem using textbook methods and later studied the work of mathematicians who had tried to prove it.
When he began his graduate studies, he stopped trying to prove the theorem and began studying elliptic curves under the supervision of John Coates.
In the s and s, a connection between elliptic curves and modular forms was conjectured by the Japanese mathematician Goro Shimura based on some ideas that Yutaka Taniyama posed. With Weil giving conceptual evidence for it, it is sometimes called the Shimura—Taniyama—Weil conjecture. It states that every rational elliptic curve is modular. The full conjecture was proven by Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor in using many of the methods that Andrew Wiles used in his published papers.
I provide the story of Pythagoras and his famous theorem by discussing the major plot points of a year-old fascinating story in the history of mathematics, worthy of recounting even for the math-phobic reader. It is more than a math story, as it tells a history of two great civilizations of antiquity rising to prominence years ago, along with historic and legendary characters, who not only define the period, but whose life stories individually are quite engaging. Greek mathematician Euclid, referred to as the Father of Geometry, lived during the period of time about BCE , when he was most active.
His work Elements is the most successful textbook in the history of mathematics. Euclid I 47 is often called the Pythagorean Theorem , called so by Proclus, a Greek philosopher who became head of Plato's Academy and is important mathematically for his commentaries on the work of other mathematicians centuries after Pythagoras and even centuries after Euclid.
There is concrete not Portland cement, but a clay tablet evidence that indisputably indicates that the Pythagorean Theorem was discovered and proven by Babylonian mathematicians years before Pythagoras was born.
So many people, young and old, famous and not famous, have touched the Pythagorean Theorem. The eccentric mathematics teacher Elisha Scott Loomis spent a lifetime collecting all known proofs and writing them up in The Pythagorean Proposition, a compendium of proofs. The manuscript was published in , and a revised, second edition appeared in Porphyry in [ 12 ] and [ 13 ] says that Pythagoras learnt geometry from the Egyptians but it is likely that he was already acquainted with geometry, certainly after teachings from Thales and Anaximander.
Polycrates abandoned his alliance with Egypt and sent 40 ships to join the Persian fleet against the Egyptians. Pythagoras was taken prisoner and taken to Babylon. Iamblichus writes that Pythagoras see [ 8 ] Whilst he was there he gladly associated with the Magoi He also reached the acme of perfection in arithmetic and music and the other mathematical sciences taught by the Babylonians Polycrates had been killed in about BC and Cambyses died in the summer of BC, either by committing suicide or as the result of an accident.
The deaths of these rulers may have been a factor in Pythagoras's return to Samos but it is nowhere explained how Pythagoras obtained his freedom. Darius of Persia had taken control of Samos after Polycrates' death and he would have controlled the island on Pythagoras's return.
This conflicts with the accounts of Porphyry and Diogenes Laertius who state that Polycrates was still in control of Samos when Pythagoras returned there. Pythagoras made a journey to Crete shortly after his return to Samos to study the system of laws there.
Back in Samos he founded a school which was called the semicircle. Iamblichus [ 8 ] writes in the third century AD that They do this because they think one should discuss questions about goodness, justice and expediency in this place which was founded by the man who made all these subjects his business. Outside the city he made a cave the private site of his own philosophical teaching, spending most of the night and daytime there and doing research into the uses of mathematics Pythagoras left Samos and went to southern Italy in about BC some say much earlier.
Iamblichus [ 8 ] gives some reasons for him leaving. First he comments on the Samian response to his teaching methods The Samians were not very keen on this method and treated him in a rude and improper manner. This was, according to Iamblichus, used in part as an excuse for Pythagoras to leave Samos Pythagoras was dragged into all sorts of diplomatic missions by his fellow citizens and forced to participate in public affairs.
He knew that all the philosophers before him had ended their days on foreign soil so he decided to escape all political responsibility, alleging as his excuse, according to some sources, the contempt the Samians had for his teaching method.
Pythagoras founded a philosophical and religious school in Croton now Crotone, on the east of the heel of southern Italy that had many followers. Pythagoras was the head of the society with an inner circle of followers known as mathematikoi.
The mathematikoi lived permanently with the Society, had no personal possessions and were vegetarians. They were taught by Pythagoras himself and obeyed strict rules. The beliefs that Pythagoras held were [ 2 ] :- 1 that at its deepest level, reality is mathematical in nature, 2 that philosophy can be used for spiritual purification, 3 that the soul can rise to union with the divine, 4 that certain symbols have a mystical significance, and 5 that all brothers of the order should observe strict loyalty and secrecy.
We know some information, but very little. Three biographies of Pythagoras have survived from antiquity, but these are all extremely late. The later lives of Pythagoras, written by the Neoplatonist philosophers Porphyrios and Iamblichos respectively, are even longer and even more inundated with legends.
The philosopher Herakleitos of Ephesos lived c. The poets Ion of Chios lived c. Our later sources are a bit thornier, so we need to be somewhat more skeptical. In the late fourth century BC, the Peripatetic philosophers Dikaiarchos, Aristoxenos, and Herakleides Pontikos all wrote accounts of the lives of Pythagoras, but none of these have survived. Pythagoras may have travelled to other lands as a young man, but the extent of his travels is unclear. In Kroton, Pythagoras established a commune in which initiates swore oaths of solemn fealty and were bound by secrecy.
In BC, Kroton won a massive military victory over the neighboring city of Sybaris and several Pythagoreans, including Milon of Kroton, were apparently major generals in the battle. The victory instigated a proposal for a new, democratic constitution—a proposal which the Pythagoreans opposed. Two supporters of democracy named Kylon and Ninon rallied the people of Kroton against the Pythagoreans and, while the Pythagoreans were gathered in one of their meeting houses, Kylon, Ninon, and their mob of supporters, set fire to the house and murdered the Pythagoreans as they attempted to escape from the burning building.
Pythagoras himself may have been killed in this purge or he may have escaped to Metaponton, where he may have survived for several more years before eventually dying. This applies both to humans and to animals and, in Pythagorean cosmology, it was possible for a human to be reincarnated as an animal or an animal to be reincarnated as a human. Hicks :. Here Pythagoras intervenes to save a dog who is being beaten by his master, saying that the dog was a friend of his in a past life and that he recognized him by the sound of his voice.
This is obviously intended satirically, but it clearly shows that Pythagoras was associated with metempsychosis from an early date. Xenophanes lived the later part of his life in Elea in southern Italy, not far from Kroton, where Pythagoras did most of his teaching, and may have either met Pythagoras himself in his old age or known people who had known him.
Pythagoras was not the first person to teach the doctrine of metempsychosis ; certainly at least Pherekydes of Syros lived c. Nonetheless, we can be sure that Pythagoras taught this idea and, in later times, Pythagoras was the one most closely associated with it. Eudoxos of Knidos takes this even further, stating that Pythagoras not only forbade eating meat, but also refused to even go anywhere in the presence of hunters or butchers.
Earlier sources, however, contradict the idea that Pythagoras forbade meat, with many of them stating that he only forbade certain kinds of meat.
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