Pythagoras was the first person to recommend music as a prescription. He connected music to craftsmanship, design, government, raising a family, fellowship, and self-improvement. He believed that music was an outflow of harmonia, the divine rule seeking to banish confusion and conflict in the cosmos. Along these lines, music was seen to have a double function as, like science, it empowered people to see into the structures of nature. The Pythagoreans also set great store by physical exercise and recommended daily morning walks and sporting activities.
Periods of self-examination at the beginning and end of every day were likewise advised. Pythagoras was the first person to suggest that the earth was a sphere, but it is not clear what led him to that conclusion. It is possibly connected to his belief that circles were the strongest shape. His experience of the universe was most likely exceptionally basic: at that time, the earth was still thought to be the focal point of the universe with everything revolving around it.
The Pythagorean view of the universe was pretty straightforward and did not take into account any observation of the movements of the planets. They held the belief that the planets all moved in giant circles, and that when they brushed against one another, they made a sound.
They believed that these sounds were melodic harmonies, and this music has come to be known as the Music of the Spheres. This music was not audible because it was a constant background noise. The Pythagoreans also held the belief that the earth, planets, and stars all circled a central flame, and night and day was caused because of this movement. Since they felt that fire was more important than earth, the focal point of the universe must be fire. Pythagoras was the first to perceive that Venus at night and Venus toward the beginning of the day were the same planet.
Pythagoras started the idea of a numerical system, and therefore the beginning of mathematics. To the Pythagoreans, genuine numbers were the most vital thing, and numbers make up the world. A soul was believed to exist in both animal and vegetable life, even though there is no proof to indicate that Pythagoras thought the spirit could be contained in a plant.
Although a number of modern scholars have speculated on what sort of proof Pythagoras might have used e. All that this tradition ascribes to Pythagoras, then, is discovery of the truth contained in the theorem. The truth may not have been in general form but rather focused on the simplest such triangle with sides 3, 4 and 5 , pointing out that such a triangle and all others like it will have a right angle.
Robson It is possible, then, that Pythagoras just passed on to the Greeks a truth that he learned from the East. If the story is to have any force and if it dates to the fourth century, it shows that Pythagoras was famous for a connection to a certain piece of geometrical knowledge, but it also shows that he was famous for his enthusiastic response to that knowledge, as evidenced in his sacrifice of oxen, not for any geometric proof.
What emerges from this evidence, then, is not Pythagoras as the master geometer, who provides rigorous proofs, but rather Pythagoras as someone who recognizes and celebrates certain geometrical relationships as of high importance.
It is striking that a very similar picture of Pythagoras emerges from the evidence for his cosmology. A famous discovery is attributed to Pythagoras in the later tradition, i. The only early source to associate Pythagoras with the whole number ratios that govern the concords is Xenocrates Fr. It may be once again that Pythagoras knew of the relationship without either having discovered it or having demonstrated it scientifically.
The relationship was probably first discovered by instrument makers, and specifically makers of wind instruments rather than stringed instruments Barker , Here in the acusmata , these four numbers are identified with one of the primary sources of wisdom in the Greek world, the Delphic oracle.
This acusma thus seems to be based on the knowledge of the relationship between the concords and the whole number ratios.
The picture of Pythagoras that emerges from the evidence is thus not of a mathematician, who offered rigorous proofs, or of a scientist, who carried out experiments to discover the nature of the natural world, but rather of someone who sees special significance in and assigns special prominence to mathematical relationships that were in general circulation.
Some might suppose that this is a reference to a rigorous treatment of arithmetic, such as that hypothesized by Becker , who argued that Euclid IX. There is, moreover, no talk of mathematical proof or a deductive system in the passage from Aristoxenus just quoted. Pythagoras is known for the honor he gives to number and for removing it from the practical realm of trade and instead pointing to correspondences between the behavior of number and the behavior of things.
The doxographical tradition reports that Pythagoras discovered the sphericity of the earth, the five celestial zones and the identity of the evening and morning star Diogenes Laertius VIII. In each case, however, Burkert has shown that these reports seem to be false and the result of the glorification of Pythagoras in the later tradition, since the earliest and most reliable evidence assigns these same discoveries to someone else a, ff.
Thus, Theophrastus, who is the primary basis of the doxographical tradition, says that it was Parmenides who discovered the sphericity of the earth Diogenes Laertius VIII. Parmenides is also identified as the discoverer of the identity of the morning and evening star Diogenes Laertius IX. The identification of the five celestial zones depends on the discovery of the obliquity of the ecliptic, and some of the doxography duly assigns this discovery to Pythagoras as well and claims that Oenopides stole it from Pythagoras Aetius II.
As was shown above, Pythagoras saw the cosmos as structured according to number insofar as the tetraktys is the source of all wisdom. His cosmos was also imbued with a moral significance, which is in accordance with his beliefs about reincarnation and the fate of the soul West , —; Huffman , 60— 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. Reis eds. Patterson, V. Karasmanis, A. Hermann eds. Festa ed. Kahn, C. Kirk, G. Laks, A. Pythagoras believed:. Because of the strict secrecy among the members of Pythagoras' society, and the fact that they shared ideas and intellectual discoveries within the group and did not give individuals credit, it is difficult to be certain whether all the theorems attributed to Pythagoras were originally his, or whether they came from the communal society of the Pythagoreans.
Some of the students of Pythagoras eventually wrote down the theories, teachings and discoveries of the group, but the Pythagoreans always gave credit to Pythagoras as the Master for:.
Pythagoras studied odd and even numbers, triangular numbers, and perfect numbers. Pythagoreans contributed to our understanding of angles, triangles, areas, proportion, polygons, and polyhedra.
Pythagoras also related music to mathematics. He had long played the seven string lyre, and learned how harmonious the vibrating strings sounded when the lengths of the strings were proportional to whole numbers, such as , , Pythagoreans also realized that this knowledge could be applied to other musical instruments. The reports of Pythagoras' death are varied.
He is said to have been killed by an angry mob, to have been caught up in a war between the Agrigentum and the Syracusans and killed by the Syracusans, or been burned out of his school in Crotona and then went to Metapontum where he starved himself to death. At least two of the stories include a scene where Pythagoras refuses to trample a crop of bean plants in order to escape, and because of this, he is caught.
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