Hippasus of Metapontium

Hippasus von Metapont (Greek Ἵππασος Híppasos ) was a Greek mathematician, music theorist and philosopher from the circle of the Pythagoreans . He lived in the late 6th and early 5th centuries BC. BC in southern Italy and is one of the most famous Pythagoreans of the early period. Three discoveries are ascribed to him: the construction of the dodecahedron inscribed into a sphere , the discovery of incommensurability and the determination of the numerical ratios of the basic consonances through sound experiments.

Life

Hippasus came from Metapont (today Metaponto in Basilicata , southern Italy). This city, where Pythagoras spent the last years of his life, was one of the most important centers of the Pythagorean movement, which Hippasus joined.

From the news that the late ancient philosopher Iamblichus passed on, it emerges that Hippasus, in the alleged split of the Pythagoreans in two directions, the mathematical and scientific researching "mathematicians" and the "acousmatics" who live according to traditional rules of behavior, one Should have played a role. The information given by Iamblichus is contradicting itself: He writes in several places that Hippasus was an acousmatic, and reports that mathematicians consider him the founder of the acousmatic tendency; elsewhere he counts Hippasus among the mathematicians and reports that the acousmatists considered him the founder of the mathematicians. The original tradition is what Hippasus calls a mathematician.

Apparently, Hippasus was a prominent figure who was highly controversial among the Pythagoreans. According to a legend that has been handed down in different versions, he betrayed a secret of the Pythagoreans, was then excluded from their community and later had a fatal accident in the sea, which was interpreted as a divine punishment. The legend is incredible in this form, but it reflects the considerable tensions connected with his appearance and the attitude of opposing circles who accused him of falsifying the Pythagorean teaching tradition.

A historical core of the legend is that there actually was a falling out among the Pythagoreans and Hippasus played a prominent role in it. The cause of the conflict, however, was not in dealing with mathematical knowledge, but in the political contrast between revolutionary democrats and conservative forces. Hippasus supported a democratic party, while the majority of the politically conservative Pythagoreans were close to the leading sexes and thus came into conflict with popular speakers. What is striking is the independent attitude that Hippasus assumed among the Pythagoreans. Maybe he was just loosely connected to them.

Teaching

Aristotle reports that Hippasus thought fire to be the basic material ( archḗ ) of the world. He is said to have also considered the soul to be fiery. In ancient sources he is often mentioned together with Heraklit , in whose natural philosophy fire also plays a central role. Hippasus taught that the universe is finite and in constant motion and that its change takes place within a fixed time frame.

Diogenes Laertios reports that Demetrios of Magnesia (a grammarian of the 1st century BC) claimed that Hippasus left no writing. According to another tradition, also communicated by Diogenes Laertios, Hippasus wrote a work entitled "Mystical Logos ", with which he wanted to denigrate Pythagoras. This message is, even if it is not correct, an indication that Hippasus opposed the authority of the school founder Pythagoras or that this was at least imputed to him by his opponents.

The incommensurability

square

It is not known whether Hippasus discovered the incommensurability of the side and diagonal on a square or on a regular pentagon . In the 4th century Plato showed in his dialogue Meno that the inner, transverse square, which is delimited by the small diagonals (picture on the left), is half the size of the whole square. However, he did not go into the relationship between side length and diagonal.

A geometric proof of incommensurability can be given on the pentagon as follows:

The pentagram , the distinctive symbol of the Pythagoreans, creates a pentagon by connecting the tips of the star.

The task is to find a common measure for two distances, i.e. a small section of which both distances are an integral multiple. One method of finding this dimension is the alternate removal , later named after Euclid : you subtract the shorter distance from the larger one until it is no longer possible. Now you take the remainder and subtract it from the smaller one. The new remainder is deducted from the old remainder, etc. If you come to an end with this procedure because there is no remainder, you have the section you are looking for. Incommensurability of the routes is proven if it can be shown that this is impossible.

In the case of the regular pentagon, the question is whether its side has a common dimension with the diagonal. First you subtract the pentagon side (or the equally long line AC, picture on the right) from a diagonal (line AD); there is a remainder (distance CD). This is deducted from the side. The new remainder is the length of the line BC. Here one encounters the initial distance ratio again because the distance BC is the side of an inner pentagon and the distance CD is just as long as its small diagonal CC '. The smaller pentagon is geometrically similar to the starting pentagon because it is also a regular pentagon. So you are back at the beginning, because AD relates to AC as CD (or CC ') relates to BC (“ golden ratio ”). In geometrically similar figures, the length ratios of analog lines are the same. The process can thus be repeated infinitely often and always leads to a smaller pentagon. This process can also be represented as an infinite continued fraction . So there is no common measure.

"Fundamental Crisis"

In ancient sources there is talk of a betrayal of secrets by Hippasus. The secret revealed was either the dodecahedron or the incommensurability. It is said that Hippasus published his discovery and was subsequently excluded from the Pythagorean community. Later he drowned in the sea, which was interpreted as a divine punishment for his iniquity.

In older research, a "basic crisis" of Pythagoreanism was assumed to be the background of this legend. It was assumed that Pythagoras claimed that all phenomena can be expressed as integer numerical ratios and that there can be no incommensurability. The discovery of Hippasus thus disproved the basis of Pythagoreanism, and the Pythagoreans resented him for this.

However, research has deviated from this interpretation. Walter Burkert and Leonid Zhmud - who otherwise take completely contrary positions - agree that there is no convincing evidence for the claim that Pythagoras dogmatically committed himself to a worldview that in principle excludes any incommensurability. Nor is there any evidence that the discovery of incommensurability was perceived as a scandal and posed a philosophical problem; rather, it was considered a brilliant achievement by the Pythagoreans. The fact that the Greek word árrhētos (literally "unspeakable"), which in mathematics meant "irrational", was ambiguous, played an important role in the development of the legend of betrayal of secrets ; “Unspeakable” could also mean “secret”, and in this sense the word was used outside of mathematics for religious secret teachings ( mysteries ). So the idea that irrationality was a secret arose from a misunderstanding.

Music theory

One of the most important areas of interest of the Pythagoreans was music theory, especially the question of how the harmonic intervals can be expressed mathematically. Hippasus is credited with an experiment in which he created intervals with four bronze discs of the same diameter and different thicknesses, thus showing that the basic consonances can be expressed by numerical ratios. The thicknesses of the four disks were 1: 1⅓: 1½: 2.

According to the late antique scholar Boethius , Hippasus and Eubulides, another Pythagorean, added two more to the already known three symphonic intervals, the double octave and the duodecime .

swell

• Maria Timpanaro Cardini : Pitagorici. Testimonianze e frammenti . Vol. 1, La Nuova Italia, Firenze 1958, pp. 78-105 (Greek and Latin source texts with Italian translation and commentary)

literature

• Walter Burkert: Wisdom and Science. Studies on Pythagoras, Philolaus and Plato . Hans Carl, Nuremberg 1962
• Bruno Centrone: Hippasos de Métaponte . In: Richard Goulet (ed.): Dictionnaire des philosophes antiques , Vol. 3, CNRS Éditions, Paris 2000, ISBN 2-271-05748-5 , pp. 753-755
• Maria Luisa Silvestre: Il mistero di Ippaso . In: Marisa Tortorelli Ghidini u. a. (Ed.): Tra Orfeo e Pitagora. Origini e incontri di culture nell'antichità . Bibliopolis, Napoli 2000, ISBN 88-7088-395-7 , pp. 413-432
• Leonid Zhmud: Science, Philosophy and Religion in Early Pythagoreanism . Akademie Verlag, Berlin 1997, ISBN 3-05-003090-9
• Leonid Zhmud: Hippasus from Metapont . In: Hellmut Flashar et al. (Ed.): Early Greek Philosophy (= Outline of the History of Philosophy . The Philosophy of Antiquity , Volume 1), Half Volume 1, Schwabe, Basel 2013, ISBN 978-3-7965-2598-8 , p. 412-415

supporting documents

1. ↑ For the dating see Bartel Leendert van der Waerden : Die Pythagoreer , Zurich 1979, pp. 74–77; Leonid Zhmud: Science, Philosophy and Religion in Early Pythagoreanism , Berlin 1997, p. 71f. Maria Luisa Silvestre advocates early dating (birth around 560–555): Il mistero di Ippaso . In: Marisa Tortorelli Ghidini u. a. (Ed.): Tra Orfeo e Pitagora , Napoli 2000, pp. 413-432, here: 421f.
2. ↑ This is testified by Aristotle, Metaphysik 984a7 and Diogenes Laertios 8,84. Your information is correct based on the current state of research. A different tradition communicated by Iamblichos, according to which Hippasus' hometown was Croton ( Crotone ), is not credible.
3. ↑ See also Walter Burkert: Weisheit und Wissenschaft , Nürnberg 1962, pp. 188f .; Bartel Leendert van der Waerden: Die Pythagoreer , Zurich 1979, p. 67f .; Bruno Centrone: Hippasos de Métaponte . In: Richard Goulet (ed.): Dictionnaire des philosophes antiques , Vol. 3, Paris 2000, pp. 753–755, here: 753f.
4. ↑ Bartel Leendert van der Waerden: Die Pythagoreer , Zurich 1979, pp. 209f .; Maria Luisa Silvestre: Il mistero di Ippaso . In: Marisa Tortorelli Ghidini u. a. (Ed.): Tra Orfeo e Pitagora , Napoli 2000, pp. 413-432, here: 415-429.
5. ↑ Aristotle, Metaphysics 984a7.
6. ↑ Diogenes Laertios 8,84 and 8,7.
7. ↑ See Leonid Zhmud: Science, Philosophy and Religion in Early Pythagoreism , Berlin 1997, pp. 174f. (argues for the square) and Kurt von Fritz : Grundprobleme der Geschichte der Antique Wissenschaft , Berlin 1971, pp. 564-569 (argues for the pentagon).
8. ↑ Bartel Leendert van der Waerden: Die Pythagoreer , Zurich 1979, p. 71f .; Leonid Zhmud: Science, Philosophy and Religion in Early Pythagoreanism , Berlin 1997, p. 171.
9. ↑ Reject the hypothesis of a fundamental crisis and a. David H. Fowler: The Mathematics of Plato's Academy , Oxford 1987, pp. 302–308 and Hans-Joachim Waschkies: Beginnings of arithmetic in the ancient Orient and among the Greeks , Amsterdam 1989, p. 311 and note 23.
10. ^ Leonid Zhmud: Science, Philosophy and Religion in Early Pythagoreanism , Berlin 1997, pp. 173–175; Walter Burkert: Wisdom and Science , Nuremberg 1962, pp. 431–440; Detlef Thiel expressed his approval: The Philosophy of Xenocrates in the Context of the Old Academy , Munich 2006, p. 94, note 65. Cf. Gustav Junge: From Hippasus to Philolaus. The irrational and the basic geometric concepts . In: Classica et Mediaevalia 19, 1958, pp. 41-72.
11. ↑ Bartel Leendert van der Waerden: Die Pythagoreer , Zurich 1979, p. 371f .; Walter Burkert: Wisdom and Science , Nuremberg 1962, pp. 355–357. See also on the music theory of Hippasus Assunta Izzo: Musica e numero da Ippaso ad Achita . In: Antonio Capizzi, Giovanni Casertano (eds.): Forme del sapere nei presocratici , Rome 1987, pp. 137–167, here: 139ff.
12. ↑ Boethius, De institutione musica 2.19.