Plato's number
Encyclopedia
Plato’s number is a number enigmatically referred to by Plato
in his dialogue the Republic (8.546b). The text is notoriously difficult to understand and its corresponding translations do not allow an unambiguous interpretation. There is no real agreement neither about the meaning nor about the value of the number. It has been called also the 'Geometrical number' or the 'Nuptial number' (the Number of the Bride’’),and designated by various other means. Comments on this passage have appeared ever since it has been written. Without any consensus in the argumentation 216
is a most frequently proposed value for it, but 3600 or 12960000 are also commonly considered.
An incomplete list of authors who mention or discourse about includes the names of Aristotle
, Proclus
for antiquity; Ficino
and Cardano during the Renaissance; Zeller
, Friedrich Schleiermacher, Paul Tannery
and Friedrich Hultsch in the 19th c. and further new names are currently added .
Further in the Republic (9.587b) another number is mentioned, known as the "Number of the Tyrant".
The 'entire geometrical number', mentioned shortly before the end of this text, is understood to be Plato's number. The introductory words mention (a period comprehended by) 'a perfect number' which is taken to be a reference to Plato's perfect year
mentioned in his Timaeus
(39d). The words are presented as uttered by the muses, so the whole passage is sometimes called the 'speech of the muses' or something similar. Indeed Philip Melanchthon compared it to the proverbial obscurity of the Sybil
s. Cicero
famously described it as 'obscure' but others have seen some playfulness in its tone .
inserted a note that it has to be skipped in his French translation of Plato's works. In the early 20th c. scholarly findings suggested a Babylon
ian origin for the topic.
Most interpretators argue that the value of Plato's number is 216
because it is the cube of 6 i.e.
which is remarkable for being also a sum of the cubes for the Pythagorean triple
3,4,and 5:
Such considerations tend to ignore the second part of the text where some others numbers and their relations are described. The opinions tend to converge about their values being 480000 and 270000 but there is little agreement about the details. It has been noted that 6 raised to fourth power yields 1296 and
. Instead of multiplication some interpretations consider the sum of these factors: 48+27=75.
Other values that have been proposed include:
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
in his dialogue the Republic (8.546b). The text is notoriously difficult to understand and its corresponding translations do not allow an unambiguous interpretation. There is no real agreement neither about the meaning nor about the value of the number. It has been called also the 'Geometrical number' or the 'Nuptial number' (the Number of the Bride’’),and designated by various other means. Comments on this passage have appeared ever since it has been written. Without any consensus in the argumentation 216
216 (number)
216 is the natural number following 215 and preceding 217.Since 216 = 3^3 + 4^3 + 5^3 = 6^3, it is the smallest cube that's also the sum of three cubes . It is also the sum of a twin prime...
is a most frequently proposed value for it, but 3600 or 12960000 are also commonly considered.
An incomplete list of authors who mention or discourse about includes the names of Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
, Proclus
Proclus
Proclus Lycaeus , called "The Successor" or "Diadochos" , was a Greek Neoplatonist philosopher, one of the last major Classical philosophers . He set forth one of the most elaborate and fully developed systems of Neoplatonism...
for antiquity; Ficino
Marsilio Ficino
Marsilio Ficino was one of the most influential humanist philosophers of the early Italian Renaissance, an astrologer, a reviver of Neoplatonism who was in touch with every major academic thinker and writer of his day, and the first translator of Plato's complete extant works into Latin...
and Cardano during the Renaissance; Zeller
Eduard Zeller
Eduard Gottlob Zeller , was a German philosopher and theologian of the Tübingen School of theology.- Life :Eduard Zeller was born at Kleinbottwar in Württemberg, and educated at the University of Tübingen and under the influence of Georg Wilhelm Friedrich Hegel...
, Friedrich Schleiermacher, Paul Tannery
Paul Tannery
Paul Tannery was a French mathematician and historian of mathematics. He was the older brother of mathematician Jules Tannery, to whose Notions Mathématiques he contributed an historical chapter...
and Friedrich Hultsch in the 19th c. and further new names are currently added .
Further in the Republic (9.587b) another number is mentioned, known as the "Number of the Tyrant".
Plato’s Text
Great lexical and syntactical differences are easily noted between the many translations of the Republic. Below is a typical text from a relatively recent translation of Republic 546b–c:
"Now for divine begettings there is a period comprehended by a perfect number, and for mortal by the first in which augmentations dominating and dominated when they have attained to three distances and four limits of the assimilating and the dissimilating, the waxing and the waning, render all things conversable and commensurable [546c] with one another, whereof a basal four-thirds wedded to the pempad yields two harmonies at the third augmentation, the one the product of equal factors taken one hundred times, the other of equal length one way but oblong,-one dimension of a hundred numbers determined by the
rational diameters of the pempad lacking one in each case, or of the irrational lacking two; the other dimension of a hundred cubes of the triad. And this entire geometrical number is determinative of this thing, of better and inferior births."
The 'entire geometrical number', mentioned shortly before the end of this text, is understood to be Plato's number. The introductory words mention (a period comprehended by) 'a perfect number' which is taken to be a reference to Plato's perfect year
Great year
The Great Year is an archaic cosmological conception, found in different cultures, which acquired new interpretations with the development of astronomical knowledge In the Western tradition Plato has been the main source for the idea, so it was also frequently called 'Platonic year'...
mentioned in his Timaeus
Timaeus
Timaeus is a Greek name, meaning "Honour". It may refer to:*Timaeus , a Socratic dialogue by Plato*Timaeus of Locri, the 5th-century Pythagorean philosopher, appearing in Plato's dialogue...
(39d). The words are presented as uttered by the muses, so the whole passage is sometimes called the 'speech of the muses' or something similar. Indeed Philip Melanchthon compared it to the proverbial obscurity of the Sybil
Sybil
In antiquity, the oracular seeresses of the Ancient Near East and the Mediterranean were referred to by the Greek term "sibyls". In modern times, when "Sibyl" is adopted for a woman's name, the conventional spelling is "Sybil".-People:...
s. Cicero
Cicero
Marcus Tullius Cicero , was a Roman philosopher, statesman, lawyer, political theorist, and Roman constitutionalist. He came from a wealthy municipal family of the equestrian order, and is widely considered one of Rome's greatest orators and prose stylists.He introduced the Romans to the chief...
famously described it as 'obscure' but others have seen some playfulness in its tone .
Interpretations
Shortly after Plato's time his meaning apparently did not cause puzzlement as Aristotle's casual remark attests. Half a millenium later however it was an enigma for the Neoplatonists, who had a somewhat mystic penchant and wrote frequently about it, proposing geometrical and numerical interpretations. Next, for nearly a thousand years Plato's texts disappeared an it is only in the Renaissance that the enigma briefly resurfaced. During the 19th century, when classical scholars restored original texts, the problem reappeared. Schleiermacher interrupted for a decade his edition of Plato while attempting to make sense of the paragraph. Victor CousinVictor Cousin
Victor Cousin was a French philosopher. He was a proponent of Scottish Common Sense Realism and had an important influence on French educational policy.-Early life:...
inserted a note that it has to be skipped in his French translation of Plato's works. In the early 20th c. scholarly findings suggested a Babylon
Babylon
Babylon was an Akkadian city-state of ancient Mesopotamia, the remains of which are found in present-day Al Hillah, Babil Province, Iraq, about 85 kilometers south of Baghdad...
ian origin for the topic.
Most interpretators argue that the value of Plato's number is 216
because it is the cube of 6 i.e.
which is remarkable for being also a sum of the cubes for the Pythagorean triplePythagorean triple
A Pythagorean triple consists of three positive integers a, b, and c, such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are pairwise coprime...
3,4,and 5:
Such considerations tend to ignore the second part of the text where some others numbers and their relations are described. The opinions tend to converge about their values being 480000 and 270000 but there is little agreement about the details. It has been noted that 6 raised to fourth power yields 1296 and
. Instead of multiplication some interpretations consider the sum of these factors: 48+27=75.Other values that have been proposed include:
- 17 500= 100x100 + 4800 + 2700 by Otto Weber(1862)
- 760 000= 750000 +10000= 19x4x10000, 19 being obtained from (4/3+5)x3 and it is the number of years in the Metonic cycleMetonic cycleIn astronomy and calendar studies, the Metonic cycle or Enneadecaeteris is a period of very close to 19 years which is remarkable for being very nearly a common multiple of the solar year and the synodic month...
, a perfect numberPerfect numberIn number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...
proposed by Cardano. It is known that such numbers can be decomposed in the sum of consecutive odd cubes, so
by Marsilio Ficino (1496)- 50405040 (number)5040 is a factorial , a highly composite number, a superior highly composite number, a colossally abundant number, and the number of permutations of 4 items out of 10 choices .-Philosophy:...
=
by Jacob Friedrich Fries (1823)
Further Reading
- Donaldson J., "On Plato's Number", Proceedings of the Philological Society, vol.1, iss. 8, p.81-90, April 7, 1843
- Adam J.James Adam (classicist)James Adam was a Scottish classicist who taught Classics at Emmanuel College, Cambridge.He was educated at the University of Aberdeen and Caius College, Cambridge, where he gained his BA as Senior Classic in 1884 and became a fellow of Emmanuel College in 1888...
, The nuptial number of Plato: its solution and significance, London: C.J. Clay and Sons, 1891. - Laird, A.G., Plato's Geometrical Number and the Comment of Proclus, The Collegiate Press, George Banta Publishing Company, Menasha, Wisconsin. 1918
- Diès A., Le Nombre de Platon: Essai d'exégèse et d'Histoire, Paris 1936
- Allen M., Nuptial Arithmetic: Marsilio Ficino's Commentary on the Fatal Number in Book VIII of Plato's Republic UCLA 1994
- Dumbrill R., Four Mathematical Texts from the Temple Library of Nippur: a source for Plato's number http://sas.academia.edu/RichardDumbrill/Papers/182981/Four_Mathematical_Texts_from_the_Temple_Library_of_Nippur