Plato's number is a number enigmatically referred to by

Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institutio ...

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 either about the meaning or the value of the number. It also has been called the "geometrical number" or the "nuptial number" (the "number of the bride"). The passage in which Plato introduced the number has been discussed ever since it was written, with no consensus in the debate. As for the number's actual value,

216 __NOTOC__ Year 216 ( CCXVI) was a leap year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Sabinus and Anullinus (or, less frequently, year 969 ''Ab u ...

is the most frequently proposed value for it, but 3,600 or 12,960,000 are also commonly considered. An incomplete listfor more names and references see Dupuis J., ''Le Nombre Geometrique de Platon'', Paris: Hachette, 1885 of authors who mention or discourse about includes the names of

Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical Greece, Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatet ...

Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophe ...

for antiquity;

Ficino Marsilio Ficino (; Latin name: ; 19 October 1433 – 1 October 1499) was an Italian scholar and Catholic priest who was one of the most influential humanist philosophers of the early Italian Renaissance. He was an astrologer, a rev ...

and Cardano during the Renaissance;

Zeller Zeller, meaning both prisoner and monk in German, may refer to: Places *Zeller Ache, a river of Upper Austria * Zeller Bach (Isar), a river of Bavaria, Germany, tributary of the Isar * Zeller Bach (Memminger Ach), a river of Bavaria, Germany, tribu ...

Friedrich Schleiermacher Friedrich Daniel Ernst Schleiermacher (; 21 November 1768 – 12 February 1834) was a German Reformed theologian, philosopher, and biblical scholar known for his attempt to reconcile the criticisms of the Enlightenment with traditional ...

Paul Tannery Paul Tannery (20 December 1843 – 27 November 1904) 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. Tho ...

and Friedrich Hultsch in the 19th century and further new names are currently added.McNamee K., and Jacovides M., '' Annotations to the Speech of the Muses (Plato "Republic" 546B-C)'', Zeitschrift für Papyrologie und Epigraphik, Bd. 144, (2003), pp. 31-50 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 46cwith 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 mentioned in his ''

Timaeus Timaeus (or Timaios) is a Greek name. It may refer to: * ''Timaeus'' (dialogue), a Socratic dialogue by Plato *Timaeus of Locri, 5th-century BC Pythagorean philosopher, appearing in Plato's dialogue *Timaeus (historian) (c. 345 BC-c. 250 BC), Greek ...

'' (39d). The words are presented as uttered by the

muses In ancient Greek religion and mythology, the Muses ( grc, Μοῦσαι, Moûsai, el, Μούσες, Múses) are the inspirational goddesses of literature, science, and the arts. They were considered the source of the knowledge embodied in the p ...

, so the whole passage is sometimes called the 'speech of the muses' or something similar. Indeed,

Philip Melanchthon Philip Melanchthon. (born Philipp Schwartzerdt; 16 February 1497 – 19 April 1560) was a German Lutheran Protestant Reformers, reformer, collaborator with Martin Luther, the first systematic theologian of the Protestant Reformation, intellect ...

compared it to the proverbial obscurity of the

Sibyl The sibyls (, singular ) were prophetesses or oracles in Ancient Greece. The sibyls prophesied at holy sites. A sibyl at Delphi has been dated to as early as the eleventh century BC by PausaniasPausanias 10.12.1 when he described local tradi ...

Cicero Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Roman statesman, lawyer, scholar, philosopher, and academic skeptic, who tried to uphold optimate principles during the political crises that led to the est ...

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 millennium later, however, it was an enigma for the

Neoplatonist Neoplatonism is a strand of Platonic philosophy that emerged in the 3rd century AD against the background of Hellenistic philosophy and religion. The term does not encapsulate a set of ideas as much as a chain of thinkers. But there are some i ...

s, 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 and it is only in the

Renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass id ...

that the enigma briefly resurfaced. During the 19th century, when classical scholars restored original texts, the problem reappeared. Schleiermacher interrupted his edition of Plato for a decade while attempting to make sense of the paragraph.

Victor Cousin Victor Cousin (; 28 November 179214 January 1867) was a French philosopher. He was the founder of " eclecticism", a briefly influential school of French philosophy that combined elements of German idealism and Scottish Common Sense Realism. A ...

inserted a note that it has to be skipped in his French translation of Plato's works. In the early 20th century, scholarly findings suggested a Babylonian origin for the topic. Most interpreters argue that the value of Plato's number is 216 because it is the cube of 6, i.e. , which is remarkable for also being the sum of the cubes for the

Pythagorean triple A Pythagorean triple consists of three positive integers , , and , 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 . A primitive Pythagorean triple is ...

(3, 4, 5): . Such considerations tend to ignore the second part of the text where some other numbers and their relations are described. The opinions tend to converge about their values being 480,000 and 270,000 but there is little agreement about the details. It has been noted that 6 yields 1296 and . Instead of multiplication some interpretations consider the sum of these factors: . Other values that have been proposed include: *, by Otto Weber (1862). *, 19 being obtained from and being the number of years in the

Metonic cycle The Metonic cycle or enneadecaeteris (from grc, ἐννεακαιδεκαετηρίς, from ἐννεακαίδεκα, "nineteen") is a period of almost exactly 19 years after which the lunar phases recur at the same time of the year. The rec ...

. *, a

perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. T ...

proposed by Cardano. It is known that such numbers can be decomposed into the sum of consecutive odd cubes, so . *, by Marsilio Ficino (1496).Allen M., ''Nuptial Arithmetic:Marsilio Ficino's Commentary on the Fatal Number in Book VIII of Plato's Republic'', UCLA 1994, p75ff. *, by

Jacob Friedrich Fries Jakob Friedrich Fries (; 23 August 1773 – 10 August 1843) was a German post-KantianTerry Pinkard, ''German Philosophy 1760-1860: The Legacy of Idealism'', Cambridge University Press, 2002, pp. 199–212. philosopher and mathematician. Biogra ...

(1823).

References

External links

Five translations of Rep. 8.546 and 9.587
* {{MathWorld, title=Plato's Numbers, urlname=PlatosNumbers

* ttp://mathworld.wolfram.com/DiophantineEquation3rdPowers.html Math world : Diophantine Equation--3rd Powers
Sum of Consecutive Cubes Equals a Cube
Integers Dialogues of Plato Greek mathematics

Plato's text

Interpretations

See also

References

Further reading

External links