An Argument in Favor of Something Like "Platonic" Influence in the Pentateuch

[Secret Alias]

I am a strange person. One of my quirks is the number 345. I know Moses = haShem/Shema and while I don't "believe" in any of these things ancient Jews and Samaritans did. The Samaritans allowed for Marqe to be the second Moses because of 345 (Marqe = 345). So too the number 666 in Christianity goes back to 6 x 6 x 6 the great Platonic number. 216 = 3(3) x 4(3) x 5(3) which all goes back to the same understanding as Moses as 345. What exactly 216 is, I have several theories. But it is not incredible to suggest that Judaism, Samaritanism and Platonism derive from a similar obsession regarding the number 345, the Pythagorean triangle.

[neilgodfrey]

What do gematria and isomorphism have to do with Plato?

[Secret Alias]

The names Pythagoras and Plato are associated with methods of generating Pythagorean triples due to their mention in Proclus's commentary on Euclid. However, as far as I have been able to determine, there is no mention of such a method in the extant works of Plato, and, as has been mentioned in the comments, attribution of any mathematical result to Pythagoras or to the early members of his school is problematic.

In his commentary on the proof of the Pythagorean theorem in Book I of Elements, Proclus describes two methods for generating Pythagorean triples, stating that one is attributed to Pythagoras, the other to Plato. Using an anachronistic algebraic formulation, the "method of Pythagoras" yields the triple
(n,n2−12,n2+12)
for any odd integer n. Examples are (3,4,5), (5,12,13), (7,24,25), (9,40,41), and (11,60,61). As two elements of each triple differ by 1, these are clearly primitive.

The "method of Plato" yields
(n,(n2)2−1,(n2)2+1)
for any even integer n. This is equivalent to the formula you ask about. Examples are (4,3,5), (6,8,10), (8,15,17), (10,24,26), and (12,35,37). If n is twice an odd number, the method yields the same triples as the "method of Pythagoras", but doubled. If n is twice an even integer, the triples are primitive as two of their elements are consecutive odd numbers.

The passage in the question starts on page 340 (location 428.6) in Glenn R. Morrow's translation of A Commentary of the First Book of Euclid's Elements. Here it is:

Certain methods have been handed down for finding such triangles, one of them attributed to Plato, the other to Pythagoras. The method of Pythagoras begins with odd numbers, positing a given odd number as being the lesser of the two sides containing the angle, taking its square, subtracting one from it, and positing half of the remainder as the greater of the sides about the right angle; then adding one to this, it gets the remaining side, the one subtending the angle. For example, it takes three, squares it, subtracts one from nine, takes the half of eight, namely, four, then adds one to this and gets five; and thus is found the right-angled triangle with sides of three, four, and five, The Platonic method proceeds from even numbers. It takes a given even number as one of the sides about the right angle, divides it into two and squares the half, then by adding one to the square gets the subtending side, and by subtracting one from the square gets the other side about the right angle. For example, it takes four, halves it and squares the half, namely, two, getting four; then subtracting one it gets three and adding one gets five, and thus it has constructed the same triangle that was reached by the other method. For the square of this number is equal to the square of three and the square of four taken together.

Morrow refers the reader to pages 356–360 of Volume I of Heath's translation of Elements, where Heath presents some speculations on how such methods might have been discovered. Starting on page 360 Heath also discusses at length the early knowledge of the Pythagorean theorem and Pythagorean triples in India that is exhibited in the Śulvasūtras.

Interestingly, in the paragraph preceding the one quoted above, Proclus discusses isosceles and scalene right triangles, echoing Plato's classification in Timaeus. But among scalene right triangles, Plato singles out the 30∘-60∘-90∘ triangle, for which, as Proclus notes in the case of the isosceles right triangle, "you cannot find numbers that fit the sides"—so nothing there about Pythagorean triples. Later in the paragraph, Proclus says that there are scalene triangles for which "it is possible to find such numbers", giving the example of "the triangle in the Republic, in which sides of three and four contain the right angle and five subtends it". In a footnote Morrow says the allusion is likely to Rep. 546c. In my reading that passage engages in a complicated numerology involving the numbers 3, 4, and 5, but does not say anything about right triangles. If somewhere in Plato there is a mention Pythagorean triples, that would be interesting, but I have not heard of such a passage.

[Secret Alias]

https://en.wikipedia.org/wiki/Plato%27s ... 20%3D%2063.

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 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 is the most frequently proposed value for it, but 3,600 or 12,960,000 are also commonly considered.

An incomplete list[1] 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 century and further new names are currently added.[2]

Further in the Republic (9.587b) another number is mentioned, known as the "Number of the Tyrant".


Contents
1 Plato's text
2 Interpretations
3 See also
4 References
5 Further reading
6 External links
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.[3]

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.[2][4] Indeed, Philip Melanchthon compared it to the proverbial obscurity of the Sibyls.[5] Cicero famously described it as 'obscure' but others have seen some playfulness in its tone.[1]

Interpretations

An illustrated interpretation of Plato's number as 3³ + 4³ + 5³ = 6³
Shortly after Plato's time his meaning apparently did not cause puzzlement as Aristotle's casual remark attests.[6] Half a millennium 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 and 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 his edition of Plato for a decade while attempting to make sense of the paragraph. Victor Cousin 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.[7]

Most interpreters argue that the value of Plato's number is 216 because it is the cube of 6, i.e. 63 = 216, which is remarkable for also being the sum of the cubes for the Pythagorean triple (3, 4, 5): 33 + 43 + 53 = 63.

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 64 yields 1296 and 48 × 27 = 36 × 36 = 1296. Instead of multiplication some interpretations consider the sum of these factors: 48 + 27 = 75.

[neilgodfrey]

It is debatable whether the Pentateuch was influenced in its composition by gematria and isomorphism. It is easier to understand gematria and isomorphism being applied by some interpreters to the Pentateuch, and perhaps to some later revisions of it. But it's more a kabbala thing, is it not?

[Secret Alias]

Really? How much thought have you given this? Like about the time it takes you to shave.

[neilgodfrey]

Really? How much thought have you given this? Like about the time it takes you to shave.

I really don't mind if you respond meaningfully and even civilly to my comment and query. What did I say wrong?

[Leucius Charinus]

But it is not incredible to suggest that Judaism, Samaritanism and Platonism derive from a similar obsession regarding the number 345, the Pythagorean triangle.

This type of geometrical /mathematical knowledge is apparently way older than Pythagoras or Plato. It seems to have been critical to the development of building skills.

Written between 2000 and 1786 BC, the Egyptian Middle Kingdom Berlin Papyrus 6619 includes a problem whose solution is the Pythagorean triple 6:8:10, but the problem does not mention a triangle. The Mesopotamian tablet Plimpton 322, written between 1790 and 1750 BC during the reign of King Hammurabi the Great, contains many entries closely related to Pythagorean triples.

In India, the Baudhayana Shulba Sutra, the dates of which are given variously as between the 8th and 5th century BC,[72] contains a list of Pythagorean triples and a statement of the Pythagorean theorem, both in the special case of the isosceles right triangle and in the general case, as does the Apastamba Shulba Sutra (c. 600 BC).[a]

https://en.wikipedia.org/wiki/Pythagore ... em#History

[Secret Alias]

I really don't mind if you respond meaningfully and even civilly to my comment and query. What did I say wrong?

I thought of a number of ways to answer this question. Let's take this approach. The Bible (the Hexateuch) is written in Hebrew. How do you think 'Western science' came to an understanding of the Hebrew text of the Bible? Do you think it was a process akin to the Rosetta Stone where a researcher had a great insight and 'broke a code'? Of course not. They consulted with the Jews. The Jewish religious 'experts' explained Hebrew to Western researchers. Hebrew is first and foremost a religious language. The earliest Hebrew texts are either the Bible or about the Bible. On some level we took 'their' understanding, their exegesis of the Bible. Part of that exegesis (whether it was told to the researchers or not) was 'tradition' (קבלה). It seems foolish to me to say 'we accept the literal meaning of the Hebrew words as passed along by Jews' but not their tradition. I don't see where one begins and one truly ends. It's an inseparable tightly knotted bundle of yarn.

[Secret Alias]

And keeping with the original OP, "Moses" is not the name of the historical figure believed to be represented by the authors of the Pentateuch. If they used Egyptian records, his name is something like Osarseph, Thutmose or some other. I suspect they chose the specific Hebrew name משה because it represented some occult principle like the 3 4 5 triangle or the Platonic number 216. They were interested in some shared mysticism with Plato and other ancient sources. I don't say this because "I believe" in these principles or kabbalah. I just find the number 345 is a reasonable way to proceed.