Plato's number, a mathematical word puzzle

Just a little brain-twister for those of you both bored and brave enough

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 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. "

Plato

I've worked on this for years, personally, but I am always interested in hearing what other people come up with

Ekleipsis said:

And this entire geometrical number is determinative of this thing, of better and inferior births. "

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This is the key. Plato believes that certain people are better than others. If you wish to understand his dialogue you must research who he believes to be superior and inferior. It is part of his work The Republic, so he tells you in that work who he feels is superior stock. I have only read brief pieces of it, so I cannot tell you. Find out what class he determines is superior and use it to determine what number he is referring to. Good luck!

Interesting take on it, thanks

So then you think his perfect number refers to a perfect number how it's classically defined in mathematics, or is it the geometrical number ?

6 is both triangular and perfect

First platonists were definitely not pythagoreans. Second it's possible to write something, as something that you oppose, be read as support of that which you oppose, and it will only be understood when read aloud by the author, that what was written was not support but caustic mocking all due to vocal prosody. So one can read this aloud as Ah ha pythagoreans, or one can read it aloud as Ah ha parody and mocking. Once again one has to look at socrates a bit closer. So crats ends up being a very very difficult character. Remember 800 years of platonists writing has a wide spectrum of views which moves it closer to metaphysics than philosophy as we understand philosophy.

"Now for divine begettings there is a period comprehended by a perfect number,
.....blah....blah...blah.... And this entire geometrical number is determinative of this thing, of better and inferior births."

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On second pass I think Plato may be talking about astrological birth charts, and all this might be authoritative gobbledygook to backup his claim that some people are superior to others. In Republic he loves to claim there are kinds of people who fit into classes in society. He likes classes.

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,

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He is getting superstitious here about some sort of universal formula that he suspects exists. 3 x 4 is twelve, which is the number of months in a year. 3 and 4 are the sides of a perfect triangle with sides of length 3,4,5. This triangle is probably what is meant by the term pempad (according to various people).

the waxing and the waning, render all things conversable and commensurable [546c] with one another,

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Waxing and waning probably refer either to moon phases or Venetian phases which repeat due to a harmonic relationship with Earth's orbit. He could be referring to both, but moon phases do seem to affect births and personality. They also affect wine production. It would be natural for him to suspect moon phases had to do with better births.

whereof a basal four-thirds wedded to the pempad yields two harmonies at the third augmentation,

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Some ancient Greeks discovered and studied the harmonics of the tetrachord, which today we consider to be half of a major music scale. There is such a thing as a perfect note, and there is such a thing as a note that is out of tune. Two tetrachords together make up 1 major scale or an octave, notes A through F. Some Greeks noted that if you plucked a particular note there were a series of natural overtones that came as a result and this determined which following notes were most pleasing to the ear. Today we refer to scales. If you play the major scale you are following natural overtones. If you play minor or jazz scales then you are choosing some dissonance. What he's doing here is comparing harmonics to some bigger formula that he associates with 4's and 3's or better births.

the one the product of equal factors taken one hundred times,

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This sounds like a 100th root -- some factor you multiply with itself 100 times like 3 to the 100th power is 3^100. What number has a hundredth root that is a simple fraction and multiplies by itself one hundred times to a ratio that Plato finds interesting? You could plug numbers into your calculator all day and not guess what number Plato is thinking.

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,

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Not sure what he means by rational diameters of the pempad minus 1 if by pempad he means a right triangle. We are guessing at what a pempad is. A rational number is a number that can be represented as a fraction, and a diameter is the cross section of something.

or of the irrational lacking two; the other dimension of a hundred cubes of the triad.

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You can take the 3,4,5 triangle with this property: 3^2 + 4^2 = 5^2. You can take it and make more perfect triangles by scaling it up: (n * 2) ^2 + (n * 3)^2 = (n * 5)^2. If you use 100 as the 'n', you get a triangle sides 300,400,500, and its still a right triangle. In a way its like a magic trick.

And this entire geometrical number is determinative of this thing, of better and inferior births."

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So a divine birth occurs as a result of certain timing which Plato is able to determine but which we are not able to comprehend the explanation of. Plato who is claimed to be Alexander the Great's teacher has some political motivations for making such a claim and for making it so hard to follow his reasoning.