PRELIMINARY REMARKS
The present approach to Plato's dialogues does not attempt to suggest
what Plato thought, meant, or implied. Instead, the material is
presented
here in the form of mathematical problems for the readers to solve for
themselves, should they wish to, for in the final analysis it is what
the
reader understands that is important, not what is proposed by others,
however
erudite they might be. What is required is a modicum of mathematical
skill,
fundamental scientific concepts and an open mind. This is not a "Royal
Road" to the "Philosopher's Stone," it is mathematics pure and simple
to
provide a useful and necessary starting point. The end will ultimately
remain with the reader and the reckoner, for this matter increases in
both
scope and complexity as it develops.
INTRODUCTION
Plato's Republic is an enduring and much admired work, but the
two
mathematical problems presented here have nonetheless confuted
hundreds,
if not thousands of inquiring minds since Plato's time [ 427 to 347
BC].
So, be forewarned, this is not a simple matter. Nor is it a matter of
simple
arithmetic, either. Clues abound everywhere, but watch for phantoms and
misdirections nevertheless. You may find that you have wandered down a
lightly trodden pathway here; one that cuts across both space and time.
Few know where or when it began, and fewer still know where it will
end.
Galileo and Kepler passed this way in their time too, although neither
were able to go the full distance. This is hardly surprising; among
other
things Plato states in Laws VII (818-819) [translation by
A.E.Taylor]
6
that: "ciphering and
arithmetic make one subject." But he also says
much of relevance in the Epinomis
(991-992)
too, including the observation that:6
To the man who pursues his studies in the proper way, all geometric constructions, all systems of numbers, all duly constituted melodic progressions, the single ordered scheme of all celestial revolutions, should disclose themselves, and disclose themselves they will, if, as I say, a man pursues his studies aright with his mind's eye fixed on their single end. As such a man reflects, he will receive the revelation of a single bond of natural interconnection between all these problems.Those already familiar with this material will no doubt demand to know who says the Tyrant's Number is 9 in the first place, who says the two problems are related in the second, and who is empowered to make such claims in the third. The answer to the first two questions is simple enough - I do. The Tyrant's Number might be considered to be 3, or even 27, but for present purposes I for one prefer 9 (Phaedrus 248c-249c); if you go the full distance you may make your own choice. The third question cannot be explained in a few sentences; the difficulty lies in prevailing attitudes towards the contents of the Dialogues themselves.
A word of caution: there is an unsuspected fork in this timeless pathway. One branch moves forward to new and dangerous territory, the other leads back to the comforting familiarity of the old. Perhaps you have begun to suspect that there may be more to this golden treasury of Dialogues; perhaps you have already dismissed that possibility; but either way, remember your choice if you attempt to connect these problems. Remember also that much space is given over to "names" in the Dialogues, e.g., Statesman (258c-259c), Laws, Theaetetus, The Republic, and especially the Cratylus. Moreover, be aware that all manner of devices tend to be applied in Plato's dialogues to inform the initiated and confute the uninitiated alike. At times this problem can be exacerbated by differences between translations; by all means consult other sources and alternate interpretations. But in doing so, recognize that this is not necessarily an either/or scenario, and that Plato presents these matters so skillfully and artfully that conventional meanings remain applicable.
Note:. The understanding of the Number of the Tyrant comes first; do not become overly concerned with harmony in the musical sense, at least initially; it is the third meaning which predominates here, not the second, or the first. Connect the two problems and find the second number, or better, all three. You need not explain anything else -- not the diagram, not the pempad, nor the basal four-thirds or any of the rest. But human nature being what it is, what is to stop you from trying? There is much work to be done with this massive compendium of knowledge. Moreover, moving forward a little distance in time, what was it that motivated the last of the great Platonists, Proclus [410 to 485 AD] to say:
"If I had it in my power, out of all the ancient books I would suffer to be current only the Oracles 4and the Timaeus.8 "What is so important about either one, and why are they so paired?
[546a]...Hard in truth it is for a state thus constituted to be shaken and disturbed; but since for everything that has come into being destruction is appointed, not even such a fabric as this will abide for all time, but it shall surely be dissolved, and this is the manner of its dissolution. Not only for plants that grow from the earth but also for animals that live upon it there is a cycle of bearing and barrenness for soul and body as often as the revolutions of their orbs come full circle, in brief courses for the short-lived and oppositely for the opposite; but the laws of prosperous birth or infertility for your race, [546b] the men you have bred to be your rulers will not for all their wisdom ascertain by reasoning combined with sensation, but they will escape them, and there will be a time when they will beget children out of season. 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.
"For a divine birth there is a divine period comprehended by a perfect number; for a human birth, but the first number in which root and square increases, comprising three distances and four limits, of elements that make like and unlike, and that wax and wane, render everything conversable and rational. Of these elements, the root four-three mated with five, thrice increased, produces two harmonies. One of them is of equal length in one way but is an oblong, one side, of one hundred rational diameters of five, lacking one for each, or if of irrational diameters, lacking two of each; one the other side, of one hundred cubes of three. This whole geometrical number is sovereign of better and worse begettings."
"There is for a divine creature a period which a perfect number contains; for a human creature (there is a number) in that figure in which first products that are squares and rectangles, equaling and being equaled, if arranged in a proportion with three intervals and four terms, the terms being the sides of the squares and the sides of the rectangles, both if they are increasing and if they are decreasing, showed all in proportion and rational to one another; of which 3-4-5 type, if the numbers are made solid, furnishes two harmonies, the one a square with its side multiplied by 100, the other equal in area to the former but oblong, one side of 100 squares of rational diameters of five, each lacking one, or irrational diameters each lacking two, the other side 100 cubes of 3. This total, a geometric number, is in control of such a creature, of better and of worse births."
[E] Translation by Thomas Taylor (pp.274-275) 15"For divine becomings, there is a period comprehended by a perfect number; but for human, by the first in which developing capacities, dominating and dominated, on realizing three stages determined by four points [in the field of these processes of] becoming like and unlike, growing and declining, make all things conversable with and rational in respect to one another. Of these [the element representing referents in] in a ratio of 4 to 3, in lowest terms, married to the pempad, produces two harmonies when thrice augmented. Each side of the former [equals] numbers produced by squaring "the diagonal lines" that represent a "rational" component of the pempad, each diminished by one; [the other side equals numbers produced by squaring the "diagonals" that represent a component of the pempad which is] "irrational," [each of these diminished by] two. The latter [harmony] equals one hundred cubes of three. And this whole thing is a geometrical number..."* Rouse adds in a footnote: "Literally, 'by third increase,' which to the Greeks meant changing a square into a cube..."
** According to Rouse, "The long Greek number and long Greek words used were no doubt part of the jest."
...It is indeed difficult for a city thus constituted to be changed. But as everything which is generated is subject to corruption, neither will such a constitution as this remain for ever, but be dissolved. And its dissolution is this. Not only with respect to terrestrial planets, but likewise in terrestrial animals, a fertility and a sterility of soul as well as of body takes place, when the revolutions of the heavenly bodies complete the periphery of their respective orbits; which are shorter to the short-lived, and contrariwise to such as are the contrary: and with reference to fertility and sterility of our race, although those are wise that you have educated to be governors of cities, yet will they never, by reason in conjunction with sense, observe the proper seasons, but overlook them, and sometimes generate children when they ought not. But the period to that which is divinely generated is that which the perfect number comprehends; and to that which is generated by man, that in which augmentations surpassing and surpassed, when they shall have received these restitutions and four boundaries of things assimilating and dissimilating, increasing and decreasing, shall render all things correspondent and effable; of which the sequitertian progeny, when conjoined with the pentad, and thrice increased, affords two harmonies. One of these, the equally equal, a hundred times a hundred; but the other, of equal length indeed, but more oblong, is of a hundred numbers from effable diameters of pentads, each being deficient by unity, and from two numbers that are ineffable; and from a hundred cubes of the triad. But the whole geometric number of this kind is author of better and worse generations
Some final points.
Firstly, although the original
Greek versions should supply the best information, we deal here with
various
English translations. Secondly, as commentators have noted from time
immemorial,
there are playful elements in the Dialogues; but even so, the matters
under
consideration are not frivolous. Thirdly, the additional translations
are
provided to demonstrate the variance that exists, and also to supply
further
insights and potential triggers. Note that solutions to these enigmas
have
been proposed in the past (see Brumbaugh, pp.143-146), but to date none
have found universal acceptance.
The fundamental problem, as Jowett (p.113) points out, is that: "
the obscurity arises from our want of familiarity with the subject."
Finally, do not become easily discouraged; the last part is especially
difficult; but nevertheless, "Seek therein, and be not weary, the
result
justifies the labour."
Moreover, take strength and guidance from what Confucius has to say
about the Superior Man, namely:
" If another man succeed by one effort, he will use a hundred efforts. If another man succeed by ten efforts, he will use a thousand."THE NUMBER? All Three? All Four? Or Five? Or the One and the Many?
John_Harris @shaw.ca