Plato's Less-Than Ideal Arithmetic

Philosophy is nothing more than footnotes to this guy, so they say.

Plato rightly deserves his central place in the Western canon.  He founded a school--the Akademia--from which we get both the word and the inspiration for the modern academy.  His insistance on the immorality of the soul so suffused the Greek-speaking world--including the authors of the New Testament--that Nietzsche dismissed Christianity as mere 'Platonism for the masses.'  Plato wrote over thirty dialogues that survive as masterpieces of argument and storytelling--a feat made all the more striking by the fact that back when Plato lived there were no paper mills, no printers, no bookstores, and no pens.  Plato pretty much set the aims, the methods, and the questions of philosophy for the next two thousand years.

But despite his heavyweight resume, Plato seems to have flubbed his math a bit.

Here's Plato calculating the exact amount that the philosopher's life is better than the tyrant's, from Book nine of the Republic.

Or if some person measures the interval by which the king is parted from the tyrant in truth of pleasure, he will find him, when the multiplication is complete, living 729 times more pleasantly, and the tyrant more painfully by this same interval.

What a wonderful calculation! And how enormous is the distance which separates the just from the unjust in regard to pleasure and pain!

Yet a true calculation, and a number which nearly concerns human life, if human beings are concerned with days and nights and months and years.

Plato's math, according to the footnotes in the second edition of the Grube translation of the Republic are "hard to follow."  Here's a try.  The tyrant experiences only two-dimensional pleasures, while the philosopher experiences three dimensional pleasures.  Additionally, the philosopher is nine times away from the tyrant in terms of pleasure, so the philosopher's pleasure is represented by a nine-unit cube, while the tyrant's pleasure is represented by a one-unit square.  But Plato flubbed things getting to the number 729, which was sacred to the Pythagoreans.  He miscounted the number of times removed the tyrant was from the philosopher (it should have been five, not six) and multiplied where he should have merely added.  Sadly, it turns out that the philosopher is only 125 times happier than the tyrant!

But we can't blame Plato for having trouble with his sums.  In Plato's time, before zero, before calculators, before arithmetic notation, math was decidedly hard to do.  Here's another example of Plato doing math, from the Republic, Book 8:

Now that which is of divine birth has a period which is contained in a perfect number, but the period of human birth is comprehended in a number in which first increments by involution and evolution, obtaining three intervals and four terms of like and unlike, waxing and waning numbers, make all the terms commensurable and agreeable to one another. The base of these with a third added when combined with five and raised to the third power furnishes two harmonies; the first a square which is a hundred times as great, and the other a figure having one side equal to the former, but oblong, consisting of a hundred numbers squared upon rational diameters of a square (i. e. omitting fractions), the side of which is five, each of them being less by one or less by two perfect squares of irrational diameters ; and a hundred cubes of three. Now this number represents a geometrical figure which has control over the good and evil of births.

What Plato's trying to say--again according the Grube Edition's footnotes--is that the human number is the product of three, four and five raised to the power of four, or (3*4*5)^4, which comes to 12,960,000.  This can be shown geometrically in two ways.  First, by the area of a square with the sides of 3600 or as a rectangle with sides 4800 and 2700.  Simple enough for us moderns.  But we have the ease of working with arabic numerals.  You can see how Plato--even Plato!--can be forgiven for messing up his math.

And you thought math was hard in high school!  Sacrifice a cock to Asclepius in thanks that you were never a math student in ancient Athens.