(This is a summary of my longer article, although a few things are added. It was condensed for classroom use and discussion, since it is quick to download you may want to use it for that purpose.)

NOTE:If interested in the Golden Ratio, see that as a separate article in this section.

In the first half of the fourth C. B.C mathematical thought was considerably more developed than philosophical logic, and that Plato, especially in the earlier half of his output, used mathematical models for his developing thinking in ways that have not been appreciated or identified.

1) The elenchus, or "refutation' is taken in the long history of Greek thinking to be a device of arguments intended to clean out the mind and establish, a la Socrates, what we do not know. But an argument in an early Platonic dialog runs like this like this;

Virtue is courage...... and yet....virtue is prudence, but since prudence and courage are not the same, in fact essentially different at times. Therefore we eradicate the whole argument, and will return to the topic another day.

This is indeed curious reasoning, without special guidance I guess that many people in any culture will reject one of the arguments as wrong, or less important, and continue with the other. This rejection of the whole cloths seems to derive from Euclid Book I Axiom 1: Things equal to the same thing are equal to each other, and by inference, things cannot be equal to two things which are NOT equal to each other. This explains by a seed of thought clearly identifiable in Plato's mathematical world. Over his school was said to be the words LET NO ONE WHO DOES NOT KNOW GEOMETRY ENTER HERE. There was an important process of logic drawn from mathematics thought in Plato's time.

2) In the Meno Socrates tries to elicit from Meno's slave anamnesis or prior knowledge of mathematics ideas in the study of an expanding square. Doubling the length of a side apparently squares the area, as the boy quickly sees. Socrates does the example on a kind of graph paper, which permits the boy to count squares and come up with the right answer. But he avoids another way of presenting the problem, which may well be as old as Thales: By constructing a new and larger square on the hypotenuse, which has a length of square root of 2 when the side is one, the new square has an area of 2, i.e. square root of two squared. Socrates deliberately avoids this way of presenting his problem, because the he could not grasp both hypotenuse, square root, and square root squared as a rational number. Again we see involvement with the mathematics of the time, however simple.

3) In the Meno a bit later the famous (unanswerable) question of the triangle put into a circle arises. Many a critic has called the passage trivial, unimportant but it is perfectly clear beside Euclid XX IV 2, where a triangle of given angularity is in fact put into a circle. But in the Meno the area (chorion) is mentioned, and the description of the problem indicates that the last angle drawn will fall short or exceed the perimeter of the circle. Plato knows that an area can only by tried in a circle by approximations, and this example indicates that the philosopher is ready to consider an approximative method as suitable for his future needs. Euclid was much clearer, throughout Book IV he inscribes every polygon he can think of in a circle, but never dares consider an area Euclid does not deal in approximations.

4) Again in the Republic the figure of the DIVIDED LINE has attracted much comment, often with the note that it is a mythic symbol. No one who has looked at Euclid's added thirteenth book, Prop. I, can fail to see that Plato is working at a figure of normal mathematical proportion, in fact the examples are absolutely parallel. The ancient scoliast to Euclid traces these five oddly located Propositions back to Eudoxus, a figure from Plato's world, nearly a century older than Euclid himself. It seems clear that Plato is speaking of proportion as such, of a sort in which a : b :: b : c, in other words a continuing series, which will vanish into infinite smallness quickly, and into infinite bigness on the other hand with equal speed. In the direction of infinite smallness Plato sees nothing, in the area of bigness he sees something like Truth and Godlike ideas, a common human spiritual response to the unlimited (aporon).

But Plato knew other things, such as the Golden Section, which apparently the Greeks derived from describing a 1 : 3 rectangle, subtracting from or adding to the hypotenuse (square root of 5) the length of the shorter side, bisecting the remainder, and swinging it back onto a side to give the ratios 1.618 : 1 .Plato historically knew this much.

But there was more he did not know: that these numbers divided by 1 give each other (1.618 : 1 : 1 :. 618) plus/minus 1. That from the ratio 1: 1.618, one can proceed by a Fibonacci Series (adding to each preceding number), to elicit a series in which after l..618,the squares equal the number following, and next (skip one) following, and next (skip two) following, and so forth.

That these numbers (starting higher up for convenience) when square rooted, and divided by the factor give the number lower in the series (skip one regularly), but only in alternative "generations"

(This last observation is not totally explicable; it is not found in the longer Plato article referred to above.)

The ratio 1: 618 has a curious way of turning up again and again in different areas --- in biological growth, most apparently in conch formations, even in clamshell growth, and in dendritic crack formation in brittle metallic alloys, in branching of trees. Le Corbusier maintained that foot to navel of an average man when compared to navel to height, approximated this ratio. And of course the use of the Golden Ratio in Greek architecture is well known.

However we have never seen how the Greeks with a primitive number system and rudimentary measuring equipment, could have developed the use of the GR so effectively in a large building like the Parthenon. I have discovered an effective way of extending GR measurements well within the capability of the ancient Greeks, which is the subject of another article in this page, which you will find titled as "Golden Ratio".

And that is has human esthetic appeal, witness the lengths of sequential passages in the Aeneid of Virgil which Duckworth discovered years ago,, storeys in Post Renaissance architecture, pleasantness of rectangles in a Pratt Institute art survey, and development of tones and rhythms in the musical compositions of Bartok.

That Plato identified this ratio as valuable is surprising, but no more surprising than the intuition of Democritus about atoms, or Epicurus about Brownian motion, conservation in chemistry, molecular bonds and Lucretius' description of evolution.

However modern scholarship has declared that Plato is philosophy and Euclid is science....a distinction that the Greeks would never have understood. Some have maintained that Plato did not really mean what his intuition says, or that Plato's brilliant mind finally ran out in his later writings as he discussed things which are trivial because intuitive. Or even that he could not have sensed something as complex as the mathematical matters which I have outlined here

Scholarship has many dark ages, and they do not all fall in the safe confines of remote antiquity.

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William Harris
Prof. Em. Middlebury College