Mathematician or Mystic ?
Plato believed at first that Mathematics would be the key to Thought, but in the Meno he abandons hope in the context of a few sentences, which we have constantly misread. This paper examine that crux.
No reader of Plato can fail to recognize the important role which mathematics plays in his writing, as would indeed be expected for an author about whom the ancient tradition maintains that he had hung over the entry to his school the words "Let No One Un-versed in Geometry Enter". Presumably it was the level of ability to work with abstract concepts that Plato was interested in primarily, but if the student really had never studied Greek geometric materials there would be many passages in the lectures which would be scarcely intelligible to him. Modern readers, versed in a much higher level of mathematical abstraction which our society can offer, have sometimes felt that Plato's famous "mathematical examples'" were illustrations rather than central to his arguments, and some of Plato's mathematical excursuses have remained obscure to the present time.
When Euclid put together his masterful compendium in the 3 rd C. B.C. he was obviously drawing on a legacy of accumulated mathematical research which went back well into the 5th C. B.C. and possibly even further. Activity can be pushed back well into the 5th C. B.C. in view of the quite reasonable ancient assignment of the Pythagorean Theorem of Euclid I 47 to the master himself; this would naturally involve understanding of squares and irrational numbers. And i4 should be noted that well back into that century Iktinos, the architect of the Parthenon was not only considering but actually using in his work the square root of the diagonal of a 1:2 rectangle in the form which later became famous as the Golden Mean, more specifically in the ratio l.6l8 : l
It is hard for us to understand how inextricably science and philosophy were involved with each other in the Hellenic world, probably because of our culture's predisposition to place these, and other disciplines, in individual and quite separate categories. Paying lip service to the ancient fusion of these disciplines, we still fail to make many important connections, and hence miss part of the meaning of thinkers and scientists alike. The purpose of this paper is to trace the development of Plato's "logic" in the early and middle periods, which I believe is patterned extensively on the mathematical knowledge of that period as we can envisage it. In his later work he abandons mathematical aids and embarks on his own quest with the Theory or Ideas, which has the kind of abstract mathematical flavor which one finds of a page of Euclid after erasing the diagram and describing the form rather than the actual stages of the argument. It was presumably in his failure to connect this form with his interests in men, morality and politics that Plato was felt to have failed in his own lifetime, as Aristotle implies.
This paper proposes to examine four types of arguments which derive directly from mathematical thinking, all of which have become one way or another thorns in our grasp of Plato's meaning. The variety of explanations attests a certain lack of clarity; one passage has quite literally defied interpretation and is often set aside as "unimportant", which is surprising for an author as scrupulously clear in his writing as Plato.
I: The Elenchus.
The elenchus or philosophical refutation became in ancient times already so much a part of the philosopher's method that it is very difficult for us to conceive it as a fresh new experimental mode of thought. In the tradition, it serves in Socrates' employ as a device which confuses auditors and routs arguments, proceeding to aporia which is taken as a healthy breathing point on the road to self knowledge. As such it aligns well with Socrates eironic conviction that he knew nothing (and hence of course could proceed to know something, perhaps in final analysis quite a lot). This is the use and purpose of the elenchus in Plato, where we find it used over thirty times in all, and frequently in the earlier half of the work, most frequently in the early dialogs.
A typical early example involves the definition of wisdom, which is first equated with courage, and then with prudence. But since courage and prudence are not the same, in fact may be quite diametrically opposed at times, we reach a logical impossibility, which results in the abandonment of the investigation for the day and the end of the dialog. Socrates in the his role of refuter and challenger of ill framed concepts has become so familiar to us that we fail to note the strangeness of this kind of argument, which
is never used then,or now, in converse between persons, nor as an argumentative tool again. In effect it says:
A is equal to B and also A is not equal to C.
Therefore B is not equal to C
Hence the whole process misfires and must be scrapped. Were it a case of a fallacy in one of the statements we could hope to isolate that fallacy and correct it. But we never go back to the arguments, it somehow very convincingly registers zero or nul. is permanently disbarred. This is indeed an unusual mode of thought. If we look now, as we should have done before, to Euclid for a parallel which might possibly shed light on our problem, we find strikingly enough in the first Axiom of the First Book the statement:
'Things equal to the same thing are equal to each other"
Hence things not equal to the same thing are presumably not equal to each other, and furthermore one thing cannot be equal to things which are not equal to each other. So it is precisely in this sense that Socrates discovers the impossibility of equating A with B and again with C, when we know that B and C are not equals
This kind of argument is thus a borrowing from the developing logic of mathematical thought, which was new and exciting and probably not wholly understandable at the beginning of the 4th c. B C. I take it that Plato was intrigued by this new mode of thinking, used it extensively in the early dialogs and gradually displaced it, especially after the Meno, in favor of more substantial modes of argumentation. The location of elenchi in his works does point to this direction. In short, I maintain that the elenchus at its inception was a conscious transfer from the mode of mathematical argument to the mode of philosophical argument, that it was probably a stunning success, as the comparison of Socrates to the electric ray implies, but that it wore thin when heavily used. This is not the sole example of such a process in the Western World.
II: The doubling the area of a square.
This problem, which Socrates sets to the slave boy who is innocent of mathematical expertise in the Meno begins in a perfectly straightforward manner. The case is so familiar that the reader can probably keep the diagrams clear in his mind. Socrates first lays out a square, which he proceeds to divide up with two crosslines into four equal squares. Thus our unit has actually four squares, and when the boy suggests doubling the sides to double the area, Socrates points out that he has produced sixteen square, not the required eight. He then tries a square with fifty percent longer sides than the original square, and gets nine squares, which again is not satisfactory. Socrates proceeds to elicit a concept of another possible line, and the boy thinks of a diagonal, so we put diagonals in each of our original four squares with ends inconveniently touching the side or rim of the figure, and behold'. we have a new smaller square tilted on its nose, which turns out to be half the area of the first square since it has half of the isosceles right triangles which the big figures embraces.
But we wanted twice, not half, so now we turn to the rejected "big" square with its sixteen little squares, draw out the same procedure on it, and it gives an interior, tilted square of eight small squares, which is what we were after all along. Considering what we have just done, we might note that it seems unduly complex. The whole procedure, dividing the first square, Working with multiples, finding a reduction, and then applying this reduction to a provable oversize unit it is strange indeed. But on the other hand it is very neat in another way, it avoids the complexities of the Pythagorean Theorem as stated in Euclid I 47, which would involve a very different set of reckonings, as follows:
Dealing with a square with a given value of each side of one, we will soon realize that the value of the diagonal is the square root of two which we need not understand as 1.4142136 at this time. Now let us purely in the spirit of experiment construct another larger square and mark across it a diagonal with the value of two in contrast with the smaller square with its diagonal of square root of two Now by skills known to the Pythagoreans back in the 5th C. we will square that diagonal, halve it and find the square root of two as the proper side for our figure. We next calculate the area of the square by multiplying the sides (sq rt 2) and get an area or two which is what we were looking for. Hence our experiment was correct, and by constructing a larger square the diagonal of which
is the square of the value of the smaller diagonal, we do in fact double the area. And this produces a clear and diagram, which can be laid out by constructing a square, then a second superimposed square with its side the diagonal of the first square. (I am not showing this since this is a text-only file at the present time....)
This also introduces us to square, roots and irrational numbers and thereby paves the road to the future.
In this light is interesting to note that in Euclid's Book II, the first eight propositions involving rectangles are worked out without aid from the Pythagorean Theorem, I 47, whereas Proposition 9
and the remainder are proved with it. This is obviously an historical relic from the earlier developmental period, and Socrates example in the Meno stems from the tradition of an earlier phase, although a person with more skill would probably have had the ability to work it out as I have demonstrated even back in the previous century.
Although we have not changed the meaning of Socrates' demonstration concerning the square, we have gained insight into the actual procedure which he is using, noting that his is the earlier and more primitive solution. But again we can remark Plato's relationship to the growing mathematical lore of his time, and inversely we can use Plato's evidence, which has relatively fixed dating, to mark stages in the pre-Euclidean development which would otherwise be unknown to us.
III: The "Geometrical Problem" in the Meno.
Further along in the Meno occurs the celebrated case of the Geometrical Example at Meno 87, which in contrast to the previous mathematical illustration, has been twisted, tortured, and intentionally passed over for two centuries. Jebb said (loc.cit.) asven over a century ago:
The hypothesis appears to be rather trivial and to have no mathematical value. . . (which Raven echoes in 1965)
Bluck however, in 1961 devotes an excursus of some sixteen pages to a complete review of views on the problem, which include an array or barely intelligible geometrical details. The passage is made more difficult of interpretation by the fact that Socrates introduces the geometrical example in a very summary manner, which some have felt was an indication or its relative unimportance.
I believe on the contrary that the almost schematic reference implies that the topic and the example were well known to the Platonic audience, and did not need explanation. Plato knows how to explain in full, and when he refrains we must understand the matter to be common knowledge. The problem as it occurs at Meno 87 a is briefly this:
We will proceed from here on like the geometer who when asked if a given triangle can be inscribed in a given circle, will say:
'I can't say, but let us proceed hypothetically or experimentally, draw out one leg, swing the other two and see if it falls short or exceeds the rim of the circle.'
In making this paraphrase I have added the word "experimentally" for obvious reasons, and I have taken the noun chorion correctly as area (not rectangle or a triangle, as has been said, which means nothing) in a sense very well attested. So apparently with these conditions, the words themselves are not obscure or really unintelligible, although as yet the meaning has not yet come to the surface.
The key comes from Euclid Book IV Proposition 5:
"Around a given triangle to circumscribe a circle.' with detailed instructions for the construction.
Now this whole book of Euclid deals with the inscribing or polygons of every sort in circles and outside circles, but in no case does he dream of trying to put a polygon of given area into a given circle, knowing that by his methods of rigorous mathematical proof, it is impossible. Rather he discusses putting into the circle a triangle with angles the same as those of a given triangle, preserving the angles and hence the proportion of the triangle, but automatically adjusting its size for a kissing three point fit. This he can do and he explains exactly how, the other he would never consider because it can be done in only one way, which is the way Plato is rather obviously hinting at: Put two points of the triangle on the circle's rim, and see if the other falls inside or outside the circle, meaning that it is respectively smaller or larger that what a perfect fit requires.
This is a method of approximation, cut and fit, trial and error. I think we can be perfectly sure that Plato was aware of what he was saying. The fact that the reference is so short, implying that the students of the school had heard all this before, indicates that this was in all probability the stock example or how not to do rigorous Greek mathematics. So if Plato now cites this as an example of how he is going to proceed in the Meno, and perhaps later, in the search after supreme truth, by a method of method approximations, this is a very important element of method and purpose, one which must be taken with great seriousness and respect. In fact it is as good an example of the master describing for us his method as Plato ever gives us. Tricked by the appearance of brevity and unwilling to follow through Plato' a thought on the road to Euclid, we have garbled or passed over a unique piece of philosophical information.
Nor is this without meaning for the Meno itself, which contains as it were two thematic outlines operating at the same time throughout the dialog. First there is the straight-line investigation of "virtue, which proceeds calmly and with purpose as usual. But below the surface there are indications of another "way", one which involves mysteries, Pythagorean catchwords, holy and wise men and women, the guide who knows the way to where he has not been. It is this second path which I believe Plato refers to as his system of approximations, a non-exact search with ready fashioned tools, leading eventually to the great myths of the Sun and the Cave and to the Timaeus. This will be the direction for the future.
IV: The Divided Line.
In this familiar, perhaps over-familiar figure of the divided line, Plato is quite explicit.
If we take a given line and divide it into four sections, with the proviso that the first two are larger than the second two, and provide that the proportion found in A together with B as compared with C together with D be reproduced also in the relationship of A to B and also C to D, then we have a pretty fair representation of Plato's plan. Incidentally B and C will be equal, despite Raven's complaint that this is "an unfortunate and irrelevant accident" By now we can see that Plato, whatever his faults, does not tend to do things by accident.
This sounds remarkably like Euclid X 12:
''Magntudes commensurable with the same magnitude are commensurable with one another "
We can see this is an extension to the new study of proportion, which our example obviously represents, the old information of Axiom 1 of Book I about things equal to the same thing being inter se equal. The second Definition of Book VI somehow sounds related, but the text is in question and any interpretation must rest on modern mathematical reconstruction, so it seems best to leave this out for the time. Euclid thus supplies background but does not offer an immediate clarification to the whole figure.
If we suppose some such figure as I have indicated above, and wish to proceed to the right or to the left with smaller or larger proportionalized segments, we will clearly proceed very quickly to the infinitesimally small and the infinitely large. We will be dealing with some sort of geometric progression involving expansion at the one end and contraction at the other, or if we were to extrapolate this example to a 1:2 proportioned rectangle, and continue the ratio out and around on a right handed circular pattern, we would produce some sort of a geometric spiral. The question now is what meaning can Plato be attaching to these forms.
If we follow his argument to the final step, he remarks that "A" is the world of abstract true ideas, "B" the world of formed hypotheses, "C" the real world of existing "things and beings", and "D" our psychological, sense-perceiving world. Therefore as we move to the right and extend to the impossibly small, we cramp our sensations of the "beings and things", until we fail in our perceptions and lose connections altogether.
Eyes and ears are not going to be the best witnesses, as Heracleitus had proclaimed, and the relationship of the "beings and things" to abstract comprehension, hypothesis, or what we moderns would call science, is going to be more stable and less influenced by a big shrinking or expansion.
On the other hand hypotheses, mind or science is going to be faced on the other side with a huge expansion, where perhaps may lie the realm of pure truth, as Plato had hoped.. He is quite sure of the knowability of this direction, we with more experience in dealing with receding frontiers of information, might well envision another great area of unknowability, the omnidirectionally expanding three dimensional halo of the unknown. Buckminster Fuller has worked out a similar kind of figure, consisting of two concentric circles, between which lie the best chances for knowability, with a restrictively small core inside hampering investigation, opposed to an infinitely large set of possibilities outside, which also hampers investigation. Best knowledge would seem to have a certain relative range.
Plato would have none of this. The outer fringe may be the place where nature stops "teaching in departments", where things stop hiding themselves, where local forces are overridden and everything works together in unified fields. Or there may be unknowability which we may eventually define and have to learn to live with.
(The proportions involved in the proposition of the Divided Line lead to the celebrated matter of the Golden Mean or Golden Section. How the Greeks, with their rudimentary counting system, with a zero concept, without much sense of irrational numbers without accurate measuring devices --- how they could have used the GM in architecture, is a mystery. I have discovered a great deal of new material on this topic, which can be found in a separate article under title GOLDEN MEAN, on this web page., for which see the index under Philosophy. It is simply toe complex to introduce here.)
In any society at every moment all the strong and developing forces work in relation to each other, sometimes unknowingly, sometimes quite obviously. As we look back into the past, where there are many chapters permanently erased from our view, we tend to see books and men and ideas as particular and individual, and it is easy to forget that intellectual fields cross all lines of human endeavor constantly. So the study of Greek medicine is necessary for an understanding of Greek drama and Aristotle's critique of it. Acknowledge of Greek history is required for understanding of Aristotle's Ethics, and as we learn more about the ancient world, I suspect that we are going to have to know a great deal more about the nature and the influences of Greek mathematics, which was perhaps the single greatest achievement of the Greek mind, and one which certainly influenced the general tenor of the society more thoroughly and in more ways than we us can yet fathom.
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