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Again, a geometrical figure may present spatial analogues of the key relationships of concepts presented in a lecture or discussion. A metaphor from algebra may give a generalized formulation of the structure proper to a certain class of formulae. This range of source and function prevents any examination of the purely mathematical properties of such a set of illustrations from yielding any agreement as to what they illustrate; their common function of illustrating a context may be realized through many forms of functional quantitative structure.

Orientation and Limits of the Present Study This study might be characterized as a "dialectical" interpretation. Two features seem to justify that characterization. The first is the consistent objective of establishing interpretations which make philosophic sense in their contexts which does not, of course, relieve them of the need to make historical, mathematical, and philological sense as well. This is basically, however, a philosophic rather than a philological or historical approach, since the "contexts" of these passages are conceived as functional parts in the exposition of a philosophy.

The second feature of the study is its recognition throughout of the respon. A functional mathematical image must reflect these contextual metllods, principles, and distinctions adequately, even if some deviation from the techniques of "pure" mathematics is required to seCllre such reflection. Past interpretations seem often to have postulated the detachability of all mathematical passages from their contexts, as though they enjoyed an independent nleaning, like theorems in a system of geometry; or to have postulated that the passages could be understood by setting them all indiscriminately into some single eclectic context.

Since the present approach is basically philosophic, the proper criterion of its adequacy is whether or not it does, without intruding historic or philological implausibilities, illuminate the passages discussed, considered as parts of Plato's exposition of a philosophy. If an alternative approach had been chosen, the criterion might have been historical: Will the postulated interpretation explain and reconcile the extant statements of all interpreters, if we correct those statements in the light of our knowledge of the convictions and style intruded into the original by each?

This is a separate and extensive enterprise, not here undertaken; in the present study antecedent interpretations are treated schematically, by criticizing or displaying the assumptions operative in their construction, not individually, except in those cases in which they have suggested some relevant. If a third approach had been chosen, the passages might have been studied as specimens of presumably meaningful classical Greek in which the meanings were expressed by means of syntactical and etymological devices analogous to those used by other Greek authors.

Such an approach would be much more direct than either of the others, if it were only possible; but in every case where it has been tried and it has been tried often it has been necessary to resort to maxims of higher criticism which have generally taken the form of intruded, unPlatonic philosophical assumptions. The history of Platonic scholarship goes far toward underscoring the need for the present mode of interpretation. The many studies of Plato of the historical and philological sort have uniformly, so far as the passages discussed in this study are concerned, contributed little or even negatively to our understanding of Platonic philosophy.

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The concept of image is used here in its strict Platonic sense, in contrast to things, names, or formulae. Passages in which mathematics tlPpears not as illustrating but as being what is talked about, as in the Philebus, Statesman, and Parmenides though they have influenced the introductory sections, treating general technique and kinds of imagery have been excluded.

Because of this limitation, the reader will feel that supplementary treatments are required to deal completely with the Platonic mathematical vocabulary the names used , and with mathematical formulae through study of which one should recover Plato's philosophy of mathematics. However, until some agreement about the images has been reached, "any man who wants to may upset any argument" put forward to explain the formulae.

Final Comment on Tactics. The tactics of presentation used in this study have been to present translations of the passages concerned, with translations of relevant scholia, then discussion of alternative interpretations and problems, ending so far as possible with final interpretations resolving these problems. It would have been impossible to construct my own translations of these passages which did not read into them my own notions of interpretation; and even if it had been possible to do so, any normally cautious reader would be rightly suspicious.

It is easy to conjure a rabbit from a hat if you have already hidden it there, and the conflicting past assertions of philologists as to "the only meaning the Greek can bear" certainly indicate that translations of these texts leave plenty of room for conflicting interpretations. I have therefore developed interpretations from other English translations, chosen for their neutrality. In almost every case, these were made by translators who actually had in mind other interpretations than those developed in this study.

Translations of scholia are included both to show what sort of footnotes ancient scholars annexed to the mathematical passages in their manuscripts of Plato, and to prove that no set of diagrams derived from Plato's original manuscripts was known to Hellenistic scholars. On occasion, the text and the scholiast's marginal diagram are sufficiently at cross-purposes to rule out the possibility of any direct tradition.

In various introductory sections, an attempt is made to define. Some concepts and notation of contemporary mathematicallogic have been used to give the desired generality and precision to these statements. This use of concepts from twentieth-century formal logic certainly appears anachronistic, and in the present study is not explicitly related to the imagery that it describes.

For the present, it is offered simply as a useful device for talking about the properties of classes of mathematical constructions and illustrations. In a later discussion of Plato's philosophy of mathematics, however, I expect to show that the modern concepts intruded here are essentially Platonic, and do not really involve what seems to be an anachronism. One final fact requires emphatic statement: To analyze or explain an illustration is different from appreciating it directly. The more a reader recognizes what this direct awareness is like, the more keenly he will feel that an abstract analysis of imagery is missing something important.

Since concrete aptness dissolves on analysis into many abstract relations of relevance, no one of which has the concrete vividness that we expect a good illustration to show, a commentator who uses such analytic technique is often accused of reading a great deal more into the text than its author intended, or of devising analyses with a neat or "slick" finish which most illustrations lack.

Of course abstractions are neater than concrete cases; and of course no author sets about creating illustrative examples by separate consideration of a tremendous number of abstract connections. But the intuitive appreciation of the materials analyzed here presupposes an ancient Greek intllition to which we no longer have the key; so the choice is between this analytic approach to interpretation and no interpretation worth mentioning. Introductory Comment: Some Simple Illustrations the most frequent and typical intrusions of mathematical imagery into the dialogues is as illustration of class-relations, particularly in connection with problems of method or definition.

The Euthyphro,l Meno,2 Phaedo,3 and Theaetetus 4: all contain such illustrations. The only problems of interpretation which illustrations of this kind pre.. In such cases, the function of the interpreter is to revive the sharpness and cogency of the intended illustration in its original setting; but since the function of these examples is, in context, perfectly clear, their elucidation will not throw much light on Plato's philosophy, and therefore very little is gained from more intensive interpretations.

Often in these cases Plato's use of mathematical imagery to clarify discussions, where we no longer feel the relevance of such imagery to do so, poses a series of complex problems; and since the reconstruction of these diagrams provides a statement of how their context is to be interpreted, we should spare no effort to reconstruct them correctly. ONE OF.

The illustrations dealt with in this first chapter are all of the first sort; therefore, apart from supplying relevant diagrams and noting certain controversies over mathematical details of interpretation, the treatments of these passages have been kept brief. Even in this section, however, each of the images selected has some relevance to the study of Plato. In following the suggested method of beginning with simple images which present no problem requiring extensive interpretation, and in postulating that the later, more complex images are similar in function to these a postulate which can be tested by seeing how satisfactory the interpretations are that it suggests , we may profitably begin with the mention of a group of "trivial" ilillstrations; cases where neither subtle analysis of the context nor any knowledge of mathematics beyond its rudiments is required for interpretation.

In Republic , for example, Socrates says that if we are looking for four things, and have found three, the fourth must be the one that is left. This is a simple way of illustrating the contextual point that in the state, if courage, wisdom, and temperance have been defined, the cardinal virtue that remains will be justice.

In Republic , Glaucon contrasts "geometrical" and "erotic" necessity, the one proceeding from reason, the other from the passions. In Republic , Socrates asks Thrasymachus how a man could define 12 if he were forbidden to name any of its factors in his definition. Thrasymachus has just challenged Socrates to define justice, without saying that it is good, expedient, beneficial, or "any such nonsense. In the SymposiumJ 11 Aristophanes represents Zeus as calculating that if he divides each man in half he will get t'ivice as many sacrifices.

In Euthyphro 7, Socrates points out that metric techniques settle disagreements over number, weigl1t, and measure. In the Meno,12 the difference between a class and its members is illustrated by reference. These six passages are offered as instances of uncomplicated mathematical allusion, functional in illustrating problems of methods in a qllantitative subject matter where their pattern will show most clearly. There is nothing about them that is obscure or mysterious, or nonfunctional in context.

Since their pedagogical intention, contextual relevance, and mathematical content are very clear, these examples have been chosen because they show most typically characteristics which, as this study is intended to establish, all Platonic mathematical images possess. Do you not agree? Then we are wrong in saying that where there is fear, there is reverence; and we should say, where there is reverence there is also fear. But there is not always reverence where there is fear; for fear is a more extended notion, and reverence is a part of fear, just as the odd is a part of number, and number is a more extended notion than the odd.

I suppose that you follow me now? The scalene triangle has three unequal sides. The isosceles, however, has two sides equal to one another, and a third unequal. Since an even number is divisible into two equal numbers, as for example 8 into two 4's , but the odd into unequal, for example 5, the one is called "isosceles," and the other "scalene. Scalene: crooked and irregular. There are three kinds of triangles -equilateral, isosceles, and scalene. These peculiar definitions of odd and even numbers, as "scalene" and "isosceles," respectively, have occasioned some puzzled comment, because they seem either capricious or inexact.

The basis of the peculiarity is dramatic, not mathematical. The Platonic dialogue is an artistic representation of a conversation in which the illustrations offered by the speaker are appropriate to the character of his listener; when they are not, the listener says so. The demands of artistic consistency require that the speaker use examples dramatically intelligible to his audience. If, in this connection, we consider the character of Euthyphro, we find a prototype of the "ageometrical" man, denied entrance by the inscription over the door of the Academy.

He consistently substitutes piety and religious precedent for independent intellectual inquiry; he is neither interested nor skilled in matters of dialectic or mathematics. Consequently, when Socrates wants Greene, Scholia. Consequently, when odd and even numbers are cited as examples of definable classes, we should expect any definition that he can understand to have the same reference to everyday experience.

The obvious analogue to the ruler and scale is the abacus or counting-board. In the former case, the pebbles outline an "equal-sided" rectangular figure; in the latter case, an irregular quadrilateral "scalene". This peculiar mode of abacus-centered operational definition does not reflect Plato's own theory of number; it is simply Socrates' concession to the mental backwardness of his companion; the images of familiar abacus pebble-configurations seem the only counterparts Socrates can find to more accurate definitions which would be artistically inadmissible, because they would be beyond Euthyphro's dramatically established powers of mathematical and abstract imagination.

See figures I and 2. The directions are so clear-cut that from the time the passage was written there seems to have been no controversy over its interpretation. Some unusually geometrically-mil'lded scholiast has carefully constructed all the figures referred to, representing each by a separate figure, rather than as subdivisions of a single, more complex figure, as do later diagrams illustrating the passage.

IT This use of separate figures makes even clearer the exact. Figure 1 is a schematization of the only Greek abacus which has been discovered the "Salaminian table". The table is discussed by J. Although the reported lettering on the table indicates that at least one of its uses was for monetary reckoning,18 it has also been suggested that the table was used as a gameboard, and its design partly dictated by this function.

In any case, the existence of such computers' aids in Plato's time is certain; and Plato's coupling of references to the game of tCE". The important point for the present study is that an abacus is a device taking advantage of spatial orientation to differentiate kinds of relation, a general technique discussed in Chapter III, Introduction, in connection with the concept of "verbal matrices. The analogy of sets to figures was basic in Pythagorean mathematics and is still retained in the terms "square" and "cube" numbers in our own mathematical vocabulary.

The use of a dot notation, representing each number as a spatially arranged set of units, facilitated the formulation of such analogies, which gave the Pythagoreans a kind of reversed analytic geometry in that they could represent continuous figures by analogous discrete sets. See also note 38, Chapter IV, following.

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These scholia figures are reproduced on pages They have been cross-referenced in the text,20 so that each of Socrates' statements is referred to the figure showing what part of the construction he is pointing to. Meno 82 fI. If you can prove to me that what you say [that knowledge is recollection] is true, I wish that you would. Suppose that you call one of your numerous attendants, that I may demonstrate on him. MEND: Certainly. Come hither, boy.

Jowett, Dialogues, I, References to figures are in brackets.

MENO: Yes, indeed. He was born in the house. MENO: I will. BOY: Certainly. BOY: There are. BOY: Yes. Count and tell me. BOY: Four, Socrates. BOY: Of eight feet. BOY: Clearly, Socrates, it will be double. MENO: Yes. MENO: Certainly not. MENO: True. To the boy Tell me, boy, do you assert that a double space comes from a double line? BOY: No, indeed. BOY: Four times as much. BOY: True. And now tell me, is not this a line of two feet and that of four? BOY: It ought. BOY: Three feet. BOY: That is evident.

BOY: Nine. BOY: Eight. BOY: No. Tell me exactly, and if you would rather not reckon, try and show me the line. He did not know at first, and he does not know now, what is tIle side of a figure of eight feet; but then he thought that he knew, and answered confidently as if he knew, and had no difficulty; now he has a difficulty, and neither knows nor fancies that he knows. MENO: I think not. MENO: I think so. I shall only ask him, and not teach him, and he shall share the enquiry with me: and do you watch and see if you find me telling or explaining anything to him, instead of eliciting his opinion.

Tell me, boy, is not this a square of four feet which I have drawn?

BOY: Four times. BOY: I do not understand. BOY: Twice. BOY: From this. And if this is the proper name, then you, Meno's slave, are prepared to affirm that the double space is the square of the diagonal? Figures 3 through 11 from Greene, Scholia PIatonica, pp. Figure 8 and Roman lettering are my additions. Nonfunctional numbers 2', 6' are omitted in Figure 5 and nonfunctional l' marks in Figure 6. Figure 4. The area of the 3 x 3 square is cOlnputed by sumnling the unit squares. Figure 9. The numbers as they stand do not make sense. Figure 12, following, represents a suggested emendation of this figure which makes it interpretable as an alternative intuitive demonstration of the relative area of squares on the diagonal and the side.

Figure The final diagram represents an original excursion of the scholiast's own; he is advancing a figure for an alternative proof of the general theorem that "the square on the diagonal is double the square on the side. The intended proof proceeds, as does Plato's, by computing areas through summation of equal triangles; eight unit triangles are combined to form a square, abou! The area of the inscribed square is indicated by numbers summing up the number of the half-unit triangles through its four quadrants: These numbers should therefore be , rather than the which are in the manuscripts.

The diagram does not correspond to any figure in Euclid, but resembles figures associated with a suggested early method of proof of Euclid I. It would be of some possible interest to inquire f. In Jowett's figure, stages are not labelled; Heath's line AN is the side of the three-foot square which represents the slave boy's second attempted solution. Either figure gives a clear picture of the diagram as it develops during the discussion, by stages which the scholia represent separately.

The purpose of this demonstration in context is to convince Meno that knowledge is recollection. Meno's association with Gorgias has lead him to accept a Sophistic notion of teaching and learning which identifies knowledge with literal recall of past experiences and precepts. It is this conviction that "teaching" is dogmatic statement that causes Meno to agree that Socrates did not "teach" anything to the boy, though later educators not in the Sophistic tradition have repeatedly cited this passage as an example of teaching procedllre.

Socrates, convinced that learning is a matter of insight, confronts Meno with a proof that if, as Meno maintains, all knowledge is past experience recollected, he must postulate both a very peclliiar "experience" and a very extended "past. The converse form of the same point is developed in the Theaetetus where no identification of knowledge with temporal psychological process can be found that will apply to mathematical error and insight.

For this demonstration, Socrates needs an example that will be something the slave boy clearly has never learned, bllt it IIlust also be capable of being presented in a form intuitively evident enough for Meno and the slave to understand. To avoid any possibility that the construction is one which the boy may have learned, Socrates chooses an example from the field of "higher mathematics" to convince Meno. The properties of the incommensurable were one of the fields of adJ. The theorem chosen has the added advantage that when the correct construction is completed, its truth is so evident from the diagranl, that even a novice like Meno can see the demonstrated relationship.

An example from quantum mechanics would be analogous in difficulty in a contemporary dialogue, as one from calculus would have been in Leibniz' time. This theorem, however, can be proved intuitively by a construction which is clear without any special mathematical training. One cannot infer that since the problem Socrates chooses is "advanced," he assumes! This fact is highly relevant to the mathematical passage in the Meno discussed in the following section, since many of the interpretations proposed seem to require that Meno be a competent mathematician.

The answer might be "I do not yet know whether this area is such as can be so inscribed, but I think I can suggest a hypothesis which will be useful for the purpose; I nlean the following. If the given area is such as, when one has applied it as a rectangle t to the given straight line in the circle the diameter , it is deficient by a figure rectangle similar to the very figure which is applied, then one alternative seems to me to result, while again another results if it is impossible for what!

Accordingly, by using a hypothesis, I am ready to tell you what results with regard to the inscribing of the figure in the circle, namely, whether the problem is possible or impossible. Heath, History, I, The interpretative slant represented by the references to "rectangles" is discussed in the comment following Figure MEND 87A concerning the inscription of the triangle in the circle. At present, apparently from the conviction that Plato.

Although this shows a comn1endably high opinion of Plato, it is also a little reminiscent of tIle Pythagoreans' attrib.. Three main interpretations of this passage have been advanced and defended as though they were mutually contradictory. Benecke and 'Gow 23 suggest that the problem is illustrated with the same two-foot square that was used in the earlier discussion with the slave boy; in which case the inscription is possible if the radius of the given circle equals the side of the given area, represented as a square.

One could well cite Socrates' tempering of the rigor of llis definition of odd and even number to the capacities of Euthyphro as an analogous concession to a nonmathematical companion. The geometer's condition then is: "If, when you erect on the diameter a square equal to the area, an identical square can be constructed on tIle remaining segment, the inscription is possible.

Greene, Scholia Platonica, pp. The inscription is possible if the area can be applied to the diameter as a rectangle in such a way that its corner point lies on the circlImference of the circle; and, by the converse of Euclid iii. This gives the theorem more generality and more mathematical interest than Benecke's interpretation, but at the expense of a presupposing more background in geometry than Meno can consistently have and b disconnecting this figure from the ones Socrates has already drawn, whereas his references to "this triangle," "this area," etc.

In the first place, one may certainly allow the correctness of the third set of interpretations insofar as the mathematical detail of the illustration falls outside of its dialectical relevance except for a point discussed in Appendix B , in a way typical of such Platonic illustrations of method. It is less plausible to assume that there was no intended technical mathematical reference at all. The figure of Benecke's choice represents a figure for the simplest case of the more general theorem. In Euclid iii. This suggestion combines the merits of Benecke's insistence on consistency of character and economy of imagery, and the suggestion of the second group of interpreters, that some more general theorem is required to make the example mathematically interesting or meaningful.

This case is thus analogous to the Euthyphro definition of odd and even number by associated intuitive configurations; only so much of a technically valid mathematical illustration is introduced by Socrates as his audience can consistently be shown to grasp. In summary it seems tilat at least three conditions are met by this illustration, of which each interpreter has emphasized one.

The passage must be "the sort of thing a geometer would say," i. It must, or at least should, be something that would hold some intrinsic mathematical interest, deriving from the relation of the method to the illustration chosen an interest that certainly would attach to the inscription problem, where the conditiollS but not the solution were known. It must also be dramatically appropriate, in not exceeding the posited ability of the audience to whom it is addressed and, if possible, in developing from antecedent figures used for methodological ilillstration in preceding context.

If we take Benecke's figure, treated as a special case of the general theorem of inscribability, the mathematical interest of the general problem is still suggested by this special case, though considered as a theorem in its own right, apart from that context, it is not very significant or mathematically interesting. The general theorem, to gain its relevance to the contextual discussion, must itself be taken as a special case of a method used by the geometer at work, and one may be able to appreciate that technique without laboring over the detail of a postlliated, specific example.

Still, there are many diagrallls which would satisfy all three of these conditions. In Appendix B, following, a further con-. This interpretation identifies the area to be inscribed with the square used in the previous demonstration so that A CDE in the present figure is identical with ABED in Figure 9.

As Socrates refers to inscribing "this area," this interpretation visualizes him as turning again to the other diagram and pointing to ABED. If he drew a circle at Meno 74E, while discussing "roundness," he might also have pointed to the circle in stating the present problem. DB equal to A CED is the same summing of half-unit triangles used in the earlier demonstration with the slave boy. This is an equation of the fourth degree which can be solved by means of conics, but not by means of the straight line and circle.

Conversely, if this condition is satisfied, E represents a point on the circle of which AB is the diameter. The theorem is that if two straight lines intersect within a circle, the area of the rectangle bounded by the segments of one is equal to that of the rectangle bounded by the segments of the other.

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The first case: If both lines pass through the center of the circle, the proof is evident; for, all four radii being equal, the rectangles bounded by them are equal as well. Compare with Benecke's diagram, Figure 14, preceding. You mean, if I ani not mistaken, something like what occurred to me and to my friend here, your namesake Socrates, in a recent discussion. Jowett, Dialogues, IV, Now as there are innumberable roots, the notion occurred to us of attempting to include them all under one name or class.

The use of incommensurables to illustrate the proper method of definition in the Theaetetus is certainly intended as a me-. The memorial, however, has been selected with a view to its appropriateness in the context of the reported conversation. The moral of the 1vhole dialogue is one that might well be symbolized by a theorem of il1commensurability; knowledge turns alIt, "vhatever unit of comparison we employ, to be incommensurable with opinion.

Various items of knowledge, particularly the mathematical, cannot be identified with or explained by any process of perception or physical construction. The relation of knowledge to opinion cannot be described simply by identifying items of knowledge and opinion, but must be stated in some other way. What is needed is to find some common classification for all these items of knowledge, and to find a statement of their relation to our perceptions and habits of operation, as Theaetetlls and Young Socrates have defined and stated the relations of magnitudes and roots.

This example has a mathematical reference which claimed great interest from Plato, and continues to attract the attention of historialls of mathematics. Its relevance to the dialogue itself is only that of the other mathematical examples introduced as paradigms of definition in the Meno and Euthyphro. The development of the treatment of irrationals, in which Plato and the Academy took a special interest,30 provided an excellent geometrical example of this difference.

Only so much of the detail of this history as is immediately relevant to the dialogue is included by Plato, but this much gives almost the only clue ,ve have for filling in a series of technical developments not discussed fully. The proof of the irrationality of the diagonal of a unit square cited by Aristotle,31 based on the demonstration that if the side and diagonal are assumed to be commensurable, the same number must be both odd and even, is usually taken as a Pythagorean starting-point in the investigation of irrationals. Somewhere in the. But the present passage in Plato, with its confirmation by a scholion on Euclid x.

Historians agree that the procedure of Theodorus seems to have had the peculiarity that each case needed to be demonstrated separately, and that what impressed Plato most in Theaetetus' procedure was his generalization of this episodic technique. From the theorems and methods of inquiry available to tIle mathematician in Theodorus' time, conjectural reconstrllctions of his proofs have been proposed, falling into three main groups, of which one seems to fit the evidence of Plato's account better than either of the others.

The principal objection is that such a line of inquiry would establish only a presumption, not a proof, of the unending character of the approximation, hence would not really establish incommensurability. With the Pythagorean demonstration of the irrationality of the square root of two as a model, Theodorus would hardly have been satisfied with a proof which was so mllch less effective, nor could we explain Theaetetus' apparent satisfaction with the validity of the proofs of the separate cases which Plato's dialogue makes him express.

When the shorter segment of such a line is laid off on the longer, the difference is itself in the same ratio to the shorter segment that the latter had to the longer. Since the two initial segments do not have an integral ratio, it follows that no common unit will be approached by the difference of the two segments, no matter how often the one is subtracted from the other.

This satisfies both conditions attaching to Theodorus' proof: a that it is a valid proof of incommensurability, not merely an empirical illustration of a presumption; and b that it requires a separate geometric figure for each case, such that the same ratio can be shown by similar figtlres to repeat for every subtraction of the shorter segment an integral length line of the figure from the longer representing the given constructed irrational.

The final form of Theaetetus' generalization of Theodorus' results was probably the source of Euclid's theorem x. This connects the ratios of the sides of similar figures with previously demonstrated arithmetical theorems about the ratios of square numbers.

Such an intermediate step may have consisted in the application of the extension of the Pythagorean proof, mentio11ed above as one of the conjectured reconstructions of Theodorus' method, to any figure in which some standard construction established the incommensurability of the relation of diagonal and sides. The proof would then have been developed as applying to the square root of any number, the geometric representation of which as an area would have been possible only under the given conditions of construction. But the squares on straight lines incommensurable in length have not to one another the ratio which a square number has to a square number; and squares which do not have to one another the ratio which a square number has to a square number will not have their sides commensurable in length either.

I I therefore also, as the square on A is to the square on B, so is the square on C to the square on D. IN HIS description of Atlantis in the Critias Plato gives the exact numbers and measures of almost every phase of its geography, public works, and political institutions. In the description of ancient Athens, in the same dialogue, there is only one numerical detail given the total fighting strength of the state.

The institutions and customs of ancient Athens can be and are adequately specified by reference to a normative standard embodied in legislative principles; the exact measurements can be summed up by the statement that they are those which are best adapted to proper functioning.

In a disordered and only loosely unified state such as Atlantis, on the other hand, institutional and technological details are not determined and coordinated by a rational unifying plan. The closest analogue to the structural statements made about ancient Athens where the structure of the society was organized around rational legislation in an account of Atlantis is, therefore, the separate description of the institutions and public works of which this social structure happens to be composed. The substitution of some set of specific figures for considerations of proper function in a total plan is peculiarly appropriate to the description of the type of disunity and disorder which.

Poseidon seems to have been an ancestor not likely to produce philosophic and mathematically minded offspring; for, if we compare his ordering of circles of lalld and sea in Atlantis to the circles of the heavens described in the Timaeus it becomes evident that, when this god geometrizes, he does it like a carpenter's apprentice. And tIle institutions preserved by the descendants of Poseidon 'VI10 rule Atlantis SilOW that, in fact, the offspring have made no improvement, philosophically or mathematically, on the insight of their ancestor.

The key to the selection of all the numbers in the Critias is the statement that these rulers "met alternately every fifth and every sixth year, paying equal llonor to the odd and to the even. Not only is the confusion of even and odd which are the basic contrary principles of the most elementary mathematical science a sign of total lack of theoretical ability, but the specific numbers cited here, which represent the even and the odd, reflect this same confusion, the one being the sum and the other the product of the first odd and the first even number.

Since in Plato's mathematical images and formulae in dialectical and cosmological contexts the basic opposition of odd and even is observed and since in contexts dealing 1;vith legislative detail ease of nlanipulation or religious propriety is the determining factor and in the latter case the basic distinction is again that of odd and even , while in mythical contexts periods and distances are poetically dismissed as "myriads" perhaps composed of lesser, proportionately related periods, which are indicated by smaller powers of ten , the absence of anything remotely resembling "alternating fives and sixes" ill other Platonic contexts is causal, not accidental.

The choice of "five and. Reflecting and leading up to this final detail, where an explicit statement is given of the underlying confusion which accounts for its selection, all the other numbers and measurements cited, however casually, are except one either a multiples of 6 or 5 or b parts of a sum, product, or ratio which ill its entirety is a multiple of 6 or 5 or c 6 or 5.

Poseidon himself begat five pairs of twin sons; 3 his statue depicts him driving six horses; 4 his engineering consists in the construction of five circles three of sea and t'vo of land 5 about a central island with a dianlcter of five stades. ConfllsioIl of the even. In this context, therefore, the representation of 10 as a sum of 3's and 2's is not really an exception to the rule of the prominence of 6'5 and 5'5. As we survey these two sets of figures, a second principle of selection is also seen to be operative: the vastness of the numbers and distances involved is reflected by the prominence of myriads as units of description.

The ratios which give a qualitative aspect and dialectical point to these various precise statements of distances and numbers, however, remain, no matter what the scale, 6's and 5's. The way in which this principle is carried out in the principal religious-political festival of the state has already been shown. It is further represented, however, in the law that no king may be sentenced to death without concurrence in the sentence of at least six of the members of the council. These are the factors emphasized by Plato's parenthetical remark in , which underscores the deviation from tradition.

But it is a deviation which merely doubles a set of 5 times 10 and which, in conjunction with the contextual mention of the number of steeds, shows in the state religion the same basic confusion that is reflected in the alternation of 6's and 5'8. In summary, the apparently random numbers so liberally interspersed in Plato's account of the Atlantean state are not inserted simply to give an impression of great size or simply to. These "random" numbers are constructed on a dialectical al1d artistic principle in such a way that each reflects some aspect of the rulers' basic and traditional confusion in mathematics and philosophy.

Plato reveals his philosophic and artistic precision and his sensitivity to the significance of detail in inventing the history of a bad state as well as in describing the archetype of a good one. Critical commentaries and studies on platos republic 8 vols. Republic plato - wikipediaWhen plato was sixty c.

## Western Philosophy Ancient

Essay about platos education philosophy -- philosophyThe republic is a socratic dialogue, written by plato around bc, concerning justice. A comparison of the ideal states of plato and aristotle, aarathi ganesanSamuel scolnicov - - studies in philosophy and education 13 2 details. Divided line plato essayEmerson education essay summary of plato. Essay on plato and education - wordsPlato was born into an aristocratic greek family between — bc. Your email. Send Cancel. Check system status. Toggle navigation Menu. Name of resource. Problem URL. Describe the connection issue.

SearchWorks Catalog Stanford Libraries. Plato in the third sophistic. Responsibility edited by Ryan C.