1a. After the cave image Socrates considers with Glaucon the actual education of the philosophers. He begins significantly: “Would you like now to see in what way such men will come to be born [in the city] and how one will lead them up into the light, just as some [e.g., Heracles] are said to have ascended from Hades to the gods?” (521c1). The sequence of learning, which follows closely the “pathos” of the cave drama, has three stages: “conversion” (periagogē, 515c7, 515c8, d4, 521c6), the “haul” toward being, effected by mathematical studies (mathēma holkon, 515e8, 52ld3, 527b9, 533d2), and the “divine sights” of dialectic (theiai theōriai, 517d4).
“Conversion” is what we are witnessing in the dialogue itself. Since it precedes all education and is more the effect of meeting a man than engaging in a study, it is not part of the explicit study plan. Nevertheless, there is an “art of conversion” (518d3). Since this first act is largely a matter of making the soul recognize the shadows on the wall as mere shadows, it is clearly an eikastic art, namely Socratic music, the persuasive imagery of truth. As we have seen, it may be said to take the place of that traditional habituating music used in the education of the warrior-guardians but so emphatically excluded from the philosophical education (522a4). Note that in the image, as in fact, the city will try to prevent such conversions and will call them corruptions (517a). It cannot be said that these philosophers-to-be are born and bred in the just city—their upbringing seems, in fact, to be conceived against a hostile background that can hardly be the education provided by the guardian city!
1b. The long “haul” into the light of day is accomplished chiefly by the “hauling study” of mathematics (522c5-53ld6). The program appears to be that of Pythagorean cosmogonic mathematics (530d8). In arithmetic, “that lowly little thing” (phaulon), the “one” and the “two” and the other numbers are distinguished; in plane geometry the surfaces of bodies and in solid geometry the bodies themselves are studied; in astronomy bodies are put in motion; and, finally, the audible relations of moving bodies are studied in harmonics. In this way, the cosmos imaged in the Myth of Er, with its heavenly bodies giving out a harmony as they revolve, is constructed. There is only one difference between this Pythagorean cosmos and the Socratic study, but one so deep that it is very hard to grasp for Glaucon, who loves physical studies, especially astronomy. He immediately identifies Socrates’ phrase about “seeing the things above” with “looking into the [sky] above” (529a2), and Socrates has to rebuke him: that kind of astronomy, the “visible music” of the Pythagoreans (cf. Theon, Mathematical Matters Useful for Reading Plato, Hiller, 5, 17 ff.), in truth makes its students “look downward altogether” (a7). Socrates demands that in the serious study of mathematics, that paradigm of every “learning matter” (mathema), not only all practical applications, but even every suggestion of an admixture of sense experience should be put by, and only those true motions and numbers and figures which are grasped by the logos and the dianoia alone should be studied (529b). Glaucon, who follows well enough the early part of the discussion, which is concerned with demonstrating the dianoetic power of arithmetic, is somewhat puzzled by the non-physical “dimensional” studies that follow (526d ff.). For indeed it is the extended effort of actually doing pure mathematics that is needed to complete the conversion from sense (533d3), and this is still before him.
Ancient commentators, such as Plutarch, Theon, and Proclus, have a standard way of referring to the use and meaning of the mathematical course of the Republic: They simply assert the elevating effects of such studies. One of the reasons why there should be this effect is of far greater consequence than the others, since it amounts to a genuine interpretation of the sequence of subjects. There is a strong ancient tradition that attributes to Plato and the Academy the theory that the whole of being arises, as it were, in a mathematical way: From the One and the Indefinite Dyad spring the Ideal Numbers; from these the sequence of dimensions is educed, until first the mathematical solids are attained; and then the sensible world itself is brought about the “dimensional soul” discussed above belongs to this context. Now aside from the internal difficulties of this view, which Aristotle discusses at length in the Metaphysics (1080b24 ff., 1085a8 ff.), and its essentially stultifying character, there is an almost insuperable difficulty in interpreting the mathematics of the Republic along these lines. For in the Republic mathematical objects, since they are among the images that reflect the realm of being and comprise the realm of the dianoia, are fitted into a scale of increasing genuineness, within which they lie be tween natural object and being. In the theory mentioned, however, the mathematical structure comprises the whole of things arranged in a scale of decreasing dimensionality and concreteness.
It therefore becomes absolutely necessary to go to the dialogue itself to see why the young philosophers must study Socrates’ mathematics, aside from its generally purifying effect, which amounts to a conversion of the merely “embodied soul” into an embodied by primarily “knowing soul” (527d8). We may expect that, as usual, Socrates is quietly presenting Glaucon with something astounding.
For it turns out, upon inspection, that the mathematical order Socrates is talking about is indeed a reflection of being and that this means that it is an inverse of the realm of being. Glaucon must use his imagination to see why this is so. A diagram picturing the realms of the Divided Line will show what needs to be seen:
The whole realm of natural objects is here pictured as a cone with the sun at its vertex. It casts a reflection or shadow (510a1) which, like most natural images (as distinct from more deceptive artificial ones), is opposite or inverted in relation to the original, and “produces a shape (eidos) that yields a sense perception the opposite of the usual sight” (Sophist 266c3). This natural fact may now be applied as imagery to the noetic realm, and the beings of the uppermost realm may be imagined as reflected by the dianoetic objects in the same way. In the diagram, mathematics is then represented by an inverted cone having its dimensionless vertex at the bottom (representing the elementary study of the unit) and opposing its base to that of the conical sector of a sphere (representing being), in order to indicate both that mathematical objects are only a small part of the dianoetic images of being and that, unlike mathematics, the dialectic ascent within being does not end in an upper limit but culminates in a kind of source and beginning which is at once central and encompassing.
We must now show the meaning of this inversion of the dialectical order in Socratic mathematics. One observation does, however, already seem warranted: Such mathematics does not lend itself to institutionalization; Socrates cannot have been announcing the study program of the Academy (not to speak of a state education), especially since “no one unversed in geometry” was allowed to enter there, and the members were evidently left by Plato to engage in all sorts of advanced study with great freedom.
Let us look, then, at the elements of Socrates’ presentation in their order.
To introduce “the study concerning the one” (525a2), Socrates brings in one finger, a finger that is simply one and seems to have nothing contradictory within itself (523d6). In it, sight sees everything “poured together,” confused, indistinct, and for this very reason self-sufficient; the finger is “absolute” for the sense of sight (524d10). But as soon as two other fingers, one greater and the other smaller than the first, are brought in, sight reports that the first finger is now both great and small, and the other senses will report similar oppositions. At this point, the soul in its perplexity calls on the “counting capacity” (logismos, 524b4) and the dianoia to determine how many objects it is really dealing with, and thus arises the distinction between the visible and the intelligible (524cl3). For the dianoia separates out the great and the small as being “each one, both together two” (524b10), a formula that, though it does not solve the problem, states it precisely, i.e., “arithmetically:” Two different items, namely the great and the small caught in speech, can be together in one, namely the physical finger. “The whole of number” is gotten analogously (525a6)—whenever something is no more one than the opposite of one (a plurality), it is an assemblage of so and so many things taken together, and that defines an arithmos, a “numbered collection.” The “second” study, which, Glaucon knows, is “next in succession,” is geometry (526c8). After plane geometry, which is the study of the “second increase” (i.e., the second dimension), must follow the study of solids taken “by themselves” (528b). “Fourth” comes the study of astronomy, which deals with solids in visible motion. Since there are at least two kinds of motion, a last, sister study is required; namely, harmonics, which is concerned with motions relative to the ear. In pure harmonics, consonant numbers are studied; these are the numerical relations presented in a theory of numerical proportion dealing with simple and compounded ratios—such relations bear on genesis, as we know from the “marriage number” (546). At the end of this course, the “community… and kinship” of these studies must be grasped, and what is peculiar to them must be collected (531d2). This is the basis upon which dialectic is finally to be laid, “just like a coping stone” (534e2).
The elements of this study in the order in which they appear are, then:
1. the sensible undifferentiated one,
2. the great and small by comparison,
3. the “hypothetical” ones that add up to two, and every other number,
4. the dimensional structure of the mathematical cosmos,
5. the community or cosmos of all these mathematical objects seen synoptically.
Although the dialectical way is not explicitly described in the Republic (533a1), we have collected enough of its elements to name them in the order in which they would become the center of attention:
5. the community or cosmos of eidē seen in a preliminary synopsis,
4. the lowest ranks of eidē,
3. the eidetic number-assemblages going back up to the eidetic Two and its constituent eidetic ones,
2. the archē of the Great and Small,
1. the One.
The dialectical objects are, as can be seen, in fact encountered in the reverse order of those that have the same name in the mathematical sequence. For the mathematical studies ended with a community of moving and related solids that had first been set out in terms of their elements of “growth” (the progress of dimensions), then “by themselves” (solid geometry), and finally in their “consonances” (astronomy and music). So, in reverse, the dialectician must survey first and foremost the front presented by those eidē closest to genesis in order to distinguish its elements. He must try to see at once “which kind is consonant with which” (Statesman 285b; Sophist 253b, where harmonics had just been used as the type of such knowledge), and whether any elements extend through all the others; “for he who can see things together is a dialectician” (synoptikos—diaektikos). Note that it was precisely the final phase of the mathematical development that provided the required training and testing of the synoptical ability (Republic 537c). After this beginning synopsis (of which Socratic music is a foretaste) it will be possible to see and order the communities of eidē and to rise among these, attaining the larger eidē, or genē (Sophist 254d4). These higher genera contain more of being but belong to smaller—or rather, prior—eidetic assemblages or numbers. Finally being itself, the greatest “genus,” (genos) (ibid. 243d1), the eidetic Two, will be reached, together with the two great genē it comprises, rest and motion (ibid. 250; cf. Note 35). At this point, it will be possible to recognize that the eidē that run through all the others are beyond Being and ought not to be called eidē but archai. Thus, two sources appear beyond the objects that were first the hypotheses of mathematics and then the beings of dialectic: the Other (ibid. 259a-b), called by many names, of which the chief are the Indefinite Dyad and the Great and Small (e.g., Metaphysics 987b20, 1087b5), and the One, the non-hypothetical “beginning of the whole” (510b7, 511b7): in the One everything again rests without opposition, just as once before, at the beginning of the road and at the opposite extreme of being, the finger combined in fact that which is incompatible in words.
Socrates himself emphasizes the opposite ways of mathematics and dialectic at crucial points. The segment of the intelligible, he says, is cut “in such a way that in one part of it the soul is forced to search from hypotheses, using the things before imitated as images, and journeying not toward the beginning (archē) but toward the end, while in the other, the one that leads toward the non-hypothetical beginning, she goes [up] from hypotheses and [proceeds] without the images used in the former section, journeying by means of the eidē and through them” (510b; also c, 511a, 533c). The mathematical way is thus in itself a deduction, moving away from beginnings. But, at the same time, as a learning process, it is an “elevation (epanagoge) of what is best in the soul to the sights seen among beings” (532c5), though, as Socrates gently and tacitly informs a too enthusiastic Glaucon, it is not knowledge of being itself but only an approach to it (527b9). How can Socrates imagine that these two motions take place at once? To begin with, Socrates shows no interest whatever in that straightforward deduction of mathematical fact which is the pride of mathematicians, and which, because of its logical rigor, becomes for Aristotle the paradigm of “apodeictic,” i.e., demonstrative, knowledge (e.g., Posterior Analytics 71a3). This form of proof, in terms of which mathematical systems like Euclid’s Elements are presented, is called “synthetic.” Socrates, oddly enough, implies that this form of proof is to be left to all the non-mathematical arts, which deal with “geneses and syntheses” (533b5). His own approach is “analytical” or, what will be shown to amount to the same thing, “hypothetical” (cf. Phaedo 100a, Meno 86e).
The discovery of analysis as a formal method of proof was attributed in antiquity to Plato, who was supposed to have imparted it to his pupils as a means of finding solutions to problems. As Pappus explains, it differs from synthesis, which leads from agreed-on assumptions to conceded consequences, in the direction of its motion, “for in analysis we assume that which is sought as if it had already come about, and we inquire what it is from which this results, and again what is the antecedent cause of the latter, and so on, until by retracing our steps we come upon something already known or belonging to the class of the first principles, and such a method we call ‘analysis’ as being ‘solution backwards’ [anapalin lysin, i.e., reduction].” Pappus goes on to distinguish two kinds of analysis, theoretical and problematical; in the former, we assume a hypothesis and trace out its consequences until we come to something admittedly true or admittedly absurd (the second is what is known as a reductio ad absurdum or negative proof), while in the latter we assume a solution or a construction as if it had been found, and trace out its consequences until we come to something known to be possible and obtainable, or the reverse.
Analysis is, therefore, a means both of returning to and thus reflecting on those beginnings that Socrates calls hypotheses (e.g., 510b). It is also a means for achieving “agreement” (533c5) concerning the consistent consequences of any “suppositions” (the precise rendering of the Greek plural of hypothesis) that happen to be offered. Now whether or not Plato was really first to recognize in this a formal mathematical method, he himself certainly regarded it as Socrates’ very own mode of conversing, which, “while he is the man he is,” remains his only way (Theaetetus 197a1). For the most striking form of analysis shown in the dialogues, which people find “not unpleasant” (Apology 33c4), is the reductio ad absurdum, of which the Socratic refutation or elenchos is the great non-mathematical example. It is used for silencing professional “eristics” by disposing of their dangerous absurdities, and for producing in those who are well-disposed that perplexity of soul which is the beginning of serious inquiry; its complement is “Socratic ignorance” (Republic 354b9). Accordingly it occurs in the Republic only in the first book, where it is used in the provoked and forcible conversion of the violent Thrasymachus and as a “prelude” to the real inquiry (cf. 349a10, 354); thereafter, Socrates uses the gentler way of his analogic “music.” We must now see how the mathematics he proposes is also, in some sense, “analytical,” consisting as it does of “conversion-aiding arts” (533d3).
To begin with, the mathematical enterprise is analytic in the same way as is any Socratic “search,” in which the object is always assumed as something known of which the consequences will emerge in the course of the inquiry (cf. Meno 80e ff.). Socrates regards mathematics as just such a search (e.g., 523a2, 524e5, 528b-c, 531c6); in fact, it differs from any Socratic conversation only in the greater commonness of its objects (522c). Thus the question “What ever is a finger?” is transformed in the course of the mathematical inquiry into the question “What ever is the one?” (523d4, 524e6), and Socrates gets Glaucon to admit that the power of counting finds in a finger opposites that are “each one, both together two,” precisely as earlier he had gotten him to agree that since the beautiful and the shameful are opposite, they are “two, and each one” (524b10, 476a2). In other words, Socrates is interested not so much in getting on with the specific mathematical science as in returning again and again to its hypotheses; a sign of this is the preposterous name he gives to “arithmetic,” which he calls “the study of the one” (525a2), as if number science consisted of nothing but its beginning principle, that “source of the whole of number” (Aristotle, Topics 141b9), which is not even included among the numbers (cf. Euclid, Elements VII, Defs, 1-2).
But Socrates intends his young philosophers to practice analysis also in a more technical sense, namely as “problematic” analysis. Each of the two final, more or less physical, sciences is to be studied, in Socrates’ strange phrase, by “rising to problems” (530b6, 531c2). To see what the nature and purpose of such problems is—besides supplanting observation—we must look at the order of Socrates’ mathematical course. Theon, for instance, observes that non-sensible harmonics, since it is the study of “pure” number relations, of logoi and analogiai, should naturally follow immediately after arithmetic in the mathematical order (Mathematical Matters Useful for Reading Plato, ed. Hiller, p. 17). It is therefore not surprising that Socrates thinks it necessary to justify his own different ordering with an appeal to the very men whose preoccupation with matters sensible and experimental he immediately disowns; harmonics is to be understood as cosmic music, the sister science of astronomy, “as the Pythagoreans say, and we, Glaucon, agree” (530d8, 531b7). By accepting this ordering, Socrates obtains a last propaedeutic study that is at the same time non-sensible and physical, namely the unheard music of the unseen heavens underlying the “eye’s sky” (Socrates’ pun: horatos—ourilnos, 509d3).
Now we can see just what it means to “use problems” in astronomy, “bidding the things in the heavens good-bye” (530b7). According to tradition, it was Plato himself who raised that epochal question of astronomical hypotheses, setting this as “a problem for those interested in these matters: By what hypotheses of regular and ordered motions can the appearances associated with the motions of the ‘wanderers’ [i.e., planets] be saved?” (Simplicius, Commentary on Aristotle’s On the Heavens 292b10). The young philosophers will, presumably, work on just such problems, contenting themselves with “saving,” not too exactly, the more basic appearances such as the daily motion of the stars and the sun. Here “saving” the appearances means first of all learning to regard them as mere appearances of something more nearly substantial (namely the hypotheses) and so “bidding them good-bye.” Using, then, the visible heavens as a nicely made but not very important “pattern” (259d7 ff.), they are in turn to make a construct such as might be “hypothesized as a pattern” (Timaeus 48e5) by some mythical demiurge who wants to bring our visible world into being. Note that this mythical demiurge is a requirement of cosmic genesis, since the mathematical hypothesis itself, being no paradeigmatikon aition, no responsible pattern, i.e., not having the power of a true source, will never yield the sensible world by itself—there exists no deductive account of the sensible world from the mathematical cosmos alone.”
The profits from working such problems will be many: The students will learn to make images that will exceed the visible original in truth, that is, they will learn to make noetic hypotheses about the whole of appearance and to deduce their consequences; in attempting to “save” the appearances they will learn what a mere appearance really is; and they will get used to living with their mathematical cosmos as a preparation for dwelling with the invisible and bodiless noēta (cf. Sophist 246b7, Statesman 286a5). It is unlikely that this study will much resemble the spherical geometry presented in those dull “Sphaerics” (like that of Theodosius) which are supposed to be realizations of Plato’s “pure” astronomy. We must rather imagine that the young philosophers will be asked to reflect on matters such as the true significance of astronomical notions like “obliquity,” “error,” “anomaly,” and their opposites. They will find these studies “useful… in the search for the good and the beautiful” (Republic 531c6), since they concern equality, symmetry, consonance, and order; they will have regained the world on a higher level. When their time has come for the pursuit of dialectic, which “takes up (anairousa) the hypotheses” (533c8), in the senses both of “removing” them and of “raising” them to a higher mode, they will “take up” first their hypotheses about the whole, meaning that they will convert them into that array of lower eidē governing the world of appearances and human excellence within it. These are the eidē close to the world, which the dialectician encounters first (500c, 484c-d, Gorgias 508a) and which, while least in being, are longest in logos (e.g., Statesman 267a-b).
But we are also told, quite incidentally, that static geometry too is to be pursued in problems (530b6). Here again the order of studies proves interesting. The geometry originally proposed was plane geometry (528d3). Next, Socrates brought in astronomy, but, right away, with an air of significance, asked Glaucon to “draw back” (528a6), and delivered a speech in praise of the study of that missing third dimension which seems “not yet to have been discovered” (b4). Solid geometry, the study between earth measurement and the motion of the skies, aiming at objects that lie between heaven and earth, is, he implies, in some special way the city’s business; it is a political affair, which ought to be carefully supervised and which is even less than the other studies to be pursued privately (b6 ff.; cf. 525c2). He might, however, have stopped himself sooner, for he had already interrupted the dimensional development of his mathematical world, in fact, at its very beginning, when he had introduced the “second increase” of plane geometry even though he had omitted any mention of the first, the linear dimension. The one-dimensional study, however, so conspicuously missing in the Republic, is solemnly introduced in the Laws (820a-b), where it is presented precisely as the necessary prerequisite of stereometry, i.e., of solid geometry. It is the study of irrational lines, the special interest of Socrates’ young counterpart Theaetetus (Theaetetus 147d ff., 144d8), who is also credited with the first cogent presentation of all the regular solids called “Platonic.” We happen to know what the classical problem in solids is: it is the problem of doubling the cube, according to legend originally raised by Minos, the underworld judge, and later again by the Delians, whom Apollo had ordered to double his altar, and who were said to have brought it to Plato for solution. Socrates himself uses the plane version of this problem to display the natural knowledge of a slave boy whose education consists simply in knowing Greek, and who is able to find, under Socrates’ supervision, the side of the double square; it is the irrational diameter, the single mean between the two squares (Meno 82b ff.). We might conjecture that as the doubling of the square, done in the sand on the earth, is the slave’s problem, requiring for its solution only the learner’s nature, so the doubling of the cube, done in the solid world of bodies, is the citizen’s problem, requiring both nature and nurture, and that it therefore represents practical politics. For stereometric problems are solved by finding two means, sometimes called “powers” (Theaetetus 147d3), which are in this case themselves irrational, though commensurable in cube (Euclid, Elements X, Defs. 1-4). Stereometric analysis then represents the work of finding ways and means to reconcile that which has no common measure by going into a higher dimension (cf. Epinomis 990d8); mechanical, that is, mere defacto, solutions are considered unacceptable. This is perhaps why Socrates mysteriously tells Glaucon not to allow his children, who will have charge of the city, to be like “irrational lines” (534d5); he means they should not be like that which has failed to rise to the higher dimension in which commensurability and consonance are possible: There is, indeed, something obvious in taking problematic analysis as the paradigm of practical planning, and, accordingly, Aristotle in the Nicomachean Ethics compares deliberation to the analysis of diagrams, for “every deliberation is a search” (1112b20). Such a use of mathematics can be quite a dangerous game (as we moderns well know), but at its best it articulates terms that serve well in the press of action, clarifying the nature of tricky inquiries and providing models for solving complex problems (cf. Meno 86e).
Socrates concludes his exposition of the philosophers’ mathematical education by suggesting to Glaucon that “the experts (deinoí) in these matters can’t seem to you to be dialecticians” (531d9). This distinction between mathematicians and philosophers is ever his theme; for instance, he observes that “inasmuch as they themselves [the mathematicians] don’t know how to use their discoveries they turn them over to the dialecticians to use, if they have any sense at all” (Euthydemus 290c3). Socrates again has a special way of characterizing the helplessness of the mathematicians’ arts: “They dream about being, and cannot behold it as if awake, [at least] as long as they use hypotheses that they leave undisturbed, unable to give an account of them” (533b8). For Socrates, mathematics will always remain a dreamlike and “phantastic” enterprise. For instance, in the Philebus, where in the course of the investigation of the human good he constructs just such a “bodiless cosmos” (64b7) as was described above, using principles like the One, the More and Less, and Number (23c), he always attributes his knowledge of these to a god, a myth, or a dream (16c, 18b, 20b, 25b). For him, mathematical objects are shadows or reflections on the surface of being, catching its mere form; they are more immediately accessible to the human understanding than being itself but have no substance of their own. They are interesting only insofar as they are in turn reflected in the sensible world, which they divide and collect and shape in a less substantial but more general way than can the more numerous, non-mathematical eidē. (These purer eidē, caught by human speech, also appear in the dianoetic realm as hypotheses.) This ordering function in the world is what gives the mathematicals their special standing as “the middle things” (Metaphysics 987b15, 1090b31), differing from sensible objects in being eternal and immutable and from the eidē in having each a plurality of instance (while each eidos is unique). This function is also why their study is capable of “hauling the soul from becoming towards being”(521d3). The mathematical part of the dianoetic realm contains, as it were, the ordering elements, the taxis, of being, where these elements, including shapes and numbers and their relations, are taken as independent objects, just as reflections and shadows may be regarded as rendering the precise, albeit intangible, shapes of the bodies to which they belong.
It might be useful to give one example of an attempt to “take up” a mathematical hypothesis in the double sense of canceling it and raising it into its eidetic original. Out of the many hypotheses of mathematics, such as units, numbers, ratios, proportions, “the odd and the even and figure and the three kinds of angles” (510c4), let us choose the most fundamental objects, the one and the first number, two.
The senses of sight and touch elicit from us merely the words “a finger,” but in comparison with other fingers, a greater and a smaller, we have to say of the same finger that it is both great and small. However, “great” and “small” are opposites and evidently not capable of being simply thrown together. We therefore say that the finger comprises both of these objects intended in speech in such a way that they are “each one, both together two” (524b), i.e., two things at once in the finger; thus arises the first “multitude composed of units” (Euclid, Elements VII, Def. 2), the number two. Since each one object of sense includes oppositions “infinite in multitude,” every number can arise in this way (525a); each unit of sense reveals itself as numerous. Now, let the inquiry concern not some unit in the world of sense, but the greatest single noetic object, Being itself. After a survey of the possible opinions (Sophist 242b10 ff.,), it is decided that Being must comprise two “most opposite” eidē, Rest and Motion (250a8), neither of which is by itself Being but both of which are indeed Being. Hence, that Being appears as “a third thing beside these” (b7). If we now try to apply the hypotheses of mathematical unit and number to these opposites comprised by Being, that is, if we try to “count up” Being, we see that they are no longer adequate. For to count up units it is necessary that they should be mere units “capable of being thrown together and indifferent” (symbletai kai adiaphoroi, Aristotle, Metaphysics 1081a6), just as “great” and “small,” being both “determinations of size… are on that common ground indistinguishable and therefore capable of addition (526a3). But Rest and Motion have no such common ground, being irreducibly different; in the sight of the whole (where there is no relativity of motion) nothing can be at once at rest and in motion. Furthermore, if we made such a reduction we would lose exactly what we are after in dialectic, which is “the thing itself,” in its very nature. Being is not, therefore, a counting number and the result of adding two mathematical units, but a community of incomparable eidetic monads, neither prior nor posterior to their “number,” namely Being, the unique Two (ibid. 1081a ff.). We can see that here the mathematical hypotheses of the one and the two, having taken us up into Being, must themselves undergo a transformation; naturally this can never be achieved by the dialogues, which do not leave the realm of human logoi and of the dianoia—within speech Being must be put down as a permanent perplexity (Sophist 250e5), and whatever solutions are offered must be somehow misnamed. For when in the Sophist the problem of Being, there pointedly formulated in terms of the Not-being of each of its “constituent” eidetic ones (251d ff.), is solved by means of “the Other” and “the Same” (254e ff.), then these two will, when presented in speech, appear to be hypotheses exactly like the other three eidē, Rest, Motion, and Being. The Same and the Other will therefore be counted with these three among the five greatest eidē (255c), although they have the nature not of beings but of that which is “beyond being,” of archai. In the context of the Republic, the more important of the archai in the Sophist is that whose eidetic name is “the Same.” For this is precisely the One, which, itself beyond all articulation, is both the wholeness of being and the source of the oneness of each being—that which makes a being “one and the same” with itself and thus makes it just what it is. Here the mathematical one, the uncuttable and indifferent minimal (525e ff.), is, as we have seen, the total inverse of the dialectical One, which is the Whole and the source of everything (511b7; on these disparate functions of the Platonic one, see Metaphysics 1084b13 ff.).
The central purpose of Socratic mathematics, then, is the examination of the hypotheses provided by human speech, a reflection about speech carried on within speech. As so often, Socrates’ very example is chosen to express the reflexiveness of the inquiry, for a finger is a pointer, and Socrates is pointing at this pointer. The purpose of such reflection is, in turn, to bring the soul up to that highest internal “power of conversation” (hē tou dialegesthai dynamis, 533a8; cf. 511b4, Parmenides 135c2). In such conversation the soul, freed from all the senses (Republic 532a6) and raised beyond the intermediate dianoetic logoi (cf. Phaedo 99e5), confronts being immediately by means of its own noetic logos (Republic 511b4, 532a7, 534b9, c3), which is simply its power of having a direct relation, a “ratio” to being.
This is the ninth essay in this series. The other essays may be found here: I, II, III, IV, V, VI, VII, VIII, and X. We highly recommend Eva Brann’s The Music of the Republic: Essays on Socrates’ Conversations and Plato’s Writings. This essay originally appeared in the St. John’s Review (Volume 39, Number 1 and 2, 1989 – 1990) and is republished here with gracious permission of the author. Miss Brann welcomes questions/comments via mail: Dr. Eva Brann, St. John’s College, 60 College Avenue, Annapolis, MD, 21401-1655 (she does not use computers).
 The Republic of Plato, ed. J. Adams, 2nd edition (Cambridge 1963), cit. II, pp. 163 ff.
 Diogenes Laertius remarks on Plato’s special use of the word phaulos, pointing out that he uses it in the two senses of “simple, honest” and “bad” (III, 63). Actually, of course, Socrates often uses it ironically to mean “the great thing that everyone else overlooks.”
 The ancient texts are collected in K. Gaiser, Platons ungeschriebene Lehre (Stuttgart 1963), op. cit., pp. 478-508. The difficulties in using these sources to reconstruct the Agrapha Dogmata are (1) that they often mention Plato and the Pythagoreans in discriminately; the longest account (Sextus Empiricus, Against the Mathematicians X, 248 ff.) even attributes the “dimensional” teaching to the Pythagoreans alone; (2) that they rarely tell by whom, to whom, or with what pedagogical purpose such a scheme was proposed. Aristotle, for instance, refers to the Timaeus in this context (On the Soul 404b16), which should give one pause—the account there is, after all, proposed as a myth (29d2). Nor can the frequent references in the ancient literature to On the Good help here, since no one knows whether that was a public lecture or a series of lectures or a kind of seminar. In other words, it is not possible to say what Plato himself thought of “dimensional generation,” whether he approved it only for certain pedagogical purposes or became himself a thoroughgoing Pythagorean. In any case, since we know that the dialogues often play on certain differences between the author himself and Socrates, it would, even if Plato’s opinion were well known, still be necessary to investigate Socrates’ understanding of the Pythagorean order within the dialogue itself.
The locus classicus for such a cosmogenic order is the Epinomis, a dialogue said to have been tacked on to the Laws by Philip of Opus (Diogenes Laertius III, 37). Its purpose was to expound that astronomical theology which was to be the wisdom of the “midnight council,” the rulers of the non-philosophical city of the Laws. Here the cosmos arises from a continual doubling of the unit, which produces a series of terms, 1:2:4:8, such that each term duplicates the ratio that the previous term has to the unit. These duplicating logoi express the relations of the dimensions of the cosmos, from the dimensionless one, through that doubling of the point which gives rise to the line, up to the solid that is the cube of the “linear” number two, i.e., eight (990e ff.). The duplication is to be understood as the effect of the dyadic arche.
 Plutarch, in his third Platonic Query (1001c1), in which he discusses the (undecidable) question which segment of the Divided Line, the uppermost or the lowest, ought to be the longest, attempts to conflate the opinion that “the dianoia is as nous among the mathematicals, which are as noeta appearing in mirrors” with the genetic dimensional theory. The difficulties into which this leads him are instructive. He first makes Plato induce the mathematical cosmos from the noetic principles: “He leads the noesis concerned with the eidē out of its abstraction and separation from body, going down in the order of mathematics;” next we reduce that cosmos by abstracting its dimensions until “we will be among the noetic ideas themselves;” and finally, the sensible world is deduced from these by reading the realms of the Divided Line roughly downward, as stages of dimensional growth. In this account it is entirely unclear in what way the noeta are the principles of mathematics on the one hand, and of the sensible world on the other, and whether the production of mathematicals is the work of human noesis or of the noeta as sources.
 A centered sphere (as opposed to Parmenides’ partless sphere) seems at least to convey more of the expressible characteristics of the Whole than the usual pyramid (cf. Sophist 244e). The center, which in our diagram marks the end of the dialectic ascent, is, according to Aristotle (Physics 265b4), at once the arche, the meson, and the telos of a sphere, so that, in a manner of speaking, it encompasses the periphery. This corresponds to the fact that Glaucon’s assumption, that the top of the dialectic ascent is “a rest… and an end” (532e3), is quite wrong—there still must follow a syllogismos that takes the argument back again to the periphery, where the power of the center as cause, aitia, is first fully apprehended (516b9; this noetic syllogismos, which is preceded by an ascent, e.g., Statesman 267a4, is not to be confused with the dianoetic deduction, e.g., Republic 510b5). It is, of course, convenient that all three of Aristotle’s terms are also terms used of the Good. The radial lines may then represent the indefinite dyadic source, which is doubly delimited by the center and its periphery to produce the actual finite sphere of Being. But extended attention to such diagrams always implies that thought has ceased.
 See Proclus, Commentary on the First Book of Euclid’s Elements, Ver Becke (Paris 1948), pp. 58-62; on the motto over Plato’s “mouseion:“ “Let no one unversed in geometry enter here,” see Elias, Commentary on the Categories, Commentaria in Arisatetem Graeca (Berlin 1900) Vol. XVIII, Pt. I, pp. 118, 13 ff.
Socrates’ resistance to the advanced and “technical” study of mathematics is attested also by the Memorabilia (IV, vii). Xenophon represents him as advocating exactly the kind of mathematics opposite to that which he repeatedly insists on in the Republic, namely useful and applied mathematics, such as earth measurement. But when Plato and Xenophon differ in a deliberately diametrical way there is usually a vital point of agreement; here it is Socrates’ deep objection to mathematics as an independent study pursued for its own sake by private experts, a study that would fit into the “doxastic” rather than the “noetic” segments of the Divided Line. The very mental virtues required and acquired by such studies, keenness and sharpness (Republic 526b, 535b5), are regarded by him with a certain suspicion because they tend to live in a little soul (519a), and his parodies of professional talk (527a, 531b) are worthy of a “large-thinking” Swiftean Laputa.
 See Klein, Greek Mathematical Thought I, 6.
 Of course, mathematical studies provide practice not only in what might be called “cosmic synopsis” but also in that preliminary “synagogif” or collection (Phaedrus 266b4) which consists in “seeing together and bringing under one idea things scattered everywhere” (265d3), i.e., in collecting particulars into one eidos.
The necessity of “cosmic synopsis” was emphasized by Speusippus, in whose opinion “it is impossible for anyone to define any of the things that are unless he knows all the things that are” (P. Lang, De Speusippi Academici Scriptis, Fr. 31b.)
 Sources in Gaiser, op. cit. (supra, 28), pp. 461-67. Aristotle (Nicomachean Ethics 1095a32) furnishes the general background by noting that Plato was concerned with the direction that the road of inquiry ought to take, whether toward or away from principles.
 See The Thirteen Books of Euclid’s Elements, Heath (New York 1956) I, pp. 136-41.
 See Klein, Meno, (supra, 25), pp. 82-87.
 For the Simplicius passage see Gaiser, op. cit. (supra, 28), p. 464; for the “pattern cause,” p. 480, note; for Theophrastus’s comment on the absence of a deduction of the visible world, pp. 493-94.
 See Euclid’s Elements (Heath, supra 50) III, 1 ff., p. 438.
 See L. Heath, A Manual of Greek Mathematics (New York 1963), pp. 154-55.
 Klein, Greek Mathematical Thought Pt. I, 7 C.