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Cosmological 'fitness' in the Timaeus.

Not long ago Donini convincingly argued that the Greek word for 'probable', 'likely', and so on--eikos--is for Plato a conception admitting of degrees, and therefore that great care must be used when considering its precise role in a given context (popular religion, cultural values, natural philosophy, theology). (1) I have recently argued (2) that (1) prosek- and eik- share at least the semantic range denoting cognition, viz., 'it is likely', 'it is reasonable to suppose', and the like, and that (2) prosek- overlaps with prep- not only in this 'objective' sense but also in a more 'subjective' one, viz., 'it makes sense'. It follows from this, first, that prosek- admits of similar degrees as does eik- in similar types of case and, second, that in general the same careful attention to context should be exercised when scholars are faced with the two roots prosek- and prep-. In other words, the question concerns not simply 'what is likely' or 'what makes sense', but more precisely how likely and how much sense something makes. This language, then, ought to be translated in light of the more precise interpretation which is itself either authorized or even demanded by its context, for then our comprehension of Greek thought generally, philosophy more particularly, philosophical cosmology still more particularly, and even more so the Timaeus will become considerably clearer than it has heretofore been. The clarity thus gained may in turn help to resolve some issues that have plagued students of this 'dialogue' (3) for centuries already.

This investigation has three main sections. First, I establish on the basis of a few texts in the Timaeus the need for this type of semantic interpretation. These passages occur in three clearly identifiable contexts, each concerning how best to think and talk about various aspects of the universe. The first passage constitutes one of two premises in the argument concerning the relation between time and eternity; the second involves an analogy pertaining to the Receptacle; the third clarifies the language for spatial directions that can obtain in a spherical universe. Having thus shown that this sense of the language is indeed present in this dialogue, there follows an extended application to and discussion of three passages in which this language is also prominent, the first of which has commanded significantly more scholarly commentary than have the other two: (1) the other premise as well as the conclusion of the argument about the relation between time and eternity, (2) the analogy between the four primary natural bodies and the letters of a language, and (3) for the second time, the number of worlds that exist. (4) The paper concludes with an argument against Vlastos' notorious claim that Plato here amounted to a scientific Luddite who set Greek science back by crushing fact with value; Plato is, on the contrary, merely indicating the epistemological dependence of the whole of empirical science upon metaphysics.

I Basis

The behavioural sense of this language in LSJ (5) ('it is appropriate', 'it behoves one', 'it is fitting', etc.) has an organizing principle which is reasonably construed in terms of expectation or prediction, articulated primarily in terms of social harmony. Thus, 'A expects B to do X in circumstances C' would be understood either as a simple prediction, which will be as accurate as A is familiar with B, or as an injunction--actually a complex prediction, since it includes A's familiarity with how independent B is from the pressure of social values, especially of harmony or 'fitting in'. A's goal in this scenario would be to aid in B's becoming maximally predictable for the sake of social harmony. The ambiguity involved here is well preserved in the English supposition: When B thinks, 'I am supposed (or expected) to do X', a certain pressure is felt by B to do X rather than to forgo doing X, or a pressure to do Y instead of doing X. The felt pressure to act in light of that supposition (or expectation) is considered by B to have originated in A, whether A is a significant individual to B or the Impersonal and Generic Other relative to B. It is worth noting that social criticism begins with the question, 'By what moral right is supposition X justified?' The distinctively philosophical answer is that nothing can have a moral claim upon us that has not antecedently won cognitive allegiance. Philosophers, that is, seek harmony not primarily in terms of more widely held social values, but rather in terms of much more narrowly and explicitly held cognitive ones (a crucial value for the philosophical subset of society); especially sensitive to contradiction, they experience disharmony where non-philosophers generally do not. With regard to my present interest, since implication is the cognitive equivalent of prediction in the behavioural domain, 'what is appropriate' or 'what is fitting' is, in the context of thinking, precisely 'what stands to reason', 'what makes sense', or 'what can be figured out'.

I begin with a clear instance from the discussion of the relation between time and eternity at Timaeus 38a. Plato's philosophy of language, detailed in the Cratylus, rests so firmly upon his theory of Forms that it is hardly an understatement to assert that without Forms nothing could be understood; (6) this view is in full force in the Timaeus as well. Timaeus in this passage complains about the deficiencies of then-current (7) linguistic usage. The target language occurs in both of the argument's premises as well as in its conclusion. For the moment only the second premise (38a4) of that argument needs to be discussed--its first premise and conclusion are addressed in the application section below--since we are here presented with the obvious exclusivity between eternity and temporality. On the basis of nothing other than logical insight (securely anchored in Forms) and impossible without it, it can safely be inferred that, despite how it might seem, it is clearly impossible that something actually eternally, changelessly self-identical (8) could also change in any sense whatsoever, and that is why it in fact does not get older or younger because of time. (9) The safety of that inference is indicated by the impersonal verb prosekei; insofar as Forms constitute the very 'stuff' of intelligibility, for Plato, it is easy to understand that nothing real can actually host contradictory predicates simultaneously, (10) and thus also that nothing can, upon careful reflection, sensibly be thought to do so. It is not that a Form has changelessly whatever age it has, in contrast to all temporally qualified things, but rather that it simply has no such thing as an age at all. This rationally grounded inference is as utterly secure as is the fabric from which it is cut, a fabric that owing to the nature of Forms themselves has the Platonic virtue of also being completely inexhaustible, and serves as a good illustrative case in which what makes sense to one person (e.g., someone dialectically uninitiated) may well no longer make sense to another (e.g., someone dialectically proficient), since the latter's thoughtful reflection is presumed to be based upon nothing other than Forms, themselves as cognitively solid as anything imaginable.

The discussion of the Receptacle (50d) provides a second case in which the target language is easily clarified by recourse to its cognitive sense. The exposition involves an analogy, and the fundamental axiological criterion for an analogy is the extent to which it increases the intelligibility of something else. Initially the receiver is compared to a mother, the sender to a father, and the product of their intercourse to a child. (11) Both the existence of the recipient and its nature are taken to be implied by the essential bulkiness of actually existing physical objects; but implication and its weaker analogical relative, here signalled by prep-, is, like all logical insight, both characteristic of rational thought and impossible without it. The rationality involved in this case is significantly weaker than in the case of 'what makes sense' to the dialectician upon reflection; it can be readily understood by the uninitiated, likely because it is so clearly pictorial. (12)

The whole of 50d4B1b1 is characterized by Lee as a 'highly metaphorical exposition' (343). The analogies in this passage, in order of presentation, are those of the nuclear family, the odourless base for perfumes, and the pliable metal which can be fashioned into a variety of objects; in each of these cases, we are to gather the same thing (52c): that the image somehow or other simply must (13) cling to being, or else it will be just nothing at all. Obviously the image is not just nothing at all, even if there are different sorts of images ('substantial' images such as a statue made of bronze on one hand; 'insubstantial' ones such as paintings of a statue made of bronze on the other, the paint itself not being made of bronze) with relevantly different grades of ontological status (353); in general, Lee argues, the relation between an insubstantial image and its substantial original parallels the one between a substantial image and its original, namely the relevant Form (362). But it is worth noting that 'clinging' (14) is itself a metaphorical expression, and so it is evidently possible for a metaphor to play 'host' to a logical entailment.

Plato's context-less spherical universe (62d) is a third clear case of this sense of the target language. Here the issue concerns how to understand the terms 'up' and 'down', since they are meaningful enough within a spherical universe but are completely unintelligible when applied to it as a whole. (15) 'Down' can, given this cosmic image, sensibly mean nothing other than 'toward the centre' of the sphere, as any even moderately rational person grasps easily--asking about absolute directions when the universe is the ultimate context of direction is no different from asking for the context of the ultimate context. The irrationality involved here is even more clearly brought out by the characterization of the cosmos as spherical, since in the case of any other conceivable shape (e.g., a pyramid or a cube) not every extreme point of the figure will be equidistant from the figure's central point; the very possibility that any one side or corner might on some ground or other be stipulated as 'up' or 'down' is absolutely undermined in the case of a sphere. (16) This would, accordingly, constitute a medium degree of strength in terms of rationality, since the imagery is not as familiar as the familial one but still relatively easy to grasp in terms of ordinary experience with spherical objects; any difficulty involved would enter in Plato's application of the spherical the imagery to the entire universe. What characterizes pure Platonic dialectic is its resolute abandonment of sensory imagery. It is worth noting that the central point of the cosmic sphere does double-duty: it is both middle and bottom simultaneously, independently of all talk of 'absolute' orientation; but double-duty is once again completely natural for Forms--indeed it constitutes the basis upon which their very existence is first intuited.

2 Applications

1 Time

The first premise of Timaeus' argument concerning time and eternity (38a1B2) holds that 'was' and 'will be' are tense-words that make sense only within the context of change. The reason given is that the future and the past are not just features of change but actually constitute it. (17) For the parts of time (37e) cannot sensibly precede the existence of time itself (considered qua whole, not qua unit), and 38bBc asserts that time itself began to be (18) simultaneously with the heavens; hence the parts of time themselves must have been generated simultaneously with the heavens. But evidently the parts of time themselves are not basic, as is indicated by night's being part of a day (39c) and not, as Zeyl prints, 'night-and-day', since this misleadingly, unnecessarily, and therefore unwarrantedly inserts a kai into Plato's text; for the light and dark portions of 24-hour periods vary in length throughout the year, but the 24-hour period does not, and so it is the latter alone which is useful, mathematically speaking. Night and day are, accordingly, each parts constituting a single full day. (A similar point regarding the illusory character of 'basic' parts will be discussed below in the context of the natural bodies.) Equally clearly, anything susceptible of part-and-whole analysis is infinitely divisible; for any limit on division, and thus any indefinite limit, would necessarily acknowledge the existence of atoms and hence also of void--but Timaeus expressly rules out the existence of void at 58b, 79b, and 80c. (This point as well is developed in the section below on the natural bodies.)

Cornford suggests that, after the generation of time and its parts, the various astronomical entities were generated 'to define and preserve the numbers of time' (105); but better sense is gained, for reasons to be discussed below, that it was after the generation of time and its infinity of parts that the Demiurge generated, on behalf of time (i.e., as an indication of its value (19)), the various astronomical entities for the sake of distinguishing and preserving numbers. It might reasonably be objected that numbers cannot require definition and protection since, being eternal, they are whatever they are; but the same cannot be said for human consciousness of numbers, or numeracy. The celestial dances in fact constitute an aid to genuinely worthwhile human education (46eB7c); the visible universe refers beyond itself to eternity just as every image points beyond itself to its original. If so, then no temporally qualified thing can get older or younger as a result of or even through (dia) something other than itself; time, being change, could not possibly retain its own identity if it changed, since the only thing it could change into would be non-time, i.e., changelessness.

This interpretation of the passage reveals an implicit philosophy of history. For Timaeus, it is changelessness which is ultimately valuable, but since the world is physical (i.e., inherently in motion) such a property cannot apply to it. (20) The Demiurge, being axiomatically not just good (29a3) but the very best of causes (29a6), did the next best thing: he made the world revolve continuously in the same place according to number, and thus it is 'an ever-changing reflection of eternity' (37d5). Time in general is precisely penultimate, and just as an image is objectively impossible without its original, so time is objectively impossible without eternity; on the other hand, eternity is possible without time just as a physical original need not cast a shadow. What characterizes time, qua future and past at any rate, is succession, (21) which is strictly absent from eternity; 'succession' can be understood in the context of eternity only analogically, in terms of the logical dependence of X upon Y. The infinite divisibility of time limits temporal measurement (the same is true of spatial measurement) to approximation, something characteristic of opinion but not of knowledge. Moreover, if there is no smallest unit of time, then the moment has no duration, and if there is no time which altogether lacks duration, then 'is' can sensibly be predicated only of the eternal. But since 'is' denotes the present alone, it follows that time is no more than analogous to eternity and, more importantly, that the present and the eternal are identical. (22) Mohr argues that although at 38a2 Timaeus mentions only the past and the future, 'he expresses himself more fully and accurately later in the same sentence ... [this more accurate expression includes the present, and thus] being-in-the-present ... [is] completely inapplicable to Forms' (72); but in fact this 'more accurate expression' of Timaeus' clearly invokes the dual eston, which can in this context plausibly refer only to the past and future.

Even if the future (the 'not-yet') cannot change in and of itself without shedding its identity, its 'contents' actually do so, and as the various events become increasingly imminent (or remote) they are not unreasonably regarded as becoming 'younger' (or 'older'). If the future and past are time, then just like images they simply do not have it in themselves to be: their existence, such as it is, is exhausted in fleeting; they exist, in other words, only as anticipations or memories respectively. (23) Now since the events constitutive of time are governed by causality, as opposed to the strictly logical facts of eternity which are governed by implication, logical necessity holds absolute sway in the eternal order while it does not does not do so in the temporal reflection of that eternal order; causally interpreted, 'necessity' is restricted to physicality. Not only is causality similar to but not identical with implication, then, but nothing in history, which is inherently governed by causality, is absolutely necessary; the necessity characterizing history is instead hypothetical or contingent upon some other event's occurrence. Moreover, the future is intrinsically related to the past, but neither past nor future are similarly related to the present; the present seems instead to have been conceived by Timaeus as being extrinsically related to the complex of future and past. While the complex of future-and-past exists either together or not at all, then, the same cannot be said about the doubly complex present and future-and-past, since while the reality of the present does not require the reality of the future and the past (indeed they have no reality), the converse does not hold.

The claim that an event has no stable identity might be objected to, for instance in the case of a parade, which might be the sort of stable thing Platonic metaphysics would posit in giving a reasonable account of the various changes (bands, floats, etc.) constituting that parade. The shortest Platonic reply to this objection can easily be reconstructed from many aporetic dialogues: an example of something presupposes its exemplar. This is not just a semantic point but more deeply a metaphysical one: the existence of an example presupposes the existence of its exemplar. The metaphysical point, first recognized on the basis of a cognitive one, namely the logical principle of identity pertinent to subjects and assertions made about them, is immediately seen (by Platonists) to have actually been the necessary condition for that very logical principle itself. In this way thought apprehends the necessities characteristic of metaphysics and, empowered by the rational insights it discovers, returns to make increased sense of the world--natural philosophy, cosmology, and so forth. The case of the parade, accordingly, is not in fact a genuine subject of change, although the confusion around its seeming to be one is adequately explained by the underlying ignorance of Platonic metaphysics. Depending upon the context in which that parade is considered, it is clearly an event when viewed as one of a number of episodes in the larger spatio-temporal context of a festival, but it is deceptively considered to be a subject in its own right, i.e., with its own internal spatio-temporal structures. The reason it is deceptive is that Platonic 'things' are immaterial units and not material (or anything dependent for its existence upon material) 'wholes' made up of 'parts'. The very language of part and whole is itself analogous to but not identical with the language of unity (e.g., Tht 204bBc), and the same holds for their extra-linguistic referents. Spatio-temporal wholes all naturally lack the capacity for sustaining themselves, which is emphatically not the case for any real unit. Timaeus does not hereby concede that the world cannot be understood at all, but only that, to the extent that we do comprehend it, we can do so only on the basis of the unitary Forms.

The conclusion of this argument (37e6B8a1) is that only 'is' can accurately be predicated of eternity, although in less rigorous, forgetful moments we also slip into predicating 'was' and 'will be' of it. (24) The present tense quite naturally goes with eternity, then; this is the only philosophically respectable way to talk about the matter, if the present itself actually constitutes eternity. Thus it becomes clear that for Timaeus (and, to judge from his other dialogues, Plato (25) as well) not only is the true telos of history itself fundamentally a-historical, (26) but this 'fact' can be fully comprehended and appropriated for oneself only in present, personal experience, whether or not that present experience is itself of an expectation concerning post-mortem existence (e.g., Phd 106e) or of a recollection concerning its pre-mortem counterpart (e.g., Men 81e-6c).

2 The Four Primary Natural Bodies

The analogy between the four primary natural bodies and the letters of a language (48bBc) poses something of a difficulty for the thesis of this paper, but one that I think can nevertheless be dealt with adequately. In this case the natural bodies and letters of the alphabet are the simplest constituents out of which nature and language respectively are built. Timaeus objects that not only are the natural bodies not like the letters of a language, it doesn't even make sense to compare them to syllables,27 although that would not prevent someone who was intellectually significantly deficient from doing exactly that. (28) Taylor says that it doesn't make sense because the 'corpuscles' should be compared to words instead (308). (29) Cornford (162) suggests that Plato may have meant either that the primary triangles can be reduced to numbers or that body is basically unknowable; his second option is identical with that of Proclus, (30) and seems highly preferable to his first option because in fact body is not analogous to language. Language is inherently intelligible while bodies are not: dogs bark, but 'dogs' do not even 'bark', for instance.

The disanalogy between physical 'corpuscles' and words can perhaps be better seen by noticing that if they were analogous, then, just like the actual words of some language, the 'corpuscles' would be no less intelligible than those words; but the mathematical 'syllables' into which the natural 'words' are in fact analyzed are for this very reason more, not less, intelligible than the original 'words' were. (31) Three related reasons may be advanced to support this judgement: first, the unit is the logical origin of the rest of mathematics; second, the principle of identity is the fundamental principle of logic (and, for Plato, it seems, metaphysics in turn of logic); and third, nothing is more intelligible than rationality. In the case of language, on the other hand, letters, syllables, words, statements, and arguments constitute an increasing, not decreasing, order of intelligibility. In other words, the mathematical analysis of natural bodies offered by Timaeus does not, and indeed cannot, make body qua physical object intelligible for the simple reason that only inherently intelligible objects are intelligible. Cornford's frustration over the 'seemingly arbitrary feature [of Timaeus's mathematical analysis], which has never been satisfactorily explained' (217), namely the obvious over-analysis of primary bodies into constituent triangles, can be satisfactorily explained, and with it Cornford's frustration resolved. The point of over-analysing the equilateral triangle constituting each of the four faces of the pyramid of fire into six fundamental 30-60-90? scalene triangles when two clearly would have sufficed shows two things: in the first place, it shows that these six are themselves precisely not fundamental at all, since obviously each one of them is similarly analysable into three smaller ones of the same type, and so on ad infinitum; (32) in the second place, it shows that anything which is infinitely analysable is inherently incomprehensible. Thus Timaeus indirectly makes plain both (1) that physical entities, unlike words, are literally unintelligible and (2) that, as well as why, only mathematics can provide real insight into nature. This explains why it makes no sense to compare the fundamental bodies of nature to letters or to syllables, and even less to compare them to words--and consequently why a genuine sage would never do so.

This interpretation of the passage may help to elucidate Plato's reasons for assigning these 'best' triangles specifically in the manner that he does. The isosceles triangle is reserved for Earth but the scalene is delegated to Fire, Air, and Water; the latter are capable of mutual transformation, but Earth is exempt. Taking Cornford's analysis (233B4) as a starting point, one interesting feature of the scalene triangle S is that, considering it as a parent, two offspring can be produced, s1 and s2, each of which has a size imperfectly communicated to it by S, since all three sizes are different, although the shape of S is reproduced perfectly. Like S, the isosceles parent I communicates its shape flawlessly to i1 and i2 but, unlike S, is considerably more successful in terms of size, though still not completely so: for while i1 and i2 differ from I in this respect, they differ from each other not at all. The theme of same-and-different evident here is reflected also in the articulation of the world-soul (35a ff.), a somewhat bewildering mixture of intermediate Sameness, Difference, and Being. (33) This interpretation of Timaeus' triangular hypothesis helps to demystify this particular aspect of the dialogue: perfection is strictly extra-cosmic, but it is nonetheless communicated all the way to earth, where we have found ourselves confronted with the problem of living, namely how best to spend our time. (34) Only the enlightened actually grasp the importance of this problem as a problem, but through thoughtful attention to the varieties of Sameness and Difference in human affairs--the sources of knowledge and opinion, respectively, both cosmically and personally--they come to see increasingly clearly that their home is actually in eternity, and the philosopher will spend most of the Great Year cycle (39d) as close to it as possible, i.e., in the intimate company of the fixed star-gods. This insight is not only obviously humanly possible (though it does require a good deal of extra work to think metaphysically), but almost as an aid toward that insight the Demiurge has seen to it that Sameness is especially strongly reflected, at any rate wherever intelligent people exist. His motive, as explained by Timaeus, was nothing other than his own goodness--being good naturally, he was not jealous of that goodness but instead was inexhaustibly generous of it; he neither could (nor, if he could, would) prevent himself from sharing his goodness in the best possible way he knew how. Cosmically, this is expressed by his reducing pre-cosmic chaos to cosmic order, though even he had to allow for the nature of that pre-existent material, if 'instability' can sensibly be thought of as a 'nature' at all. Timaeus the character, and Plato his author, in what may be plausibly interpreted as a pure bestowal of intellectual generosity (consider the sheer length of uninterrupted discourse), here reflects the rational goodness of the Demiurge.

But none of this yet answers the question posed by Scolnicov, namely 'whether ... [the exemption of earth from the cycle of transformation] is a consequence of Plato's inability to accommodate the theory or an aspect of the theory introduced to account for certain phenomena' (62n42), the intended 'theory' presumably being that the cycle of transformation 'occurred according to the mathematical laws inherent in the structure of each element' (62). But this may be a false dilemma, for the dodecahedron (incapable of exhaustive analysis into these primary kinds of triangle) is assigned to the cosmos as a whole, and that cosmos is not a 'phenomenon' in any ordinarily accepted sense of the term. Now if the philosophical assignment of geometrical solids depends in part upon considerations other than the actual phenomena, even if in some fairly clear cases it surely appeals to them, e.g., the thoroughly familiar water cycle, then a similar consideration might well account for the exemption of 'elemental' earth. If it was on primarily philosophical grounds that earth was located at the centre of the universe, then it would be extraordinarily important that it not transform into anything else, and no better way to achieve this end is imaginable than ensuring (by fiat) that it is built out of a species of triangle which, being irreducible to the other species, is so to speak completely immune from transformation into fiery, airy, or watery Receptacle. The same reasoning ensures that Timaeus' pentagonally defined cosmos will not be dissolved into its triangularly defined components (thus overcoming geometrically the cyclical cosmology of Empedocles), and in this way his universe is everlastingly secure.

3 Five Worlds or One?

Timaeus has already answered the question of the uniqueness of the world in 31aBb; his return to the issue at 55cBd, especially since his answer remains what it was, has needlessly puzzled both Taylor and Cornford. (35) If arguments can be made invincible by revisiting them on the basis of maximally secure ones, as Timaeus seems to hold at 29b8, this passage constitutes a highly dramatic example of that very point. Taylor remarks that 'if only we knew more about fifth-century Pythagoreanism (36) we should probably find that there is an allusion to some division of opinion in the school itself which accounts for this sudden return to a question already disposed of' (378). Cornford does not fare much better himself, however: 'either our passage must be given up as inexplicable, or we must see in it a veiled allusion to the possibility of a fifth form of body [made explicit only in the Epinomis]' (221). (37) The cognitive sense of the target language is quite helpful in this case. Timaeus says that only an intellectually juvenile person would think it were reasonable for there to be an actual infinity of worlds--on the other hand, as to whether it makes sense (prosekei) to say, taking nature as one's criterion, that there are really one or five worlds, the juvenile's breath would likely be completely taken away if, the moment that question arose, he stopped right there and just reflected [sc. on it]. (38)

Timaeus' casting his vote for his own account implies that he considers it to be more likely than any other rival and therefore that all rivals are judged to be less likely. But likelihood is a matter of degree, as Donini has shown in the case of eikos. In the present case, accepting the likelihood of any account will be a function of specifying a reason, and estimating relative likelihoods will be a function of the strength of the reason appealed to. Consequently they will range from maximally unlikely to maximally likely: the former is instanced by materialism, which naturally goes with--i.e., is implied by, but need not imply--any proposed infinity of worlds; the latter by Timaeus's own account. The former is maximally unlikely not because it is so fundamentally opposed to Timaeus' espoused teleology but rather because no actual infinity can in fact be thought, even if it can be vocalized; the latter is maximally likely since it is anchored securely in inherently intelligible Forms (cf. his metaphysical argument for cosmic uniqueness indicated at 31a2Bb3). What is needed is some plausible reason for his suggestion that five rather than any other definite number is not entirely unreasonable, and it is just such a reason that Taylor and Cornford have sought.

But there is no need to look as far back as Taylor would have it, and it might be helpful to consider the matter much less concretely than Cornford suggests. While it is probably hopeless to plumb the bottomless pit of Pythagorean numerology, it seems clear that Plato took it--and likely also Philolaus' application of it to cosmology--very seriously. Our comprehension of the Timaeus, consequently, will be enhanced to the degree that we, unlike the absent number four in the dialogue's opening line, are not too exhausted to be able and willing to do the same. Whatever else might be said about the 'one, two, three' with which the Timaeus curiously begins, they refer to something other than persons who will give accounts since, as Critias makes abundantly clear at 27aBb, only he and Timaeus will be speaking thematically: for all his attested competence to follow the ensuing discussion, Hermocrates is and remains a witness. Since the accounts to be presented evidently require no more for their adequate execution than 'one, two, three', the answer to the one-or-five question posed by Timaeus may be hidden, so to speak, right before our eyes.

To actualize this possibility we need to consider briefly Pythagorean number theory, according to which the first even number is two, the first odd number is three, and the principle of even and odd number generically, though not itself a number, is one. (39) It is not hard to see, using pebble arithmetic, a close association between odd numbers and squares having the same shape but different sizes, on one hand, and, on the other, even numbers and rectangles having different shapes as well as sizes. (40) As the primary 'oblong' and 'square' numbers, then, 'two' and 'three' respectively may be taken as perfect mathematical expressions of Difference and Sameness, and in this regard they permeate Timaeus' entire cosmos, although the articulation of the world-soul involves the mathematical operations of squaring and cubing, i.e., multiplication not simply by these numbers but by these numbers to these same powers, essentially a highly efficient alternative to counting the old-fashioned way: one, two, three, etc. But Plato as author is not here simply recycling old-school Pythagoreanism: he is, rather, updating it metaphysically, and this is significant. The difference between metaphysical and non-metaphysical Pythagoreanism may be clarified by contrasting their respective conceptions of nature: for the Platonist, nature includes not just immaterial principles governing the cosmos (i.e., Being, Location, and Becoming, 52d) but actual immaterial entities as well (e.g., Forms, 27d and 48e)--indeed, in keeping with his preference for the immaterial, materiality ends up not really being very 'natural' at all, given the deep connection between the natural and the stable (49d); (41) on the other hand, for the pre-Platonic Philolaus, our best source for Pythagorean cosmological speculation and very well discussed by Huffman, nature includes no such immaterial entities even if it involves something close to immaterial principles, e.g., Harmony and Limiters (though probably not Unlimiteds). (42) Thus Timaeus' conception of nature is considerably wider, more complex, and he would surely say more meaningful than is its Philolaic ancestor, since for Timaeus something strictly 'supernatural' is literally part--in fact the chief part--of the natural package. Accordingly, Plato can, without too much trouble, have Timaeus interpret the old-school Pythagorean physical conception of Limit non-physically as Sameness (= three) and, similarly, Unlimited as Difference (= two), rather than simply associate them. If Philolaus' Harmony was replaced by the Demiurge, then Sameness and Difference might be almost literally joined together (although they were naturally 'hard to mix', 35a8) by him to produce the physical world. Mathematically, the resulting harmony would be expressed quite simply by adding two and three to get five: the 'second-best option', for Timaeus. (43) It is second-best on the pre-Platonic Pythagorean cosmological assumption that the physical world exhausts reality, an assumption Timaeus does not himself share, despite the evidently hypothetical status of the Forms and the Demiurge. (44) But that some Pythagorean cosmology would be the best one seems quite beyond question, for Timaeus, on this interpretation of the passage.

The sentence immediately following the text cited is especially interesting in that it invokes the 'likely account' together with everything that has been said about it so far; the 'reflective' account, as it might more accurately be regarded in this context, and as I have suggested above is deeply related to reflection as a mental process, is most profoundly conceived as a reflection upon reflection itself, or thinking about thinking. It is after all only careful thinking that has been able to distinguish clearly between time and eternity and (as argued above) revealed that the moment is eternity. No wonder the intellectual juvenile's breath would be taken away: s/he would have suddenly realized that, not only can there be only a single world because, given the metaphysical machinery earlier invoked--and subsequently applied to the very accounts involved--real unity effectively governs multiplicity, but also, and more importantly, s/he would have discovered the value of the physical universe as a signal, by virtue of its status as an image, which refers beyond itself to its original, i.e., to ultimate value and thus to his/her personal, rational kinship with that ultimate value.

3 Conclusion

If this application of the cognitive sense of the target language should win scholarly acceptance, then for Timaeus it is mathematics which alone yields cosmological knowledge, and this in two related senses. In the first place, as a moving image of the eternity upon which it is ontologically dependent, the cosmos might be understood as the expression of the derivative numerical principles of Even and Odd, reflected as they are in physical bodies in terms of Difference and Sameness. In the second place, these derivative numerical principles, like the cosmos itself, point directly to that from which both they and it are derived, viz., the Pythagorean 'Even-Odd', updated by Plato via Timaeus to the 'One', i.e., non-physical, eternal Being. In this way, it does not seem too much to claim that in this dialogue mathematical knowledge has an essentially moral, even religious, aim: it is, as Timaeus makes clear at 47b-c, precisely owing to our numeracy that we might actually propel ourselves, through imitation of the celestial revolutions, toward the best of lives available to us. As a result of astronomical observation we have philosophy, he says, 'a gift from the gods to the mortal race whose value neither has been nor ever will be surpassed' (tr. Zeyl).

But Plato as author knows that most people are not philosophers, and thus will not be able to properly appreciate the value of such a gift, and so to make adequate use of it. He is evidently not much concerned with humanity at large but much more specifically with his own students (and perhaps philosophical colleagues as well), and Donini's thesis about the varying intensities of eik-, which I have argued above suggests similar varying intensities of prep- and prosek- in the Timaeus, may help to illuminate why this is so. There are two distinct ways in which a proposal might be considered 'stronger' or 'weaker': either 'it is likely' (compared to its weaker counterpart, 'it makes sense'), or 'it is more (or less) likely'. It is important to note that while likelihood could never sensibly function as a premise for reasonableness, the latter could easily do so for the former, and it seems to me in fact always does do so, since the generally accepted criterion of likelihood is nothing more than relevant, informed, and unbiased professional opinion. For nonprofessionals, professional opinion would feel more like knowledge than what it actually is: opinion. Those who are in the best position to know the difference between knowledge and opinion, and so to tell whether their own views count as one or the other, are just the experts. This expert opinion in turn is measured in terms of how much sense a given proposal makes of the relevant data to those professionals, and is (however cautiously and provisionally) expressed in the essentially democratic vote of those who, by nature or training, have become competent to cast and arbitrate votes worth taking seriously. There is consequently an irreducibly subjective component to the practice of empirical science.

This result can shed some light on a complaint leveled at Plato's cosmology by Vlastos, who claimed that Plato through the Timaeus impeded the development of Greek science by deriving fact from value: for Plato, he says, 'it would be better, more beautiful, if things were thus and so; ergo, they are thus and so' (30). Plato's argument is not, according to Vlastos, properly inductive but improperly deductive, and as such has no rightful place in empirical science. He regards Plato's contribution to cosmological inquiry ironically as a 'retrograde turn' (29) in comparison with that of his pre-Socratic ancestors. As I have argued above, however, the point specifically in relation to the operations of logical inference made in the Timaeus is that induction is strictly impossible unless it is grounded in deduction, for any judgement concerning likelihood presupposes a judgement concerning certainty. In other words, the requirements for knowledge--whether or not one accepts Platonic Forms--alone can form a secure standard against which likelihood of being true can be ascertained.

This holds for all of empirical science, as the above investigation of the cognitive or epistemic sense of prep- and prosek- strongly suggests, and Vlastos' complaint is accordingly short-sighted. For while there is no question that for Timaeus (and in this respect also Plato, the 'mouthpiece problem' notwithstanding) empirical facts are not ultimate ones--in this way empirical claims are to be evaluated by a strictly nonphysical, non-empirically discernible standard--and that the ultimate fact is related to the ultimate value, if this is the substance of Vlastos' objection, it amounts to a rejection of the view that metaphysics or value has any positive role to play in empirical science. However, there is no good reason to suppose that Plato via Timaeus is interested here in abolishing or even disparaging empirical science; rather, he is simply articulating the metaphysical and logical basis upon which alone empirical science is able to do what it actually does: to understand some contingent sequence of events more or less well. The scientific method, we may note, is itself an expression of this value insofar as empirical data count for nothing without a coherent, organizing concept explaining the data in a cognitively satisfying way; no scientist could accept a theory in his or her own field of expertise as being likely to be true while not even understanding it. But if empirical science is limited to contingent truth, two further related issues arise: first, whether there could actually be nothing other than contingent truth; and second, if contingent truth implies necessary truth, whether necessary truth is actually related to anything besides other groups of words (cf. the possibly 'vacuous talk' about Forms). If the argument articulated above is cogent, then the first issue must be answered in the negative, supposing that the whole of science is ultimately based upon logic, and logic in turn upon the principle of identity (e.g., P has whatever truth-value it has). But the second issue can be answered in the affirmative only if the fundamental logical principle of identity is related to real identity (e.g., X is whatever it is). That there actually exists such a relationship in at least one case might be shown relatively easily by premising that the logical principle of identity really is exactly what it is, i.e., nothing other than the logical principle of identity; but how far this might be extended is well beyond the scope of this paper, though one might reasonably surmise that its extendability would crucially depend upon the premise that logic describes non-linguistic reality accurately. It is in some such way as this, it would seem, that Platonists might understand (I do not say 'discover', nor for the reasons indicated above do I believe Plato would) the truth about their stable stellar homes and, buttressed by this insight, focus their resources as much as possible upon living wisely.


Calvo, T. and L. Brisson, eds. Interpreting the Timaeus-Critias. Sankt Augustin: Academia Verlag 1997.

Cornford, F. M. Plato's Cosmology. London: Routledge and Kegan Paul 1937.

Dombrowski, D. A. Plato's Philosophy of History. Washington DC: University Press of America 1981.

Donini, P. 'Il Timeo: Unita del Dialogo, Verisimiglianza del Discorso.' Elenchos 8 (1988) 5-52.

Guetter, D. 'The Cognitive Sense of prep- and proshk-.' Phoenix 60 (2006), forthcoming.

Johansen, T. K. Plato's Natural Philosophy. Cambridge: Cambridge University Press 2004.

Lee, E. N. 'On the Metaphysics of the Image in Plato's Timaeus'. The Monist 50 (1966) 341-68.

Levin, S. B. 'Greek Conceptions of Naming: Three Forms of Appropriateness in Plato and the Literary Tradition'. Classical Philology 92 (1997) 46-57.

Liddell, H. G. and R. Scott, eds., rev. H. S. Jones, with rev. suppl. P. Glare. A Greek-English Lexicon, Ninth Edition. Oxford: Clarendon Press 1996.

Mohr, R. D. The Platonic Cosmology. Leiden: Brill 1985.

Mortley, R. J. 'Plato's Choice of the Sphere'. Revue d'Etudes Grecs 82 (1969) 342-5.

O'Brien, D. Theories of Weight in the Ancient World, Vol. II: Plato, Weight, and Sensation. Paris: Les Belles Lettres and Leiden: Brill 1984.

Plato. Timaeus, tr. D. J. Zeyl. Indianapolis: Hackett 2000.

Plutarch. de Defectu Oraculorum, in Moralia 5, tr. F. C. Babbit. Cambridge, MA: Harvard University Press 1936.

--. de E apud Delphos, in ibid.

--. de Musica, in Moralia 14, tr. B. Einarson and P. H. De Lacy. Cambridge, MA: Harvard University Press 1967.

Press, G. A., ed. Who Speaks for Plato? Studies in Platonic Anonymity. Lanham, MD: Rowman and Littlefield 2000.

Proclus, Commentary on the Timaeus of Plato, Vol. I, tr. T. Taylor. London: [the author] 1820 and Ann Arbor, MI: University Microfilms 1967.

Racionero, Q. 'Logos, Myth and Probable Discourse in Plato's Timaeus'. Elenchos 1 (1998) 29-60.

Runia, D. T. 'Plato, Timaeus 30b6-c1'. Elenchos 10 (1989) 430-43.

--. 'The Literary and Philosophical Status of Timaeus' Prooemium', in Calvo and Brisson 101-18.

Scolnicov, S. 'An Image of Perfection. The Good and the Rational in Plato's Material Universe'. Revue de Philosophie Ancienne 10 (1992) 35-67.

Smyth, H. W. Greek Grammar, Second edition, rev. G. M. Messing. Cambridge, MA: Harvard University Press 1956.

Taran, L. Academica: Plato, Philip of Opus, and the Pseudo-Platonic 'Epinomis'. Darby, PA: Diane Publishing Company 1975.

Taylor, A. E. A Commentary on Plato's Timaeus. Oxford: Clarendon Press 1928.

Trepanier, S. 'The Structure of Empedocles' Fragment 17'. Essays in Philosophy: A Biannual Journal 1 (2000) <> accessed 15 Feb. 2007.

Vlastos, G. Plato's Universe. Seattle: University of Washington Press 1975.

(1) Donini 42n73 also broaches the possibility of a relationship between opinion ([TEXT NOT REPRODUCIBLE IN ASCII.]) and likelihood ([TEXT NOT REPRODUCIBLE IN ASCII.]). Runia argues that Donini's highest and lowest levels are questionable, the former because Demiurgic goodness, for instance, is introduced already in the prelude, evidently not part of the 'likely account' ([TEXT NOT REPRODUCIBLE IN ASCII.]) but rather 'established on the secure basis of true knowledge' (1989, 440), the latter because Timaeus does not actually use traditional mythology, for instance, in his cosmological account (ibid., 437). Eight years later, Runia observed that not everything subsequent to this prelude is to be taken as more or less likely; for instance, of the true account ([TEXT NOT REPRODUCIBLE IN ASCII.]) at 52c6 he remarks, 'There is nothing likely about it' (1997, 112, emphasis added). This is no doubt correct, but such things as logical entailments within the [TEXT NOT REPRODUCIBLE IN ASCII.] (see, e.g., 38a on time and eternity below) are decisively more than maximally likely: they are as certain as things can get for humans, even if the premises are not themselves known to be true. An interesting question, but which cannot be pursued here, is whether or not the logical phenomenon of validity can form a sufficient basis for soundness. See also Johansen 50B63 for degrees of certainty in the Tim. in general, and in particular his identification (50) of truths relative to generation at 37b9, 51d6, and 53e3.

(2) Guetter (2006). The argument there is essentially an economic one: when the context clearly warrants a more precise sense (a particular sort of noticing, e.g., smelling or hearing, vs. noticing by means of sensation more generally), then there is every reason to interpret accordingly. It is, I note, not hard to extrapolate from noticing in a sensory manner to noticing in a cognitive one; see Pind Pyth 10 67, where gold is compared to an upright mind, the latter being susceptible to testing by a touchstone of the right sort.

(3) Johansen argues that the near-total (and, for Plato, highly unusual) monologue form of the Timaeus vastly increases Timaeus' chances of maximizing 'proportionality' in his account of the universe (193). It must not be forgotten, however, that the Timaeus is a work of heavily Pythagorean cosmic poetry which re-creates in words Plato's speculations about the natural world. The significance of the wellknown 'back-and-forth' character of the dialogue has often enough been advanced to show that Plato did not intend the cosmology as a literal cosmogony, but it might also reasonably be taken to represent Plato's attempt to modify theistically Empedocles' materialistic evolutionary cosmogony: Trepanier, for instance, has recently argued that Empedocles B17 presents not a series of 'false starts' but instead a significant correlation between world-growth and poem-growth.

(4) The 'target language', as I shall refer to it, also occurs in the methodological section concerned with the metaphysical principles governing natural science in general and cosmology in particular, but this large topic deserves more space than can responsibly be given to it here.

(5) Liddell, Scott, and Jones, A Greek-English Lexicon, s.v. [TEXT NOT REPRODUCIBLE IN ASCII.] and [TEXT NOT REPRODUCIBLE IN ASCII.]

(6) See Levin 49B50 for the reliance of accurate etymologies upon Forms.

(7) See on this point Taylor 189.


(9) [TEXT NOT REPRODUCIBLE IN ASCII.], 38a3-4. The case could also be put hypothetically: if there is anything changeless, it could not change in any sense, and if it could not change in any sense, then it would not change in any sense.

(10) The simultaneous predication of tallness and shortness to Simmias at Phd 102d is clearly restricted to perception. It is the recourse to the Forms of Tall and Short, however, which provides the ontological basis for perception (i.e., organized sensation) in the first place.


(12) For Plato's use of metaphor (and myth more generally) as didactic, see Racionero.

(13) [TEXT NOT REPRODUCIBLE IN ASCII.], 51c4; a very strong, logical sense of implication seems a significantly more reasonable construal in this case than does anything like an obligation. It is an interesting question, but beyond the scope of this paper, regarding the possibility of Plato's having derived ontological obligations from logical implications.



(16) Mortley argues that Plato chooses a sphere because no other shape will work, i.e., preserve the phenomena of celestial rotations, given his denial of intra- and extracosmic void.


(18) I.e., was generated ([TEXT NOT REPRODUCIBLE IN ASCII.], 38b6 and 38c6).

(19) See Smyth 319 #1336.

(20) Taylor 188 says that 'time is not [thought of here as] something existing before or after or besides the events which make up the life of nature, a sort of frame into which the events are put, which might still be something with a structure of its own, if there were no events to fill it'. Cf. Cornford 102: time is 'a feature of ... [the physical] order'.

(21) I do not mean here post-cosmic measurable units of time, i.e., ordered succession (for which see Mohr 19 and especially 53-81), but instead bare succession. Such succession is naturally susceptible only to wildly unstable measurements, but these 'seeds' of measurable time nonetheless admit of a not completely useless distinction between 'before' and 'after', especially if, as would seem likely, the pre-cosmic shaking (52e-3b) was envisioned as cyclical. The imposition of structure (e.g., year, month, day, hour) upon generic succession (future/past), like the imposition of structure (e.g., metre, centimetre, millimetre, nanometre) upon space, does not rule out pre-cosmic 'seeds' (less biologically conceived as ikne, 'traces', 53b2) of either space or time. What is certain is that in Plato's estimation, without divine nurturing--a type of change--these seeds will never develop but on the contrary will remain everlastingly dormant, no matter how much change of any non-divine sorts there might be.

(22) Cf. Dombrowski 150: '[the] self-sameness [of sensible things] is only a derived sort of self-sameness'.

(23) The anticipation or memory is itself neither in the future nor in the past, respectively, but entirely in the present, a point to which I shall return below.


(25) For the 'mouthpiece problem' in Plato, see Press. The general consensus of the collection's contributors is that claims to the effect that any particular character (e.g., Socrates, Timaeus, the Eleatic Stranger, the Athenian Stranger, or some other 'leading speaker') expresses Plato's own specific dogmatic views are very hazardous at best.

(26) This not only enables a certain philosophical detachment from the ordinary goings-on endemic to life, it might also propel one into the realm of the absolutely true, good, and beautiful, and from perfection it would be extraordinarily difficult to take one's leave, as is implied by the prospect of having to force the philosopher-king to go into political life (R 499c).The business of shaping the habits of other people directly would likely be tedious for him, but he might be persuaded to consent to serve as a model for the others (while--perhaps even by--engaging himself in philosophy), just as the Demiurge serves as the model for Timaeus and the Form of Living Creature as the model for the visible universe and its various parts.

(27) See Tht 202dB8b for an analogous discussion of the inadequacy of defining knowledge by the method of resolving X into its component parts: the initial syllable of [TEXT NOT REPRODUCIBLE IN ASCII.] is first considered (SV-), then [TEXT NOT REPRODUCIBLE IN ASCII.]- is resolved into its constituent letters S and V. Clearly the more minute the analysis, the less intelligible the result, as would be even more apparent had they considered next only the upper horizontal stroke (or, to update the point, a single pixel) of [TEXT NOT REPRODUCIBLE IN ASCII.]. The same problem can be found in Arist Protrepticus B33.


(29) See Tht 205e in particular for a refutation of Taylor's suggestion here. It is true that knowledge is never finally defined in this dialogue, but Socrates is clearly aware (209dB10d) that the third of his possibilities, viz., the statement of something's defining characteristic, is question-begging.

(30) '[Intelligence] is confuted by the [physical] thing itself, as not being able to comprehend the nature of it, such as it really is' (Vol. I 288, tr. T. Taylor).

(31) In short, volumes are wholes and wholes are not units. The inherent clarity of a mathematical example is a Platonic commonplace; for its role in the Theaetetus, see 204bBd.

(32) See also O'Brien 84n14.

(33) Cornford (59-66), appealing to Proclus, explains the cognitive soul's nature--a mixture of the Indivisible and Divisible varieties of Being, Sameness, and Difference--in terms of its proper task: to produce true judgements concerning Sameness and Difference in the contexts of permanently existing, ungenerated intelligible objects and non-permanently existing, generated perceptible objects. Since Plato accepts the thesis that like knows like, he continues, the cognitive soul could not have both knowledge and opinions unless it had as constitutive features of itself both intelligibility and generation respectively. However, this account may need modification, if it is not simply an opinion but rather knowledge that perceptual object X is after all perceptual object X rather than Y. Perhaps, then, the intermediate nature of the mixture constituting the soul is more adequately conceived as necessary given that it is to be able to tell the difference, not simply between the members of the classes of knowledge and opinion, but more generally between the natures of these respective classes.

(34) As Johansen has also observed, without a mathematical understanding of matter, one might as well be living in the pre-cosmos (124): no (Platonic) knowledge will be possible, and this will effectively undermine the overarching ethical goal of the dialogue (22).

(35) In de Defectu Orac 421f-2a Plutarch reports Plato's selection of anywhere up to five worlds as 'a reasonable probability [[TEXT NOT REPRODUCIBLE IN ASCII.]] to those who postulated one world to correspond to each element' (tr. Babbitt; cf. de E 390a, where a similar point is made). But this 'elemental' interpretation is only one of a number which Plutarch considers: at de E 389d-f, for instance, a 'musical' interpretation is discussed, according to which there are five elements of melody, namely the quarter-tone, the half-tone, the tone, the tone-plus-half-tone, and the double-tone; the author of de Musica emphasizes the relation between harmony and both sight and hearing at 1140b as well, at 1147, as that between harmony and the entire universe, listing by name Pythagoras, Archytas, and Plato. It is hard to imagine that Philolaus B6 (see below) should not be counted as relevant here.

(36) Taylor's peculiar interpretation of the dialogue, which has won no scholarly acceptance, is otherwise a treasure chest of erudition.

(37) Taran argues at length that the Epinomis is not Plato's.


(39) See Plut de E 387fB8a.

(40) For the table of opposites, see Arist Metaph A 5, 986a22.

(41) illustrated, for instance, by the tenuous grasp of physical bodies--i.e., images of Forms in the Receptacle of Becoming--clinging to existence in whatever way they can at 52c, discussed in the 'Basis' section above

(42) See Philolaus B6 for Harmony, Limiters, and Unlimiteds. Cosmogony begins with the work of Harmony, which I would argue is 'updated' by way of Plato's Demiurge, i.e., given something resembling a face.

(43) See Plut de E 387fB8a: 'since two makes the first of the even numbers and three the first of the odd, and five is produced by the union of these numbers, very naturally [[TEXT NOT REPRODUCIBLE IN ASCII.]] five has come to be honoured as being the first number created out of the first numbers' (tr. Babbitt); cf. Plut de Defectu Orac 429a: if the one, the very measure of counting, is done away with, so is the rest of number. This has important consequences: 'two is the first of the even numbers [associated with corporeality], and three the first of the odd [associated with form]; from the two combined comes five, which in its composition is common to both numbers' (tr. Babbitt). The Pythagorean assignment of five to marriage (e.g., Plut de E 388a) makes a good deal of sense given their association between three/odd/male and two/even/female in the familial analogy discussed above.

(44) Prefacing his account of Being and Becoming by 'as far as I'm concerned' ( [TEXT NOT REPRODUCIBLE IN ASCII.], 27d5) ensures this. See also 51c regarding the possibility that Form-talk is, after all, vacuous. The hypothetical status of the Demiurge is supported by his being replaced by Reason or Intellect at 47d, but curiously he seems reintroduced, though only by implication from the world's generated status, at 92c; cf. 28b-c.

David L. Guetter

Department of Philosophy

University of Windsor

Windsor, ON N9B 3P4
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