Intelligible Forms, Mathematics, and the Soul’s Circles : an Interpretation of Tim. 37a-c
- 1 Here I take up in greater detail part of my article Il Timeo e la matematica embodied, in La sapien (...)
- 2 See, for example, L. Brisson, Come rendere conto della partecipazione del sensibile all’intellegibi (...)
1In this article I propose a solution of a problem that seems to me fundamental for understanding the Timaeus and Plato’s onto-cosmological perspective in general1. I shall be trying to give an account of the sense of the claim as to the mathematical character attributed to the relation between the sensible and the intelligible. I shall try to make explicit what in Plato’s text remains mostly implicit, and to give an answer to the following question : how is the order of intelligible ideas, which is essential or formal in character, translated into the mathematical order (arithmetical or geometrical) of sensible things ? That this is what happens in the Timaeus has been, as far as I can see, amply demonstrated2 ; what seems to be lacking, though, is the last link in the explanatory chain of this interpretive model, and that is what I shall try to bring to light here.
- 3 See on this point G. Vlastos, Plato’s Universe, Parmenides Publishing, Las Vegas 2005 (first ed. Ox (...)
- 4 See for example Resp. I 353d ; Soph. 248b-e ; Leg. X 896e-897a.
2After describing the constitution of the world-body by the demiurge, Timaeus points out that the world needs to have a soul too if it is really to be a completely “living” being. That is why the world-soul is introduced, and its constitution follows that of the world-body in Timaeus’ exposition, even though, according to the actual sequence of the demiurge’s actions, it is certainly produced before the body, so that, being older and thus superior to the body, it can dominate and direct it for the best (34a – 40d). The world-soul is produced from a mixture of being, sameness and difference, but not of the intelligible ideas of being, sameness and difference, nor of their corresponding sensible properties, but of a mixture of both : in this merely negative way Timaeus illustrates the intermediate nature of the world-soul and its middle place in the hierarchy of reality. There is nothing accidental in the mixture of being, sameness and difference that the demiurge effects to produce the world-soul, as it follows a precise mathematical order, establishing rigorous numerical proportions for the combination of the three elements. Once this mixture has been achieved, the demiurge divides the material obtained into two equal strips, which he first lays out in the form of a X, and then bends to bring together the extremities of the two strips, obtaining two concentric circles, one tilted in relation to the other, like the equator in relation to the ecliptic, intersecting at two opposite points. The first, the circle of the same, rotates outside the other, the circle of the different (34a-36b, see fig. 1) ; the movement of the circle of the same is perfectly regular and uniform, single and undivided, while that of the circle of the different, though regular and uniform, is divided six times, creating seven circular, concentric trajectories, each characterised by different movements (36d ; see fig. 2). Immediately after, in 36e-40d, the question of the functions pertaining to the world-soul is introduced, as the demiurge attributes it with a moving function from which a cognitive function also derives. This means that a single hypothesis, that of the existence of the world-soul, can explain two kinds of phenomena. First, by virtue of the circular movements of the two circles, the world-soul impresses and maintains the regular movement of the heavenly bodies, but also the movement of the earthly bodies, which is an irregular movement that can nevertheless be assimilated to the movements of the world-soul, insofar as the geometrical-mathematical structure characterising the world-soul corresponds, in gradually descending degrees, to that pertaining to all bodies, heavenly and earthly, that have a soul and body3. Secondly, as a soul not only lives and moves, but is also intelligent and has cognitive attitudes – insofar as intelligence and knowledge consist in a series of movements produced or received4 – cognitive attitudes depend on the moving capacity of the world-soul. When knowledge is achieved through the movement of the circle of the same, the world-soul has true knowledge of the intelligible objects which the circle of the same is always turned to and closely related with ; but when knowledge is achieved through the movement of the circle of the different, the world-soul only has (true) opinions, and perceives the sensible objects which the circle of the different is always turned to and closely related with (37a-c). The circle of the same produces intellection and science (νοῦς ἐπιστήμη τε) because, as it is the circle of the same, the sameness that it ensures between the knowing subject (the soul) and the object known (the intelligible) guarantees the absolute and immediate truth of the knowledge obtained, while the circle of the different produces true opinions (δόξαι καὶ πίστεις ἀληθεῖς) because, as it is the circle of the different, it is the difference that it establishes between the knowing subject (again the soul) and the object known (the sensible) that thus determines that discursive judgement that is opinion. In other words, through the circle of the same the world-soul is identified with its intelligible objects ; while, through the circle of the different, it preserves its difference from the sensible objects with which it enters into relation, a relation that remains always partial to the extent that it does not allow an effective identification of the soul with its objects, thus producing mere opinions.
- 5 For this interpretation see F. Fronterotta, Platonismo e scienze della mente: cosa è l’intuizione?, (...)
3But how is the identity between the circle of the same and its intelligible objects achieved, and why does the circle of the different remain different from its sensible objects ? It is plausible that this can be explained by observing that the circle of the same, which produces intellection by contact (ἐφάπτηται) with intelligible objects, moves by rotating on itself with a perfectly regular motion, which leads it to meet its objects, not suddenly and on the wave of a disorderly movement that produces an unseemly collision, but with a contact produced by a regular and uniform movement of approach. This ensures that the objects “touched” are impressed on it, or rather literally subsumed under it, in an orderly way ; while the circle of the different, which produces opinion by contact with sensible objects, moves by rotating on itself with a movement that is also pure, regular and uniform, but less so than the movement of the circle of the same : the latter is called single and undivided (μίαν…ἄσχιστον), while the former is affected by division and disparity (σχίσας…κύκλους ἀνίσους), and so by multiplicity and difference (36d), which leads it to meet its objects in a violent and unseemly collision, so that the objects “touched” are not impressed on it, or subsumed under it, in a stable and steady order (see fig. 3). This cognitive mechanism is probably extended from the world-soul to the individual, embodied souls5.
- 6 In this context I am silently assuming that the world-soul has the double task of reproducing and p (...)
- 7 I know that there is something paradoxical in this term when used to refer to a non-sensible realit (...)
- 8 On the progression and propagation of this movement, see also Laws X 893c-d and 894c.
4This explanation of the moving and cognitive functions of the world-soul takes us to the heart of the question concerning (1) how the world-soul operates in mediating between the intelligible and the sensible or effecting the transposition of the perfect order of the intelligible into the partially regulated organisation of the sensible ; and as this mediation is implemented by virtue of the mathematical model or, still more, this very transposition is mathematical in character, (2) the actual, concrete role of mathematics in the Timaeus6. The mechanism illustrating the cognitive attitudes of the world-soul also allows us to explain how it manages to reproduce the formal order of ideas in sensible things and why this reproduction has the features of a geometrical-mathematical arrangement of these sensible things. As we have said, the world-soul and every individual embodied soul have two circles, the circle of the same and the circle of the different, which make contact with intelligible and sensible objects respectively. However, as we are now concerned with the transposition of the intelligible order in the sensible universe, we can limit ourselves to considering the world-soul (because we are examining only its macrocosmic and cosmological activity and not the microcosmic and psychological activity of each individual embodied soul) and its circle of the same (because we are examining how the intelligible order is transposed in the sensible, and that presupposes exclusively knowledge of ideas through the circle of the same). And here is the question we started from : what happens when the circle of the same “touches” an idea ? Although this contact is not sensible and material, and so is in some sense metaphorical, it has to be sufficiently real to produce what we call knowledge or intellection, which is an affection of the soul, a passion, and so, in some sense, a modification of it7. I am inclined to think that, although we have here an intelligible and immaterial object – the idea – and a subject that is not sensible and is equally immaterial – the soul – contact between them produces such an effect that the subject is modified, and that knowledge or intellection consists precisely in this modification. But, keeping to the model illustrated in the Timaeus, the circle of the same consists of a circular trajectory that, by virtue of its uniform rotation, moves the circle of the different and, through it, the other six concentric circles that it governs (36d). The transmission of this movement too is naturally by contact, since the circle of the same and the circle of the different were laid out by the demiurge in two strips overlapping at one point, and then folded in the form of a X, bringing their extremities together (36b-c). They therefore intersect at two points, and it is obviously through these two points of intersection that the transmission of movement takes place. Later, the circle of the different, whose circular movement embraces and accompanies that of the six concentric circles that it includes beneath itself, transmits to them in turn this movement, which then propagates itself, gradually declining through the inferior and inner circles, until it reaches the centre of the soul, which is united at the centre of the world-body ; and that is how the movement of the soul, starting from the perfectly regular and uniform movement of the circle of the same, gradually descends, multiplying and diversifying itself, finally reaches the body and can thus actually animate it8.
- 9 For eternity, because the rational and immortal soul’s knowledge of ideas seems to have the charact (...)
- 10 Participation should have the double character (1) of eternity, regarding the overall arrangement o (...)
5But what is actually transmitted in this transmission belt ? Certainly movement, in the ways we have just illustrated, but not only that. When the circle of the same “touches” an idea, it sustains its contact, which leaves an imprint in it of the idea that has been “touched”, and is thus modified by it. The circle of the same thus continues on its perfectly regular and uniform trajectory, but the point where it has “struck” the idea is modified by the essential form that the idea has impressed on it. It is certainly not a figure in the strict sense, because there is no material, in the strict sense, that can receive its features, but, as the circle of the same, like the whole soul, has a geometrical-mathematical structure, the modification produced by contact with the idea will appear as a geometrical-mathematical modification, supposing, for example, that the circumference of the circle of the same is subject to a numerical variation or a quantitative oscillation of the elements that make it up at the point in which the contact with the idea took place. Now, continuing in its rotation, the circle of the same intersects the circle of the different at two given points, in each of which, in the course of the rotation, it will pass on (for as long as it preserves a trace of it, i.e. very probably for eternity) the modification undergone by the circle of the same as a result of its contact with the idea9. This means that the circle of the same not only transmits to the circle of the different the regular and uniform movement that characterises it, but also impresses on it the modification that it has undergone through contact with the idea and whose impression it still has – so it actually transmits a circular trajectory that has at one point a modification corresponding to the essential form of the idea “touched” by the circle of the same, i.e. to the numerical variation or quantitative oscillation of its components that has occurred at that point. It is not difficult to imagine what follows in this process of transmission through the concentric circles marked inside the circle of the different, and then, by virtue of the contact between the world-soul and the world-body, in the world-body. It is an extension or radiation of a trajectory and a movement, but also of the modification of this trajectory which has the form of a numerical variation or quantitative oscillation that, impressed on a sensible material (the world-body), appears as a concrete and visible form, deriving from a particular distribution of the geometrical elements that make up the body of all things and from their consequent spatial arrangement (see fig. 4). Applying this explanatory model on a large scale, extended to the totality of ideas “touched” by the world-soul, which are known partly simultaneously (thanks to the probably synoptic cognitive attitudes of the world-soul), and partly in succession, it will be possible to justify the complex arrangement of the universe, its stable and eternal condition as well as its transformation and becoming10 ; just as it will be possible to explain why and how the purely formal order of the intelligible is translated, in the sensible world, into a geometrical-mathematical quantitative order.
- 11 One also needs to ask how the soul can turn to knowledge of the intelligible, by recollection or re (...)
6As far as I can see, nothing prevents the same mathematical model from explaining, at the level of the individual embodied soul, how each soul directs its body, functioning in turn as a transmission belt (1) of the intelligible order, through its circle of the same which entails the proper functioning of the body, but also, being an embodied soul placed in the sensible world, (2) of sensible disorder, through its circle of the different which entails the malfunctioning of the body. Naturally, as it is in this case an embodied soul, the strongest and most immediate cognitive signal reaches it now through and from the body so that, being in touch with external sensible reality, the body is affected through one of the five senses, which undergoes a physical deformation, whose effects are transmitted to the soul (through special corporeal transmitters, i.e. through the marrow, a particularly subtle substance that is steeped in the soul, to reach the inferior species of the soul), finally touching its circle of the different (43a-44b, 61c-68d, 73b-d). Dealing with signals coming from sensible objects that are changeable and unstable, the circle of the different that is affected by them will suffer repercussions and deviations of its movement, thus disturbing or even upsetting the whole soul, and through it, the body of which it is the soul, overturning its natural equilibrium. But when the individual soul is turned to knowledge of the intelligible (41c-d, 44b, 47a-c etc.)11, then a virtuous mechanism is set off analogous to that previously described for the world-soul : the non-sensible impressions left by contact with ideas on the circle of the same, transmitted to the circle of the different, and so to the lower orders of the soul and then to the body, impress on the whole psycho-physical organism a regular movement and an orderly disposition that re-establish the equilibrium of the whole.
7It follows that, since, unlike the world-soul, the disposition of the individual soul can be unbalanced or overturned, we need to distinguish between the condition that occurs when the circle of the same is in control, with the intelligible acting as a model for a properly conducted life, and the condition that occurs when the circle of the different is dominant, with sensible realities acting improperly as a model for a life that is irregular and unhealthy : it is thus the circle of the same that should always be in command, thanks to the exercise of philosophy and the contemplation of the intelligible (access to which comes through the study of the geometrical-mathematical structure of the visible universe), while the circle of the different, with its tendency towards opinion and the sensible affections, should always be subjugated.
1 Here I take up in greater detail part of my article Il Timeo e la matematica embodied, in La sapienza di Timeo, ed. L. Napolitano Valditara, Vita e Pensiero, Milan 2007, pp. 173-206, where some bibliographical references can be found.
2 See, for example, L. Brisson, Come rendere conto della partecipazione del sensibile all’intellegibile in Platone?, in Eidos-Idea. Platone, Aristotele e la tradizione platonica, ed. F. Fronterotta & W. Leszl, Academia Verlag, Sankt Augustin 2005, pp. 25-36.
3 See on this point G. Vlastos, Plato’s Universe, Parmenides Publishing, Las Vegas 2005 (first ed. Oxford 1975), pp. 50-54.
4 See for example Resp. I 353d ; Soph. 248b-e ; Leg. X 896e-897a.
5 For this interpretation see F. Fronterotta, Platonismo e scienze della mente: cosa è l’intuizione?, in Platonism and Forms of Intelligence: Proceedings of the International Symposium of the Institute of Philosophy, Zagreb, Hvar 2006 (forthcoming). There are of course some important differences between individual souls and the world-soul : embodied individual souls (1) are weighed down by a kind of inferior, irrational and mortal soul that weakens them (42a-b, 43c-44b) ; they (2) are in the body (42a, 43a) and are hindered and made impure by it ; (3) as these bodies are mortal and corruptible, they interrupt and make discontinuous the eternal course of the vicissitudes of the immortal soul. In the framework of the functional analogy between the cognitive attitudes of the world-soul and embodied individual souls, one might ask how it is possible for the opinion produced in the contact between the circle of the different of the world-soul and sensible objects to be always true and, above all, steady (δόξαι καὶ πίστεις βέβαιοι καὶ ἁληθεῖς). Actually, the world-soul is in some way a “super-soul”, different from the individual soul, in that (1) inferior, irrational and mortal functions are not added to it, (2) it does not enter a body that dims and binds it, but it envelops a body, dominating it entirely (34b, 36d-e), and also (3) the body that it governs is in turn a particular body, that of the whole world, which, though sensible like all bodies, is nevertheless eternal and incorruptible because of its good constitution (33a, 36e, 37d, 38b-c). Yet, despite these differences, it is still surprising that, in this case, opinion, which, according to the lesson of Meno (97c-98a) for example, is fundamentally changeable and instable, is said to be always true (which might be considered exceptionally difficult, but not impossible), and also steady and stable. Even though this problem is beyond the scope of the present work, I believe that the solution lies above all in (3) : inasmuch as the world-soul’s opinion is determined in contact between the circle of the different of a pure and purely rational soul (the world-soul) and objects which are sensible, but also stable, eternal and incorruptible (because, clearly, the only sensible object which the world-soul can deal with is the world-body), it appears to have the rather strange features of truth and stability. Although it is represented as an opinion, it is distinguished from the intellection of the intelligible only by the corporeal nature of its sensible objects.
6 In this context I am silently assuming that the world-soul has the double task of reproducing and preserving the ideal order in the sensible universe, so suggesting a metaphorical interpretation of the mythological figure of the demiurge, who would be an image of the pure, noetic function in which the world-soul consists : this question is, however, much debated, see F. Fronterotta, Questioni eidetiche in Platone: il sensibile e il demiurgo, l’essere e il bene, in “Giornale critico della filosofia italiana”, 85 (2006), pp. 412-36.
7 I know that there is something paradoxical in this term when used to refer to a non-sensible reality, but Platonic and Aristotelian epistemology make the point clear enough : even if the actions and passions, contacts and deformations produced by a contact take place between sensible and material things (because one can register a contact and its consequences between corporeal realities, with their reciprocal actions and passions), we must suppose that the same processes can take place on the intelligible level of immaterial realities, like the soul and the ideas it “touches” : in this case too we shall have a contact, by which the soul is affected by the idea that it has “touched”, which leave in it an imprint corresponding to the form known, a psychic trace that remains in the soul and is the basis of knowledge and reasoning : see particularly Phaed. 79d, in which φρόνησις is described as a πάθημα of the soul, when it is turned towards what is « pure, eternal, immortal and immutable » and « remains always immutable in relation to its objects, because so are the objects that it touches » ; but there are many verbs referring to physical contact in the context of intelligible knowledge (see for example Symp. 212a4 ; Phaed. 65b9, 79c8, d6 ; and, in general, the closing pages of book VI of the Republic, especially 511b-e, in the context of the noetic section of the image of the line). We know that Aristotle, in De an. III 4, 429a10-20, and 5, 430a10-20, draws a parallel between thinking and perceiving, between which there is an analogy, by which thinking is the passion suffered by the rational soul for the action of the intelligible.
8 On the progression and propagation of this movement, see also Laws X 893c-d and 894c.
9 For eternity, because the rational and immortal soul’s knowledge of ideas seems to have the character of eternity, even though the soul may temporally forget its possession of them, as is illustrated by the doctrine of recollection introduced in the Meno, the Phaedo and the Phaedrus (see F. Fronterotta, ΜΕΘΕΞΙΣ. La teoria platonica delle idee e la partecipazione delle cose empiriche. Dai dialoghi giovanili al Parmenide, Edizioni della Scuola Normale Superiore, Pisa 2001, pp. 91-100). Actually, the explanation above of the mechanism of intelligible knowledge may enable us to clarify this point. The imprint left by a sensible object on a material substratum (1) tends not to last in time, as a material distortion is subject to time and so to gradual dissolution, and (2) various imprints can be superimposed on the material, gradually becoming confused with each other. By contrast, the imprint left by an intelligible object on a non-material substratum remains (1) eternal and (2) clear, precisely because (1) what is immaterial cannot suffer degeneration, and so remains immutable over time ; (2) nor does it have a spatial dimension which entails a physical distribution of the impressions registered that may improperly coincide.
10 Participation should have the double character (1) of eternity, regarding the overall arrangement of the cosmos and the foundations of sensible nature, and (2) temporality, regarding the specific changes of state that pertain to each particular sensible being in the course of its individual becoming. Concerning (1), the world-soul has access to simultaneous knowledge of all the ideas whose order it reproduces in the cosmos as a whole ; concerning (2), however, the world-soul has access to a selective knowledge of individual ideas whose order it reproduces each time in each individual sensible being in the course of its becoming.
11 One also needs to ask how the soul can turn to knowledge of the intelligible, by recollection or reflection or in some other way. In any case it is likely that sensation is the first and most immediate instrument of knowledge for the individual embodied soul ; and that is why Plato explains that, on entering the body, every individual soul is agitated and upset (42a-b, 43a-44c) : outside the body it was essentially ruled by the circle of the same and turned towards intelligible knowledge, while in the body its new condition favours sensible knowledge and so a pre-eminence of the circle of the different. But the circle of the same can and must retrieve its supremacy, supposing that it is possible to go back to the exercise of intelligence and intelligible knowledge from sensible knowledge : because, returning to our explanatory model, the circle of the same of individual embodied souls does not need to, or perhaps cannot even really “touch” the ideas ex novo, as happened to the world-soul without a body, but can bring back to light those modifications already undergone when it was outside the body, and preserved impressed in it. And that is recollection, the only form of intelligible knowledge that pertains to embodied souls.Haut de page
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Francesco Fronterotta, « Intelligible Forms, Mathematics, and the Soul’s Circles : an Interpretation of Tim. 37a-c », Études platoniciennes [En ligne], 4 | 2007, mis en ligne le 01 septembre 2016, consulté le 24 avril 2017. URL : http://etudesplatoniciennes.revues.org/906 ; DOI : 10.4000/etudesplatoniciennes.906Haut de page
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