Dominic j. O'Meara

In Book V of On Pythagoreanism Iamblichus attempted to show in detail the paradigmatic function of mathematics in understanding the material world; the result was an unusual work on 'arithmetical physics'. According to the reconstruction of the book that I have proposed in Chapter 3, this arithmetical physics consisted in a mathematized revision of Aristotle's Physics in which the principles of physical explanation presented in the first part of the Physics were shown to be exemplified in the properties of various sorts of number: numbers exhibit the formal, efficient, and material causality at work in the immanent organizing principles of nature, 'physical numbers'. While following the pattern of physical explanation provided by the early books of Aristotle's Physics Iamblichus did not ignore Aristotle's criticisms of the Pythagoreans, but attempted to meet such criticisms and improve Aristotelian physics so as to turn it into a true 'Pythagorean' science of nature.

Proclus also wrote works in mathematical physics, the Elements of Physics and his Commentary on the Timaeus. In this chapter I shall discuss these two works with a view to determining what they show about the function of mathematics in physics. In this way inferences may be made concerning the philosophical standpoint of Proclus' physics vis-à-vis Iamblichean Pythagoreanism. At the same time the very bulk of the material in Proclus, as compared to the fragmentary remains of Iamblichus, will permit of a better view of precisely what a later Neoplatonic mathematical physics consists of, at least in the form it takes in Proclus.

1. Aristotle's Physics Geometricized

The geometrical form given to Aristotle's physics in the Elements of Physics is so prominent as to seem almost exaggerated. Each of the two books of this short work is headed by a list of definitions in the Euclidean manner. These are followed by a series of propositions with

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demonstrations and corollaries often using geometrical principles. Euclidean style is adopted even in the language: imperatives, postulations, the QED formula. ¹ 

¹ Cf. e.g. El. phys. 2, 17; 4, 12, 15, 20-1, 25; 10, 15; 12, 4.

The imposition of geometrical form is all the more appropriate in that the Aristotelian texts used already incorporate much recourse to geometrical argument. However Proclus shows ingenuity in reworking Aristotle: Aristotle's points are organized into a strict sequence, often clarified and supplemented. In particular strict demonstrative (i.e. syllogistic) form is insisted on in the recasting of Aristotle's arguments in support of his points. Proclus' rewriting of Aristotle here could be compared in its skill to his rewriting of Iamblichus in the first prologue of his Commentary on Euclid. However the approach of the first prologue is somewhat different: it is a broad and not too demanding introduction to mathematics, whereas the Elements of Physics imposes itself as a demonstration or exercise in scientific rigour. Indeed the purpose Proclus seems to have had in geometricizing Aristotle's physics was to make Aristotle's arguments stronger from the point-of-view of scientific method. Geometry's contribution to physics is then methodological, and this method is essentially that of syllogistic argument.

The tendency to identify syllogistic method with the method of geometry has already been observed in Proclus' Commentary on Euclid, as has been a preference for geometry among the mathematical sciences in transpositions into physics and metaphysics. If a comparison in these respects between Iamblichus' arithmetical version of Aristotelian physics and Proclus' geometrical version of the same can serve to indicate how Proclus is inspired by, and yet moves away from the precedent set by Iamblichus, it should also be noted that Proclus' Elements of Physics is not to be considered simply as a geometrical substitute for Iamblichus' arithmetical physics. Iamblichus' book covers the principles of physical explanation presented in the first part of Aristotle's Physics, whereas Proclus' work is based on the second half of the Physics, in particular Books VI and VIII supplemented by passages from De Caelo I. Proclus deals, not with the question of physical explanation, but with motion and change with particular reference to the inference to the 'unmoved mover' that the analysis of motion produces. Aristotle's argument in Physics VIII that motion in the world ultimately presupposes an unmoved cause appears to be Proclus' chief interest in the Elements of Physics; the more general

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propositions he first extracts from Physics VI function as the groundwork logically presupposed by this argument. And in reformulating Aristotle's argument Proclus reveals his Platonic point-of-view in stressing at the end the immaterial nature that Aristotle's unmoved mover must have. ² 

² El. phys. 58, 20-7 (compare Aristotle, Phys. 267 b 19-26). Dodds, in his edition of the El. theol. (xviii, 201, 250), has disposed of Ritzenfeld's claim in his edition of the El. phys. that the work was written early in Proclus' career while studying Aristotle and before reading Plato and becoming a Platonist.

The practice of taking definitions, axioms, and demonstrations from Aristotle's physics for purposes which are Platonic in inspiration is not confined to the Elements of Physics. It may be found elsewhere in Proclus. ³ 

³ Cf. In Tim. II 93, 30-94, 15 (compare El. phys. 30, 10-11; 34, 3); In Remp. II 9, 26.

However, the value of Aristotle's physics was limited. Like Iamblichus and Syrianus, Proclus considered Aristotle's writings in physics to be defective imitations, produced in a spirit of rivalry, of Platonic physics as revealed in particular in the Timaeus. ⁴ 

⁴ In Tim. I 6, 21 ff.; 237, 17 ff.; III 323, 31 ff.; cf. I 7, 15-16; 295, 26-7.

A measure of Aristotle's heretical inclinations is to be found in the criticisms he makes of the Timaeus, and Proclus devoted a work (since lost) to responding to these criticisms. The Timaeus was then a far better source for true physics. Proclus produced early in his life a vast Commentary on the Timaeus of which (about) the first half survives. ⁵ 

⁵ Cf. Festugière (1966-8), V 239-48.

In the rest of this chapter 1 I shall discuss the conception of Pythagorean mathematical physics that Proclus develops in this Commentary in connection with the intrepretation of the physics of the Timaeus.

2. Plato's Timaeus as 'Pythagorean' Physics

At the beginning of his Commentary Proclus announces his view that Plato's Timaeus is inspired by a work of the Pythagorean Timaeus of Locri:

And indeed the book by the Pythagorean Timaeus On Nature, written in a Pythagorean manner, 'starting from which Plato embarked on writing the Timaeus', according to the writer of sylloi. We have placed this book at the head of our commentary so that we will be able to see what Plato's Timaeus says that agrees , what Plato adds, where there is disagreement. (I 1, 8-16; cf. 7, 18-21)

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There is some indication in the manuscript tradition that the supposed Pythagorean original was indeed placed before Proclus' Commentary. ⁶ 

⁶ Cf. Marg (1972), 2.

Furthermore it will be noticed that Proclus' project—to comment on his text by measuring it against its Pythagorean source—is the same as that which Iamblichus had adopted before in commenting on Aristotle's Categories, when he had measured Aristotle against the putative original, (pseudo-)Archytas. Indeed it is likely enough that Iamblichus had used the same approach in his (lost) Commentary on the Timaeus, comparing Plato with Timaeus of Locri, and that Proclus here is following his lead. ⁷ 

⁷ So Harder (1926), xvi-xvii; cf. above, pp. 96, 99. Marg concedes that his doubts on this are a little exaggerated, (1972), 92.

There are some differences, however, between the way Proclus uses Timaeus of Locri in his Commentary on the Timaeus and Iamblichus' use of (pseudo-)Archytas in his Commentary on the Categories. Iamblichus keeps Archytas very much in mind, quotes him frequently, and vigorously compares Aristotle (to his disadvantage) to Archytas. On the other hand Proclus refers to Timaeus of Locri comparatively little in his Commentary on the Timaeus, and then in a way that assumes or argues for agreement between Timaeus and Plato. ⁸ 

⁸ In Tim. II 79, 4-11; 101, 9-14; 188, 9-190, 11; III 138, 3-11.

May we therefore conclude that Proclus gradually abandoned or forgot the Iamblichean approach to which he subscribed at the beginning of his work? ⁹ 

⁹ Cf. Harder, loc. cit.; Marg (1972), 91-2.

Could this have to do with the retreat of Pythagoreans in favour of Plato that has already been observed in Proclus?

Appealing as these possibilities may be, I believe that they are open to some serious objections. First of all the range of text spanned by Proclus' references to Timaeus of Locri (see note 8 ) shows that Proclus did not abandon or forget him. Then it must be stressed that for Iamblichus (and Proclus) Aristotle was something of a deviant requiring active and constant correction, whereas the harmonious relations between a true Pythagorean and Plato would scarcely call for such critical vigilance. Thus the difference in exegetical practice resulting from the different relationships between Archytas and Aristotle and between Timaeus of Locri and Plato might suffice to account for the less prominent role of Timaeus of Locri in Proclus' Commentary as compared to that of Archytas in Iamblichus' Commentary on the Categories. Thirdly, although Proclus does indeed distinguish

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between a Pythagorean revelatory approach and the Socratic demonstrative method, he also subordinates the figure of Socrates to that of Timaeus in analogy to the subordination of lower to higher beings (I 354, 5-16), and the figure of Timaeus in Plato's dialogue he characterizes as a Pythagorean whose discourse functions and is to be interpreted according to Pythagorean principles. ¹⁰ 

¹⁰ In Tim. I 129, 31-2; 223, 5-6; III 168, 7-9; cf. In Parm. 723, 19-20; Theol. Plat. IV 88, 2 ff.

Finally, Proclus indicates that he regards Plato's Timaeus as a Pythagorean work (I 15, 23-5), exemplifying a Pythagorean approach to physics (204, 3-5). Indeed Plato alone, as compared to other physicists (one thinks of Aristotle), followed Pythagorean methods and doctrines. ¹¹ 

¹¹ In Tim. I 1, 25-6; 2, 29-3, 4; 33, 7-11; 262, 10-11; 267, 1-2.

I believe consequently that the Commentary on the Timaeus shows a tendency to Pythagoreanize in the manner of Iamblichus and Syrianus, a tendency less pronounced in Proclus' other works. A telling sign of this can be found in the fact that the Commentary makes use of a good number of specific 'Pythagorean' authorities and texts, whereas the usual practice in Proclus' other works is to refer on occasion and in a vague way to 'the Pythagoreans' (above, ch. 7 n. 26). The reason for the particular Pythagorean emphasis of the Commentary on the Timaeus can be surmised: we know that it was one of Proclus' first major works, completed probably shortly after Syrianus' death, and it is a fair guess that it reflects in some respects, especially in its earlier parts, the influence of Syrianus. ¹² 

¹² Thus, for example, the ontological subordination of the figures of the dialogue appears to be Syrianus' idea (Proclus, In Tim. I 20, 27 ff.: Timaeus as the 'monad' of the group). Cf. Sheppard (1980), 34-8.

3. The Geometrical Method of Plato's Physics

It remains to determine the extent to which Proclus' programmatic claims for the Pythagorean character of Plato's Timaeus affect the actual interpretation of physics to be found in his Commentary. To what extent, in particular, does Proclus develop the Timaeus into a work of mathematical physics? In what sense does Proclus show the physics of Plato's dialogue to be mathematical? Plato himself brings in so much mathematics that it might at first appear impossible for a commentator not to find mathematical physics in the dialogue. Yet Plato's text presents more than enough ambiguities and difficulties to

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leave open for the interpreter a wide range of possible approaches and options.

In the first section of this chapter the chief contribution of mathematics to physical argument in the Elements of Physics was found to be methodological: Aristotle's text was given the strict demonstrative form Proclus assumed to be characteristic of geometric method. In his Commentary on the Timaeus Proclus describes the method of Plato's Timaeus also as 'geometrical', which is to say rigorously syllogistic in form. However, Timaeus' discourse is preceded in the dialogue by Critias' recounting of the myth of Atlantis. This follows, according to Proclus, the Pythagorean practice of fore-shadowing with images the scientific account that begins with Timaeus' discourse. ¹³ 

¹³ In Tim. I 4, 8-9, with 30, 2-10; 33, 1-11.

Timaeus then proposes five hypotheses or axioms, from which he derives physical conclusions of which the first three are: that the world is constituted; by a demiurge; who made use of an eternal paradigm. In the following commentary Proclus frequently emphasizes the geometric method that he believes characterizes Timaeus' speech. As this motif in Proclus' Commentary has been fully examined elsewhere, ¹⁴ 

¹⁴ By Festugière (1966-8), II 7-9, also (1963), 565-7; to Festugière's evidence one could add In Tim. I 265, 3 ff.; 283, 15-19; 332, 6-9; 345, 3-4; II 7, 19-20; 114, 14-15.

it will be sufficient merely to note that the assimilation of scientific (demonstrative) to geometric method is characteristic of Proclus not only here and in the Elements of Physics but also, as we will see, in his metaphysical (theological) works, and that this assimilation is related to the attitude to geometry expressed in his Commentary on Euclid.

4. Is Physics a Science?

Mathematics in the form of geometry lends much more to physics than merely its syllogistic method. It also helps to resolve for Proclus a major problem posed by the physics of Plato's Timaeus, namely that concerning the questionable scientific status of physics. The problem is introduced by Timaeus towards the beginning of his discourse (Tim. 29 b c):

And in speaking of the copy and the original we may assume that words are akin to the matter which they describe; when they relate to the lasting and permanent and intelligible, they ought to be lasting

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and unalterable, and, as far as their nature allows, irrefutable and immovable—nothing less. But when they express only the copy or likeness and not the eternal things themselves, they need only be likely and analogous to the real words. As being is to becoming, so is truth to belief. (Transl. Jowett)

This passage casts a shadow over the cosmology that follows, for it suggests that true and abiding, i.e. scientific, statements can be made only of the Forms and that all that can be claimed for what is said about the physical world is likelihood, not truth. In this sense then there can be no science of nature, strictly speaking, nor can Timaeus' account be taken to be such a science. The interpreter of Plato seems faced with a choice between ignoring or glossing over this passage, so as to be able to regard the cosmology of the Timaeus as scientific; ¹⁵