33 Cf. Annas (1976), 151.

What would create greater difficulty today would be the overall metaphysical view according to which these constructions are (indirectly) models of the universe, in so far as the universe is the work of divine soul using principles from which derive the projections of the mathematician. On the other hand, this overall metaphysical view certainly gives some account of the mathematical background of physics. However, the sense in which physics is mathematical is no doubt very different from what is the case in modern mathematical physics. One obvious difference consists in the kind of

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mathematics used by Iamblichus and Proclus. This mathematics is dominated by Nicomachus' arithmetic which, apart from its mathematical limitations, tends to treat numbers as if they each possessed particular qualities or properties of a rather metaphysical character. This attitude to arithmetic, as a sort of disguised metaphysics, also affects Proclus' view of Euclidean geometry, since here also geometrical figures are assigned individual metaphysical properties. When then it comes to physics, which in Iamblichus and Proclus takes the form of an Aristotelian natural philosophy revised so as to conform to Neoplatonic metaphysics, a correspondence can be established between it and mathematics for the simple reason that both reflect the same metaphysical assumptions. This is surely very different from the mathematical symbolization and analysis of abstract quantitative relations that might be regarded as characteristic of modern mathematical physics. ³⁴ 

³⁴ Cf. Sambursky (1965), 3; Brunschvicq (1929), 60.

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10 Mathematics and Metaphysics in Proclus

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Dominic j. O'Meara

In the preceding chapter I have argued that Proclus' conception of the relations between mathematics and physics, as expressed in his Commentary on the Timaeus, can be regarded as representing quite closely Iamblichus' Pythagoreanizing programme with regard to physics. In this programme Iamblichus also pointed to the use of mathematics in other sciences, in ethics and psychology for example, and materials relating to such a mathematical ethics and mathematical psychology can also be found in Proclus. ¹ 

¹ Cf. In Tim. II 193, 7-27; 195, 11-24; 212, 3-9; 237, 11-240, 2; 245, 23-246, 9; III 337, 17-24; In Remp. I 212, 23 ff.; II 47, 23 ff.; 136, 19 ff.; 169, 4 ff.

However, mathematics, for Iamblichus, was of most importance as a preparation for and anticipation of the highest level of science, that of the study of pure intelligible being and of the gods. Proclus, as has been seen, uses the same approach in his Commentary on Euclid, and in other works, in particular in the Elements of Theology, the Commentary on the Parmenides, and the Platonic Theology, both discusses mathematical transpositions upwards to the divine and develops in detail an account of the science of the divine, dialectic or theology.

According to the reconstruction proposed above in chapter 3 of On Pythagoreanism VII, Iamblichus may possibly have reacted to Aristotle's attacks on the Platonic theory of intelligible Forms and numbers, anticipating the fuller response given by Syrianus in his Commentary on the Metaphysics. It seems at any rate that Iamblichus in his book suggested analogies between the properties and flow of numbers from the monad to the triad (and possibly to the decad) and the gods, i.e. an ultimate principle, the One, and an order of unities intervening between it and the level of intelligible being, the same principles being traced in their lower forms on the intellectual, psychic, and physical levels. At the same time Iamblichus stressed the limitations of such theological transpositions of mathematics, their inferiority as compared to a higher approach to the divine. This higher

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approach was presented, it seems, not in On Pythagoreanism itself, but in other primarily theological works such as that On God which may have made use of (pseudo-) Pythagorean authorities.

In the following pages I shall examine some of the ways in which Proclus adopted and developed Iamblichus' Pythagoreanizing approach to the highest levels of philosophy. In particular I would like to bring out the distinctions Proclus makes between the science of the divine proper and the theological transpositions of mathematics, distinctions which will entail a brief account of Proclus' conception of the science of the divine. Although insisting, I believe, even more than did Iamblichus on the subordination of theological mathematics to the science of the divine, Proclus betrays in certain areas of his account of this science the influence of Iamblichean theological mathematics, as will be shown towards the end of the chapter.

1. The Elements of Theology: A Geometrical Metaphysics?

If the prologues of Proclus' Commentary on Euclid were to be well known and highly valued by mathematicians, historians, and philosophers of mathematics, his Elements of Theology was to have, in other circles, no less great a success. Already in the Middle Ages it was translated into a number of languages, Georgian, Arabic, Hebrew, Armenian, and Latin, ² 

² Cf. Dodds's edn., xxix-xxx; Boese (1985).

and the Latin translation of an abbreviated and modified Arabic version, the De causis, became a basic textbook of metaphysics in Western Medieval philosophy. The Elements of Theology remains the first book of Proclus to be read by philosophers and historians of philosophy. This success is due in part no doubt to the pedagogic qualities of the work, a (relative) clarity and simplicity of exposition which takes a form that is also intellectually appealing and impressive: the work consists of a series of propositions, each proposition being accompanied by one or more demonstrations, the premises constituting the demonstrations sometimes deriving as conclusions from earlier arguments, the whole appearing in this way as a systematic and rigorous deduction of a series of abstract metaphysical truths. The impression of scientific rigour is reinforced by the Euclidean character of some of the arguments and language. ³ 

³ Cf. e.g. 6, 30; the scholium on 58, 11; Lowry (1980), 37.

It is not then

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without reason that Proclus' work has been seen as a metaphysical system demonstrated more geometrico, in the manner of Spinoza, the first such attempt in the history of Western philosophy. ⁴ 

⁴ Cf. the references given by Hathaway (1982), 123-4.

But in what sense is the metaphysics of the Elements of Theology geometrical? Is geometrical demonstrative method applied to metaphysics in the way that Proclus applies it to physics in the Elements of Physics? If one compares the two works one soon notices some important differences. Firstly, the Elements of Physics, as indicated in Chapter 9, does not deal with physics in general, but is concerned with motion and change as pointing to the existence of an immaterial principle of motion. On the other hand, the Elements of Theology covers the whole range of the science of the divine ('theology'), going from the highest principle of all reality, the One, down through the lower levels of the divine to the lowest, the level of soul. ⁵ 

⁵ On soul as the lowest level of the divine, cf. Theol. Plat. I 114, 23-116, 3; Plotinus V 1, 7, 47-9. Dodds points out in his edn. of the El theol. (X, 187) that the word 'theology' in the title is to be understood in its Aristotelian sense (= metaphysics, first philosophy, science of the divine), which Proclus and Syrianus (cf. above, pp. 120-1) identify with the highest Platonic science, dialectic.

Secondly, the Elements of Physics is much nearer to Euclid in form than is the Elements of Theology, in that its demonstrations are preceded by definitions, whereas the Elements of Theology simply presents a succession of propositions which function as the conclusions of the demonstrations supporting them. ⁶ 

⁶ On the fact that the El. theol. is not simply a geometricized metaphysics, cf. also Lowry (1980), 37; Hathaway (1982); Lohr (1986), 59-60.

This last point is sufficient to show that the Elements of Theology is not a geometricized metaphysics in the way that the Elements of Physics is a geometricized physics; Proclus does not simply apply Euclidean demonstrative method to metaphysical doctrine. How then account for the undeniable geometrical aspects of the Elements of Theology? Our study of the Commentary on Euclid in Chapter 8 has led to conclusions that are relevant here. One of these conclusions was that Proclus tends to see geometry as exemplifying particularly well scientific reasoning on the discursive level. Hence geometrical method is assimilated by him to what we would call logical rigour. In whatever context this rigour is applied we will find in consequence a tendency in Proclus to allude to geometrical procedures and terms. Although closely associating geometrical with scientific method, Proclus also insists, in his revision of Iamblichus in his Commentary on Euclid, that geometry

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derives its method from a higher science, Platonic dialectic: the parts of this method (definition, division, analysis, and demonstration) are already found in dialectic. In his theological works Proclus also brings out these four constituents of dialectical method. ⁷