27 Cf. Nicomachus, Intro. Arith. 126, 12-128, 19.

Proclus then brings out the property or feature characteristic of geometric proportion, namely the identity (ταυτ της) of relation that the middle term produces in regard to the extremes, binding them together and communicating to each the same relation. This identity he regards as superior to the 'equality' characteristic of arithmetical proportions and to the 'similarity' he attributes to harmonic proportions, and he describes it as leading up to and depending on unity ( νωσις).

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Proclus then moves (b) to the physical interpretation of the Platonic passage (II 23, 9). He stresses that the mathematical theory must not be confused with the physics of the passage, for the transition is 'to another genus', ²⁸ 

28 II 23, 30-2; this is Aristotle's caveat, An. Post. I 7, 75 a 38.

but that the mathematical theory can throw light on the physics. What then in physics is the proportion binding the elements of the world? For Proclus it is the cosmic life that penetrates the world giving it the unified life of a living animal. There are other principles of unification—the demiurgic causes 'above' this life and the particular forms and qualities of the elements 'below'—but Proclus takes Plato to be referring to an immanent (rather than transcendent) and causative (rather than caused) 'physical proportion', the life-principle of the universe. The living texture and rhythm of the universe reflect the work of this physical analogue of geometric proportion: it gives physical substances 'rational and homologous' ²⁹ 

29 On these mathematical terms cf. Festugière ad loc. (III 52 n. 2); cf. In Tim. I 17, 4-6.

relationships to each other, i.e. the same proportional relations link them as terms in a series; it produces the development out of unity and return to its cause, the multiplying and reuniting of the life of the world, a pattern that recalls the progressive and regressive aspects of geometric proportion: the subsequent terms of a geometric series are produced by the same proportion; the middle term 'brings back' the third to the first term. The participation of the elements in each other in the world also has its analogue in geometric proportion, in which extreme terms can function as means (e.g. 2:4 = 1:2). Proclus finally attributes identity to the cosmic life, a feature corresponding to the property he found to be characteristic of geometric proportion.

Proclus does not propose then a direct mathematical reading of the physics of Plato's passage. For him Plato is not simply thinking of a mathematical proportion linking the elements of the universe. Rather, a physical cause must be found that possesses a nature and function analogous to what is characteristic of geometric proportion. The parallel between this physical cause (the cosmic life-principle) and geometric proportion is developed in such a way as to avoid mathematizing the specific physical character (or categorial domain) of this cause and its effects. The geometric proportion of the mathematician is not the binding power of the universe, but it illustrates in its own field features which also are found in its 'sister' (above, p. 186) in the natural realm. The parallel between geometric proportion and the cosmic life-principle depends, we can assume, on the fact that they

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represent projections of the same original psychic principle, the one in the discursive thought and imagination of the geometer, the other in the material world. Proclus does not tell us what geometric proportion purified of its projected (discursive, extended) aspect might turn out to be, but one can imagine that it would coincide essentially with the cosmic life-principle. He does, however, allow for the distortions that the material world can produce in the action of this life-principle: sensible objects, as divisible and 'other', attain identity and unity only in that, through the action of the life-principle, they reach an 'affinity to identity and, through it, to unity' (27, 10-13).

(ii) The second case of mathematical physics I propose to examine comes a little later (II 28, 14-42, 2), in commentary on the next lines of the Timaeus:

If the universal frame had been created a surface only and having no depth, a single mean would have sufficed to bind together itself and the other terms, but now, as the world must be solid, and solid bodies are always compacted not by one mean but by two, God placed water and air in the mean between fire and earth. (Tim. 32 a b, transl. Jowett)

In explaining this passage Proclus first argues, very much as Aristotle does, that change takes place between contraries and that there must therefore be two contrary elements of the world, fire and earth (II 28, 14-30, 8). He then tries to show why two intermediate elements are needed to link the extremes. The demonstration of the Platonic postulate is provided first in mathematical and then in physical terms.

(a) The mathematical demonstration begins with numbers, for they are what is 'originative' ( χο ιδ ς) and 'self-existent' (α το υ ς) as compared to 'geometrical necessity' (30, 13-15); i.e. the objects of arithmetic, hence its demonstrations, are higher than those of geometry, thus nearer the source in psychic nature. ³⁰ 

³⁰ Cf. also 34, 4-27, where it is claimed that what Plato says is truer in arithmetic than in geometry.

Proclus then argues that 'solid' numbers (cubes), as distinct from 'plane' numbers (squares), require two middle terms in a proportion: e.g. in the series 8, 12, 18, 27, the cubes that are the extremes 8 (2³) and 27 (3³) are linked by 12 (2² × 3) and 18 (3² × 2), whereas only one middle term may suffice to link two plane numbers, as in the series 9 (3²), 12 (3 × 4), 16 (4²). The fundamental feature of the middle terms linking two solid numbers (cubes) is the fact that each middle term is made up of two 'sides' of one extreme and one side of the other (thus 2² × 3 = 12, 3² × 2 = 18).

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(b) The same principles hold for Proclus in physics: physical planes may be linked together by one middle term, whereas physical solids require two middle terms. Thus two elements, water and air, are needed to connect fire and earth. On the subject of physical planes Proclus cites the view of Iamblichus—whom he praises as alone having gone beyond the mathematical problem to consider the physical one—that they are the powers and forms of solids. As for solids, in this case the elements, the extremes (fire and earth) must each contain three powers or qualities in order to yield the one different and two common qualities needed by the two middle terms, as demonstrated in the case of cubic numbers (39, 19-27). This produces the physical series:

Fire (fineness, sharpness, mobility) Air (fineness, bluntness, mobility) Water (thickness, bluntness, mobility) Earth (thickness, bluntness, immobility).

In this series the properties of the 'middle' elements link the extremes much as the 'sides' of the middle terms link cubic numbers: each middle element is composed of two qualities of one extreme and one quality of the other. The same principle of mediation governs physical and arithmetical solids.

It appears then that the physical elements are linked to each other according to a structure of double mediation echoing that which links two solid numbers. This represents but an instance of the general principle that 'physicals are images of mathematicals' (39, 18-19). However, Proclus does not regard his demonstration of why two middle elements are required as merely an application of mathematical theory in physics. His argument is based rather on analysis of the physical nature of the elements themselves, an analysis which shows up the three aspects that each element must have if it is to function with the other elements in the constitution of physical bodies. And it is with physical, not mathematical, arguments that Proclus rejects theories according to which fewer than four elements might be required or which would find in each element fewer than three powers or qualities required to link together the elements. ³¹ 

³¹ Cf. 28, 20-30, 4; 37, 14-41, 9.

If then the same structure of double mediation occurs both in mathematics (for solid numbers) and in physics (for the elements), this is supposed to result,

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I would suggest, not from direct application of mathematics to physics, but rather from the fact that mathematics articulates in a clearer and more accurate way a principle which governs and is expressed also in a more imperfect way in the physical structure of the world.

In this second case of Proclus' use of mathematics in relation to physics, the approach seems to be as indirect and oblique as that observed in the first case. Proclus does not simply settle issues in physics by reference to mathematics. However, mathematics shows up more clearly principles that also obtain in physics and that can be formulated on the basis of physical analysis. In both cases I believe we can conclude that Proclus' approach to mathematics in connection with physical theory is consonant with what one would expect given the places assigned to mathematical and physical principles within the structure of reality. Proclus' laudatory reference to Iamblichus' attention (in his lost Commentary on the Timaeus) to the physical dimension raises the question as to the extent of Proclus' debt, in his way of using mathematics, to Iamblichus. Certainly Iamblichus was as clear about the need to avoid confusing mathematics with physics, as about the paradigmatic value of mathematics in physics. And it appears to be the case that Proclus here (37, 14 ff.) is following in general Iamblichus' physical approach as described at 36, 20-37, 14, even if he is expanding, clarifying, and perhaps altering some of the details of this approach.

(iii) The third case that I wish to mention concerns Proclus' use of 'opinative' mathematics in physics. As we might expect, calculations based on sense-perception have little value for him. Thus, when astronomical calculations lead to a theory of the precession of the equinoxes that conflicts with Plato's system, Proclus does not believe such calculations are a match for the divine authority of Plato and the Chaldaeans. Even the observations made by the Chaldaeans suffice, Proclus believes, to discredit the theory. ³² 

³² III 124, 19-125, 4; cf. III 62, 6-63, 24 (Festugière ad loc. also refers to In Remp. II 220, 4 ff.). On this subject cf. Segond's full study (1987).

6. Concluding Comment

The results of the above study of Proclus' treatment of physics show both his debt to Iamblichus and his originality. His preference for

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geometry among the mathematical sciences expresses itself in his geometricizing of Aristotle's physics and in his identification of the scientific method of Plato's Timaeus as 'geometrical' (i.e. syllogistic). However he shares Iamblichus' views on the relative value of Aristotle's and Plato's physics and, in his Commentary on the Timaeus, treats Plato's physics as essentially 'Pythagorean' in inspiration, a Iamblichean approach due perhaps to Syrianus' influence on the young Proclus. In interpreting the 'Pythagorean' physics of Plato's Timaeus, Proclus makes considerable use of mathematics as a source of paradigms for physics. However, he distinguishes between the domains of physics and mathematics and does not simply impose mathematical solutions to problems in physics. His use of mathematics is rather oblique, reflecting, I believe, the fact that mathematical principles are not for him the direct model of the universe, but are quantitative discursive projections by the human soul of higher principles which, on the divine level, guide the making of the world. In a Pythagorean physics such as that suggested in Nicomachus' works, a more direct application of mathematics to physics is assumed, and this is also the impression given by the fragments of Iamblichus' On Pythagoreanism V. However, Iamblichus is clear about the need to avoid confusing mathematics with physics, and Proclus praises him for having paid attention, in his interpretation of the Timaeus, not only to the mathematical, but also to the physical dimension.

A final comment might be ventured on Iamblichus' and Proclus' Pythagorean mathematical physics as seen from a modern stand-point. Their 'realism' with respect to the existence of mathematical objects is certainly a plausible position and is combined in an interesting way with another plausible approach in which such objects are explained as constructions ('projections') for which the mathematician is responsible. ³³