15 In the strong Greek sense of science of course. To the extent that modern physics regards its claims as probable, it seems to be no more ambitious than Timaeus' discourse.

abandoning the hope of finding physical science in the Timaeus; or determining the way in which an account of nature, in being 'likely', attains a certain degree of truth and to this extent approaches, without being, science.

Proclus resolves the problem along the lines of the third option, using a key provided by the structure of geometrical demonstrations. Such demonstrations rest on axioms and hypotheses, which can thus be regarded as the principles or 'causes' from which conclusions in geometry result or are derived. In physics the world and its contents are the results ('conclusions') to be explained, to be related back, that is, to their constitutive first principles. But for a Platonist such principles (both of the world and of scientific discourse) must be transcendent, immaterial, and unchanging. Platonic physics must then begin and be derived from meta-physical causes. But this also means that it is based on immaterial unchanging truths and in this way is related to science. Thus discourse about the world can be true to the extent that it refers to transcendent principles. And it is the relation between axioms/hypotheses and conclusions in geometry that serves to bring out the scientific basis of physics. ¹⁶ 

16 Cf. I 337, 29-338, 5, with 346, 29-347, 2; 348, 23-7.

An example will help to illustrate and specify further the extent of the scientific claims of physics. Propositions held to be universally true occur in astronomy. They are true in this way, however, only to the extent that they express the universal mathematical laws to which the heavens are subject. In so far as they apply to particular heavenly

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bodies they are liable to be inaccurate. Thus if the astronomical law that 'the celestial circles bisect each other' is universally true, this has to do with the geometrical principles at issue, for in the case of particular celestial bodies there can be no point of bisection in the geometrical sense required by the law: a physical point of bisection is divisible, therefore defective, as compared to the indivisibility of the geometrical point. ¹⁷ 

¹⁷ In Tim. I 349, 6-350, 1; 346, 29-347, 2; cf. In Parm. 796, 26-39. For this in Plato cf. Allen (1970), 163-4.

The deficiency of such laws when applied to physical bodies relates not only, however, to the difference in ontological character between these bodies and the immaterial principles from which the laws derive. It is also due to the fact that we know of physical objects through sense perception, which limits our scope—our access to the heavens, for example—and makes us prey to all sorts of perceptual error. ¹⁸ 

¹⁸ I 351, 2-352, 27; 346, 12-347, 2; cf. II 51, 6-27; Blumenthal (1982), 9-10.

In fact the scientific aspect of physics and (yet) its failure to attain anything more than probability reflect its hybrid character: in introducing mathematical laws it starts from a basis inspired by universal and unerring principles; but the formulation of these laws by, and in relation to, physical bodies diminishes their universality and accuracy, given the 'errant' nature of these bodies and the fallibility of our perceptual access to them.

The 'mixed' character of physics ¹⁹ 

¹⁹ Cf. also I 410, 11-19; III 356, 17-22.

is found by Proclus also in the fact that much of Timaeus' discourse is concerned, not with the world, but with its transcendent 'divine' causes: the demiurge, the Forms, the lesser gods, soul. In this sense the Timaeus can be regarded as a 'theology': not a theology, or science of the divine per se such as Proclus finds in the Parmenides, but a discourse on the divine as producing the world:

Thus Socrates shows that he wishes the physics to be Pythagorean, one that begins from the divine cause, and not one such as he rejected in the Phaedo, which blinds the eye of the soul, holding as responsible causes winds and ether according to Anaxagoras. For true physics must be attached to theology, as nature depends upon the gods. ²⁰ 

²⁰ I 204, 3-10; cf. also I 8, 4-13; 84, 22-85, 4; 217, 25-8; III 152, 7 ff.; 168, 7-20.

The reference to Pythagoreanism reminds us that Iamblichus, according to Proclus, emphasized the theological aspects of the Timaeus. ²¹ 

²¹ Cf. Iamblichus, In Tim. frs. 25, 54. Compare Iamblichus Comm. 56, 3-4 (quoted above, p. 162).

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But it is not clear how much of Proclus' explanation of the semi-scientific status of physics he owed to Iamblichus.

These ideas also explain for Proclus the superiority of the physics of Plato to that of Aristotle. Aristotle concerns himself mainly with immanent causes, matter and form in nature, and pays less attention than does Plato to the transcendent efficient, paradigmatic, and final causes (I 2, 15-3, 20). It follows that to the extent that he neglects the divine presupposites of nature Aristotle diminishes that which gives physics what scientific value it can have (cf. I 295, 26-7).

5. The Application of Mathematical Truths in Physics

It can be seen from the preceding section that the contribution of mathematics to physics is not confined to the area of method. Geometry provides a scientific form of demonstration to physics. But physics also introduces specific mathematical principles and laws in its explanations. What might be called the substantive (as compared to the methodological) contribution of mathematics to physics is no less important for Proclus than it is for Iamblichus. A particularly striking aspect of Proclus' Commentary on the Timaeus is the somewhat complicated, oblique way in which mathematical truths are sometimes applied to physics. They are not simply read into nature, but are transformed in various ways in the course of being applied. In order to reach a better understanding of this procedure it will be useful to examine once more the place of mathematical objects in the scheme of reality, in particular vis-à-vis natural objects. For it is a reasonable assumption that the ontological relations between these different sorts of objects imply certain conditions governing the application of the one sort to the other.

The nature of mathematical objects as projections of principles innate in the soul has been discussed above in Chapter 8 in connection with the Commentary on Euclid. However, the place of this nature in the structure of reality, as determined by Proclus in the Commentary on the Timaeus, remains to be identified. We might begin with the simplified three-level system—sensible/mathematical/intelligible objects—that Iamblichus attributes (Comm. 10, 7-11, 15) to the 'Pythagoreans'. The same 'Pythagorean' system reappears in Syrianus

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(above, p. 131) and in Proclus' Commentary on the Timaeus, at I 8, 13-27, and again at II 23, 9-25:

After the mathematical grasp of these words (Tim. 31 c-32 a) we must turn to a physical interpretation. For we must not digress by remaining in mathematics (for the dialogue is about physics), nor must we neglect mathematics, seeking out only the perceptible, but we must join the two and always weave together mathematicals and physicals, just as the realities themselves are woven together and are homogeneous and of the same parent ²² 

²² A Plotinian image (cf. IV 3, 6, 10-15) that recalls also Archytas as quoted by Iamblichus (Comm. 31, 4-7 = In Nic. 9, 1-5).

according to the derivation from Intellect. And indeed if in general the Pythagoreans place mathematical reality between intelligibles and sensibles, as more unfolded than intelligibles and as more universal than sensibles, why then must we think only of physics, neglecting mathematicals? For how is the sensible organized? According to what principles? From what principles did it come if not mathematical ones? Such principles then obtain first in souls, having descended from Intellect, and then are found in bodies having come from souls.

Proclus indicates at various points in his Commentary that certain distinctions must be introduced in the three-level scheme. (i) Firstly, for the human soul as knower, we must distinguish between the objects of its mathematical thought and the mathematical principles constituting its essence: although the monadic quantitative numbers of arithmetic are images of the essential psychic numbers, the former are not to be confused with the latter. ²³ 

²³ Cf. In Tim. II 164, 17-165, 12; 166, 4-12; 193, 17-27; 212, 5-9. Cf. above, p. 133 (Syrianus).

In the Timaeus Plato introduces geometrical and harmonic elements in the constitution of the soul: Proclus indicates that these also are not to be interpreted according to ordinary mathematical usage (as the objects studied in mathematics), but are to be understood in a higher sense (II 245, 23-246, 4). Indeed all four of the mathematical sciences are pre-contained essentially at a higher level in the constituents of soul (II 238, 10-239, 16). (ii) A distinction must also be made between psychic essential number, the mathematical unfolding or discursive image of this, and the empirical or 'opinative' number produced in the perceptual observations of astronomy. ²⁴ 

²⁴ Cf. In Tim. I 255, 7-26; III 85, 23-31; 89, 7-13.

Both mathematical and opinative number are generated from psychic number, but opinative number is derived indirectly, through perception of enmattered images ('physical numbers') of

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psychic number. (iii) Other distinctions need to be made in connection with divine soul as organizer of the world. Proclus makes use of the Iamblichean notion of 'physical numbers' as immanent formative causes, and distinguishes them (as images) from the mathematical principles according to which soul organizes the world. ²⁵ 

²⁵ Cf. In Tim. II 25, 1-3; 39, 18-19; 161, 7-12; 164, 21-8; 166, 4-12; 193, 17-27; cf. In Parm. 869, 3.

The latter must be closely related or identical to the psychic mathematicals with which the demiurge composes soul's nature. In this work the demiurge imitates the Forms, which in consequence can also be referred to as 'number'. ²⁶ 

²⁶ In Tim. II 194, 4-17; cf. I 41, 25-42, 2; II 161, 3-32; III 19, 14-26; 32, 26; Hypo. astron. 2, 1-5.

The subject of 'ideal' and 'divine' number will be examined in the next chapter and need not delay us here.

This scheme of different sorts and levels of numbers, which is essentially the same as that proposed by Syrianus, may be used to determine the conditions under which mathematics is applicable in physics roughly as follows. To be used in physics, the objects with which the mathematician deals must, on the one hand, be purified of the features produced through their being projected from the soul's nature. These features include quantity in arithmeticals and extension in geometricals. By removing such additions in mathematicals we can approach nearer to the principles innate in the soul which, as divine demiurgic principles, guide her in her organization of the world. Allowance must be made, on the other hand, for the fact that the principles of the soul are not expressed perfectly in the world: the materiality of the world diminishes the universality and accuracy of such principles as reflected in the world. These two requirements show that an indirect and qualified, rather than a direct and simple application of mathematics to physics is appropriate. Our reliance on perception for matching mathematical principles with their physical analogues introduces a further limiting factor. The results of 'opinative' mathematics, such as astronomical calculation based on observation, must, it appears, be far inferior to those of 'projected' mathematics since they do not derive from discursive articulation of the psychic principles governing the world, but depend on perception mediating particularized and fleeting instantiations of such principles. Such qualifications and conditions for the application in physics of mathematics can plausibly be inferred, I believe, from Proclus'

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scheme of reality. Whether he does in fact observe them may be determined at least in the following examples of the use of mathematics in his Commentary.

(i) In the first case Proclus is commenting (II 20, 19-28, 7) on the passage of the Timaeus concerning the proportion ( ναλογ α) that the demiurge uses to link and unify the elements of the world:

Wherefore also God in the beginning of creation made the body of the universe to consist of fire and earth. But two things cannot be rightly put together without a third; there must be some bond of union between them. And the fairest bond is that which makes the most complete fusion of itself and the things which it combines, and proportion is best adapted to effect such a union. For whenever in any three numbers, whether cube or square, there is a mean, which is to the last term what the first term is to it, and again when the mean is to the first term as the last term to the mean—then the mean becoming first and last, and the first and last both becoming means, they will all of them of necessity come to be the same, and having become the same with one another will be all one. (Tim. 31 c-32 a, transl. Jowett)

As in other similar instances, Proclus discusses first the mathematical aspects of this passage and then turns, in a clearly marked transition, to consider the passage with reference to its physical doctrine.

(a) The mathematical question the passage poses for him concerns the kind of proportion—arithmetic, geometric or harmonic—to which Plato is referring. Proclus agrees with Nicomachus (Intro. arith. 129, 14-20) that Plato is thinking of geometric proportion. For Plato speaks of the same relation (λ γος) between extreme and middle terms (in geometric proportion the proportion is the same, e.g. double in 1:2 = 2:4), not of an equal quantitative difference between the terms (as in an arithmetic proportion such as 1:2 = 2:3), nor of the same fractions of the extremes (as in a harmonic proportion such as 3:4 = 4:6, where the middle term exceeds and is exceeded by a third of the extremes). ²⁷