37 For this division in Iamblichus, cf. Above, p. 44 (Iamblichus' text is very probably the source of inspiration of Syrianus' tripartite division of reality).

At each level Forms are to be found that match the particular ontological characteristics of the level (81, 38-82, 2). The intelligible Forms are 'by the gods' (πα θεοι ς) and 'complete' the ranks of the divine (82, 3 and 12-13)—these vague formulations will be clarified later. The discursive Forms imitate the intelligible Forms, assimilating the psychic order to the intelligible (82, 14-15). As contemplated by divine and demonic souls, discursive Forms function as demiurgic principles. But for souls that have fallen, as in the Phaedrus, from contemplation and thus from the power of making, they are no more than objects of knowledge. ³⁸ 

³⁸ 82, 15-20; cf. Praechter (1932), 1746.

We have access to them in virtue of the fact that the demiurge of Plato's Timaeus constructed soul through geometric, arithmetical, and harmonic analogies (82, 20-2). In this way discursive Forms are innate in us and make possible the recollection of Forms in the fallen soul (82, 25). These Forms are 'universals' (καθ λου λ γοι), not in Aristotle's sense, as abstracted from sensible objects, but as existing a priori as

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part of the being of souls (82, 26-8). However for higher souls, as noted above, they are also demiurgic principles guiding the causes immanent in and organizing nature (82, 28-9). These immanent causes (λ γοι α τια), which are inseparable from sensible bodies, represent the lowest level at which form is found (83, 5-7). To all levels 'the Pythagoreans' applied mathematical terms, not because they were unable to distinguish between levels, but because of the relation of image to model linking each level to the level above it. ³⁹ 

³⁹ 83, 14-26; Syrianus regards much of Aristotle's criticism as showing only a failure to grasp this point (cf. 180, 17-25; 186, 30-6).

In what follows I shall attempt to fill out this general scheme with what can be gleaned from other parts of Syrianus' Commentary, beginning with the intermediate, 'discursive' level—where numbers, properly speaking, are found—and then considering how numbers function in paradigmatic extrapolation in physics, as the causes of sensible bodies, and how they anticipate and express the objects of metaphysics, true being and the divine.

(III) Number and the Soul

The relation between the objects of mathematics and soul is, in what remains of Iamblichus' On Pythagoreanism, somewhat unclear. This is not the case in Syrianus: intermediate between intelligible and sensible reality, the objects of mathematics are part of the nature of soul (4, 5-11). As we have seen, mathematicals have a double function, as principles guiding the demiurge action of unfallen souls on the world ⁴⁰ 

⁴⁰ These are subordinates of the demiurge proper of the Timaeus (cf. 82, 15-22).

and as innate universals in fallen souls allowing them to regain their lost knowledge. The correspondence between the demiurgic principles of unfallen souls and the innate universals in fallen souls explains how we (the latter) are capable of scientific demonstrations concerning the physical heavens and other material objects: by developing demonstrations applying to physical objects from first principles, or universals, that we possess innately and cognitively (γνωστικω ς), we rehearse the actual constitution of these objects from the same principles functioning 'demiurgically' (δημιου γικω ς). ⁴¹ 

⁴¹ Cf. 88, 24-32; 97, 1-5; 82, 36-83, 1; 27, 30-7; Duhem (1914), II 102-3; also Proclus, In Tim. II 236, 23-7.

Indeed, Syrianus argues, scientific demonstration, as Aristotle understands it, is not possible if we follow Aristotle in

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treating 'universal principles' as abstracted (a posteriori) from physical objects. For how could such principles be what is prior scientifically, what is most clear and well known, as Aristotle claims, if they are derivatives of sensible particulars? Aristotle's (anti-Platonic) view in Metaphysics M of 'universals', especially mathematical universals, as derived from particulars is inconsistent with his own understanding of the nature of scientific demonstration. Consistency is recovered if such universals are regarded as principles existing prior to the sensible world, functioning cognitively in our souls and demiurgically in the souls that organize the world. ⁴² 

⁴² 90, 8-23; cf. 95, 29-96, 6. On the 'universals' in the soul, cf. 4, 37-5, 2; 12, 5-15; 163, 6-8; 91, 20-1; 95, 13-17.

The relation between soul and mathematical objects requires further specification, however. Mathematicals constitute but a portion of the universals in the soul (84, 1-3). What the other universals are is not quite clear in Syrianus, and we must turn to Proclus for further information on this point (below, p. 201). Furthermore, mathematicals appear at several levels in the soul. In producing soul the demiurge gives it number which may be called 'ideal' ( δητικ ς) number—although it is not properly speaking ideal number, i.e. intelligible Form—to distinguish it from the number it produces from itself, mathematical or 'monadic' (μοναδικ ς) number. ⁴³ 

⁴³ 123, 19-24; cf. 88, 8-9; Praechter (1932), 1759-60; mathematical number is 'monadic' because it is made up of units, monads; cf. Iamblichus as quoted by Simplicius, In cat. 138, 10-11 (= fr. 45 Larsen); Merlan (1965), 171-2; Gersh (1978), 139.

Some light is shed on soul's generation from its ideal or essential number of mathematical number at 132, 14-23, and 133, 10-15, where Syrianus represents soul as producing mathematical numbers from two principles that it possesses within itself, a monad and a dyad. Thus we do, after all, 'think up' mathematical number, but not in Aristotle's sense: our souls contain essential numbers, images presumably of intelligible Forms, in particular a monad and dyad that pre-contain all of the formal features of mathematical numbers and from which we generate mathematical numbers. A similar psychic generation takes place for geometrical objects: geometrical figures are produced from essential, indivisible principles in the soul (ο σι δ ις, με ει ς λ γοι) when geometry, in its cognitive weakness, projects these in imagination, and thus in extension, so as to grasp them more easily. ⁴⁴ 

⁴⁴ 91, 25-92, 5 (cf. Philoponus, In de an. 58, 7-13); Praechter (1932), 1752.

There are several important advantages for the Platonist in this distinction between levels of mathematicals in the soul and in the idea

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that the lower level is a projection of the higher: a place is found for the creativity of mathematics that reconciles it with a foundation that is not invented, but 'given', a priori, universal, and necessary; furthermore quantity and extension in mathematicals can be accounted for without sacrifice of the indivisible, unextended character of their originals in the soul. ⁴⁵ 

⁴⁵ Cf. Charles (1982), 192 (on Proclus). The theory that mathematicals are projections, imagings forth, of intelligible principles may go back to Iamblichus (cf. Comm. 34, 7-12; 43, 21; 44, 9; Steel [1978], 63, 67-8) and appears to have been adopted by Plutarch of Athens (in Philoponus, In de an. 515, 20-9; cf. Beierwaltes [1985], 260, Blumenthal [1975], 134-6). Its background must be the interpretation of what precisely might be the medium responsible for the image status of mathematicals in Plato's Rep., and the identification of the imagination ( αντασ α) with this medium. Charles (1971), 251, refers to Plotinus, IV 3, 30, where imagination acts as a mirror showing forth, articulating thought. Cf. below, p. 168.

(IV) 'Physical Number'

The relation between psychic mathematicals, as both causal and epistemic principles, and the organization of the physical world has already been noted. To understand physical phenomena we must relate them back ( να ο ) to their paradigmatic mathematical principles (cf. 98, 16-31; 155, 36-156, 6). Indeed the world is organized by causes that, as patterned after number, can be described as 'physical numbers'. ⁴⁶ 

⁴⁶ Cf. 122, 25-7; 142, 27 and 32; 188, 5 and 10; 190, 22; Hermias refers briefly to δοποιο ιθμο (In Phaedr. 16, 4-5).

Syrianus speaks of a science corresponding to this Iamblichean idea of physical number, 'physical arithmetic' (189, 13; 192, 2-3). The differences between physical numbers and mathematical numbers (in particular monadic numbers produced from essential numbers in the soul) must be carefully observed, for confusing them is the source of some of Aristotle's criticisms in the Metaphysics. ⁴⁷ 

⁴⁷ Cf. 122, 25-9; 143, 4-10; 190, 35-7; Proclus, In Tim. I 16, 25-17, 4.

Thus the dyad that produces mathematical number is not that which produces physical solids (180, 22-5). Yet the connections between physical and mathematical numbers justify Pythagorean analogies between physics and mathematics; mathematical numbers exemplify, they bring out formal properties that explain physical phenomena (cf. 143, 9-10; 122, 28-9). Such indeed is the 'physical arithmetic' developed in Iamblichus' On Pythagoreanism V and which Syrianus in part reproduces (above, p. 130). However, Syrianus' treatment of the subject is limited: he is concerned with it to the extent that it involves distinctions

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that serve to dismiss some of Aristotle's assumptions about Pythagorean-Platonic number-theory.

(V) Number in Metaphysics

Aristotle's polemic with Platonism-Pythagoreanism in Metaphysics MN has to do mainly, of course, with the theory of transcendent immaterial Forms and ideal numbers ( δητικο ιθμο ). Syrianus consequently has a good deal more to say about this metaphysical level of reality. He understands the relation between Forms and numbers in a 'Pythagorean' way: ideal numbers represent a Pythagorean way of speaking of Forms. Forms themselves are not (mathematical) numbers, ⁴⁸