48 Cf. 103, 15 ff.; 186, 30-5; 45, 33-46, 5; for the difference between Forms and universals in the soul cf. 105, 37-106, 5.

but by referring to Forms as numbers the Pythagoreans expressed through mathematical analogy significant ontological (formal and dynamic) features of Forms. Forms can be described as numbers in that as causes they act as measures for their effects, fitting them together and unifying them (103, 15-104, 2; 134, 22-6). The Forms constitute a decad in the sense that they are a comprehensive paradigm for the world, just as the decad contains all numbers (147, 29-148, 9). And just as number is produced from a monad and dyad, so there are two principles producing Forms which, by reason of their analogous function, can be referred to as monad and dyad (cf. 132, 14-20; 134, 22-35; 48, 23-5). However the fact that the realm of Forms is being approached through analogies starting from their images, mathematical numbers, must not be ignored, since mathematical numbers operate with units and quantity, features not relevant to a proper account of Forms (131, 37-132, 2; 186, 30-5). The difference between a mathematical foreshadowing of the formal characteristics of pure Form and a true knowledge of the Forms is found also in the distinction Syrianus makes (as did Iamblichus) between the demonstrative method of mathematics and the higher, direct insight ( πιβολ ) whereby metaphysics attains its objects (4, 29-5, 2). Yet even when it comes to expressing the relation between metaphysics and the sciences subordinate to it (mathematics and physics), Syrianus cannot avoid mathematical analogy. In reply to Aristotle's puzzle in Metaphysics B 1, as to whether metaphysics should study (only) the first causes of substances, or also the principles of all demonstrations, Syrianus defends the latter alternative on the grounds that metaphysics,

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like a monad, pre-contains the principles of all the sciences subordinate to it (3, 17-30).

It is clear that Syrianus identifies Aristotelian metaphysics, the science of being, with 'dialectic', the study of pure intelligible being (or the Forms) of Plato's Republic. ⁴⁹ 

⁴⁹ Cf. 55, 3-32; 58, 13-14; Verbeke (1981); O'Meara (1986), 5-7.

Both in turn relate in some way to the Pythagorean science of the divine which is to be found also in the theology of Plato's Parmenides (cf. 121, 5; 126, 15-16). This theology deals with the divine characterized as 'divine number' (θει ος ιθμ ς) in the same way as the Forms are called 'ideal numbers'. ⁵⁰ 

⁵⁰ Cf. 124, 24-5; 146, 9; 166, 16-24; 177, 19-27; 179, 23-30.

Pythagorean-Platonic theology, it seems, concerns more than the Forms: higher principles come within its scope. Before such principles are introduced, an important differentiation within the realm of Forms must first be noted.

In Chapter 3 above a distinction was found in On Pythagoreanism VII between 'intelligibles' (νοητ ) and 'intellectuals' (νο ). It reappears in Syrianus' Commentary (cf. 4, 17-18) and expresses itself as a distinction between 'intelligible' and 'intellectual' numbers (122, 31-2). Indeed Syrianus refers us to On Pythagoreanism for the difference between intelligible and intellectual monads (140, 11-15). Syrianus also introduces the distinction in his exegesis of some (Pythagorean? Orphic?) verses at 106, 15 ff.: Forms come to be 'intelligibly' and 'tetradically' in the 'ideal animal' (τ α τοζ ον) and then 'intellectually' and 'decadically' in the demiurgic intellect. The language indicates that this is in fact an interpretation of Plato's Timaeus in which the problem of the relation between demiurge and model is solved by subordinating the demiurge to the model. ⁵¹ 

⁵¹ That this is in fact Syrianus' interpretation of the Timaeus is confirmed by Proclus, In Tim. I 322, 18-323, 22; cf. Gersh (1978), 139-40.

Mathematical expression is given to the subordination: the demiurgic intellect comprises the Forms fully articulated as are the numbers in the decad, whereas the 'ideal animal' contains the Forms in the way that the tetrad contains the essential features of numbers; the ideal animal is to the demiurge as the tetrad is to the decad. Thus mathematical relations can serve to formulate the structure internal to the realm of Forms.

The 'intellectual decad' of Forms is subordinate and pre-contained then in a more unified way in the 'intelligible tetrad' which itself derives from two more ultimate principles, a monad and dyad. Indeed

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the monad and dyad play a central role in the constitution of reality. The identity and diversity by which they produce Forms are echoed at each succeeding level of being, more and more faintly, down to the organization of nature, each level 'unfolding' from a monad-dyad pair correlative to that level but deriving ultimately from the first and highest pair. ⁵² 

⁵² Cf. 5, 16-31; 48, 23-31; Praechter (1932), 1756; and especially Sheppard (1982) on the importance of monad and dyad as principles.

This first pair is not to be identified with the first members of the intelligible tetrad, the 'monad itself' and 'dyad itself', but it transcends the tetrad, and therefore pure intelligible being. The supra-intelligible monad, as a unity, itself presupposes a higher principle, the ultimate origin, the One. Although Syrianus' need to invoke theological authorities sometimes blurs rather than sharpens these distinctions between the ultimate One, the supra-intelligible monad and dyad and the intelligible monad and dyad, they emerge with sufficient clarity from the following two passages:

It seems to me that Empedocles supposes 'Friendship' to be nothing other than the One, not the One that is not co-ordinate with anything , but that which is co-ordinate with the unlimited dyad, which he refers to as 'Strife', from which what primarily is and all the intelligibles and the sensible world come to be. (11, 28-31; cf. 43, 11-20)

Metaphysics N 1> one should say that these men include in the first causes, not such opposites as are non-beings in the sense of being inferior to being, but . . . such as are non-beings in the sense of superior to being. For the first principles of beings must transcend being. And yet these men did not begin from opposites , but they were aware of what transcends the two opposite orders, witness Philolaus who says god produced limit and the unlimited. ⁵³ 

⁵³ 165, 31-166, 1; cf. 112, 35-113, 5 (δυ ς χηγικ /α τοδυ ς); Sheppard (1982), 2-4.

Some of the elements of this metaphysical structure can be found in Hermias' Commentary on the Phaedrus. Reality, in Hermias, also culminates in an ultimate One (121, 19). He too assumes a distinction between the 'intelligible' and 'intellectual' orders (134, 3-4; 89, 8-15). And he alludes to a supra-intelligible level in the following text:

But if you were to contemplate the beautiful, the wise and the good according to a mode transcending the gods' intellect, you should say that there is a light ⁵⁴ 

⁵⁴ Or. Chald. 49, 1; cf. Proclus, In Tim. III 13, 29-14, 4.

proceeding from the Good . . . Plato says in the Philebus that it is not possible

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for human thought to grasp the Form of the Good with simple insight. Since thus that light proceeds immediately from the Good, it still remains above Form and simplicity. It is thus impossible to grasp it by simple insight ( πιβολ ). (134, 12-18)

(VI) Syrianus and Iamblichus

If one compares the metaphysical landscape presupposed by Syrianus' Commentary and Hermias with that reconstructed above in Chapter 3 from what remains of Iamblichus' On Pythagoreanism, one will find a large measure of agreement, but also, it appears, some differences. One difference regards the level intervening between the ultimate One and intelligible being (the Forms). In Iamblichus it seems that this level included not only monad and dyad, but also triad and (probably) the following members of a supra-intelligible decad. These 'divine numbers' constitute the most unified order of members who are themselves unities. Yet in Syrianus it appears that only a monad and dyad inhabit this level. This may, however, be a matter of emphasis and context. Hermias refers to the principles that follow the ultimate One as 'the first henads' (121, 19). Syrianus also speaks of 'monads or henads which proceed from the very first cause' (In Met. 183, 24-5) and he makes a distinction, for Forms prior to the demiurgic (i.e. 'intellectual') Forms, between 'unified' ( νιαι ος) and 'essential' (ο σι δης) numbers (126, 17)—since the latter constitute the intelligible order, the former must represent the order above it. And if the supra-intelligible level in Syrianus contains 'henadic numbers', it seems reasonable to expect that it extends beyond monad and dyad to the following (properly speaking) numbers of a decad. ⁵⁵ 

⁵⁵ This allows us to explain the relation between Syrianus' supra-intelligible monad-dyad pair and the doctrine of divine henads attributed to Syrianus by Proclus (In Parm. 1062, 20-34; Theol. Plat. I lxxxvi). The problem posed by this relation is raised by Sheppard (1982).

Clear disagreement between Iamblichus and Syrianus can be detected however on some specific points. A particularly revealing case is found in Hermias. At issue is the interpretation of the company of twelve gods, led by Zeus, in the Phaedrus (246 e). Some possible exegeses are examined and rejected and then that of Iamblichus is introduced (136, 17-29):

However the divine Iamblichus, noting the name Zeus, relates the passage to the one demiurge of the universe who is spoken of in the Timaeus <>

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'intellectual' demiurge>. But we, here also admiring the man's insight ( πιβολ ), would add only this: that by Zeus is not to be understood simply the one and transcendent demiurge, for this transcendent demiurge does not have a particular place in the 'army' Phaedrus>, . . . but is present equally in all ranks. . . . This is then our view, following Plato and the theologians: after the demiurgic monad, the one and transcendent Zeus, come three Zeuses . . . each of whom has subordinate to him four further gods.

Zeus is found at three levels: the transcendent demiurge proper on the 'intellectual' level; below him a triad; and below this twelve gods, led by the Zeus of Plato's Phaedrus. ⁵⁶