18 Cf. In Parm. 926, 16-29.

For they show the images in numbers of the characters of what transcends being, revealing in discursive objects the powers of intellectual figures. Thus

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Plato explains to us many wonderful doctrines about the gods by means of mathematical forms, and the philosophy of the Pythagoreans conceals its secret theological initiation using such veils. Such also is the entire Sacred Discourse, the Bacchae of Philolaus, and the whole approach of Pythagoras' exegesis concerning the gods. (21, 25-22, 16)

With this Proclus ends his theological section. What is additional here in Proclus, it is clear, is a recapitulation of the major point concerning the theological relevance of mathematics as expounded by Iamblichus, with the addition of some references. The contrast implied in the latter between Plato's explanations and the secretiveness of Pythagorean initiation will not appear fortuitous if seen in connection with Proclus' distinction, found elsewhere (above, p. 148) between Plato's scientific and the Pythagorean mystical approach.

(b) Physics. Proclus lists most of the mathematical aspects of physics that occur in Iamblichus. However Iamblichus' 'analogy that travels throughout everything in nature' (55, 25) appears in more correct Platonic form in Proclus as the 'analogy that binds together everything in the world, as Timaeus says somewhere . . . ' (22, 19-21). A little later Proclus returns to the Timaeus to illustrate further the mathematical structuring of nature (23, 2-11), whereas Iamblichus ends his section with a vague reference to 'noble students of nature who start from first principles' (56, 3-4). Iamblichus' general reference, intended presumably to cover all sorts of ancient Pythagorean cosmology, is narrowed in Proclus so as to refer specifically to Plato's Timaeus.

(c) Politics and Ethics. The chief difference here is the use made of Plato in Proclus (23, 12-24, 20) to illustrate and elaborate on points made more briefly in Iamblichus (56, 4-13).

(d) The Arts. Proclus' treatment of theoretical arts (24, 23-7) is much shorter than the section on this in Iamblichus (57, 9-22), but it makes Iamblichus' point more clearly and names the art Iamblichus is discussing, rhetoric. As for productive arts Iamblichus says merely (57, 22-3): 'for productive arts has the rank of paradigm', whereas Proclus tells us: 'for productive arts has the rank of paradigm, generating before in itself the principles and measures of what are produced' (24, 27-25, 3). The all-too-brief statement in Iamblichus becomes, with the further words in Proclus,

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Intelligible. Finally, on the subject of the practical arts, Proclus makes explicitly (25, 6-7) the use implicitly made in Iamblichus (57, 26-7) of Plato's Philebus.

The differences then between Proclus' chapter 8 and the corresponding part of Iamblichus' book can be accounted for as deriving from a desire on Proclus' part to clarify, explain, and document Iamblichus' text. The reasons for his systematic introduction of passages from Plato will only become fully clear when his views on the relations between Euclid and Plato are examined. But already a distinctive tendency of Proclus that has been noticed elsewhere, namely his inclination to see Plato as more scientific and the Pythagoreans as more mystical, has emerged in our comparisons. This result indicates that the differences in Proclus' text vis-à-vis Iamblichus' book are to be taken as expressions of Proclus' own preferences. The study of chapter 6 of Proclus' prologue, in the critique of the Geminus theory above, also shows that this chapter is to be understood as an elaboration of Iamblichus' text in the light of what Proclus had learnt from Syrianus and developed further.

We may conclude then that the supposition that Proclus used Iamblichus directly in composing his prologue yields satisfactory results in the case of the two chapters examined. From this viewpoint the similarities and differences between the texts of Proclus and Iamblichus can be explained. Indeed some of the differences reflect ideas characteristic of Proclus and his teacher Syrianus. The revisions that these differences imply are also compatible with what is otherwise known of Proclus' skill as a philosophical writer. And his use of Iamblichus is made in any case extremely likely through what has been shown concerning Syrianus' use and recommendation of Iamblichus' book. On the other hand the attribution of some chapters in Proclus corresponding to Iamblichus to an earlier source has been shown to be untenable. ¹⁹ 

¹⁹ I do not mean to deny that Proclus uses Geminus, merely that the chapters in Proclus corresponding to Iamblichus' work are taken, not from Iamblichus, but from Geminus. It is worth paying attention to the way in which Proclus introduces Geminus in the first prologue: he is named as a proponent of a division of the mathematical sciences other than that of the Pythagoreans (38, 1-4), i.e. that of Nicomachus reproduced by Iamblichus (Comm. ch. 7) and summarized by Proclus in the preceding pages (35, 19 ff.). The former division is alluded to by Iamblichus (95, 22-4), but Proclus gives it a representative, Geminus. Again Proclus is clarifying and documenting Iamblichus' text. More generally on the liberties taken by Proclus with his sources, cf. Heath (1956), 133-4.

These conclusions deserve perhaps some emphasis. They indicate

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that Proclus had studied (at least) Book III of Iamblichus' On Pythagoreanism as carefully as Syrianus could have wished, and that Proclus' prologue can justifiably be used as evidence for his attitude to Iamblichean Pythagoreanism as expressed in On Pythagoreanism III. ²⁰ 

²⁰ In In Parm. 758, 37-8 Proclus appears to be using On Pythagoreanism IV (In Nic. 20, 14-17) rather than Nicomachus Intro. arith. 15, 4-10. Proclus' references outside In Eucl. to Pythagorean mathematical ideas are often not precise and full enough to enable us to be sure that his source is Iamblichus rather than some other text, for example Nicomachus.

This makes the prologue a privileged source for the study of the evolution of Proclus' ideas in relation to those of Iamblichus.

(II) Proclus' Revisions of Iamblichus

The analysis above has indicated some of the ways in which Proclus altered the Iamblichean original: he documented the Iamblichean material by naming and quoting more accurately and fully from Iamblichus' sources, and clarified Iamblichus by simplifying some passages and expanding others. It can be seen that he accepts the essentials of Iamblichus' position on the importance of mathematics: consequent on the intermediate position of its objects in the structure of reality, mathematics has a pivotal and far-reaching function, for it constitutes a paradigm for physics, ethics, politics, and the arts, and fore-shadows the objects of theology or metaphysics. The latter aspect expresses a function of mathematics that Proclus documents (20, 14 ff.) with the text of Plato's Republic that Iamblichus (55, 8 ff.) silently uses, namely the function of leading the soul up to immaterial being. The one important doctrinal difference between Proclus and Iamblichus noted so far concerns the relation between mathematicals and the soul. The somewhat confusing situation in Iamblichus' text is not allowed to persist in Proclus: he firmly places mathematicals in the soul in the way Syrianus had done, that is, they are 'projections' by the soul of intelligible principles innate in it. This clarification is of some importance, for it determines, as I shall show later, the precise way in which mathematics may be used in physics and metaphysics. In what follows I shall supplement these results with comparisons between some other chapters of Proclus' prologue and Iamblichus' text.

The effort observed above to name and quote more accurately and fully the sources Iamblichus incorporated in his text can be found elsewhere in Proclus' prologue. One might note for example his use of

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Aristotle to this effect. ²¹ 

²¹ Compare Proclus, 32, 23-6 and 33, 21-34, 1, with Iamblichus 84, 21-5 (with the scholium ad loc., 103, 22) and 86, 2-6.

Especially noteworthy, however, is his use of Plato: he not only supplements Iamblichus' text with quotations from Plato, but also substitutes the original Platonic passage for the (Pseudo-) Pythagorean plagiary cited as an authority by Iamblichus. Thus Iamblichus' 'Archytas' text in Comm. 12, 9-13, is replaced in Proclus (3, 14-16) by a reference to the original text of the Republic. The same happens elsewhere, ²² 

²² Compare also Iamblichus, Comm. 44, 10 ff. ('Archytas') with Proclus 45, 15 ff. (Plato's Meno); the substitution of Platonic for Pythagorean texts is noticed by Mueller (1987b).

most notably at 10, 15 ff., where Iamblichus' authorities, Brontinus and Archytas (Comm. 34, 20 ff.), are not reproduced; Proclus refers rather to Plato's Republic (the source of Iamblichus' authorities), saying: 'Let us examine after this what the criterion of mathematics might be and let us place before us Plato as guide to this revelation (π οστησ μ θα κα τη ς το του πα αδ σ ως γ μ να τ ν Πλ τωνα)' (10, 16-19). The language is Iamblichean. ²³ 

²³ Cf. above, Ch. 4 n. 20.

But it also indicates, I believe, how Proclus' substitution of Platonic for 'Pythagorean' texts is to be understood. It is not the case that Proclus suspected that Iamblichus' Pythagorean authorities were forged, copied from Plato. Proclus does not hesitate after all to cite himself in his prologue the Sacred Discourse (above, p. 162). Rather he prefers Plato to Archytas, Brontinus, etc., as his guide to mathematics. ²⁴ 

²⁴ Cf. the opening of Proclus' second prologue: Πλ τωνι συμπο υ μ νοι (cf. Phaedr. 249 c), 48, 3.

This preference has to do primarily with the fact that he regards Euclid, as we shall see, as a faithful exponent of Plato, and this in turn makes Plato into the principal authority of his Commentary on Euclid. But we may suspect also that the distinction for Proclus between Plato's scientific explanation and Pythagorean revelation (above, p. 162) is responsible at a deeper level for his choice of Plato's dialogues, rather than Iamblichus' Pythagoreans, as appropriate sources for the study of the nature and function of mathematics.

Greater attention to Plato leads, however, to a problem which Proclus discusses in chapter 10 of his prologue, a chapter coming between two chapters corresponding respectively to Iamblichus' Comm. chapters 26 and 27. The problem is the following: a major theme in Iamblichus' book (following Nicomachus) is the scientific character of mathematics. Yet some members of Proclus' school (29,

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14 ff.) found that Plato after all does not regard mathematics as scientific. In the Republic in particular Plato regards mathematics as defective since it does not examine the hypotheses from which it proceeds. Mathematics is therefore subordinate to dialectic, a science that works back to an ultimate unhypothetical first principle upon which all knowledge depends. In defending the scientific character of mathematics Proclus fully recognizes its subordination to dialectic, which he calls 'perfect' and 'true' science (31, 19-23). As receiving its first principles from dialectic, mathematics is science in a secondary way. ²⁵ 

²⁵ 32, 3-5. Cf. also De prov. §§ 50, 28-9; I. Hadot (1984), 123-4 and (on earlier discussions of the problem) 77-9.

As if feeling, however, that this might still not be sufficient to save the scientific claims of mathematics, Proclus attempts to defend further these claims. It is not the case that mathematics is ignorant of its first principles, but it receives these principles (as concepts innate in the soul) and demonstrates what follows from them. In this way mathematics elaborates a scientific knowledge of its matter (32, 5-7 and 16-18). Proclus' argument in fact gives to mathematics the characteristics of science as identified by Aristotle. Plato's own approach in the Republic is hardly as positive. The distance separating Plato from Proclus' interpretation of Plato can be measured by the fact this interpretation arises from the attempt to reconcile Iamblichus' Nicomachean emphasis on the scientific character of mathematics with Plato's critique of the deficiency of mathematics, a reconciliation based on Aristotle's characterization of science.