Number in the metaphysical landscape
The question of the relation between number and metaphysics forms one of the sharpest conceptual divides between Plato and Aristotle. Consistently throughout his dialogues, Plato envisions number as a building block of the universe: from the arithmetical psychogony in the Timaeus, to the mixture of Limit and Unlimited in the Philebus, to the second hypothesis in the Parmenides, to the Indefinite Dyad in the “unwritten doctrines”.1 The systematic pursuit of number earns Plato the recognition that he “Pythagorizes”.2 In his own words, the significance of Pythagoras’ teachings equals only Prometheus’ gift of fire to humanity (Phlb. 16c5–10). Aristotle, on the other hand, negates any ontological value to number. His anti-Platonic polemic in the Metaphysics rejects the idea of number as an intermediary ontological class between sensibles and non-sensibles, and the conception of the Indefinite Dyad as the originating principle of substantiation.3 For Aristotle, number is simply intellectual abstraction of empirical experience.
The title of this chapter itself declares on which side of this Platonic–Aristotelian divide the Neoplatonists stand. Despite Aristotle’s virulent criticism, Plato’s view of number finds fertile soil to grow and flourish in the propitious climate of Neopythagorean philosophy and the exact sciences in late antiquity. This chapter examines the Neoplatonists’ expansive interest in the constitutional role of number and maps the metaphysical landscape according to the ontology of number in the thought of its most avid proponents: Plotinus, Iamblichus, Syrianus and Proclus.
The times have passed when one could cautiously study the “relation” between number and metaphysics with the palpable intent to preserve the analytical purity of Neoplatonism from the shadow of numerological mysticism. In the last three decades, the concept of number has grown out of its cast as “the ugly duckling” in the study of Neoplatonic ontology.4 Our advanced understanding of its Neoplatonic treatment requires restating the question about the relation between number and being directly, without its Aristotelian baggage.5 The recent trend of taking Neoplatonic metaphysics, in its dazzling diversity, as a dominant framework which integrates all other philosophical branches, including physics, psychology and ethics, makes the re-examination of the import of number timely and relevant. If we welcome the assessment of Chiaradonna and Trabattoni – as we should – that “physics is but a part of metaphysics”, we should also agree that this is even more so true in the case of mathematics.6
NUMBER THEN AND NOW
To the Neoplatonic mind, the beauty and multiplicity of the visible world are but pale imperfections of the beauty and order of its invisible underlying principles. As Linguiti elicits in Chapter 22, below, physicality, as if an optical illusion, distorts the holistic perfection of the intelligible principles. Seeking to understand the paradigm of the natural world, the Neoplatonists, following their Platonic predecessors, grapple with ways to transgress the illusory knowledge gathered by the senses. Just as in everyday empirical demonstrations such as when we look at a spoon in a glass of water from below and it appears crooked, so, from a Neoplatonic perspective, do observation and examination of physicality obscure and misinform our understanding of the true essence of reality.
Now, if we look at the same spoon from above, it does not appear crooked any more but only slightly magnified. The top-down approach yields a more accurate perception of the spoon and as long as we take into account the magnifying effect of the water, we acquire a better understanding of its shape. The example illustrates why the Neoplatonists prefer the top-down approach for learning about reality.7 As long as we realize the impermanency and the ontological inferiority of physical reality in comparison to its intelligible model – just as we realize the magnifying distortion in the appearance of the spoon in the water – we can grasp the Neoplatonic view of the natural world as “a sort of ‘physical instantiation’ of metaphysical principles”, albeit distorted and illusory.8 In Plotinus’ words, “even here below a thoughtful life is majesty and beauty in truth, though it is dimly (amydrōs) seen” (Enn. VI.6[34].18.22–4; cf. VI.6[34].8.11).
Rilke’s statement that “we are the bees of the invisible” resonates with the principal Neoplatonic view that our human task is to find the origin of that which surrounds us, to understand the truth of absolute existence and, especially in regard to humanity, to restore our ontological kinship with the divine.9 Like the bees, we labour, purposefully, to collect all possible data from physical reality in order to understand systematically and comprehensively its underlying principles.
For scientists – ancient as well as modern – the honey the bees collect is the quantifiable information measuring every physical property. According to Michael Psellus’ report, Iamblichus understands sensation as “solid number” (Psellus, On Ethical and Theological Numbers 18–19 [O’Meara]). Syrianus attributes to Iamblichus the even more modern view of number as “the ‘place’ of the universe which may be regarded as empty when taken by itself”, but when taken phenomenally, the Monad, with its generation of numbers, does not seem to leave “any place empty, filling as it does all arithmetical receptacles with an uninterrupted succession of numbers”.10 Syrianus’ own contribution to the understanding of arithmetical numbers involves the idea of number as the matter upon which the soul imposes the form of the particular number it inherits from the intelligible.11
From both phenomenological and epistemological perspectives, we try time and again “to rectify” the perception of the “crooked” spoon in the water by measuring all of its quantifiable aspects and by theoretically modelling, based on the collected data, the “perfect spoon”.12 But even then, regardless of how many times we measure the length, width and weight of the spoon, the numbers do not produce absolute but approximate measurements which, for practical purposes, “are good enough” to our senses. Statistical data and mathematical analysis provide a way out for us to quantify the physical world but ultimately they do not furnish true understanding of the inner workings of reality and leave us reaching out to theoretical modelling, speculation and dialectical enquiry.13 This is where the jobs of the scientist and the philosopher converge.14 This convergence is best summarized in Socrates’ advice to the budding mathematician Theaetetus to follow the example of mathematical proof in searching for “a single formula that applies to the many kinds of knowledge” (Tht. 148d4–7).15
Mathematics studies the relation between number and multiplicity and as such it intersects with one of the main domains of philosophy: the study of one and many in the structure of the universe. Rilke’s metaphor about the bees and the invisible acquires a stunningly literal meaning in Socrates’ examination of the question in the Meno. While searching for a single definition of virtue, he encounters a “swarm” (smēnos) of examples. Baffled by this plurality, he observes:
I seem to be in luck. I wanted one virtue and I find that you have a whole swarm of virtues to offer. But seriously, to carry on this metaphor of the swarm suppose I asked you what a bee is, what is its essential nature (peri ousias hoti pot’ estin), and you replied that bees were of many different kinds … what is that character in respect of which they don’t differ at all, but are all the same?
(Men.72a4–b2, trans. Guthrie)
With Socrates, the Neoplatonists also look for the single essence (ousia) of things that are many. Multiplicity, discrete or continuous, is the first most apparent feature of physicality. If unexamined, it appears chaotic, random and infinite. But for Socrates and the Neoplatonists, behind the overwhelming diversity of physical magnitude and multitude, there is a permanent intelligible order. The concept of number is in the privileged position to be both “the honey the bees collect” when measuring the quantifiable properties of multiplicity and “the bee” itself in modelling the paradigm of reality.
The philosophical debate whether number is quantifiable or ontological in nature is still ongoing. The modern permutations of the Platonic–Aristotelian divide are many, from Kant’s utilitarian arithmetic, to Frege’s view of number as a priori analytic judgement, to Field’s nominalism.16 The latest Aristotelian instalment in it is Leng’s (2010: 258–60) defence of mathematicals as “representatively useful fictions” of mathematically stated hypothesis of empirical theories. Her premise supports Quine’s quip that “[t]he philosopher and the scientist are in the same boat” (Quine 1960: 3). Quine’s boat can be happily named “Naturalized Ontology”, in Leng’s phrase, and it is sailing away from the Platonic paradise of understanding numbers not only as expressions but also as constituents of underlying ontological principles (Leng 2010: 36–43).
In late antiquity, Quine’s boat would still have had scientists and philosophers on board. They would have been a quite diverse crew whose differences, however sharp, would not have led to mutiny. Even in Euclid’s Elements, the absence of specific treatment of the relation between number and ontology is not strong enough to make an ex silentio argument for his rejection of the idea. Anyone acquainted with the common Neoplatonic analogy of point–line–circle for visualizing and explaining the structure of the universe would recognize the Platonic and Neoplatonic sub-context of the opening entries in the Elements.17 A stronger ontological flavour of the Euclidian definition of point is detected in Moderatus’ distinction between a monad as the first principle of number (tōn arithmōn archē) – homogenous and indivisible – and the arithmetical number one as the principle of enumerating individual things (tōn arithmētōn archē).18 Taken even further, Moderatus’ conception of number as “a system of monads” (systema monadōn), which progresses from a monad into multiplicity and regresses from multiplicity into a monad,19 finds its fully developed metaphysical parallel in the Neoplatonic view of procession (proodos) and return (epistrophe) as the ontological movement of intelligible substantiation.20 It foreshadows the future Neoplatonic understanding of the ontological dynamics between one and many microscopically in the individual substantiation of beings, macroscopically in the Monad of the Supreme Living Being and supra-cosmically in the absolute unity of the One as the foremost originative principle of existence transcending existence itself.
It is unnecessary to look for possible tension between Euclid’s strictly arithmetical treatment of number and Moderatus’ ontologically suggestive definition of number.21 The physical and metaphysical dimensions of the concept are mutually informative even when a philosopher or a mathematician still favours one over the other because the other, although absent, saliently remains in the background of discussion.
At the same time hard-core Neopythagoreans like Nicomachus toy with ways to articulate sharper terminological differences in the entangled fabric of the concept. In his Introduction to Arithmetics, Nicomachus talks about intelligible number (noētos arithmos) and scientific number (epistēmonikos arithmos). The former operates at the cosmological level: “In the thought of this demiurgic god”, there is intelligible number (noetos arithmos) which is completely immaterial (aulos), eternal (aidios) “according to which … everything is completed: time, motion, heavens, stars, revolutions of all kinds”. The latter “embraces the essence of quantity, … odd and even”.22 This explanation does not, explicitly or implicitly, distinguish the scientific number from the intelligibles nor does it connect the scientific number with its physical representations, as one familiar with the later Neoplatonists’ “Platonized” interpretations of Nicomachus may expect.23 Instead, it seems that this kind of number is still part of the intelligible and Nicomachus is working out here the conceptual difference between its cosmological role and its intrinsic property of expressing quantity.24 Nicomachus’ search for “the essence of quantity” in scientific number (ousian … tēs posotētos, Ar. I.6.4) and its proper place among the universals faintly anticipates Plotinus’ principal question about number; that is, how it relates to primary substance (ousia). Nicomachus’ notion of ousia in the definition of scientific number does not share the Neoplatonic meaning of the term, but it does mark an evolutionary stage between Socrates’ use of the term in the Meno (72b1), cited above, and its fully fledged Neoplatonic content by juxtaposing the questions of essence of being and essence of quantity.25
NUMBER IN PLOTINUS’ METAPHYSICS
Plotinus’ understanding of number stands out in its simplicity in comparison, as we will see, with the later developments of the concept in Iamblichus, Syrianus and Proclus. He conceives of two kinds of number: substantial (ousiōdēs) and monadic (monadikos). The former, non-discursive and non-quantitative, belongs to the intelligible realm. The latter, discursive and quantitative, measures or enumerates the physical world (Enn. V.5[32].4–5; VI.6[34].9). The two are in a Form-and-image relation.
Underneath this simplicity, there are four undercurrents foreshadowing the future evolution of the concept: Aristotelian, Platonic, Neopythagorean and genuinely Plotinian. The Aristotelian undercurrent is recognized in the initially crude distinction between intelligible and arithmetical number. Plotinus’ singular treatment of number in Enn. VI.6[34] offers an anti-Aristotelian defence of Plato’s stance.26 It systematically refutes Aristotle’s misconceptions of the “number of infinity”, the Indefinite Dyad and the nature of the arithmetical number as intellectual abstraction by soul. Ultimately, Aristotle’s terms for the intelligible and the mathematical number – respectively formal (eidētikos) and arithmetic number (arithmētikos) – are reconstructed and renamed as substantial (ousiōdēs) and monadic number (monadikos).27
The anti-Aristotelian and pro-Platonic undercurrents in Plotinus’ view are, of course, given. His most original work on the concept does not concern Plato’s exoneration but his concentrated effort on explicating the nature of substantial number. Instead of focusing on the relation between intelligible and monadic number (as one would if following the Platonic–Aristotelian divide), he shifts his attention to the relation between number and substance (ousia). It is in this aspect that the Neopythagorean undercurrent in his concept is most visible (Charles-Saget 1980: 9–17, 52; Slaveva-Griffin 2009: 42–53). Because the ultimate Neopythagorean answer to the question of how number relates to the One is the equation of number with the One as the Father and the Monad, Plotinus has to work out a solution in which the One, as the Neoplatonic first originative principle of existence, retains its transcendence even when it comes to the origin and role of number in the intelligible:28 “For number is not primary: The One is prior to the dyad, but the dyad is secondary, and originating from the One, has it as definer, but is itself of its own nature indefinite; but when it is defined, it is already a number, but a number as substance (arithmos hōs ousia); and soul too is a number” (Enn. V.1[10].5.6–9).
Plotinus’ solution to the problem is this (cf. Enn. V.5[32].4; VI.6[34].9). Since the One is prior to any existence, the One is ontologically superior to the Indefinite Dyad – the principle of potentiality – and the substantial number. The Indefinite Dyad, however, is not ontologically superior to number but the two of them form an ontologically equal pair in which the One defines the Dyad and this defined Dyad is number “as substance” (arithmos hōs ousia). The Indefinite Dyad and the substantial number are the two sides of the same ontological process. The Dyad as an absolute principle of potentiality is indefinite by nature but, as a recipient of the productive power of the One, it is limited and thus defined by “number as substance”. Number – and not the Indefinite Dyad – orders intelligible multiplicity.
What does Plotinus mean by “number as substance” and why should there be a relation between number and substance? If he is going to argue that number in the intelligible does not have anything in common with arithmetical number in the physical world: first, he has to deny its association with quantity, and second, if number is going to participate in the intelligible, he has to explain how it relates to substance as a primary kind of being.29
Plotinus’ first answer – concerning quantity – is Platonically straightforward. The main characteristic of intelligible existence is incorporeality and the main characteristic of incorporeality is the complete lack of quantity.30 Even the Form of quantity (eidos he posotes, Enn. II.4[12].9.7) itself is bodiless and thus without quantity.31 Physical quantity and the arithmetical number are excluded from the intelligible.
The second answer – about the relation between number and substance – is in the heart of Plotinus’ ontological work on number. Insisting that the One is not number, he is quick to explain that “number has an existence from itself” but also that “substantial number is that which continually gives existence (to einai aei parechōn)”.32 If number is the defining expression of the One onto the Indefinite Dyad, then number must be intrinsic to substance:
And certainly the beings were not numbered at the time when they came to be; but it was [already clear] how many there had to be. The whole number therefore, existed before the beings themselves. But, if numbers were before beings, they were not beings. Now number was in being, not as the number of being – for being was still one – but the power (dynamis) of number which had come to exist divided being and made it, so to speak, in labour to give birth to multiplicity. For number will be either the substance or the actual activity of being (energeia), and the absolute living being is number, and Intellect is number (to zōion auto kai ho nous arithmos). (Enn. VI.6[34].9.22–9)
The passage explicates what Plotinus means by number acting “as substance”, cited earlier (Enn. V.1[10].5.6–9, above, p. 204). The relation of number to substance is such that number is its power (dynamis) and actuality (energeia). On this account, he does not call this kind of number “intelligible” (noetos) with Nicomachus, nor “Formal” (eidetikos) with Aristotle, nor even “true” (alethes) with Plato, but “substantial” (ousiōdēs).33
As power and actuality, substantial number then permeates the structure of the intelligible in four specific aspects: Being is “unified number” (to on arithmos henomenos),34 Intellect is “number moving in itself” (nous arithmos en heautōi kinoumenos), beings are “number that has unfolded outward” (ta onta arithmos exeleligmenos) and the Complete Living Being is “encompassing number” (to zōion arithmos periechōn).35 These four aspects converge in Plotinus’ definition of existence as “separation from the One” (apostasis tou henos) (Enn. VI.6[34].1.1; Slaveva-Griffin 2009: 42–53). The beginning of the centripetal motion in Intellect is thinking itself, which makes its first ontological stop at the “unified number” of Being through the self-moving/thinking number of Intellect. Then it unfolds outwardly into the individual beings, just to fold itself inwardly as the “encompassing number” of the Complete Living Being. Since all these are only aspects of the activity of substance as number, they are ontologically equidistant from the One (see Corrigan [Chapter 24], below). Together the four aspects of substantial number compose the map of Plotinus’ metaphysics.36
Since for Plotinus there are only two layers of reality, excluding the transcending One, he is concerned only with two kinds of number: substantial and monadic. His concept of monadic number is deliberately straightforward Platonic, rejecting the Aristotelian notion of abstraction and underplaying the Neopythagorean idea of intermediacy. Since the monadic number pertains to the physical world, it is an image of substantial number. This is perhaps the reason why Plotinus does not busy himself with the subject of mathematicals – a well-established Platonic topos, which flourishes in the later developments of the concept – as intermediary entities between the intelligible and the physical world.37 Plotinus understands Plato’s “true number” to mean “number as substance” which “has an existence from itself and does not have its existence in the numbering soul” (Enn. V.5[32].4 and Enn. VI.6[34].4.21–3 respectively). The notion of intermediateness between the two layers of the structure of the universe is carried out by the concept of Soul, the third underlying principle of existence, which he conspicuously leaves out from having a specific aspect in the substantial number (Slaveva-Griffin 2009: 112–30; Maggi 2010: 89–93).
The internal complexity of substantial number reflects the dynamic relations within the intelligible. The non-linear nature of the concept, contrary to its physical counterpart, creates tension with its monolithic surface. This opens the door for future conceptual proliferation, the seed for which is already planted by the Neopythagoreans, but it could not grow to fruition without Plotinus’ metaphysics.
NUMBER IN IAMBLICHUS’ METAPHYSICS
The first signs of proliferation of the concept are found in Iamblichus. His strong interest in a more precise stratification of reality focuses on the outer edges of the ontological ladder by positing, respectively at the top and at the bottom, a divine number superseding the intelligible and a self-moved number superseding the physical. Thus the divine number (theios/theologikos) leads the catagogical sequence of intelligible/eidetic number (noētos or eidētikos), intellectual number (noeros), self-moved number (autokinētos) and physical number (physikos).38
This variegated complexity differs starkly from the two-toned nature of the concept in Plotinus. It obviously suits better Iamblichus’ pronounced Neopythagorean interests. With the Pythagoreans around Hippasus, he views number “as the first paradigm of creating the universe” and mathematicals as useful for understanding both “physical and theological matters” (physika ē theologika) (In Nic. 10.20–21; Comm. Math. 92.19–20). But matters of ontological considerations are equally important to him (Shaw 1995: 33; 1999: 129; Van Riel 1997: 44–5; Maggi 2012: 83). While Plotinus says that all gods are in the intelligible, without further elaboration, Iamblichus sorts out the intelligible paradigm to fit in the ontologically superseding gods.39 He shares the Pythagorean view that divine number “is fitted to the substance of the gods and power, order and activities” and commends the search for “which numbers are similar (syngeneis) and related to which gods (homophyeis)”.40 For him, the last question reveals a genuine – and not symbolic – ontological affiliation between numbers and gods.
In Iamblichus’ ontological ladder, the divine number is succeeded by the intelligible/eidetic number (Comm. Math. 64.1–2). This kind of number, he clarifies, is contemplated in harmony “with the purest substance”, the substance of the intelligible realm.41 Unfortunately, he is not as specific as we would have liked him to be about the ontological relation of this kind of number with soul in the way he is about the relation between the divine number and the gods. His only assertion is that it concerns “the self-moving substance (autokinētos ousia) and the eternal rational principles (aidioi logoi)” (Comm. Math. 64.4–6). It is not clear, however, how this kind of number, if it concerns soul, is different from its successor, the self-moved number (autokinētos). If we accept O’Meara’s judicious association of the self-moved number with soul, as the so-called psychic number,42 the self-moved number completes the sketch of Iamblichus’ metaphysical map.43
For him, the kinds of number are altogether ontologically different and do not express the internal relations of the intelligible realm. While Plotinus focuses exclusively on the inner workings of substantial number within their ontological environment, Iamblichus is interested in ironing out the conceptual wrinkles surrounding the relation of number with what is beyond and what immediately succeeds the metaphysical realm. The difference in their approaches yields different results, with not much ground for comparison.44 There are still two observations worth making.
First, Plotinus time and again carefully dances around the question about the relation between number and the One. The crux of the problem lies in his understanding of the One as simultaneously comprising the ideas of being the first originative principle of all existence as well as of transcending all existence.45 Iamblichus separates the two ideas into an Ineffable One that is beyond existence and One that oversees a dyad of Limit and Unlimited.46 The latter is responsible for mixing Limit and Unlimited in the production of the Unified (to hēnōmenon), which, in its turn, is responsible for the existence of the intelligible (Iamblichus, in Ti. frag. 7 [Dillon]). This Unified is conceptually closest to Plotinus’ “unified number” as an aspect of substantial number, expressed in Being (see above, p. 205). But while Plotinus insists on keeping the unified number of Being separate from the One, although he considers it ontologically superior to the multiplicity of beings, Iamblichus delegates the absolute unity of the Unified to the productive power of the One. He diffuses the tension between ontological superiority and intelligibility, detected in Plotinus’ “unified number” of Being, by distancing the Ineffable One from the productive unified One.47
Second, Plotinus defines Intellect as number “which moves in itself” (arithmos en heautōi kinoumenos) in the intelligible, while Iamblichus talks about “self-moved” number (autokinētos arithmos) as ontologically successive to the intelligible (see above, p. 205). Iamblichus’ placement is ontologically more accurate in the context of the Pythagorean and Platonic notion of soul as number moving in itself (Xenocrates, frags. 181–2). It also makes Plotinus’ concept of Intellect as “number moving in itself” stand out. There is also a difference in the understanding of agency. In his view of Intellect as substantial number moving in itself, Plotinus emphasizes the active agency of motion in Intellect while Iamblichus does not determine the agency of motion in his self-moved number: it is only secondarily implied through its association with soul.48
The reason for the above differences stems from and demonstrates the principal difference between Plotinus’ and Iamblichus’ view of number. We have already noted that the primary focus of Plotinus’ treatment of number lies exclusively on the place of number in the intelligible. By transposing the traditional idea of a self-moving number from the Soul to the Intellect, Plotinus conveys successfully the characteristic of Intellect as self-thinking and thus completes the map of the intelligible as charted by the four aspects of substantial number.
Plotinus conspicuously skirts the question of intermediacy between the two kinds of number and since this is where the role of soul comes to play, it does not receive a corresponding aspect in substantial number. Although he does say that soul is number “if it is a substance”, the modality of his statement betrays his hesitation. The duality of soul, with its descended and undescended part, makes soul unsuitable to his understanding of substantial number. Soul’s relation with the physical world precludes it from gaining a place among the aspects of substantial number and therefore from receiving more attention in his treatment of substantial number.49
Iamblichus, on the other hand, as we have already seen, is interested exactly in the upper and lower edges of the intelligible and their relation with their adjacent realities. His placement of the self-moved number after the intelligible number portrays his commitment to the idea of soul and mathematicals as intermediaries between the metaphysical and the sensible. He embraces the Pythagorean stand on the analogical use of number in understanding the structure of the universe and expands the system of different kinds of number to all echelons of reality. Iamblichus’ number, as put by Maggi (2012: 80), is omni-extensive.
An indicator of this omni-extension is his equally profuse interest in the physical applications of number. The physical number represents the principles that are mixed in bodies (enkekramenoi logoi tois sōmasi), enmattered rational principles (enyloi logoi) and enmattered images (enyla eide) (Comm. Math. 64.8–12; Psellus, On the Physical Number 1–98). They are explicitly distinguished from mathematical numbers (Sheppard 1997). The former “is seen in common concepts”, while the latter “is found in the lowest things, generated and divided in bodies”.50 The terminological difference between Iamblichus’ physical number and Plotinus’ corresponding monadic number reflects Iamblichus’ openly Neopythagorean agenda in examining the physical structure of the universe. Since for him mathematics busies itself with describing the nature (physiologein) of things in generation, he busies himself as well to enumerate the pervasive presence of number in nature (Comm. Math. 64.17–18).
The copious conceptual proliferation of number in Iamblichus embodies his attempt to sharpen the contours of the metaphysical realm, with specific attention to its periphery. With the help of Proclus’ Commentary on the Timaeus, we can note that, for Iamblichus, numbers “are symbols of divine and ineffable truths” (in Ti. II.215.5–7).
NUMBER IN SYRIANUS’ METAPHYSICS
Almost a century after Iamblichus, Syrianus joins Quine’s boat sailing the deep Platonic waters of metaphysics (D. O’Meara 1989: 119–41; Wear 2011). Two examples from his hermeneutical work on Homer suffice to illustrate his intolerance for Aristotle’s obstinate failure to accept Plato’s ontological view of number. To characterize the depth of Aristotle’s arrogance in the matter, Syrianus quotes Poseidon’s rebuke of Zeus’ threat to destroy him: “excellent though you may be, you have spoken defiantly”.51 And, to expose the futility of Aristotle’s criticism, he adduces Hektor’s puzzlement at Glaukos’ accusation of cowardice for not fighting Ajax: “why did a man like you speak this word of annoyance”.52
The quotations point the direction of Syrianus’ interest in number. With Plotinus’ anti-Aristotelian fervour, he fully extends the Platonic model of substantial and monadic number to interlink consecutively all levels of reality (in Metaph. 83.14–26; D. O’Meara 1989: 132; Longo 2010: 623–4). At the same time, with Iamblichus’ Neopythagorean passion, he directly correlates numbers with the Forms by supposing the Forms as causes the ontological effects of which are measured, analogically, by corresponding numbers (in Metaph. 103–4; 134.22–6). Thus the map of Syrianus’ universe is charted successively by divine/henadic (theios or heniaios), intelligible (noetos, eidetikos or ousiōdes), intellective (noeros), psychic (psychikos), mathematical (mathematikos) and physical number (physikos).53
The Iamblichian structure of this scale is apparent but of most interest to us is what Syrianus does with it. He begins by clarifying the conceptual stratigraphy at the top of the ontological hierarchy. He reformulates Iamblichus’ Ineffable transcendent One into a simple unparticipated One, which “is not co-ordinate with anything”, “related to itself” and “superior to being”.54 In the same vein, he understands Iamblichus’ productive One as a participated One which “is co-ordinate with the unlimited dyad”.55 The latter presides over the cosmic principles of Limit (peras), causing unity and sameness, and Unlimited (apeiria), causing procession and multiplicity (in Prm. frag. 5 [Wear]).
Parting further from Iamblichus, Syrianus equates the participated One with Limit and envisions the first relation between Limit and Unlimited as the Unified (to henomenon).56 Thus construed, the Unified constitutes the henadic realm, which is responsible for bridging the gap between the unparticipated transcendent One and the intelligible realm. Regardless of whether the Unified consists of the henads or is the totality of the henads, in any case the Unified and the henads enact the productive power of the participated One in the emanation of Intellect (Wear 2011: 8–9).
Plotinus, almost two centuries before Syrianus, calls the henads holding places (protypōsis) “for beings which are going to be founded on them”.57 Iamblichus, who is most often credited with the origination of the concept, places the henads at the level of the participated One and not at the level of the Unified as Syrianus does.58 Visually speaking, if we were to discern nuances in the internal stratigraphy of the One, the henads occupy the lowest ring of the One in its participated aspect. They represent the absolute unities of beings before they have unfolded, to use Plotinus’ language, into the intelligible and thus they are supra-noetic (Wear 2011: 8). By inserting what could be called a henadic realm or even a henadic number between the One and the intelligible, Syrianus brings to fruition the long-term project of Plotinus and Iamblichus to explain how exactly the One imparts its generative power to Intellect without compromising its absolute unity and transcendence.
“Syrianus,” as felicitously put by Wear, “postulates a new layer of reality for every difficulty he finds in the text of the Timaeus” (ibid.: 4). This, we should add, is true for Plotinus and Iamblichus. The consecutive Form-and-image relation between the kinds of number constructs a tightly knit map of reality in which the outer edges of the metaphysical further emerge: first, through a more specific articulation of the concept of henads, and second, through an elaborate account of the intermediary status of numbers (D. O’Meara 1989: 133). Concerning the latter, Syrianus sharpens the distinction between mathematics and theology found in Iamblichus. Both arithmetical numbers and geometrical figures are ontologically innate to soul and this kinship allows them to reveal, more than analogically, the higher realities.59
NUMBER IN PROCLUS’ METAPHYSICS
The Neoplatonist who first comes to mind on the subject of number and metaphysics is Proclus, both with his Commentary on the First Book of Euclid’s Elements and his exhaustive opus reaching farther than the mention of a single title. In his thought, number outgrows the restrictive division between physical and metaphysical to embroider holistically the fabric of the universe. For Proclus, as for Plotinus, the real meaning of the Platonic axis between physics and metaphysics is primarily ontological. Sensitive to the elevated status of dialectics, Proclus, unlike Iamblichus and perhaps even Syrianus, draws the unexpected conclusion that mathematics veils, not unveils, the hidden principles of reality.60 Number fossilizes and thus distorts the ever-flowing universal processes. How does Proclus reach such a profoundly different but still programmatically Neopythagorean conclusion?
Recent studies of Proclus’ metaphysics have ever so sharply delineated the distinction between “theologizing mathematics and theology or dialectic proper”.61 The heart of the matter for Proclus is that the Greek term analogia, despite its connotation of “vagueness” in English, authentically in Greek denotes “precision”, particularly in mathematical contexts.62 The material world is informed by mathematics not only because arithmetic and geometry quantify, measure, and spatially relate or mentally extrapolate physical properties but also because mathematicals occupy a middle ground between the intelligible and the sensible. But the elements of precision and intermediacy do not suffice to convince him that mathematicals unequivocally reveal the true principles of existence: “For the reasons (logoi) that govern Nature are not receptive of the accuracy or the fixity of mathematicals. …Therefore it is not possible to consider physical things arithmetically.”63 To finish the job of mathematics, Proclus employs the method of “analogical reasoning” (even when expressed more geometrico), built upon the premise that the mind puts together the similarities between different objects from different layers of reality not because of its dialectical capacity but because the layers themselves are ontologically related.64
His realism about the constraints of mathematicals in the study of the natural world inherits a certain Plotinian nuance. With the four aspects of substantial number, Plotinus completely divorces number from mathematics and physicality in order to explain the dynamics within the intelligible. Proclus too sees, on his own terms, the main constraint of mathematicals in that they do not enact but freeze, in a frame, the self-sustainability of both the intelligible and the living organism of the universe:
There is the first proportion through which nature puts harmony into its own works and through which the Demiurge organizes the universe (logos) running through itself primarily and then through all things … sympathy or co-affection (sympatheia) comes to be among all the things in the cosmos, inasmuch as all things are guided by one life and a single nature.
(in Ti. II.24.2–7)
This life consists of individual instantiations of the tripartite cycle of procession (proodos), remaining (mone) and reversion (epistrophē) re-enacted at each ontological level.65 The physical world, including the crooked spoon we mentioned in the beginning of the chapter, actively participates, according to its own ontological purpose, in the constant cycle of energy which builds the universe. Every element in this dynamic system, even the seemingly lifeless spoon, in order to exist, imitates actively its higher emanating principles (Chlup 2012: 67).
The centre of this dynamic metaphysics is Proclus’ concept of the henads which crowns his predecessors’ unceasing efforts to grasp and resolve the tension between ontology and transcendence. The concept explicates – as much as possible for an ineffable and unknowable entity – what Iamblichus and Syrianus refer to as divine number (theios arithmos) (ET prop. 123). The caption of ET prop. 113 states: “the whole series of gods (theios arithmos) has the character of unity” (heniaios).66 The proposition demonstrates Proclus’ principal conceptual change in the term “number” to denote, in “a broad, non-mathematical sense”, a series, group or class (D. O’Meara 1989: 205). More specifically, this divine number does not express the ontologically informed but still mathematically analogical idea of gods as a procession of numbers, but the altogether non-mathematical idea of causation in which the gods are absolutely unitary henads “pre-subsisting” at the level of the One:
For if the divine series (theios arithmos) has for antecedent cause the One, as the intelligible series (noeros) has Intellect and the psychical series (psychikos) Soul, and if at every level the multiplicity is analogous (analogon) to its cause, it is plain that the divine series has the character of unity (theios arithmos heniaios estin), if the One is God.
(ET prop. 113.2–5, trans. Dodds, modified)
The first two lines of the passage run as the inventory lists of the different kinds of number found in Iamblichus. A closer look, however, does not discern an inventory of the ontological variations of number but an analytical map of the series of ontological causations (divine, intelligible and psychical) substantiating the three principal layers of reality. The passage unfolds around the key word “analogous” which explains the relationship between multiplicity and its source, more accurately cause. By “analogous” Proclus does not mean a mathematical “fitting”, as Iamblichus would say, between the different kinds of number and their ontological match but an intrinsic kinship.
Since the One is the immediate antecedent cause of the divine series containing the henads, the divine series therefore has to be akin to the One and since the One is unitary, the divine series too has to be unitary: “Thus if a plurality of gods (plēthos theōn) exist, this plurality must be unitary” (heniaion esti to plēthos).67 “Unitary” here particularly means the unmediated closest possible relation to the One (to hēn) and differs distinctly from the idea of “brought together multiplicity”, conveyed by the term “unified” (hēnomenon).68 It is precisely from their closest proximity to the One that the henads derive their name. At the level of the One, they are indistinguishable from the One and in this sense unitary (heniaios): they “share nature with the One”, are “one-like, ineffable, supra-essential and altogether similar to its cause”.69 From the levels proceeding from the One, however, they appear as many.70 From this upward perspective, Proclus abandons the language of mathematics in their description. They are not a series of a monad, dyad, triad and so on, but “paternal”, “generative”, “perfect”, “protective”, “zoogonic” (ET prop. 151–5). In other words, they are life giving. To understand the full extent of their primary ontological role, we should put them in the spotlight of Chlup’s metaphor of the productive power of the One as “a chewing gum bubble coming out of our mouth and yet being only kept in existence by the constant stream of air we are blowing into it”.71 The henads are the “constant stream of air” which keeps the bubble of the universe alive.
Proclus’ dynamic model culminates in the Platonic enterprise in explicating the structure of the universe. If we are to redirect Socrates’ wish in the Timaeus from gazing at the distinctive physical qualities of “magnificent-looking living organisms” to gazing at them alive in their original ontological environment, Proclus – not Critias or Timaeus, Iamblichus or Syrianus – fulfils Socrates’ wish completely, about ten centuries later.72
For the Neoplatonists the bottom-up approach of examining reality is illusory to the naked eye and the Platonically uninformed mind. As Plato professes at the end of the Timaeus, the birds “descended from innocent but simpleminded men, men who studied the heavenly bodies but in their own naïveté believed that the most reliable proofs concerning them could be based upon visual observation” (Ti. 91d6–e1, trans. Zeyl). Starting with Iamblichus, the later Platonists indefatigably strive to close the gap between the applied sciences of mathematics and its ontological dimension.73 From the quantifiable heterogenic multiplicity of the physical world, to the unquantifiable homogenic multiplicity of the metaphysical realm, to the highest productive principle of existence, the Neoplatonists embody to an absolute degree Rilke’s vision that “we are the bees of the invisible”. The metaphysical landscape is the meadow from which they purposefully and laboriously collect the essence of life, from the “crooked” spoon to the divine henads.
NOTES
1. Respectively Ti. 34b10–36d7, Phlb. 16c5–10, Prm. 142b1–151e2.
2. Pseudo-Plutarch, Placita Philosophorum 887c4; Eusebius, PE XV.37.6.3; Cicero, de Fin. 5.87.4–9. Cf. Burkert (1972: 15).
3. Metaph. 987a29–988a17; 1085b5. Cf. Syrianus, in Metaph. 187a. On Aristotle’s position, see Annas (1976); Turnbull (1998: 74–82).
4. It has received special attention in the works of D. O’Meara (1975, 1989), Charles-Saget (Bertier, Brisson & Charles-Saget et al. 1980), Horn (1995a), Nikulin (1998a, 1998b, 2002), Radke (2003: 234–41), Slaveva-Griffin (2009), Maggi (2009, 2010).
5. To be fair, Aristotle’s rejection of Plato’s view is one of the catalysts behind the Neoplatonists’ interests in the concept and should not be completely omitted. If this is not apparent in Iamblichus and Proclus, it is certainly so in Plotinus and Syrianus.
6. Chiaradonna & Trabattoni (2009: 14). The latest instalment is Wilberding & Horn (2012).
7. Although the top-down approach easily lends itself to the idea of hierarchical vertical organization of the ontological layers and has dominated the discussion of Neoplatonic ontology for quite some time (Wagner 1982b; 2002: 301), it is more accurate to think of them as simultaneous horizontal threads. The idea of vertical hierarchy is a “side effect”, intrinsic to logical thinking. The constraints of the up-and-down approach have long been noted in D. O’Meara (1996) and freshly reinstated by Linguiti (Chapter 22) and Corrigan (Chapter 24), below.
8. Chiaradonna & Trabattoni (2009: 5). For a more redeeming view, see Martijn (2010a: 297–302).
9. Rilke (1963: 157) in B. Mazur (2003: 4, 235). For a timely critique of the anthropocentric focus in the study of metaphysics and rationality, see Corrigan (Chapter 24), below.
10. Syrianus, in Metaph. 149.31–150.4, hereafter trans. O’Meara. This view anticipates most, if not all, developments in modern science and physics, not the least the binary language of computing.
11. In Metaph. 13–14, 133.4–14. See Mueller (2000); Dillon & O’Meara (2006: 3).
12. An ancient example for the misleading nature of the senses is the famous Epicurean claim that the sun is only as big as we perceive it.
13. Enn. V.1[10].5.10–13: “masses and magnitudes are not primary: these things which have thickness come afterwards, and sense-perception thinks they are realities. Even in seeds it is not the moisture which is honourable, but what is unseen: and this is number and rational principle.” Hereafter the Greek of the Enneads follows Henry & Schwyzer (1964–82); the translation, with alterations, follows Armstrong (1966–88). For a more utilitarian approach, see August Comte in Serres (1982: 85): “For in renouncing the hope, in almost every case, of measuring great heights or distances directly, the human mind has had to attempt to determine them indirectly, and it is thus that philosophers were led to invent mathematics.”
14. Syrianus, in Metaph. 13–14, 27–32: “For either astronomers and all mathematicians, and indeed all physical philosophers, must give up hope of scientifically demonstrating any proposition, and abandon the idea that proofs derive from the causal principles, not just of the conclusion, but of reality itself, or, as long as both of these is maintained, it must be the case that the causal principles of all things that are produced both in the heavens and in the whole of nature pre-exist in some kind of universal reason-principles.”
15. Cornford’s translation (1935). Cf. R. 510b–11b, 522c, 534a; also D’Ooge (1926: 23–6), D. O’Meara (1989: 17–23). For an anagogical interpretation of mathematics and geometry, see Epinomis 990c–91b, and Vitrac & Rabouin (2010). Burnyeat (1987) is sceptical but admissive of the idea.
16. Respectively, van Atten (2012: 4); Frege (1950: 99); Field (1980). For a recent reassessment of the status quo of pure mathematics, see Pincock (2012: 279–99).
17. Def. 1: Σημεῖον ἐστιν, οὗ μέρος οὐθέν. Def. 2: Γραμμὴ δὲ μῆκος ἀπλατές. For Plato’s possible influence on Euclid’s replacement of Aristotle’s term στιγμή for line with σημεϊον, see Heath (1956: 4). Cf. D’Ooge (1926: 48–52).
18. Theon, Expos. p. 18, lines 3–8. Cf. Stobaeus, Anth. I.21. See Dillon (1996a: 346–50), Mueller (2000), Slaveva-Griffin (2009: 42–6).
19. According to Iamblichus, Thales first defines the monad (in Nic. 10.9). See Bulmer-Thomas (1983: 384).
20. Plotinus, Enn. V.2[11].1.7–21; Proclus, ET prop. 30.12–14, 33.1–6, 35.1–2, 39, 42. There is also a strong ontological resonance in his conception of the monad as a complete privation of multiplicity (στερηθεῖσα), onlyness (μονή) and stability (στάσις), Theon, Expos. p. 18, lines 7–8. Cf. Stobaeus, Anth. I.21. See Dillon (1996a: 350).
21. The passage of time from Euclid to Moderatus has not diluted but condensed the ontological notions in the concept.
22. Ar. I.6.6–12, trans. D’Ooge. See also Helmig (2007: 140). A distant echo of Nicomachus’ idea of revolutions (έξελιγμοί) is found in Plotinus’ concept of beings as “number which has unfolded outward” (έξεληλιγμένος ἀριθμός), Enn. VI.6[34].9.30. Below, p. 205.
23. Such as by Philoponus and Asclepius, but not Iamblichus. See Helmig (2007: 140–42).
24. On the relation between scientific number and the true objects of philosophy, see Helmig (2007: 141–2).
25. A further distillation of the future ontological layers of the concept is found in Nicomachus’ threefold definition of number as “a combination of units” (μονάδων σύστημα), “limited multiplicity” (πλῆθος ώρισμέμον) and “a flow of quantity made up of units” (ποσότητος χύμα έκ μονάδων συγκείμενον) (Ar. I.7). For the ontological relation between χύμα and ὔλη, see Enn. VI.7[38].12.19–26.
26. Number is also mentioned in passing in Enn. 5.1[10].5, V.5[32].4–5, VI.2[43].13.31.
27. Enn. VI.6[34].2, 4, 9.34–5, 17; Aristotle, Metaph. 1086a2–5; 1083b16–17. See Slaveva-Griffin (2009: 58–68, 93).
28. Enn. V.5[32].4.16–18: The One “does not even belong to the category of substantial (ουσιώδης) number and so certainly not to that which is posterior to it, the quantitative number”. I amend here Armstrong’s rendering of οὐσιώδης as “substantial”, not “essential”, in light of its reoccurrence, a few lines below, which Armstrong also translates with “substantial”.
29. Narbonne’s treatment of matter and evil in Plotinus below (Chapter 15) amply justifies Plotinus’ position on number here. For the nature of ousia, see Chiaradonna (Chapter 14), below.
30. Enn. II.4[12].9.2–4: “Certainly that which exists is not identical with that which has quantity. … One must regard all bodiless nature as altogether without quantity.” See Slaveva-Griffin (2009: 109–12).
31. Enn. II.4[12].9.5–7: “For quantity itself is not a thing which has quantity; that which has quantity is that which participates in quantity; so it is clear from this, too, that quantity is a form.” Consequently geometrical figures in the intelligible are also without quantity and as such they are “unfigured figures” (ἀσχημάτιστα σχήματα, Enn. VI.6[34].17.25–6).
32. Enn. V.5[32].4.18–19; Cf. Enn. VI.6[34].4.21–2. The verb παρέχω means “cause” when referring to incorporeal entities, as in the quoted phrase, and “produce” when referring to sensible things.
33. Νοητὸς ἀριθμός appears only once in Plotinus (Enn. V.9[5].11.13); ἀληθής ουσία (Enn. VI.6[34].8.10) distantly echoes Plato’s ἀληθής ἀριθμός. See Horn (1995a: 235–6).
34. Enn. VI.6[34].9.29–32. Plotinus’ “unified number” of Being is an early prototype of the Unified in Iamblichus and Syrianus: see below, pp. 207–8 and 209, respectively. It finds a parallel in the concept of Protophanes in “the Platonizing Sethian” Treatises: see Turner (Chapter 5), above.
35. Horn (1995a: 235). For exeleligmenos, see above, note 22.
36. These four aspects also correspond to the four primary kinds of substance: the “unified number” of being corresponds to rest, the “number moving in itself” of Intellect to motion, the “number unfolded outward” of beings to otherness, the “encompassing number” of the Complete Living Being to sameness. Cf. the originative principles of number in Nicomachus, Ar. II.17.1; 18.1; 19.1; 20.2. On the relation between substantial number and the primary kinds, see Slaveva-Griffin (2009: 95–130); on the relation between substance and the primary kinds, see Santa Cruz (1997: 105–18).
37. This does not mean that Plotinus discounts the importance of mathematics in the philosopher’s training in dialectics, see Enn. I.3[20].3.6–10. Also Maggi (2009: 57–61).
38. According to Comm. Math. 63.23–64.14. There is no specific order for the different kinds of number in Psellus, On the Physical Number and On Ethical and Theological Arithmetic. He also refers to intelligible number variously as noetos, noeros, ousiōdes, eidetikos, On the Physical Number 4–5.
39. Enn. V.8[31].9.15–17. Aside from the use of θεολόγοι (Enn. III.5[50].2.2, 8.22), the term or its derivatives do not occur in the Enneads.
40. Comm. Math. 63.24–9: θεώρημα πρώτόν έστι τὸ θεολογικόν, τῇ τῶν θεῶν οὐσίᾳ καί δυνάμει, τάξει τε καί ἐνεργείαις συναρμοζόμενον. With D. O’Meara (1989: 79) and contrary to Maggi (2012: 80), I take the lines to refer to divine number and not to all mathematicals in general.
41. It is, at times, distinguished from the intellectual (noeros) number. The intelligible number represents the ontologically prior numbers in the intelligible, while the intellectual number belongs to the ontologically secondary demiurgic activities, associated with soul. Syrianus, in Metaph. 140.10–15. See D. O’Meara (1989: 79).
42. At times it is also followed by yet another kind of number of even more obscure and problematic nature, the hypostatic number. It is absent from Comm. Math. 63–64 and is mentioned only passingly in Psellus. D. O’Meara (1989: 79) speculates a possible identification with mathematical number which is not on Iamblichus’ list either. Maggi (2012) omits it also.
43. Xenocrates, frags. 181–5. It is tantalizing, in view of his commentary on the de Anima, that Iamblichus does not explicitly identify the self-moving number with Soul here. We do not have a strong enough reason here to suspect any possible influence of Plotinus’ treatment of soul in relation to substantial number. But the reluctance of both Plotinus and Iamblichus on the matter is conspicuous and merits further investigation. See Slaveva-Griffin (2009: 112–18), and D. O’Meara (1989: 62), followed by Maggi (2010: 172).
44. But we should note that Iamblichus’ comment on the relation of intelligible number to the purest substance supports Plotinus’ choice of naming this kind substantial and not intelligible or eidetic.
45. Enn. I.8[51].2–5; III.8[30].10; V.5[23].4.6–7, 5.2–11; VI.7[38].32.21–3.
46. As documented in Proclus, in Prm. 1114.1–10 and Damascius, Pr. I.87.8–10. For a detailed discussion, see Halfwassen (1996).
47. Dillon (1993: 50–53) even argues that Iamblichus makes the productive One an object of intellection, thus placing it closer to Plotinus’ “unified number” of Being. See Wear (2011: 9).
48. D. O’Meara (1989: 79) also renders it as self-moved.
49. When counting, soul externalizes the internal non-quantitative number into a specific quantitative expression (Enn. VI.6[34].16.47–54). See Slaveva-Griffin (2009: 114–30).
50. On the Physical Number 6–8. In Psellus, the physical number is associated with each one of the Aristotelian causes and not with the Aristotelian view of numbers as abstractions from material quantity.
51. Syrianus, in Metaph. 13–14, 170.27 in relation to Aristotle, Metaph. 1088b35–89a7. Cf. Il. XV.185 [hereafter Lattimore’s translation with alterations].
52. Cf. Il. XVII.170. Manolea cautions that, while Syrianus is openly scornful of Aristotle’s take on the Platonic concept of number, his “tone does not exclude the respect he feels for Aristotle” (2004: 222–3). Other Homeric quotations with similar purpose include Syrianus, in Metaph. 13–14, 168.9–12; 168.35–8; 194.59. For a complete discussion, see Manolea (2004: 218–31).
53. Most characteristic references are found in in Metaph. 122.31–2; 130.24; 142.27; 146.9.
54. In Prm. frag. 4.5; in Metaph. 11.29, 165.33. Wear (2011: 247–9).
55. In Prm. frag. 12.4–5; in Metaph. 11.30. Wear (2011: 316–18).
56. In Prm. frag. 11. Iamblichus’ Unified supersedes Limit. There is even a touch of Plotinus’ “unified number” in Syrianus’ idea of a specific power (dynamis) between the Unified and Being.
57. Enn. VI.6[34].10.2–4 and 20–29. See Horn (1995a: 248–50). Plotinus’ role in this development has been overlooked, see Slaveva-Griffin (2009: 91–2, 94, 113). Most recently Mesyats (2012: 161–2) partially acknowledges Plotinus’ contribution. For the history of the debate about the origin of the idea of the henads, see Wear (2011: 9) and Longo (2010: 620).
58. According to Dillon (1993: 50), the One-Being, including the multiplicity of the henads, coincides, in Iamblichus, with the highest level of Intellect and is thus object of intellection.
59. In Metaph. 82.20–25, 4.29–5.2. D. O’Meara (1989: 131, 133, 135).
60. In Ti. IV.15: “By way of concealment of the words Plato used mathematicals, as veils (παραπετάσμασιν) of the truth about reality, as the theologians use their myths, and the Pythagoreans their symbols: for in images one can study the paradigms, and through the former make a transition to the latter,” trans. Martijn (2010a). Cf. Iamblichus’ somewhat mechanical view of numbers “fitting” the corresponding layers of reality, Comm. Math. 63, above, note 40. See Martijn (2010a: 197).
61. D. O’Meara’s phrase (1989: 204). Also see Martijn (2010a: 186–92); Steel (2010: 635); Chlup (2012: 47–111).
62. As Martijn argues (2010a: 190–91) with support from Gersh (1973: 87) and Baltzly (2007).
63. In Ti. II.23.29–33. For analysis, see Baltzly (2007: 69).
64. In Eucl. 51.9–56.22, 78.20–79.2. On phantasia and mathematical projection, see Sheppard (1997); Martijn (2010a: 190). On the strong didactic and dialectic foundation of Proclus’ much-famed “geometrical method” of exposition, see Martijn (Chapter 10), above.
65. ET prop. 35.1–2: “Every effect remains in its cause, proceeds from it, and reverts on it” (trans. Dodds). See Steel (2010: 639–41); Chlup (2012: 62–82).
66. By translating arithmos as number, Dodds interprets the syntactical function of theios attributively. But the agreement between the two words also suggests a conceptual entendre between “number of the gods” and “divine number”.
67. ET prop. 113.8–9, although the absolute unity of the henads still drives from within, not from the One (ET prop. 114). See Dillon (1972, 1993); D. O’Meara (1989: 82–3, 204–7); Butler (2005, 2008); Chlup (2012: 114). Cf. Iamblichus, Myst. 59.1–60.2.
68. ET prop. 115: “Being, Life and Intelligence are not henads but unified groups” (οὐχὶ ἑνὰς ἀλλ᾽ ἡνωμένον). See Chlup (2012: 115–16).
69. PT III.3.12.21–23. Cf. ET prop. 115.
70. ET prop. 64: At the level of the One, the henads are self-complete (αὐτοτελεῖς), when contemplated from below, they are irradiating (ἐλλάμψεις). This is another way of looking at them respectively as non-participating (άμέθεκτος) and participating entities (μετεχόμεναι, μεθεκταί) (ΕΤ prop. 116). See Mesyats (2012: 152–3).
71. Chlup (2012: 67) continues: “By way of analogy, all of our world is such a ‘bubble’ that may seem steady and firm at first sight, and yet would immediately burst and collapse if the higher levels stopped pumping their energy into it. Luckily enough, such a thing can never happen, for the energy flows from the higher levels as a spontaneous by-product of their perfection.”
72. Ti. 19b. As suggested by Smith in Chapter 8 above and by Steel (2010: 652), for Proclus, the conceptual culmination of number in the henad even becomes a tool of ideological defence of pagan polytheism against Christian monotheism.
73. See above, p. 202. The Neoplatonists’ effort prompts Psellus to note (On the Physical Number 2–3) the great diversity (poikilia) of number which ever expands the map of ontological realities.