1.1 Proclus’ EP
While older scholarship has mostly focused on Neoplatonist metaphysics, leaving the impression that late-antique Platonists were only interested in this and related areas, more recent publications have also shed light on their preoccupation with natural philosophy.Footnote 1 These publications have demonstrated that the Neoplatonists had a genuine interest in thinking about the physical world, while still emphasising its dependence on the intelligible realm.Footnote 2 Crucially, Proclus regarded the study of physics as an indispensable preparation for metaphysics (PT 1.2.10.25–11.4). A good example for Proclus’ interest in this area is – besides the commentary on the Timaeus – his work on kinematics, that is, the study of physical motion,Footnote 3 Elements of Physics. Although the latter has been little studied, it is here that Proclus’ engagement with Aristotle’s natural philosophy strongly manifests. It allows us to assess Proclus’ views on the latter which elsewhere are quite critical and dismissive, such as in his infamous statements on Aristotle in the prologue of the commentary on the Timaeus (see Introduction).
EP is based on Aristotle’s Physics 6 and 8 as well as De caelo 1 and deals primarily with the question of motion and its ultimate origin.Footnote 4 Hence its alternative name On Motion which – judging by the content of the treatise – seems to be the more accurate title. Interestingly enough Proclus neither names nor discusses Aristotle’s definition of motion from Phys. 3.1 in the treatise. To my knowledge, the definition is entirely absent from his extant oeuvre. Demonstrating that the eternal motion of the cosmos is caused by an unmoved mover is the culmination of the treatise – which smoothly connects physics with metaphysics. The goal is to sum up the fundamentals of Aristotelian kinematics, which it presents as a unitary and independent body of knowledge, and to make them accessible to beginners. Thus, EP was presumably related to the Aristotelian curriculum in the Neoplatonist school in Athens and was studied while reading Aristotle’s works on natural philosophy.Footnote 5 This does not entail that Proclus endorses unqualifiedly everything he includes in this work; some aspects need to be explained in depth in a more advanced and Platonist context. Additionally, the propaedeutic character of the work accounts for the lack of references to it in other Proclean works (just as in the case of ET). Due to this, a confusion about its dating has arisen. While earlier scholarship tended to regard EP as a youthful work, stemming from a ‘pre-Platonic’ phase of the philosopher when he was only versed in Aristotelian philosophy,Footnote 6 it is nowadays understood to be chronologically, and not just stylistically and formally, related to ET to which it is in a certain way complementary.Footnote 7 While it seems likely that EP was written before ET, as Proclus would have used the axiomatic method for natural philosophy – for which Aristotle was in a restricted way an example – and then would have moved on to apply the method to metaphysics, this issue is impossible to settle. Like ET, EP is designed as a textbook written in an axiomatic or geometrical manner (more geometrico) that is influenced by Aristotle and Euclid. That is, it posits certain principles and builds on these various theorems.
Due to its proximity to the Aristotelian original and perceived lack of originality, EP has received little attention in scholarship. The text has been established by Ritzenfeld (Reference Ritzenfeld1912), although Boese (Reference Boese1958: 13–14) has proven the latter’s stemma codicum to be wrong and incomplete. The first article on this work by Nikulin (Reference Nikulin, Cleary and Gurtler2003) provides a helpful overview but lacks clarity and intelligibility at some points. The most comprehensive discussions remain Opsomer (Reference Opsomer, Chiaradonna and Trabattoni2009: 193–203) and, especially, (Reference 236Opsomer, Baghdassarian, Papachristou and Toulouse2020b). While the former paper focuses specifically on Proclus’ argument for the existence of the unmoved mover in EP, the latter concerns its method and formal characteristics. Most recently, Kiosoglou (Reference Kiosoglou2022) offered an analysis of Proclus’ usage of theorems and analysed more closely EP §§1.1–2 and §2.7.Footnote 8 While these articles have deepened our understanding of the treatise, a closer analysis of Proclus’ exegetical background and method is still outstanding, as is also an English translation. Such an analysis clarifies Proclus’ engagement with Aristotle’s kinematics and its influence on his natural philosophy. It also offers us an excellent example of Proclus’ use of the axiomatic method which he employs also in ET.Footnote 9 From a more general perspective, a closer look at EP allows us to draw conclusions regarding the exegesis of the Physics and the De caelo before the significant commentaries of Simplicius and Philoponus and the relation of EP to these later works.
To this end, I focus on Proclus’ use of Aristotle in EP to elucidate the presence of Aristotle both regarding the content as well as the presentation and the method of the treatise. Thus, I offer first an analysis of the content as well as of the exegetical and philosophical background of this treatise (Section 1.2). Here, I show how Proclus – far from a mindless compiler – was influenced by a specific exegetical tradition and certain philosophical convictions in restructuring and interpreting Aristotle’s work. I also emphasise the importance of De caelo for Proclus’ project, which has not been appreciated in scholarship so far. Moreover, I show how he circumvents more contentious issues which he chooses to discuss in an advanced work, such as the commentary on the Timaeus. In the third section, I focus on Proclus’ method and argumentative structure (Section 1.3). As I demonstrate, Proclus uses Aristotle’s scientific ideal outlined in the Posterior Analytics as a model for presenting non-mathematical material in an axiomatic manner. I then offer a close analysis of the principles and some of the theorems mentioned.
1.2 The Content and the Background of EP
In the following, I briefly outline the content of EP (1.2.1), before moving on to an analysis of its exegetical background (1.2.2) and its omissions from the Aristotelian original as well as its relation to other Proclean writings (1.2.3).
1.2.1 Summary of EP
In EP Proclus restructures in two books Aristotelian doctrine from Physics 6 and 8 and De caelo 1. Two main motivations guide Proclus. He arranges the arguments in an axiomatic manner in order (a) to provide an accessible overview of Aristotle’s theory of motion and (b) to prove the necessary existence of the unmoved mover as cause of the cosmos’ motion.Footnote 10 Each book starts with a series of definitions and hypotheses, followed by a number of theorems. In a theorem, a proposition precedes a short proof, mostly a reductio ad absurdum/impossibile, whereby the logical impossibility of the contrary proposition is demonstrated. The result is then summarised in a conclusion.
The first book opens with six definitions which are regarded as self-evident and indemonstrable. The definitions are discussed in the then-following thirty-one theorems or propositions which are – for the most part – deduced from them. Book 1 mainly summarises central insights from Physics 6, dealing with topics such as divisibility and continuum (§§1–7), velocity (§§8–10), time (§§11–18) and motion (§§19–31). Common Aristotelian doctrines such as the infinite divisibility of magnitude, motion and time (§11) or the indivisibility of the now (§16) are discussed. In treating these topics, Proclus preserves the order of Physics 6, whereby each proposition deals with a different chapter.Footnote 11 Proclus occasionally adds theorems that have no obvious counterpart in Aristotle in order to strengthen the axiomatic structure (e.g., §1; §3) just as he adduces new arguments to buttress the propositions (e.g., §14.14.24–6; §16.16.21–6). Yet, he also omits certain passages from Physics 6, such as the anti-Zenonian excursus. Book one concludes with the proposition that the indivisible is unmoved: ‘Everything quantitatively without parts is in itself unmoved’ (§31.26.29–30: πᾶν τὸ ἀμερὲς ἐν ποσῷ ἀκίνητόν ἐστι καθ’ ἑαυτό).Footnote 12 This already points towards the goal of EP, preparing the argument for the existence of the prime mover.
The second book consists of six hypotheses,Footnote 13 eight definitions and twenty-one theorems. It focuses mainly on the properties of motion, particularly on the eternal circular motion of the cosmos or heaven (§§1–6) and shows via a lengthy discussion of infinite power (§§7–8; §§11–15) that the first mover of this motion cannot be a body. Rather, the cause must be an eternal (§18) and indivisible (§21), that is, non-physical, substance, possessing infinite power (§21) in order to cause the cosmos’ eternal motion. In doing so it remains itself unmoved (§19).Footnote 14 The work thus concludes with the fundamental proposition: ‘The prime mover of the circular motion is indivisible’ (§21.58.11–12: τὸ πρῶτον κινοῦν τὴν κύκλῳ κίνησιν ἀμερές ἐστιν). Since what is indivisible is unmoved (§1.31) it follows that the prime mover is unmoved. Proclus thereby reaches his goal.
Book two adheres less strictly to its Aristotelian source by mixing content from De caelo 1.2–7 and Physics 8 as well as rearranging the order of the arguments of these books. For instance, Proclus demonstrates first that circular motion has no opposite (§4) and then that the thing in circular motion is ungenerated and indestructible (§5), whereas Aristotle proceeds the other way around. Yet, Proclus proves here to be an attentive reader, as Aristotle actually presupposes in his argument for the indestructibility of the heaven the claim that circular motion has no opposite (DC 1.3.270a19–20). Proclus also intervenes in the Aristotelian material by omitting more narrative passages (e.g., Aristotle’s insistence on the importance of a discussion of the infinite at 1.5.271b1–17) and certain arguments (e.g., in proving that something moving in a circle is finite Proclus uses only two of the six arguments Aristotle provides in DC 1.5). Some of these differences can be explained by Proclus’ exegetical and philosophical background, as I make clear in the next section.
From this summary it emerges that EP is primarily a physical work where the role of the unmoved mover is essential but only dealt with at the outset. This is because the unmoved mover belongs to the metaphysical realm and should be properly discussed in a metaphysical work, as in fact Proclus does elsewhere.Footnote 15 This σκόπος is also useful for bringing Aristotle and Plato together, given that Proclus elsewhere makes extensive usage of Aristotle’s doctrine of the unmoved mover.Footnote 16 By focusing less on the nature of the prime mover, Proclus remains true to his Aristotelian model, Physics 8, where the prime mover receives a mostly negative description – unlike the more positive account in Metaphysics 12 – and primarily serves to explain the origin of eternal motion.Footnote 17 Thus, although the scope of the work is to prove the existence of the unmoved mover, the argumentation shows a genuine interest in physical issues, especially related to kinematics, and is not just a means to an end. It is thus misguided to try to explain the form and content of EP by focusing solely on the unmoved mover, as Nikulin (Reference Nikulin, Cleary and Gurtler2003: 187) seems to suggest: ‘Only if justification of the existence of the prime mover, in which Proclus closely follows Aristotle, is the ultimate purpose of the whole treatise, can it be explained why the treatise has the form that it has’. Clearly, the reader is supposed to grasp basic doctrines of Aristotle’s theory of motion. In this way, the proof of the unmoved mover’s existence as well as the accessible and comprehensive presentation of Aristotelian kinematics are Proclus’ twin concerns in EP.
1.2.2 Exegetical Background
Proclus’ EP did not arise out of thin air. Rather, the treatise is the product of an – overwhelmingly, but not exclusively, Platonist – exegetical tradition on Aristotle’s Physics and De caelo. To some extent, Proclus follows in the footsteps of his master Syrianus who wrote commentaries on both works.Footnote 18 The main studies on EP are mostly silent on its exegetical context. While Nikulin (Reference Nikulin, Cleary and Gurtler2003) says almost nothing about the exegetical background of the work, already Ritzenfeld (Reference Ritzenfeld1912) in his commentary had pointed to some overlaps with Simplicius. Unfortunately, the latter did not further develop these findings. Most recently, Opsomer (Reference 236Opsomer, Baghdassarian, Papachristou and Toulouse2020b) has taken up Ritzenfeld’s lead, emphasising the similarity between Proclus and Simplicius: ‘[S]everal close parallels with Simplicius’ Commentary on the Physics can only be explained by the fact that both works belong to the same tradition of commentary and exegesis (it cannot be excluded that Simplicius read Proclus’ text, but that is impossible to prove)’ (84).Footnote 19
While Opsomer is content in his article to point out a few similarities between Proclus and Simplicius with regard to specific passages, I attempt in the following to go beyond this by explaining how a common exegetical tradition shaped the content and form of EP. This rather general background which transcends more specific parallels should not be underestimated, as it allows us to situate EP in a certain context and to shed light on some of its striking features, such as the combination of the Physics and the De caelo. Let us briefly consider the exegesis of both works in Neoplatonism.
Regarding the Physics, it is only with Neoplatonism that it achieves its status among Platonists as one of Aristotle’s most significant works.Footnote 20 Plotinus makes copious references to the work and engages with it critically, adopting some of its basic concepts, while simultaneously putting emphasis on the physical world’s dependence on metaphysical principles. The latter claim is seen to be – to certain degrees at least – at odds with Aristotle.Footnote 21 His successor Porphyry was the first Neoplatonist to engage in proper Aristotelian exegesis, as evidenced by his commentary on the first four books which were known in antiquity as Peri archōn (Περὶ ἀρχῶν).Footnote 22 After Alexander, he is the most important source in Simplicius’ commentary on the Physics.Footnote 23 Most importantly, however, it seems that Porphyry also wrote a summary (sunopsis/σύνοψις) of the last four books known as Peri kinēseōs (Περὶ κινήσεως).Footnote 24 This summaryFootnote 25 was, I believe, in a certain regard a precursor of Proclus’ EP since we find already there the idea of summarising the theory of kinematics of the last books of the Physics (although Proclus focused only on books 6 and 8).Footnote 26 Porphyry’s ‘agenda’ in the summary thus proved influential. But since no fragments survive it is impossible to establish the exact relationship between these works. Given that Proclus arranges his treatise as a stoicheiōsis (στοιχείωσις) I assume that the presentation and formal structure of the two works differed significantly but not, however, the fundamental idea of presenting the content in a succinct and accessible manner.
Moving on to the De caelo, it can be established that, while it never acquired the prominent role of the Physics among the Neoplatonists, it garnered significant attention. Again, Plotinus seems to have played a key role in popularising the De caelo by making extensive use of some of its central doctrines, especially in Enn. 2.1–2.Footnote 27 At any rate, he influenced Proclus who cites Plotinus’ interpretation of the De caelo (e.g., In Tim. 2.48.1–5 [1.237.22–7]). Moreover, Proclus was acquainted with the Peripatetic works on the De caelo, as his knowledge of, for example, Xenarchus shows (In Tim. 2.321.1 [1.425.22]). While he himself never wrote a commentary, it is clear from the references in Simplicius’ commentary on De caelo – the only fully extant commentary on the work – that he commented on some passages in detail, probably insofar as they were in tension with Platonic doctrine (especially DC 3.7–8).Footnote 28 These seem to have been part of Proclus’ early Investigation of Aristotle’s Objections to the Timaeus (see Chapter 2).
This exegetical tradition accounts for some of Proclus’ choices in respect to the form and the content of EP. Most importantly, Proclus regards the prime mover in Physics 8 as being consistent with the elemental theory of the cosmos’ circular motion in De caelo – a claim which has been contested in more recent scholarship. This agreement of both works is not a Proclean innovation. In fact, according to Simplicius (In DC 3.9–10, 5.35–8), all ancient exegetes agreed that the De caelo followed the Physics – based on Aristotle’s own claims at Meteor. I.1 – and complemented its account.Footnote 29 Thus, both treatises essentially expressed the same doctrine which is in line with the Neoplatonists’ systematic understanding of Aristotle.Footnote 30
From a modern perspective, this position is shaky at best, as many scholars not only tend to assume a development between the De caelo and the Physics but also an irreconcilable difference between the two.Footnote 31 This developmental interpretation is so widespread that it is simply called ‘the traditional view’ by Judson (Reference Judson, Gill and Lennox1994: 155). According to this reading, the unmoved mover is absent in the De caelo.Footnote 32 Instead, the circular motion of the heaven in De caelo 1–2 is self-caused and, hence, a type of self-motion. As such the heaven’s motion requires no further explanation and is causally responsible for all other manifestations of motion in the cosmos. This position would be then rejected in Physics 8 where Aristotle asserts that self-motion stricto sensu is impossible and that the motion of the heaven must be ultimately caused by an external, separate unmoved mover. In a next stage the theory of the unmoved mover is further elaborated in Met. 12.6–10, especially in regard to the unmoved mover’s causality.Footnote 33
The Neoplatonists were neither oblivious to discrepancies in or between Aristotle’s works – arguably this is the main reason for writing commentaries – nor were they completely immune to developmentalism. Simplicius, for instance, is happy to admit that Physics 7 likely represents an earlier stage in Aristotle’s writing of the Physics (see In Phys. 1036.17–1037.3).Footnote 34 Yet, at the same time their systematic fervour is strong, and they maintain that Aristotle has a consistent stance and a uniform doctrine. That the unmoved mover of Physics 8 is ultimately responsible for the circular motion of the heaven as outlined in De caelo 1 seems obvious to Proclus in EP, where he combines doctrines from both works and stresses that there needs to be an unmoved principle as cause of the heaven’s motion (§2.19).Footnote 35 The same goes also for Simplicius who claims that the unmoved mover is referred to in De caelo (In DC 271.5–21) and, thus, part of Aristotle’s doctrine there. The interest in systematicity is especially pronounced in Proclus’ EP since the main motivation was to offer a succinct summary of Aristotle’s theory of motion for beginners. Thus, the form and goal of the treatise do not permit elaborate or subtle discussions of the relationship between Physics 6 and 8 as well as between Physics 8 and De caelo 1.
Although De caelo 1 is a constitutive element of EP 2, where almost all principles (i.e., twelve of fourteen) and two-thirds (i.e., fourteen) of the theorems are based on it, it remains unclear why it is used so extensively there. Focusing just on Physics 8 seems to be the more natural option, given that one purpose of EP is to prove the existence of the unmoved mover. This is evidenced also by Simplicius who in commenting on Physics 8 makes a few passing references to De caelo without dwelling on its arguments. I suggest that Proclus integrates the De caelo material for two main reasons. On the one hand, the form and the issues of De caelo 1 coincide significantly with those of Physics 6 and 8. Formally, De caelo 1 includes many arguments written in a mathematical style just like Physics 6 and, to a lesser degree, Physics 8.Footnote 36 Proclus himself is aware of this formal similarity, as becomes clear from In Tim. 2.47.16–48.11 [1.237.17–238.4], where he mentions Aristotle as imitating in the Physics and the De caelo the axiomatic method of Plato. In regard to content, there are also important overlaps. For instance, Proclus takes Aristotle’s proof from DC 1.6 that infinite bodies (would) have infinite powers in EP §2.7 and then adds as a following proposition ‘finite bodies have finite powers’ based on Phys. 8.10. More generally, while De caelo establishes the circular motion of the cosmos, Physics 8 tackles the problem of the eternity of that motion as well as its origin. On the other hand, I want to emphasise that Proclus’ apparent intention was to give his audience a suitable overview of Aristotelian kinematics. This pedagogical element of comprehensiveness and accessibility guides Proclus. Given the significance of theories of simple motion, the cosmos’ circular motion, and the cosmos’ finitude in antiquity, it is less surprising that Proclus turned to De caelo 1 where these issues are extensively dealt with. Due to the assumed doctrinal coherence between the two works it seemed quite natural to Proclus to combine material from De caelo 1 with Physics 6 and 8.
Thus, starting from the very idea of writing a summary of Aristotle’s kinematics to the combination of the material, Proclus’ EP has been shown to be significantly dependent on a long commentary tradition. This impression will then be substantiated by the content of the treatise in Section 1.3. In the following, I analyse more closely how Proclus’ own views shaped what he excluded from EP.
1.2.3 Proclus’ Editorial Choices
While remaining relatively close to the original, Proclus also leaves out crucial discussions from the two Aristotelian treatises. More contentious issues such as self-motion (see Chapter 2) or the precise nature of the fifth substance/element are not part of EP, although they are, in fact, extensively treated by Aristotle in Physics 8 and De caelo 1. Since Proclus stops including material from DC 1.8 onwards, he makes mention neither of Aristotle’s arguments for the uniqueness of the cosmos at DC 1.8–9 nor of his lengthy elaboration at DC 1.10–12 on the terms ‘(un)generatedness’ and ‘(in)destructibility’ which was clearly directed at Plato. Notably, these issues are dealt with elsewhere, especially in his now lost Investigation of Aristotle’s Objections to the Timaeus and in In Tim. which preserves some of Proclus’ objections.
Since scholarship has mostly overlooked these omissions in EP and the discussion of these issues elsewhere, the relationship between EP and other Proclean works appears somewhat obscure. Although Nikulin (Reference Nikulin, Cleary and Gurtler2003: 190) and Opsomer (Reference Opsomer, Chiaradonna and Trabattoni2009: 195–6, n. 32) discuss the case of self-motion, they otherwise offer no analysis of why Proclus omits other topics and what this, in turn, tells us about EP and its connection to other writings. This is the first attempt at offering a more comprehensive discussion of this relationship. How is EP connected to other Proclean treatises and what relevance does it have for them? Is the doctrine of EP really an ‘integral part of [Proclus’] physics’ (Opsomer Reference Opsomer, Chiaradonna and Trabattoni2009: 193)? As I demonstrate in this section, Proclus presents to the student in the introductory EP those parts of Aristotle’s kinematics that can be embraced by a Platonist, while simultaneously leaving out more controversial and complex passages. However, in more advanced works such as his commentary on the Timaeus, he points out the limitations of some of the doctrines in Physics 8 and De caelo 1, and even the tension between them and the correct Platonist teachings. This will be shown by evaluating the evidence on the nature of the heaven (1.2.3.1) and its ungeneratedness and indestructibility (1.2.3.2) in EP and the commentary on the Timaeus.
Before I consider the examples, it is worth bearing in mind that this procedure of subordinating Aristotle to Plato fits Proclus’ more general views on their relation. As I have shown in my discussion of the prologue of his commentary on the Timaeus (1.9.14–10.18 [1.6.21–7.16]), Proclus explains that Aristotle’s natural philosophy is inferior to Plato’s since the former imitated the latter in his physical works but focused too much on material explanations. Proclus mentions Aristotle’s discussion of motion and its origin favourably which explains why he summarised it in EP (9.17–18 [6.25–6]). Most importantly for my current purposes, Proclus states here that Aristotle agrees with Plato on the heaven’s ungeneratedness and the fifth substance (10.2–3 [6.31–2]). The impression we get from this passage is one of agreement on some central issues but negligence in studying them properly due to Aristotle’s non-metaphysical approach to them.
1.2.3.1 The Fifth Element
Let us first consider Proclus’ treatment of Aristotle’s claim that the heaven is constituted of a distinct fifth element (e.g., DC 1.2.269a13–18). Surprisingly enough, Proclus makes no reference to this in EP. Instead, he employs the intentionally vague expression ta kuklō kinoumena (τὰ κύκλῳ κινούμενα) when referring to the heaven (e.g., §§2.1–2, §2.5.). At no point does he identify τὰ κύκλῳ κινούμενα with a distinct fifth element, aether, as Aristotle does in DC 1.3, although he offers an extensive list of characteristics such as bodily simplicity (EP §2.1), difference from bodies moving with rectilinear motion (§2.2), or lack of weight and lightness (§2.3). Why does Proclus leave out its elemental constitution which plays an important role in De caelo? While one reason for choosing this expression is certainly the kinematic context of EP, I argue that the main motivation for this omission is made manifest in Proclus’ commentary on the Timaeus. As Proclus demonstrates there, he disagrees with Aristotle on the heaven’s constitution – just as numerous Peripatetics and Platonists did before and after him.Footnote 37 In this way, the conciliatory attitude of EP, where less controversial features of the heaven are listed, can be contrasted with some passages in his commentary on the Timaeus where Proclus strikes a different tone in reference to Aristotle’s elemental theory. Proclus explicitly criticises Aristotle’s concept of a fifth element:
[Objection] But perhaps the marvellous (θαυμαστός) Aristotle will contest our account by positing that not all visible things are so through participation in fire (πᾶν τὸ ὁρατὸν πυρὸς μετουσίᾳ), for the chorus of stars and the mighty sun itself are not [in his view] things composed of fire even though they are visible.
[Response] But one might respond to him by saying that enmattered (ἔνυλον) fire is one thing but immaterial (ἄυλον) fire is another – that is, it is immaterial because compared to the matter of the things in the sublunary sphere it is immaterial – and the one kind is destructible (φθαρτόν) while the other is indestructible (ἄφθαρτον). While one kind is mixed (συμμιγές) with air, the other is pure (καθαρόν). And generally speaking, because fire has many forms (πολλὰ εἴδη), perhaps Aristotle will concede to this account and listen to the theologians who call the sun ‘fire, channel of fire’ and ‘dispenser of fire’ and all other such names.
Proclus here turns to Aristotle’s claim that the visibility of the stars and heaven is not due to their composition of fire but, we are to add, due to being composed of aether – the fifth element.Footnote 38 This view is wrong according to Proclus. In fact, Proclus responds, the element of fire differs in the sublunary sphere from the fire in the heaven, since there are different kinds of fire (πολλὰ εἴδη πυρός). That is, the element of fire has different gradations, depending on whether it is part of the sublunary sphere or of the heaven. Not only is the fire of the heaven indestructible (ἄφθαρτον) and in a certain sense immaterial (ἄυλον),Footnote 39 but it is also pure (καθαρόν). Hence, the stars and the heaven are in part made up of this pure and superior form of fire. Proclus then recommends to Aristotle – in an almost comical way – to consider the sayings of the ‘theologians’, that is, the Chaldaean Oracles (fr. 60 des Places), who apparently back up Proclus’ and Plato’s view of the elemental constitution of the heaven.
Objections like these amount later in the commentary to an explicit rejection of the fifth element, based on Proclus’ exegesis of Timaeus and more general reflections.Footnote 40 For instance, Proclus points out the problem of explaining the diversity of appearances when positing only one element as constitutive of the heaven: some parts of the heaven are transparent, others solid or luminous and so on (In Tim. 3.60.1–61.11 [2.43.20–44.23]). Aether could not account for these different phenomena. According to Proclus, the heaven, including the stars and planets, is made up of a purer or a higher form of fire in which the other three elements pre-exist κατ’ αἰτίαν since fire is their cause (3.60.1–3 [2.43.20–3]).Footnote 41 He thus rejects Aristotle’s assertion of a distinct and unmixed fifth element.
However, this does not mean that Aristotle was completely wrong in Proclus’ eyes since he correctly pointed out that the heaven’s substance does differ significantly from the four sublunary elements. In this way, even Proclus accepts talk of a fifth substance, insofar as the term refers to the purity and specific combination of the four elements in the heaven:
Therefore, the heaven is a fifth substance (οὐσία) besides these four elements, since it is a combination from the simple elements. For in the heaven the elements are not the same [as they are here] but are rather the highest forms of them and the four elements of all things are unmixed and are bounded in relation to one another by their appropriate forms.
It is in this specific sense that Plato and Aristotle agree according to Proclus, as he states at In Tim. 1.10.2–3 [1.6.31–2] ([Aristotle] τῷ Πλάτωνι συμφώνως, καθόσον ἀγένητον τίθεται τὸν οὐρανὸν καὶ πέμπτης οὐσίας).
From this presentation of Proclus’ complicated discussion in his commentary on the Timaeus it emerges why Proclus decided to stick to the superficial term τὰ κύκλῳ κινούμενα in EP instead of outlining his own elemental theory and complex views on what Aristotle precisely gets right or wrong. While the latter discussion goes beyond the limited purpose of EP, it is necessarily complementary to it as it shows us how the characteristics of the heaven established in EP are further elaborated in a more advanced work on Plato. In this sense, one can establish a continuity between these two works.
1.2.3.2 Ungeneratedness and Indestructibility
Besides Aristotle’s treatment of the elemental nature of the heaven, Proclus also leaves out in EP the discussion of DC 1.10–12 about the notions of ‘(un)generatednes’ and ‘(in)destructibility’. This is in spite of his adoption of Aristotle’s insight that τὰ κύκλῳ κινούμενα are ungenerated and indestructible (§2.5).Footnote 43 Aristotle’s discussion was in part directed at Plato’s Timaeus, as he makes explicit (DC 1.10.280a27–32). Aristotle understands the Timaeus as proposing simultaneously a temporal generation and indestructibility of the cosmos.Footnote 44 This is absurd, Aristotle maintains, as every generated being has the potentiality to be destroyed or to perish and it is impossible for this potentiality not to be realised in an infinite stretch of time. Again, in the commentary on the Timaeus Proclus shows us what he really thought about Aristotle’s analysis by objecting to it in at least two passages (2.70.2–73.10 [1.252.11–254.18]; 2.133.4–16 [1.295.27–296.12]).Footnote 45 I briefly discuss the second one which is part of a larger section (2.131.11–133.16 [1.294.28–296.12]).
In this substantial section, Proclus contrasts Plato and Aristotle and points out their differences, while simultaneously showing that they do not conflict with each other entirely: ‘in this respect at least, the two men do not engage in conflict, but they do differ’ (2.131.11–12 [1.294.28–9]: καὶ ταύτῃ γε οἱ ἄνδρες οὐ μάχονται, διαφέρουσι δ’ ὅμως).Footnote 46 He goes on to say:
Since the splendid (δαιμόνιος) Aristotle copiously prattles all over the place (ἄνω καὶ κάτω θρυλῶν) about the reciprocations of the generated and the destructible and of the ungenerated and the indestructible, we should remind him that much earlier Plato too agrees with these fundamental propositions (τοῖς ἀξιώμασιν) when he writes in the Republic on the one hand that ‘for everything that has come into being destruction follows’, and in the Phaedrus on the other that what is ungenerated is also immortal. How, then, is it possible that he [Plato] would ascribe generation to the universe and not introduce destruction [for it] as well, or that he would ascribe destruction to that which is moving in a disharmonious and disorderly manner without also giving it generation before its destruction? But in actual fact he has devised for the universe [a form of] generation which was different and has adapted [a form of] everlastingness [for it] which was appropriate for the manner of its generation.
Proclus’ strategy in defending Plato against Aristotle is as simple as it is common: Aristotle was not an attentive reader of Plato. The latter already pointed out in earlier works, namely in the Republic and the Phaedrus, that generation implies destruction just as ungeneratedness implies indestructibility.Footnote 47 Since the cosmos is corporeal and generated – albeit not in time like sublunary beings but rather in the sense of having an external causeFootnote 48 – it is essentially destructible. However, as Proclus clarified earlier (2. 130.17–131.11 [1.294.9–28]), through the demiurge’s providence the cosmos acquires its eternal existence. Thus, Proclus claims that in one sense the cosmos is generated and destructible, that is, by being dependent on a cause and corporeal, while in another it is not, that is, by not being generated in time and by being kept in eternal existence by the demiurge.
The tone of the passage is rather condescending towards Aristotle, as particularly the expression ἄνω καὶ κάτω θρυλῶν and the reminder ὑπομνηστέον of Plato’s works prove. I thus disagree with Baltes (Reference Baltes1978) who calls this dispute ‘very polite’ (70: ‘sehr höflich’). Proclus’ use of δαιμόνιος in addressing Aristotle – as above thaumastos (θαυμαστός, 3.13.2 [2.9.7]) – has also an ironical tinge.Footnote 49 Proclus seems to be interested to show that Aristotle and Plato actually agree on the reciprocal implication of ungeneratedness and indestructibility, as he makes clear with the expression συγχωρεῖ here, in In Tim. 1.10.2–3 [1.6.31–2], and in endorsing Aristotle’s view on this issue in EP §2.5. Nevertheless, he does not shy away from maintaining that Aristotle misunderstood Plato. This is made clear by the tone of the passage and, especially, by the rhetorical question asked at 2.133.10–13 [1.296.6–9].
Proclus’ attitude can be contrasted with Simplicius’ who in his comments on DC 1.10–12 is at pains to point out that Aristotle did not, in fact, misunderstand and criticise Plato, as Proclus and Alexander suggested, but objected only to a superficial and fallacious reading of the Timaeus:Footnote 50 ‘Aristotle’s objections affect neither the theologians nor Plato, but rather those who interpreted the doctrines of the ancients in such a way as to suppose that, while the world was generated at a particular time, it was none the less indestructible. This is really absurd and well refuted by Aristotle’ (In DC 296.26–30; tr. Hankinson).
Although Proclus and Simplicius both effectively endorse and were influenced by Aristotle’s teaching on this issue – albeit with certain qualifications – they disagree on whether Aristotle actually meant to criticise Plato. This is a significant difference between Proclus’ and Simplicius’ approaches to Aristotle which we encounter often in their reactions to Aristotle’s criticisms of Plato.Footnote 51 Moreover, Proclus even points out that there is a certain danger emanating from Aristotle’s discussion in DC 1.10–12, as people might be misled into believing that Plato actually does not regard the cosmos as eternal.Footnote 52 This, again, makes clear why the discussion of the cosmos’ (un)generatedness was kept to a bare minimum in EP.
Proclus’ and Simplicius’ view can be contrasted with earlier Platonists, such as Plutarch (De an. procr. 1013D–1017C; Quaest. Plat. 4) and Atticus (fr. 4.41–2), who were less influenced by Aristotle and had no qualms with holding the view that the cosmos is temporally generated and indestructible.Footnote 53 A view that Proclus and Simplicius reject based on their Aristotelianised reading of the Timaeus. Proclus subjects both to criticism in his commentary on the Timaeus (esp. 2.106.3–11 [1.276.30–277.7], 2.257.5–19 [1.381.26–382.12]) for their doctrine of the cosmos’ temporal generation (2.106.4–5 [1.277.1]: κατὰ χρόνον τὴν γένεσιν). I discuss similar examples, such as the relationship of the unmoved mover and self-mover and the nature of self-motion in the next two chapters. There too it emerges that the Middle Platonists were less knowledgeable of Aristotle’s doctrines and his objections to certain Platonic views. Additionally, even if they knew these objections, they did not regard Aristotle as high an authority as the Neoplatonists and consequently did not saw the same pressing need to respond to Aristotle’s criticism. This clearly shifts with the Neoplatonists who not only integrate Aristotle’s teaching much more consistently into the Platonist system than before, but also take Aristotle’s criticisms against Plato more seriously.Footnote 54 How they deal with these criticisms differs however, as seen in the case of Proclus and Simplicius.
1.2.3.3 Conclusion
What do these two passages from the commentary on the Timaeus tell us about Proclus’ use of De caelo in EP? Above all, they exemplify how Proclus in EP makes a conscious choice as to what is – and what is not – worth summarising in an introduction to Aristotelian kinematics. In his adoption of the Aristotelian material, Proclus leaves out certain issues which he either rejects (e.g., the fifth element as an unmixed distinct element) or cannot properly address (e.g., the precise meanings of the terms ‘ungeneratedness’ and ‘indestructibility’) in a work such as EP.Footnote 55 It would be less useful and even counterproductive to include in EP doctrines which Proclus views critically and which require further qualification from a Platonist view. This emphasises the introductory character of EP, which is not suited for more subtle discussions and, hence, limited in its explanatory power. But, on a more fundamental level, it also points towards the limitations of Aristotle’s own works, as Proclus believes that his natural philosophy is in important regards inferior to Plato’s (In Tim. 1.9.14–10.18 [1.6.21–7.16]). Instead, EP offers the student a foundation from which she can later progress to intricate issues which are treated in their proper context (e.g., in the commentaries on Plato).
For instance, in EP Proclus establishes that τὰ κύκλῳ κινούμενα κατὰ φύσιν – that is, the cosmos – is ungenerated and indestructible (§2.5), based on an argument from DC 1.3.270a12–24. While Proclus embraces this insight, it clearly needs to be qualified, as the cosmos is in an important sense generated and destructible as well, as seen. Such an explanation, however, falls outside the limited scope of EP and is presented in his commentary on the Timaeus which was the penultimate dialogue in the Platonic part of the Neoplatonist curriculum. Another example is the eighth proposition from book 2: ‘Powers of bodies which are finite in magnitude are not infinite’ (i.e., finite bodies have finite power).Footnote 56 Again, in EP Proclus makes use only of the Aristotelian arguments, but in the advanced commentary on the Timaeus he further backs it up with Platonic doctrine (e.g., 2.91.15–18 [1.267.12–14], 2.109.3–5 [1.279.7–8.], 2.131.17–18 [1.295.3–4], 3.169.17–19 [2.123.2–4]).Footnote 57 This procedure of deepening or qualifying certain basic doctrines resembles the relationship of his other στοιχείωσις, ET, with the commentary on the Parmenides and Platonic Theology.Footnote 58 In the latter two, teachings from ET can appear quite differently and much more elaborate. Just as in the case of EP, the reason is, of course, the introductory character of ET and the more advanced nature of the other treatises.
1.3 The Form and Method of EP
The selection of the content clearly illuminated Proclus’ exegetical background and own views. But what about the form and the method of EP? Proclus chose for EP three specific models in devising these: (1) Euclid’s Elements, (2) Aristotle’s Posterior Analytics, and (3) Physics and De caelo. Proclus consciously followed these in order to achieve a high degree of systematicity and accessibility in presenting the material. While the impact of (1) and (3) has been well-established, the influence of (2) has not been properly investigated. Opsomer (Reference Opsomer, Aubry, Brisson, Hoffmann and Lavaud2021: 138) already made clear that Aristotle played an important role, next to Euclid. My aim in this section is to further develop this insight by considering the Aristotelian background – with emphasis on the Posterior Analytics – of Proclus’ axiomatic method in EP.Footnote 59 Although Proclus already encountered in Physics 6 – to a certain degree at least – an application of this method, he followed Euclid’s example as well as the general scientific theory of the Posterior Analytics to further axiomatise the text. Establishing the precise origin of Proclus’ axiomatics helps us understand the history of this method, as Proclus was the first to write completely axiomatised works on physics and theology, that is, metaphysics.Footnote 60 Crucially, he was later to be followed in these two domains by Newton and Spinoza. I would like to note here that this historical picture does not necessarily coincide with Proclus’ understanding of the origin of the axiomatic or geometrical method: in his commentary on the Timaeus he often credits Plato with employing a geometrical method in the dialogue.Footnote 61 This is because – as Proclus clarifies in the proem – the Timaeus is conceived as a Pythagoreanising work by Plato which allows him to use a more mathematical approach to the subject matter.Footnote 62 Now before I discuss the Aristotelian influence, it is necessary to mention briefly the Euclidean background.
1.3.1 Euclidean Influence
In naming his work stoicheiōsis (στοιχείωσις), Proclus already provides us significant cues about his intentions. In one sense, Proclus uses in his title the term στοιχείωσις in the more general meaning of ‘introductory handbook’ or ‘summary’ of a certain philosophical subject.Footnote 63 Works like these included a systematisation of elementary doctrines – just like Proclus’ EP and ET.Footnote 64 More importantly, however, Proclus specifically alludes to the Στοιχεῖα of the mathematician Euclid whose geometrical-axiomatic method he formally adopts in his presentation.Footnote 65 Euclid’s work is part of a genre of mathematical literature with strict formal and methodological characteristics.Footnote 66 Proclus associates himself consciously with this genre and adopts its conventions. His acquaintance with Euclid’s works is exemplified by his commentary on Euclid’s Elements 1 (In Eucl.) which supplemented his teaching activity in mathematics.Footnote 67
To put it simply, a mathematical στοιχείωσις or στοιχεῖα – there is no clear terminological distinctionFootnote 68 – contains besides certain preliminary principles, which will be discussed in Section 1.3.3.1, several elements (στοιχεῖα).Footnote 69 These στοιχεῖα are most basic theorems, from which further theorems or propositions are deduced. The different propositions form discrete (i.e., linguistically not directly connected) entities and are presented in a standardised and impersonal language. As pointed out by Asper (Reference Asper2007: 114–34), these three characteristics of discreteness, standardisation and impersonality are the main stylistic features of the genre and are found in EP (as well as ET).
As Proclus emphasises in In Eucl. 71.13–17, the theorems of a στοιχείωσις must be ordered according to their simplicity, fundamentality and proximity to the first principles. These most basic theorems are foundational for the development of further theorems in the work. It should be made clear that not all theorems in a στοιχείωσις are στοιχεῖα but only the most fundamental ones which are being ‘implicated in them [i.e., theorems] all and providing demonstrations for many conjunctions of qualities’ (72.11–13; tr. Morrow). The other theorems are only στοιχειώδη (elementary) and not relevant for the whole domain of a science.Footnote 70 In EP it is hard to distinguish between these kinds of theorems, as they are not explicitly marked as such. Presumably, one way of distinguishing them would be to separate theorems that are involved in other theorems from those that are otherwise not used and merely corollaries.
The purpose of a στοιχείωσις, according to Proclus (70.19–71.21), is to offer a systematic presentation of a certain science as well as to present it in an accessible manner to the student.Footnote 71 Proclus adopts the features described above to a significant degree in his own στοιχειώσεις, ET and EP, which – unsurprisingly – share argumentative and terminological similarities.Footnote 72
1.3.2 Aristotle’s Axiomatic Method and Proclus’ Commentary on Euclid’s Elements
There are aspects about Proclus’ application of the axiomatic method in EP which are absent in Euclid. While Euclid uses the method for mathematics, Proclus employs it in different domains and believes it can be universally used for any science. Moreover, Proclus actively reflects on the nature of the method – a self-consciousness which is missing in Euclid. One can legitimately ask whether these elements are derived from Aristotle. As I argue, Proclus’ grounding in Aristotle’s Posterior Analytics accounts for the universal applicability of the axiomatic method in Proclus by devising a general understanding of what science is and how it should be presented. Its influence on EP was decisive – just as Euclid’s Elements, and Aristotle’s Physics 6–8 and De caelo 1.
Proclus’ engagement with Aristotle’s theory of science in the Posterior Analytics has been neglected and not properly investigated in scholarship on EP.Footnote 73 The Posterior Analytics was generally important for the Neoplatonists, being regarded as the ‘culmination of the logical treatment’ (τέλος τῆς λογικῆς πραγματείας) (Philop. In APo. 1.5; tr. McKirahan), that is, Aristotle’s Organon, which in turn was considered preparatory for Platonic dialectic.Footnote 74 Proclus’ knowledge of the Organon is well attested both by his biographer Marinus (VP §9.33–6) and his extant works. As often, Proclus’ reading of the Posterior Analytics was possibly shaped by Syrianus with whom he read Aristotle’s works.Footnote 75 We know that Proclus had written a commentary on the Posterior Analytics which accounts for his use of it in his surviving works.Footnote 76 Most important among these is again his commentary on Euclid’s Elements where Proclus uses Aristotle to clarify the principles of science.Footnote 77
While Euclid uses a purely mathematical method, Aristotle bases his axiomatics on his syllogistic logic which has consequences for the structure of a science and its principles.Footnote 78 The fundamental idea, however, remains in both the same: a science x should be based on certain indemonstrable principles from which the (whole) domain of x can be deductively disclosed. It is in this sense that I use axiomatic and axiomatisation in the following. The requirement for indemonstrable principles is supposed to prevent an infinite regress which would occur if a given science had to demonstrate everything (APo 1.2–3; Met. 4.3). As Proclus puts it:
[N]o science demonstrates its own first principles (τὰς ἑαυτῆς ἀρχὰς ἀποδείκνυσιν) or presents a reason for them; rather each holds them as self-evident (αὐτοπίστως), that is, as more evident than their consequences. The science knows them through themselves, and the later propositions through them. This is the way the natural scientist (φυσιολόγος) proceeds, positing the existence (ὑποθέμενος εἶναι) of motion and producing his ideas from a definite first principle.
As Proclus’ example of the φυσιολόγος makes clear, he is here not just referring to the specific case of mathematics and therefore is not describing Euclid’s procedure in the Elements. Rather, his general reflections on the principles of science as being indemonstrable presuppose Aristotle’s theory in the Posterior Analytics and Met. 3.2 with which they agree. For instance, in On Providence §6, Proclus cites Aristotle approvingly for claiming that one must proceed from common notions to the theorems to be demonstrated.Footnote 79 It should be noted that Proclus does not refer here to an absolute indemonstrability of principles (i.e., it is generally impossible to demonstrate scientific principles) but rather to a relative indemonstrability (i.e., a specific science does not demonstrate its own principles).Footnote 80 Moreover, his proximity to Aristotle is also evidenced by the claim that the natural scientists posits the existence (ὑποθέμενος εἶναι) of an object in his field of study which fits Aristotle’s understanding of a hypothesis as making an existence claim (APo 1.2.72a18–24).
What then are these principles from which a science proceeds? Both Aristotle and Euclid are commonly assumed to accept a similar threefold division of principles:Footnote 81
| Euclid (Elements 1) | Aristotle (APo 1.2.72a14–24Footnote 82) | |
| (1) definitions (ὅροι) | : | (1) definitions (ὁρισμοί) |
| (2) common notions (κοιναὶ ἔννοιαι) | : | (2) axioms (ἀξιώματα) |
| (3) postulates (αἰτήματα)Footnote 83 | : | (3) hypotheses (ὑποθέσεις) |
In both Euclid and Aristotle, (2) common notions/axioms are general (e.g., law of non-contradiction or law of equals from equals), while (1) definitions and (3) postulates/hypotheses are science specific.Footnote 84 While the notions of axioms and definitions are quite straightforward and overlap, hypotheses and postulates differ. According to Aristotle, hypotheses are existence claims that posit the existence of something, for example, points (APo 1.2.72a23–4; 1.10.76b3–5). This is not the case with Euclid’s postulates. For instance, Euclid’s first postulate in Elements 1 ‘Let the following be postulated: to draw a straight line from any point to any point’ (tr. Heath) clearly presupposes the existence of points as well as lines and does not assert their existence. However, the similarity of Euclidean postulates and Aristotelian hypotheses lies in their function as both ‘introduce things into the realm of discourse of a science’ (McKirahan Reference McKirahan1992: 139).
Regardless of the relationship of these classifications, more relevant for my current purpose is what Proclus thought about their relation. Crucially, he seems to allude to a congruence between the two (In Eucl. 76.4–77.2) by using his own terminology for the tripartite division:
(1) axioms (ἀξιώματα)
(2) hypotheses (ὑποθέσεις)
(3) postulates (αἰτήματα)
Proclus’ unusual terminology in referring to the principles matches neither directly Aristotle’s nor Euclid’s use. This, in turn, has led to an extensive debate in scholarship.Footnote 85 Crucially, Proclus appeals to Aristotle as an authority for distinguishing these three principles (e.g., In Eucl. 182.14–20), clearly having the discussion of the Posterior Analytics in mind. But how well can Proclus have understood Aristotle who plainly states that definitions are not hypotheses, since definitions point out what something is, but do not make existence claims (APo 1.2.72a23–4; 1.10.76b35–77a4)? Prima facie, it seems that Proclus mistakenly takes Aristotelian definitions to be ‘hypotheses’.Footnote 86 Connected to this, is then Proclus’ choice of calling Aristotelian hypotheses ‘postulates’. Has Proclus simply misunderstood Aristotle’s teaching on scientific principles or is his terminology a result of a specific philosophical background?
I take it that the latter is the case. Proclus presents us in this and related passages a Platonising interpretation of Aristotle and not an objective summary of Aristotle’s theory in the Posterior Analytics, as some scholars have assumed.Footnote 87 The names of the principles are indicative of this, as Proclus simply uses his own terminology and imposes it on Aristotle.Footnote 88 This Platonic background explains Proclus’ peculiar terminology in the passage above. Proclus’ aim is to find support in Aristotle for his tripartition of principles. As he shows numerous times in his commentary, the ancients struggled to distinguish between different principles which is reflected in differing usage of the terms axiom, definition and so on.Footnote 89 Hence, this terminological instability and lack of clarity in Proclus is also due to the conflicting evidence of his authorities. This is made clear by the apparent hypothesis – definition mix up.
Proclus regards definitions – like other principles – in the specific sciences as necessarily hypothetical, that is, provisional.Footnote 90 For Proclus almost all sciences, including their principles, are hypothetical: ‘for there is only one unhypothetical (ἀνυπόθετος) science, the other sciences receiving their first principles (τὰς ἀρχάς) from it’ (75.9–10; tr. Morrow).Footnote 91 This unhypothetical science is – based on Plato’s Republic (e.g., 6.510b–511d; 7.533b–d) – dialectic which Proclus treats in ET and whose role lies in furnishing the principles of lower sciences (In Eucl. 8.21–10.14). Unlike other sciences which are characterised by discursive thinking, dialectic is only accessible through νοῦς and νόησις (In Eucl. 11.9, 42.12–18). To a certain degree this accounts for why Proclus often – but not exclusively –Footnote 92 employs the term ‘hypothesis’ for ‘definition’ (ὅρος, ὁρισμός) in In Eucl. All definitions are hypothetical and can be thus called ‘hypotheses’, but not all hypotheses are definitions since the term hypotheses can also refer to other principles. This explains why Proclus sometimes uses hypotheses for all three types of principles and not only for definitions.Footnote 93
Let us briefly conclude. Proclus shows in the preface to his commentary on Euclid’s Elements an awareness not just of Euclid’s but also of Aristotle’s axiomatic procedure in the Posterior Analytics. Proclus’ engagement with the latter is seen in four areas where Proclus’ practice in EP differs from Euclid and/or is directly based on Aristotle. (1) Proclus takes over the idea from the Posterior Analytics of applying axiomatics to non-mathematical sciences. This ultimately culminates in his axiomatisation of theology in ET. (2) Moreover, Proclus engages with Aristotle’s theory of principles and takes over his understanding of hypotheses as existence claims (In Eucl. 75.14–20) – even though he sometimes calls them postulates. In the next discussion two more points will emerge clearly. (3) Following Aristotle’s advice that axioms can be sometimes left out, Proclus does not mention them – unlike Euclid. (4) In his reflections on the nature of reductiones – which he uses throughout EP and even supplements (if they were not included in the original) – Proclus is inspired by the account of the Posterior Analytics. These characteristics offer evidence for my thesis that Aristotle’s Posterior Analytics had a lasting and productive impact on Proclus’ axiomatic project and should be considered in treatments on the origin of the axiomatic method in Proclus.
1.3.3 Aristotle’s Mathematical Style and Proclus’ EP
Besides his acquaintance with Aristotle’s model of science – particularly with his theory of scientific principles and model of scientific discourse from the Posterior Analytics – Proclus also encountered in Physics 6, 8 and De caelo 1 texts which already contained mathematical style arguments and evidenced a certain level of axiomatisation.Footnote 94 This ‘mathematisation’ is especially pronounced in Physics 6, on which Proclus mostly bases EP 1, and to a lesser degree in Physics 8Footnote 95 and De caelo 1, which Proclus uses in EP 2. The reason why Aristotle adopted this style there is not entirely transparent. For instance, Owen (Reference Owen1986: 164) explains the mathematical and axiomatic character of the De caelo and the Physics – partly – with Aristotle’s proximity to the Academy (and thus regards these works as early). Yet, the most attractive explanation for the presence of these features is proposed by Jope (Reference Jope1972) who sees in Physics 6 an example of the axiomatic-deductive system envisaged in Posterior Analytics 1. He focuses on the fact that Aristotle explains in Physics 6 motion and time in reference to continuity which is a mathematical characteristic. But, Jope argues, insofar as Aristotle studies motion and time qua continua and, thus, physical things in relation to a mathematical property, his approach is that of a subordinate, and not ordinary, science. Subordinate sciences, such as harmonics or optics, study the physical qua mathematical, and are only able, according to Aristotle, to provide demonstrations of the ‘that’ (τοῦ ὅτι) but not of the ‘why’ (τοῦ διότι). That is, they describe that something is the case but are unable to point out why that is. The latter is only reserved to mathematics itself which studies the mathematical qua mathematical.Footnote 96 That Aristotle offers primarily demonstrations τοῦ ὅτι, not τοῦ διότι, will be important in Section 1.3.3.2.
It is apparent that Proclus was aware of the theory of an axiomatic science as well as of the axiomatic structure of the Aristotelian original. Proclus only had to accentuate already present features and add logical rigour to the arguments. He then structured the text into discrete entities, that is, propositions, as in Euclid. In many cases, Proclus’ restructuring of the original text did not require great interventions. He thus axiomatised even less structured parts of the Aristotelian text – which does not mean that we have a flawlessly axiomatic work, as will become clear.
In the following I show how in EP Proclus puts this axiomatic model of science into practice by comparing some examples from EP with their Aristotelian original. I first discuss some peculiarities of the principles which have not been sufficiently discussed in scholarship (1.3.3.1) and then turn to the theorems (1.3.3.2).
1.3.3.1 Principles
In EP Proclus posits in total twenty principles: six definitions in the first book, six hypotheses and eight definitions in the second. Almost all are taken directly from the Aristotelian original, the precise references being provided by Ritzenfeld (Reference Ritzenfeld1912). In my analysis I focus on those aspects which are either relevant for Proclus’ axiomatisation of Aristotle’s kinematics or constitute significant departures from the original source. The six definitions of the first book are:
(Def. 1.1) Συνεχῆ ἐστιν, ὧν τὰ πέρατα ἕν.
Continuous are those things whose limits are one.
(1.2) Ἁπτόμενά ἐστιν, ὧν τὰ πέρατα ἅμα.
Contiguous are those things whose limits are together.
(1.3) Ἐφεξῆς ἐστιν, ὧν μηδὲν μεταξὺ ὁμογενές.
Next-in-succession are those things which have nothing of the same kind in between them.
(1.4) Πρῶτός ἐστι χρόνος κινήσεως ὁ μήτε πλείων μήτε ἐλάττων τῆς κινήσεως.
The primary time of a motion is the time which is neither more nor less than the motion.
(1.5) Πρῶτός ἐστι τόπος ὁ μήτε μείζων τοῦ περιεχομένου σώματος μήτε ἐλάττων.
The primary place is the place which is neither more nor less than the encompassed body.
(1.6) Ἠρεμοῦν ἐστι τὸ πρότερον καὶ ὕστερον ἐν τῷ αὐτῷ τόπῳ ὂν καὶ αὐτὸ καὶ τὰ μέρη.
Resting is that which itself and its parts is before and after in the same place.
First, it should be remarked that the first three definitions are not formulated in accordance with the subject matter, that is, the γένος of the specific science, as demanded by Aristotle himself (APo 1.7, 1.28). That is, since the subject matter is physical science it requires appropriate principles (in this case definitions) that do not stem from another science.Footnote 97 But defs. 1–3 are actually definitions used in mathematics and lack the specific, physical character that one would expect. Since they are close to the Aristotelian original (Phys. 5.3 and 6.1) – Proclus directly takes them over – this problem can be traced back to Aristotle who himself refers to defs. 1–3 explicitly as definitions (Phys. 6.1.231a22). However, this does not mean that Proclus is not conscious of the requirement for science-specific definitions. When he discusses whether Euclid’s definition of a point is adequate, he claims that ‘the scientist in a special area … has the responsibility of examining and expounding only that indivisible nature which is appropriate to his first principles’ (93.11–15). For instance, the geometrician studies the point, whereas the arithmetician the monad. Proclus presumably takes 1.1–3 over in this form because these broader definitions allow him to insert the relevant physical term, that is, ‘[a] magnitude/time/motion is continuous when its limits are one’. Additionally, these first three definitions also situate Proclus safely in the Neoplatonist exegetical tradition since the wording of defs. 1.1–3 is, in fact, closer to Simplicius than Aristotle.Footnote 98 For instance, the expression ὁμογενές in def. 1.3 is found in Eudemus, Alexander and Simplicius, but it is absent in Aristotle who uses the term συγγενές (which Simplicius also correctly cites).Footnote 99
Defs. 4–6 concern time, place and rest, and have their own peculiarities. Unlike the earlier definitions, they are clearly subject-specific and less general. While def. 6 is straightforward, a few remarks on the other two are necessary. Def. 5 on primary place is a combination of two principles established by Aristotle at the beginning of his discussion of place: the first expresses the idea that place is the ‘first thing surrounding that of which it is place’, while the second claims that ‘primary place is neither less nor more than’ the thing it surrounds. Def. 4 on primary time is not found explicitly in Aristotle’s text and has been modelled after def. 5. I take it that Proclus, following Aristotle, uses the expression πρῶτος χρόνος/τόπος in these two definitions to indicate that he talks about the time or place which includes neither more nor less than the object or process to which it belongs. That is, a time which stretches more or less than the duration of a process, such as walking from a to c instead of a to b, is only secondary. Analogously, the same applies to place where, for example, a dog can be primarily located in a seat and secondarily in car and so on.
It is noteworthy that defs. 4 and 5 are not taken from Physics 6 but from book 4.Footnote 100 This is peculiar, since Proclus’ primary concern in EP 1 is to offer an axiomatisation of book 6. He adds, I submit, these definitions because he believes the three definitions listed by Aristotle at the beginning of Physics 6 are insufficient for deriving all doctrines of the book. This shows us how Proclus takes over the already axiomatic structure of the Aristotelian original and further perfects it. Although EP deals primarily with motion, it does not contain a definition of motion (or of place). Likewise, while time is defined in def. 2.7 (‘Time is a number of the motion of the heavenly bodies’), its definition remains dependent on the concept of motion which we lack. In this way, the concepts and definitions of motion and place are already presupposed, since they are included in the defs. 1.4–6 and 2.7.
I now turn to the principles of book 2 which are more numerous.
(Hyp. 2.1) Πᾶν σῶμα φυσικὸν κινητόν ἐστι κατὰ τόπον.Footnote 101
(2.2) Πᾶσα κίνησις τοπικὴ ἢ κύκλῳ ἐστὶν ἢ ἐπ’ εὐθείας ἢ μικτὴ ἐκ τούτων.
Every locomotion is either circular or linear or a combination of both.
(2.3) Πᾶν σῶμα φυσικὸν μίαν ἐκ τούτων κίνησιν κινεῖται.
Every physical body moves with one of these motions.
(2.4) Πᾶν σῶμα φυσικὸν ἢ ἁπλοῦν ἐστιν ἢ σύνθετον.
Every physical body is simple or composite.
(2.5) Πᾶσα κίνησις ἁπλῆ ἁπλοῦ σώματός ἐστιν.
Every simple motion is of a simple body.
(2.6) Πᾶν σῶμα ἁπλοῦν μίαν κατὰ φύσιν κινεῖται κίνησιν.
Every simple body moves according to its nature with one motion.
(Def. 2.1)
Λόγον ἔχειν πρὸς ἄλληλα τὰ τάχη λέγεται, ὃν τὰ διαστήματα ἔχει, δι’ ὧν τὰ κινούμενα κινεῖται. The relation of the velocities to each other is the relation of the distances through which the moving things move.
(2.2) Βαρύ ἐστι τὸ ἐπὶ τὸ μέσον κινούμενον.
Heavy is that which moves towards the middle.
(2.3) Κοῦφόν ἐστι τὸ ἀπὸ τοῦ μέσου κινούμενον.
Light is that which moves away from the middle.
(2.4)
Κύκλῳ κινεῖσθαι λέγεται τὸ ἀπὸ τοῦ αὐτοῦ πρὸς τὸ αὐτὸ φερόμενονσυνεχῶς. Circular motion means moving continually away from the same point towards the same point.
(2.5) Ἐναντίαι κινήσεις εἰσὶν αἱ ἀπὸ τῶν ἐναντίων εἰς τὰ ἐναντία.
Contrary motions are from the contrary to the contrary.
(2.6) Ἓν ἑνὶ ἐναντίον.
One thing is (only) contrary to one other.
(2.7) Χρόνος ἐστὶν ἀριθμὸς κινήσεως οὐρανίων σωμάτων.Footnote 102
Time is a number of the motion of the heavenly bodies.
(2.8) Μία κίνησίς ἐστιν ἡ κατ’ εἶδος ἀδιάφορος καὶ ἑνὸς ὑποκειμένου καὶ ἐν συνεχεῖ χρόνῳ γινομένη.
A single motion is a motion which is unchanged in its form, directed at one object, and takes place in a continuous time.
According to the standard edition by Ritzenfeld and the opinion of most scholars,Footnote 103 Proclus mentions in EP only definitions – six in the first book and fourteen in the second. He thus leaves out the other two principles – axioms and postulates/hypotheses – which he clearly recognised, as seen above. But, as I argue here, this impression is wrong. Based on their content, the manuscript tradition of EP, and other remarks in Proclus’ and other Neoplatonists’ oeuvre, it can be safely established that Proclus conceived the first six principles in fact as hypotheses.
If we look at the content of the first six principles, it becomes quite obvious that these are not definitions, as they do not define what something is – unlike the other principles.Footnote 104 Rather, they seem to have the character of hypotheses by making certain assumptions, that is, they posit that something is or, more precisely, that every being of a certain class is in a certain way. The quantifiers πᾶν/πᾶσα clearly separate them linguistically from the other principles and provide the impression that they are structured like premises for (Barbara-type) syllogisms. For instance, hyp. 2.1 claims that every physical body is moveable in space without offering a definition of a physical body by answering the question ‘What is x?’. However, the proper definitions of EP 2 do exactly that. For instance, ‘What is heavy?’ – ‘Heavy is that which moves towards the middle’ (def. 2.2).Footnote 105
Evidence from the manuscript tradition also suggests that the first six principles should be termed ‘hypotheses’. Ritzenfeld (Reference Ritzenfeld1912: 70) remarks on the definitions of book 2 that (some) editions call the first six hypotheses. He refers here primarily to the editio princeps by Simon Grynaeus (Reference Grynaeus1531), called b in his critical apparatus. Grynaeus’ edition is based on codex (Z) and attests on page 28 ὑποθέσεις instead of ὅροι for the first six definitions. It is thus possible that (Z), which Ritzenfeld does not seem to have consulted (presumably because it was no longer extant),Footnote 106 offered this version as well. Unfortunately, since Ritzenfeld’s stemma codicum is incomplete – he has consulted only eleven of the over thirty codices known to him – as well as faulty, as demonstrated by Boese (Reference Boese1958: 13–14),Footnote 107 it is impossible to determine here the consequences for a reconstruction of the original text.
The strongest argument, however, for calling these six principles hypotheses instead of definitions, which has been – to my knowledge – ignored so far, stems from Proclus himself. In his commentary on the Timaeus he claims that Aristotle demonstrates the indestructibility of the cosmos based on certain hypotheses (2.48.2 [1.237.23]: ὑποθέσεις) and cites Plotinus approvingly who also employs the term hypotheses for these principles.Footnote 108 Proclus mentions five hypotheses (2.48.5–8[1.237.27–238.1]) of which the first three correspond to the principles 2.5, 2.6 and 2.2. Simplicius adopts this usage of the term as well in his commentary, counting six hypotheses which Aristotle uses to determine the heaven’s eternity. Some of these match Proclus’ hypotheses.Footnote 109
There is thus a unanimous awareness in Plotinus, Proclus and Simplicius of the fact that Aristotle builds his demonstrations in De caelo 1 (in part) on hypotheses and not just on definitions. This Neoplatonist interpretation is based on Aristotle’s own usage of the term, as he refers to some of the propositions appearing as definitions in EP 2 repeatedly as hypotheses (e.g., at DC 1.3.270b3 and 1.7.274a34, and b11: τὰς πρώτας ὑποθέσεις). While Aristotle uses the term hupothesthai (ὑποθέσθαι) and its variants in a more general sense of statements that have to be assumed without being demonstrated, it is clear that the Neoplatonists were inspired by his terminology. Together with an analysis of their content as well as the manuscript tradition, it seems highly probable that the first six principles of EP 2 should be considered as hypotheses and not definitions, as commonly assumed.
It is, however, beyond doubt that Proclus leaves out axioms in EP. Why? Presumably, Proclus found axioms too obvious to be stated explicitly, just as Euclid does not list, for example, the law of excluded middle as a common notion. Aristotle himself claims that not all principles have to be necessary stated, mentioning specifically the case of axioms which can be left out due to their familiarity (APo 1.10.76b16–21). Moreover, while some principles are necessary for Proclus, it does not mean that their threefold division must be respected in each science. In this way, Proclus sticks rather to Aristotle’s precepts in the Posterior Analytics than to Euclid’s Elements, where some axioms occur.
1.3.3.2 Theorems
Based on these definitions and hypotheses Proclus then constructs several theorems. The adherence to the axiomatic method is less strict than one might expect since most theorems include unstated premises which are neither based on the principles nor on the preceding theorems.Footnote 110 Interestingly, this lack of axiomatisation can be already traced back to Aristotle’s original text. But as we have seen, Proclus has also added principles to make the presentation more axiomatic. To make this clear I discuss a few examples. As I show here, Proclus (1) further completes the formalisation already present in the Aristotelian original and, specifically, (2) adds reductiones ad impossibile (εἰς τὸ ἀδύνατον/διὰ τοῦ ἀδυνάτου ἀπαγωγαί) which serve to foster the axiomatic structure of the text.
Let us start with discussing a few formal features by looking at EP §1.19 which is based on Phys. 4.4.234b10–20:
Πᾶν τὸ κινούμενον μεριστόν ἐστιν.
Ἔστω γάρ τι κινούμενον ἐκ τοῦ Α εἰς τὸ Β. ἢ οὖν ἐν τῷ Α μόνον ἐστὶν ἢ ἐν τῷ Β ἢ ἐν ἀμφοτέροις ἢ ἐν οὐδετέρῳ ἢ τὸ μὲν αὐτοῦ ἐν τῷ Α, τὸ δὲ ἐν τῷ Β. ἀλλ’ εἰ μὲν ἐν τῷ Α, οὔπω κινεῖται· εἰ δὲ ἐν τῷ Β, οὐκέτι κινεῖται· εἰ δ’ ἐν ἀμφοτέροις, καὶ οὔπω κινεῖται καὶ οὐκέτι κινεῖται· εἰ δ’ ἐν οὐδετέρῳ, οὐκ ἔσται ἐκ τοῦ Α εἰς τὸ Β ἡ κίνησις. [οὐδὲ μεταξὺ αὐτῶν] ἀνάγκη ἄρα τὸ μὲν αὐτοῦ ἐν τῷ Α εἶναι, τὸ δὲ ἐν τῷ Β· διαιρετὸν ἄρα τὸ κινούμενόν ἐστιν.
Every moving thing is divisible.
For let something be moving from A to B. Then either it is in A alone or in B or in both or in neither or one part is in A and another in B. But if it is in A, it is not yet in motion. If it is in B, it is no longer in motion. If it is in both, it is both not yet and no longer in motion. But if in neither, there will be no motion from A to B. Necessarily, it is partly in A and partly in B. Therefore, the moving thing is divisible.
Based on his Aristotelian source, Proclus wants to establish here that every entity in motion must be physically divisible.Footnote 111 Two things strike the reader at first sight. First (1) is the repetition of the proposition at the end of the passage which is common to mathematical works and present also in Aristotle. Proclus, thus, imitates here Aristotle directly and only accentuates a tendency already encountered in the latter.Footnote 112 A conspicuous difference is (2) the shift from μεταβάλλον in Aristotle to κινούμενον in EP which can be explained by Proclus’ preference for terms derived from κινέω and not from μεταβάλλω.Footnote 113 Since there does not seem to be a conceptual difference between the terms in EP, it seems that Proclus simply tries to make the terminology more unitary. It should be also mentioned that Proclus adds a diagram at §1.19.18.8–9 which is missing in Aristotle. It is a matter of debate whether there were diagrams in the original manuscripts of the Physics.Footnote 114 Aristotle’s use of lettered variables certainly suggests a visual model in form of a diagram. It comes thus to no surprise that Proclus makes extensive use of diagrams in EP which are transmitted to us.
Not only diagrams but also variables seem absent in the Aristotelian original. Yet, this is not generally the case in the Physics (or in the De caelo, for that matter), as Proclus only completes the formalisation that is otherwise present in Aristotle. This is evidenced by the sentence following the Physics passage quoted above where he states that ‘if the whole AC is in motion, its parts AB and BC will also be in motion (234b23–4). And then Aristotle himself uses the third person imperative which is a typical feature of the mathematical works to prove his point: ‘accordingly, let the movement of the parts be M1–2 of AB and M2–3 of BC’ (234b24–5).
Thus, two of the three central characteristics of a στοιχείωσις (see Section 1.3.1) – standardisation and impersonality – occur already in the Aristotelian original. Proclus only had to accentuate these formal characteristics by completing the formalisation and add logical rigour to the arguments by spelling them out in a clear manner. In many cases Proclus’ restructuring of the original text did not require great interventions. This resemblance is also reflected in the types of arguments employed. In EP Proclus uses in his proofs primarily reductiones. So far, this strong reliance on reductiones in EP – even in the passages and arguments added by Proclus – and their purpose in this work have not been sufficiently explained. Although Opsomer (Reference 236Opsomer, Baghdassarian, Papachristou and Toulouse2020b: 93–6) discusses this important aspect, he focuses in his explanation exclusively on evidence from Aristotle’s Prior Analytics 1.23.41a21–37 and 1.44.50a29–38.Footnote 115 While this is undoubtedly right, in the following, I argue that the background of the Posterior Analytics played a more significant role – as witnessed also above in the case of principles.
The first example of a reductio is encountered in §1.2. Proclus had established in §1.1 that two points are not contiguous since contiguity requires the ends of two (or more) things to be together.Footnote 116 This, however, is not possible in the case of points since they have no parts and thus no ends which could be together. Now, in §1.2 Proclus wants to establish that two points can also not be continuous:
Δύο ἀμερῆ συνεχὲς οὐδὲν ποιήσει.
Εἰ γὰρ δυνατόν, ἔστω δύο ἀμερῆ τὰ ΑΒ καὶ ποιείτω συνεχὲς τὸ ἐξ ἀμφοῖν. ἀλλὰ πάντα τὰ συνεχῆ ἅπτεται πρότερον· τὰ ἄρα ΑΒ ἅπτεται ἀλλήλων ἀμερῆ ὄντα, ὅπερ ἀδύνατον.
Two partless things will not form something continuous.
For if it were possible, let A and B be two partless things and let them form something continuous from each other. But all continuous things touch each other earlier; then, A and B touch each other, although they are partless, which is impossible.
The argument is simple. Proclus first assumes the positive hypothesis that the two points A and B are continuous. However, this cannot be the case since continuity requires contiguity, that is, A needs to be contiguous with B in order to be continuous. But, as §1.1 has shown, it is impossible for two points to be contiguous. Therefore, they are also not continuous.
This argumentative style is again influenced by the Aristotelian original – here Phys. 6.1.231a24 – as well as Euclid’s Elements since both use reductiones extensively.Footnote 117 Yet, the use of demonstrations by reductiones is puzzling since both Aristotle and Proclus deny their explanatory power. As is well known, Aristotle claims in the Posterior Analytics that knowledge is attained by demonstrations in the form of deductions. (In)famously, the premises of these demonstrations have to meet certain criteria among which are ‘prior to and explanatory of the conclusion’ (APo 1.2.71b22). Neither criterion, however, is met by reductiones, as is shown in a condensed argument in APo 1.26,Footnote 118 where Aristotle argues that direct negative demonstrations are superior in their explanatory power over indirect negative demonstrations, that is, reductiones. Direct negative demonstrations proceed from premises prior to the conclusion, while reductiones proceed from premises posterior to the conclusion. The sense of priority is priority in nature (87a17). But if the premises of a reductio are not prior to the conclusion, it does not meet the necessary requirements of a scientific demonstration laid out in APo 1.2.Footnote 119 Still, a reductio can be considered a demonstration in a more general sense. For, in APo 1.13, Aristotle points out that besides the genuine form of demonstration which provides an explanation of the reason why (τοῦ διότι), that is, Why does A hold of B?, there is another type of demonstration which reveals a property or fact (τοῦ ὅτι): Does A hold of B? While the former is to be preferred, the latter plays an indispensable role in acquiring knowledge and is often presupposed by the demonstration of the explanation, as he makes clear in APo 2.1.89b27–31.Footnote 120 Consequently, Aristotle holds that knowledge of facts is temporally prior to knowledge of causes.Footnote 121 Thus while reductiones are not demonstrations stricto sensu, since they do not provide explanations, they are demonstrations in a more general sense, since they produce knowledge (i.e., provide proof) of the fact.Footnote 122
A reductio ad impossibile does not provide the cause of a conclusion and is thus not explanatory. That was and still is the common understanding of Aristotle’s views.Footnote 123 Proclus shares this view, displaying again his knowledge of the Posterior Analytics:
When geometers reason through the impossible, they are content merely to discover the property (τὸ σύμπτωμα μόνον) [of a given subject]. But when their reasoning proceeds through a principal demonstration, then, if the demonstrations are partial, the cause is not yet clear, whereas if it is universal and applies to all like things, the ‘why’ at once becomes evident.
Why then adopt an argumentative style where reductiones are virtually omnipresent? Partly, Proclus’ motivation is based on emulating the Aristotelian originalFootnote 124 and Euclid’s Elements. Moreover, as mentioned, knowledge of the fact is a requirement for knowledge of the cause – a doctrine with which Proclus was evidently acquainted.Footnote 125 In this sense, the style fits quite well to the propaedeutic character of EP: it establishes first the facts of kinematics before providing a reason.Footnote 126 Moreover, through the reductiones it becomes clear that earlier propositions were correct. For instance, in the example discussed above, §1.2, the reason why the hypothesis is impossible can be found in the preceding proposition, §1.1.
This example proves why reductiones are so useful for Proclus in a work such as EP. They point backwards to earlier propositions and fortify the axiomatic structure of the treatise. In this way, the role of the reductiones is to buttress the arguments of earlier propositions as well as to hint at the reason behind the reductio. Additionally, I would like to emphasise that Proclus chooses consciously to adopt this argumentative method, as the argument of §1.2, for instance, is not found in the form of a reductio in Aristotle. He thus adds further reductiones to the text. The scientific theory of the Posterior Analytics accounts for the presence of this argumentative feature in the Aristotelian text and, hence, in EP. Considering Aristotle’s understanding of reductiones in the Posterior Analytics illuminates not just his practice in the Physics and the De caelo but also Proclus’ familiarity with them and use in EP.
1.4 Conclusion
In this chapter, I offered a comprehensive discussion of the content and structure of Proclus’ little-known treatise EP. This provides us an insight into the reception and the place of Aristotelian kinematics in Proclus. In Section 1.2, I situated EP in a larger exegetical tradition of Aristotle’s Physics and De caelo which has been little explored so far, although it explains some of the work’s peculiarities. In designing EP, Proclus is dependent on this tradition. I also emphasised how, due to its argumentative and conceptual similarity to Physics 6 and 8, as well as its importance for kinematics, Proclus believes it is necessary to include material from De caelo 1. Moreover, I demonstrated how Proclus excludes certain topics from EP. This is mainly due to its introductory nature, but it is also in line with the axiomatic structure of the treatise. These more controversial topics are then discussed in advanced works such as his commentary on the Timaeus which further develop the issues treated in EP by providing a stronger metaphysical fundament. This emphasises the connection of EP to other Proclean works.
Section 1.3 focused on the form and method of EP. By analysing the principles of EP I demonstrated that Proclus also includes hypotheses in EP and not just definitions. An analysis of the theorems showed how Proclus took over the Aristotelian text and further axiomatised it. More generally, I argued for attributing a more important role to Aristotle’s Posterior Analytics – besides Euclid’s Elements and Aristotle’s Physics and De caelo – in Proclus’ development of the axiomatic method in EP. This claim was supported by certain features which are absent in Euclid but can be explained by referring to the Posterior Analytics and also by Proclus’ frequent discussions of the Posterior Analytics in his commentary on Euclid’s Elements. Proclus’ use of the axiomatic method in EP is thus not only based on the Aristotelian original but also on his own theoretical reflections on its features, such as the nature of principles, the derivation of theorems from principles, use of reductiones and so on. Most importantly, there is an awareness that the method can be applied to different non-mathematical sciences, which later paved the way for the use of axiomatics not just in natural philosophy but also in theology.
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