An old chestnut in Proclus research is “the geometrical method”. This notion refers, of course, to Euclid’s method in his Elements. In Proclus, it is primarily associated with the presentation of the basics of Neoplatonic metaphysics in the Elements of Theology, a work which in different ways summons the mos geometricus. Although it has become clear that Proclus does not exactly follow Euclid’s method of presentation in his Elements of Theology (see § “First case: Elements of Theology” below), the importance of the suggestion that he does cannot be underestimated: it is this chestnut which is primarily responsible for the rehabilitation of Proclus’ thought in the past century. Taylor, when writing a defence of Proclus in 1918, states: “We have learned that the Neo-Platonists were neither magicians nor emotionalist schöne Seelen, but systematic philosophers addressing themselves to the philosopher’s task of understanding the world in which he lives as seriously as Aristotle or Descartes or Kant.” His main evidence for this thesis is Proclus’ Elements of Theology, as displaying a “manner and method” similar to those of “the great rationalists”, rather than “ecstasies and other abnormal psychological wonders” (A. E. Taylor 1918: 605–6).
The key terms in Taylor’s analysis are “systematic”, “seriously” and “rational”. And the names of Aristotle, Descartes and Kant summon the same image: of serious, systematic, rational and methodic philosophers striving for true understanding of the world. As a vindication of Neoplatonism against verdicts of mysticism and vagueness, such a statement works quite well. It does not, however, tell us what Proclus’ geometrical method is, or what its aims are.
The same criticism holds for most descriptions of Proclus’ supposed method in the Elements of Theology, or of the stereotype more geometrico has become. The general, but too limited, picture is that it moves from principles to their consequences, via deduction – it is a “synthetic” method of “a priori” deduction (Dodds 1963: xi), or an “axiomatic-deductive” method,1 or also a “hypothetical deductive” method.2 However, as Lloyd says (concerning Galen, but it holds for anyone subscribing to mathematics as an ideal of science), “quite what that ideal comprises is more difficult to pin down than is generally recognized” (G. E. R. Lloyd 2005: 110).
In the following, we will try to capture how Proclus himself understands the “geometrical method”. The chapter falls into four parts, not counting the introduction: “The elements of Elements” contains an analysis of Proclus’ description of “Elements” in his commentary on book I of Euclid’s Elements, “Methods of geometry” an overview of the relevant methods of geometry as identified by Proclus, “Are Proclus’ Elements written more geometrico?”3 discusses the two cases of Elements among Proclus’ own works: the Elements of Theology and Elements of Physics, and “More geometrico in the commentaries” three cases of “geometrical method” in Proclus’ Platonic commentaries. As we will see, there are four aspects in which the earlier-mentioned descriptions of Proclus’ geometrical method fall short: first, they ignore the didactic aim the mos geometricus has for Proclus. Second, they focus on its synthetic side, but, as we will see, analysis is just as important.4 Third, they lack a reference to the necessary metaphysical background of the method. And finally, there is not one geometrical method: in different contexts the method is elaborated differently.5
For reasons of space, we will not go into the details of Proclus’ sources of inspiration. Let me merely point out that Euclid cannot be Proclus’ only source,6 mainly because in this work there are no hypotheses, and it does not reveal ontological causal relations, which is, however, one of the main reasons for Proclus to use the method in the first place (cf. Harari 2008). Among possible other sources are Zeno, the method of hypothesis from Plato’s Phaedo, the hypotheses of the Parmenides, the description of geometry in the Republic, the classical model of science of Aristotle’s Posterior Analytics,7 and Iamblichus and Syrianus.8
THE ELEMENTS OF ELEMENTS
Assuming that for Proclus more geometrico means “in the manner of Euclid’s Elements (stoicheiōsis)”, the best place to start a proper assessment of the method as he sees it is his analysis of the title “Elements”.9 That analysis is part of a discussion of the two aims Proclus distinguishes in the work, one with reference to the subject matter (kata ta pragmata), namely constructing the five regular solids or “cosmic figures”, and one with reference to the student (kata ton manthanonta, in Eucl. 71.20–26):10
Of the aim of the work with reference to the student we shall say that it is to lay before him an elementary exposition (stoicheiōsis, as it is called) and a method of perfecting his thinking (teleiōsis tēs dianoias) for the whole of geometry. If we start from the elements, we shall be able to understand the other parts of this science; without the elements we cannot grasp its complexity, and the learning of the rest will be beyond us. The theorems that are simplest and most fundamental and most like first hypotheses are assembled here in a suitable order (taxis prepousa), and the demonstrations of other propositions take them as the most clearly known and proceed from them. … Such a treatise ought to be free of everything superfluous, for that is a hindrance to learning (mathēsis); the selections chosen must all be coherent and conducive to the end proposed, in order to be of the greatest usefulness for scientific understanding (epistēmē); it must devote great attention both to clarity and to conciseness, for what lacks these qualities confuses our thought (dianoia); it ought to aim at the comprehension of its theorems in a general form, for dividing one’s subject too minutely and reaching it by bits make knowledge (gnōsis) of it difficult to attain.
(in Eucl. 71.4–75.5, trans. Morrow, slightly modified)11
From this passage and the rest of the discussion, it is quite clear that the term stoicheiōsis should be understood as functioning primarily in a didactical context (D. O’Meara 1989: 171; Nikulin 2003: 195). It is this context which determines what a good stoicheiōsis is according to Proclus: any work which presents the basics of a science and trains our discursive intellect,12 and which provides us with the basic tools to further elaborate the parts of the science that are not described (cf. in Eucl. 71.22–3). It does so by bringing together in a suitable or proper order primary theorems and demonstrations from them.
The proper order, as Proclus finds it in Euclid’s Elements, follows a particular sense of stoicheion as the simplest thing into which the composite can be divided,13 where that which is “more of the nature of a principle” (archoeidestera, in Eucl. 73.8) is the element of that which is “placed (tetachmenōn) in the account of the result” (i.e. the conclusion, 73.8–9) – or, as he later describes it: “that from which everything else proceeds and into which it resolves” (analuetai, 73.18). Note that this latter aspect of resolution already hints at the presence of analysis in Proclus’ geometrical method (see below in this section); moreover, the starting points that have to be distinguished are the definitions/hypotheses,14 not the axioms and postulates.15 This shows from Proclus’ remark that a geometer has to distinguish between starting points and what follows from them, because geometry is a hypothetical science – as opposed to the only unhypothetical science, that is, metaphysics, which provides the principles to the other sciences (in Eucl. 75.5–14).16 This also implies, as we will see, that any Elements of metaphysics does not distinguish principles.17
It is difficult, says Proclus, to properly select and order the elements (in Eucl. 73.15–18). The proper ordering should not contain anything superfluous, should be coherent and to the point, clear and concise, and have a high level of universality in the theorems. These characteristics are emphatically each connected to a didactic aim (didaskalia) of the text: learning (mathēsis), scientific understanding (epistēmē), our thought (dianoia) and knowledge (gnōsis) respectively.18 All proofs should “result from the principles”, that is, use nothing other than the foregoing,19 and anything that seems to be missing either can be derived using the given methods, or is too complex for the present purpose, or can be composed on the basis of “given causes”.20 This implies that hidden assumptions, that is, assumptions that have not been explicitly stated as deriving from a definition or argument, should be avoided.
In sum, then, any Elements is a concise introductory teaching, which, for didactic reasons, consists of first principles (if any) and derivations (or deductions), presented in the proper order:21 any deduction should be made using only the principles and what has already been established.
So far, Proclus’ description of what makes a good stoicheiōsis matches modern descriptions of more geometrico quite well, except for the emphasis on didactics. Today the axiomatic method tends to be seen in a context of justification (Beierwaltes 2007: 82–3); that is, of showing how certain propositions are true because they are grounded by ultimate ones, which are considered as already known to be true.22 But that is not its primary aim for Proclus: grounding, as well as coherence, consistency, clarity, being to the point, conciseness and universality, are all subordinate to the didactic aim of the text.23
If anything, Proclus’ view is closer to the context of discovery sometimes related to the axiomatic method (De Jong & Betti 2010: 193). For Platonists, discovery should of course be understood in the Meno’s sense, that is, as anamnēsis, and hence as part of the didactic aim of science. And although it is not in the foreground in the description of what makes a good stoicheiōsis, it does play an important part in Proclus’ “geometrical method”: and this is where the analytic side comes in.
Proclus’ motivation for emphasizing the proper order is primarily epistemological: he who does not follow it upsets all cognition by putting together things that do not belong together (in Eucl. 75.23–5). Behind the epistemological motivation, however, lies the well-known assumption of ancient epistemologies that the structure of knowledge reflects that of reality.24 As a consequence, the order of presentation of knowledge should follow the order of reality: cause and immediate effect belonging together, but not cause and more remote effect. It is telling in this regard that Proclus, in one of the first pages of his first prologue, already speaks of the “principles” of mathematics from which everything else comes forth in the “proper order”, in the context not of epistemology, but of metaphysics:
To find the principles of mathematical being (ousia) as a whole, we must ascend (animen) to those all-pervading principles that generate everything from themselves: namely, the Limit and the Unlimited. For these, the two highest principles after the indescribable and utterly incomprehensible causation of the One, give rise to everything else, including mathematical being (physis). These principles bring forth all other things collectively and transcendentally, and the things which proceed receive their procession in appropriate degrees and in the proper order (metrois tois prosēkousi kai taxēi tēi prepousēi).
(in Eucl. 5.14–24, trans. Morrow, modified)
This ontological order is the very same order we have seen Proclus advocate in the presentation of knowledge.25 There is an important caveat to be made here, however, with regard to the didactical context of Elements. In a Platonic schema of anamnēsis, the direction in which the teacher takes the student is up, in an analytic movement from consequences to causes, from participations to Forms, from the many to the One, from composite to elements – not down, in a synthesis from the first principle to its consequences. As Proclus says in the quoted passage, “we must ascend”. Whenever that which is, to use an Aristotelian distinction, most clear and familiar by nature (physei) is not most clear and familiar with respect to us (hēmin, Phys. 1.1.184a16–18); that is, when the first principles in the order of reality are not the starting point or “the point of entry” in the order of knowledge, analysis leading to the first principles is necessary.26
It is significant in this context that in his discussion of the ordering of material in a stoicheiōsis, Proclus explicitly mentions reductio or proof per impossibile and analogic reasoning (in Eucl. 73.20–22). The fact that he notes the occasional omission of these two methods suggests that he considers them to be more or less standard components of the geometrical method. And indeed, in Proclus’ own works, as in Euclid, they are (see § “Methods of geometry” below). A reason for their separate mention might be that they are closely related, for Proclus, to the crucial analytic, ascending aspect of the method. When we turn to his own applications of the geometrical method, we see that, rather than (synthetically) starting from principles and moving to consequences, as Euclid does, they indeed commence with a short analytic move from principles to even higher principles (see §§ “Are Proclus’ Elements written more geometrico?” and “More geometrico in the commentaries” below).
METHODS OF GEOMETRY
To understand in what manner Proclus’ more geometrico includes the analytic road to principles as part of anamnēsis, let us take a closer look at the relevant methods of geometry. Proclus is hardly interested in the various operations on figures and ratios; that is, the constructive work (drawing a straight line between points, describing a circle, dividing and compounding figures, etc.).27 He considers the constructions a necessary, but inferior and preparatory part of geometry, more closely related to the perceptible than to the Forms, and better at home in productive, than in theoretical science.28 Instead, as a Platonist he prefers the theorems and their demonstrations concerning the essential properties of geometrical entities (in Eucl. 77.11–12).
The primary methods involved, then, are the dialectical methods needed for direct, syllogistic demonstration (in Barbara) on the basis of first principles: division of genera into species and definition “in the case of the first principles”, and demonstration “in the case of their consequences”, to show how the more complex proceeds from the simpler.29 But of course, Proclus also involves the fourth dialectical method, analysis, “to revert again to the [simpler]” (in Eucl. 57.25–6) – the method, that is, which shows how the (until then) unknown is actually “resolvable” into known principles, thereby affirming (thetikai) those principles.30 Analysis is the “upward” method; that is, the method that brings the student to knowledge of causes. As shown below, in his application and recognition of “the geometrical method”, Proclus tends to give analysis a special place and to emphasize other geometrical methods and formulae which contribute to or signal this upward direction (see §§ “Are Proclus’ Elements written more geometrico?” and “More geometrico in the commentaries” for examples).
Take, for example, the two methods Proclus mentioned as sometimes omitted from the presentation of Elements: indirect demonstration per impossibile (tropos di’ adynatou), and demonstration by analogia. The former starts from diaeresis, dividing the genus proposed for examination into its species, and eliminating the parts irrelevant for the proof, and then refutes the contrary of what is sought on the basis of its impossible consequences, indirectly establishing the truth of the thing sought (in Eucl. 211). Although it is as such not specified among the four dialectical methods, it is, of course, a very dialectical method in the sense that it is part of the actual practice of Socratic dialectic as the refutation of false opinions. It is analytic, to the extent that it proves the higher on the basis of the lower (cf. Heath 1956: 140).
The method of analogia, that is, demonstration using proportional relations, draws conclusions using the common notion “if four entities are proportional, they will also be proportional alternately”; that is, the alternation of equal ratios (A:B = C:D).31 The mathematical variety of this method is closely related to the metaphysical one (for Proclus they are the same method on different levels of reality) (the same goes for the literary variety: see Martijn 2010a: ch. V). In metaphysics, the proportional relations between cause and effect guarantee the continuity of emanation, and as a consequence, in epistemology, they guarantee the possibility of reversion of the soul.32
A related method is geometrical conversion (in Eucl. 69.19–24): the exchanging of (parts of) theorems (described at in Eucl. 251.24–259.14 & 409.1–17). Theorems consist of a hypothesis and a conclusion, in the sense of an antecedent and a consequent, for example, “if in a triangle two angles are equal, the sides which subtend the equal angles will also be equal”. This method is not a method of demonstration, but it is part of demonstration to the extent that it generates theorems, which can be demonstrated and used as premises in subsequent demonstrations. Geometrical conversion is a crucial element of Neoplatonic argumentation, because it is based on the fact that the relation between the subject in the antecedent and the attribute in the consequent is primary and per se (in modern terms: the propositions are actually double implications). As a consequence, and as opposed to logical conversion, it renders universal affirmative propositions (cf. Heath 1956: 256). This method is importantly related to analysis, to the extent that, for analysis to be possible, propositions need to be unconditionally convertible33 – one of the reasons, no doubt, why geometry has its paradigmatic role.
As mentioned above, even some at first sight superficial geometrical practices and formulae contribute to “the geometrical method”. Although they are not all considered necessary components of good Elements, Proclus does use them in his applications of the geometrical method, as well as when referring to it in the commentaries. Anticipating the results of the case-studies in the following sections, let me mention the most important ones. First is the division of the work into separate propositions and, within them, the division of the mathematical proposition into enunciation, setting-out, definition of goal, construction, proof and conclusion.34 Of the latter, most important for our purposes is the repetition or reformulation of the “enunciation” at the end of the demonstration, by way of conclusion, and the corresponding formula quod erat demonstrandum (hoper edei deixai etc.). For Proclus they indicate the two directions of geometry; that is, from starting point to end and back.35 Second is the use of imperatives signalling postulations, as a sign of the didactic or dialectical context;36 third, the corollary (porisma), that is, an extra result of the demonstration of another theorem, or a “lucky find”,37 as indicating the synthetic side of Elements and the completability of a system; fourth, variables as signs of the universality of the demonstrations, another requirement of good Elements;38 fifth, “a fortiori” (pollōi ara meizōn/pleon), a formula which Proclus identifies as originating in geometry (see below) and which refers to the method of demonstration by analogia:39 owing to analogic relations between layers of reality plus the principle that a cause has a property to a higher degree (cf. ET prop. 18), it is possible to ascend to causes on the basis of knowledge about the caused;40 and finally, “geometrical necessity” (geōmetrikai anankai): this is in itself not an aspect of methodology, but Proclus does connect it to the method of demonstration, and ascribes it to (the drawing of) conclusions. The source of the necessity differs from one context to the next: in in Eucl., it is said to be due to the subject matter of geometry,41 and in non-geometrical contexts, to the universality of demonstration (in Ti. I.346.31–347.1, see below, § “First case: physics and the commentary on the Timaeus”), or to the logical relation between premises and conclusion (see below, § “Second case: metaphysics and the commentary on the Parmenides”). These three features are interrelated, and are necessary conditions for any ascent to causes.
ARE PROCLUS’ ELEMENTS WRITTEN MORE GEOMETRICO?
First case: Elements of Theology
In his famous and influential introduction to metaphysics, entitled the Elements of Theology (Stoicheiōsis Theologikē), Proclus sets out his metaphysics in 211 propositions, each followed by what looks like a demonstration, or at least an elaboration supporting it.42 In roughly the first half of the work (props. 1–112), the fundamental oppositions of Neoplatonic metaphysics (between unity and plurality, cause and consequent, etc.) are introduced (see also Dodds 1963: x). In the second half, these oppositions are elaborated with regard to the ontological hierarchy consisting of the henads (props. 113–65), intellects (props. 166–83) and souls (props. 184–211). Within each of these subsections, the propositions are roughly ordered from the ontologically prior to the posterior.43 For example, the first proposition about the henads regards their overall unity, the last one regards the lowest, encosmic henads.
The title and form of the Elements of Theology indicate that the method of exposition Proclus chose for this work follows Euclid’s example. And, as said above in the introduction, this is how the work is traditionally viewed. On the other hand, that view has been challenged. On the face of it, the primary demands of a stoicheiōsis are met: the work starts from a principle – in a broad sense – and works its way down in the proper order. Moreover, within the argumentation for the propositions, we find the methods Proclus regards as standard (although not always used) components of geometrical Elements: indirect demonstration per impossibile is very frequently used (props. 1, 5, 22, etc.); the method of analogy for reasoning on the basis of the similarity of different levels of reality is both argued for and used.44 And finally, Proclus uses some of the above-mentioned primarily geometrical practices and formulae, repeating or reformulating the “enunciation” at the end of the demonstration by way of conclusion (e.g. prop. 1, 2, 7, etc.; the formula quod erat demonstrandum does not occur, but we do find comparable expressions at props. 6 and 25),45 and he often gives corollaries.46
Whether the ET contains anything superfluous, and is coherent, to the point, clear and concise, is to some extent a matter of taste. Compared to Proclus’ other works, however, the ET is a model of parsimony, to the great relief of Dodds (1963: xi). This shows most of all from the fact that it does not contain any appeal to philosophical or religious authorities. Proclus does not inform us what the work is about, or what his aim is (besides the telling title), but this is just another aspect of its conciseness. Proclus’ intention seems quite clear, since, as Lowry (1980: 90) says, “all the fat is trimmed away”. Likewise, although completeness is here a relative term, the ET seems at least intended to be complete – as an introduction to theology or metaphysics; that is, to true causes – in the sense that it discusses all transcendent levels of reality.
On the other hand, one can easily criticize the ET qua stoicheiōsis (Lohr 1986: 59–60). It has often been argued that the ET only ostensibly follows the Euclidean example. One of the main arguments adduced for this modification is the absence of definitions or other starting points in Proclus’ introduction of metaphysics (cf. D. O’Meara 1989: 197). For him, however, this would not be a convincing argument, since, as we saw, the separate treatment of starting points belongs to hypothetical sciences. But metaphysics is an unhypothetical science (see also Nikulin 2003: 200–201). Stronger arguments have been adduced, however. For example, the argumentations are not flawless: the “demonstrations” are quite often elaborate repetitions of the enunciation, phrases from Plato or Oracles are presented as a priori truths, and the work contains some logical errors (Dodds 1963: xi–ii). Second, “it is not obvious that there is a real dependency between [the propositions] as there is, for instance, between propositions of Euclid book I”.47 That dependency does exist in the sense that the lower hypostases, discussed in later propositions, metaphysically depend on earlier ones, but the concomitant logical dependence should be reflected in the argumentations, as it is in Euclid. And finally, the ET seems to violate the criterion of “proper order” in different ways. Most interesting is the fact that the first proposition, “Every manifold in some way participates in the one”, is not the most fundamental one (see also Gerson & Martijn forthcoming). On the basis of parallels with in Prm., it has been shown that not the first, but the fourth proposition is the foundation of all the others. The first four propositions form a short dialectical analysis, followed by a synthetic ordering of propositions derived from the first principle (D. O’Meara 2000: 285–88; Opsomer 2013). The first proposition can be read as a general characterization or even something resembling a definition of the manifold, as always containing some “one”, either in its parts considered separately or in the manifold as a whole. The subsequent argument “demonstrates” the first proposition by showing that “the ultimate constituents of any plurality … need to be atomic” (Opsomer 2013). Proclus then goes on to show that, since any manifold is both one and not one (prop. 2), it is different from, posterior to and dependent on oneness. Before there can be “many”, there has to be “one” (prop. 5). Moreover, this “one” has to be “the one itself” (prop. 4), which is separate from the thing that becomes “one” by participation, as well as from the “one” which it participates in (prop. 3). It is not until the fifth proposition that Proclus has concluded to the necessity of the One by itself, prior to a plurality of participated ones, on the basis of the general characterization of the manifold expressed in the first proposition. Thus the ET follows the geometrical method, but commences with a short analysis,48 based on what is technically speaking not a definition of “manifold”, although it does resemble one, and which functions as a “point of entry” or “cognitive pilot”, directing our thought to the un-hypothetical principle.49
Second case: Elements of Physics
The Elements of Physics, Proclus’ presentation of the necessity of an incorporeal unmoved mover on the basis of Aristotle’s physical works, is closer in form to Euclid than is the ET (cf. D. O’Meara 1989: 197). Apart from propositions, demonstrations using reductions (Nikulin 2003: 185) and corollaries, which we find also in the ET, the Elements of Physics moreover uses definitions, variables and formulae such as imperatives for postulations and the formula quod erat demonstrandum:50 a phrase Proclus otherwise uses only in his commentary on Euclid.
On the other hand, it is less of a stoicheiōsis, as it does not obey the criteria of “proper order” in two respects: (a) it is not complete, since, contrary to what its title promises, it does not treat all of physics, but only certain aspects of motion and change (D. O’Meara 1989: 197). More damaging is that it could nonetheless be complete regarding its apparent aim, that is, showing the necessity of an unmoved mover, but it is not, since: (b) like the ET it contains hidden premises.51
Let us, however, again consider Proclus’ supposed intention and tentatively put forward a thesis the confirmation of which actually requires detailed analysis of the Elements of Physics. Since its demonstrations lead upwards from bodies in motion to knowledge of the cause of corporeal motion, what the introduction of physics gives us is a version of the geometrical method, which is primarily analytic.52
Proclus’ own Elements, then, are both written more geometrico, but this does not mean that they obey all criteria, nor does the method come down to the same in both cases. The latter goes also for the geometrical method as it occurs in the Platonic commentaries. There is not one “geometrical method”, but its details depend on the science or subject matter at hand.
MORE GEOMETRICO IN THE COMMENTARIES
In commenting on Plato’s text, Proclus often imposes a structure on it, which we would qualify as logical. In some dialogues, however, he himself identifies it as a geometrical structure, or describes elements of it in terms borrowed from Euclid (we will only look at explicit references to geometry in connection with methodology). The extent to which he does this differs from one commentary to the next, depending on the given structure and subject matter of the dialogue in question, and ranging from an occasional remark to an extensive “elementification”.
First case: physics and the commentary on the Timaeus
On one extreme we find the commentary on the Timaeus in which, more than in any other commentary, Proclus labels the method applied by Plato as “geometrical”.53 There are obvious reasons for the strong presence of the geometrical method in this commentary. First of all, the main speaker of the dialogue, Timaeus, is a Pythagorean, so it is no surprise to find Proclus emphasizing the presence of mathematics in its various guises in the commentary. Second, there is a clinching argument for Proclus to think of geometry as a methodological paradigm in the context of the Timaeus, and that is that philosophy of nature, like geometry (but unlike dialectic), is a hypothetical science (Martijn 2010a: 90–99).
The geometrical method is very present in this commentary, but only in a specific part of it: not the comments on the passages of the dialogue which already contain geometry (Ti. 31b–32c and 34b–36b),54 but those on the prooemium (Ti. 27d5–29d3), in which Timaeus sets out the principles of his cosmology. The prooemium seems already to have a “geometrical” (in the sense of hypothetico-deductive) structure (Finkelberg 1996; Runia 1997), but Proclus takes it to extremes: to answer the general problem of cosmology (whether the universe is “becoming”), Plato, says Proclus, presents a kind of division, “a distinction of two genera (Being and Becoming), in order for the exposition to proceed as from geometrical hypotheses to the examination of the consequents” (in Ti. I.226.24–7); he “defines the always Being, assuming that it exists, as the geometer defines the point, assuming its existence”, because “[philosophy of nature] is also a hypothetical science and its hypotheses should be assumed before the demonstrations” (in Ti. I.228.26–229.3, cf. in Ti. I.236.15, 30), and “the geometer would no longer be a geometer if he discussed his starting points” (in Ti. I.236.33); he then “truly according to the geometrical method presents … axioms after the definitions” (in Ti. I.258.12–13), introducing the efficient and paradigmatic causes of Becoming; he presents another axiom which “imposes a name on the subject in a geometrical fashion”: that is, “the heavens or the cosmos” (in Ti. I.272.10–11); he shows “demonstratively that the cosmos has become, from the definition, according to the conversion of the definition: for geometers also use such proofs” (in Ti. I.283.15–18); and subsequently demonstrates, among others with a reductio, the necessity of the demiurgic and paradigmatic causes of the cosmos; finally, in what Proclus with hesitation calls “the fourth demonstration”, he recognizes analogic reasoning, namely “geometrically adding the alternate ‘as Being is to Becoming, so truth is to belief’” (in Ti. I.345.3–4).55 Definitions, hypotheses, axioms, demonstrations, conversion, proof per impossibile and analogy, just as in Euclid:56 the intention is clear.
An important additional feature of Proclus’ exegesis of the prooemium is that, in the presentation of the “axioms”, Proclus discerns a short analytic ascent to the causes of Becoming (Ti. 28ab; in Ti. I.258.9–272.6). Elaborating the axioms into a number of syllogisms (also involving reductio), he shows that on the basis of the definition of Becoming one has to conclude the necessity of the demiurgic and paradigmatic causes.57 As in the ET, then, the “geometrical method” involves a short analysis leading to the principles of the science being presented. This analysis fits Proclus’ view of the Timaeus as a “teaching” (didaskalia) (see Martijn 2010a: esp. 84–5 & ch. V).
The harvest of the analysis is that the universe as “most beautiful object” was made after a “most divine paradigm” by “the best maker”, that is, the demiurgic and paradigmatic causes of Becoming. Proclus buttresses this harvest using a number of reductiones, to conclude with yet another reference to geometry: “let these … statements be laid down as demonstrated by ‘geometrical necessity’, as they say”.58 The geometrical necessity of the conclusions, which Proclus explains as due to their universality and the intelligibility of the transcendent (in Ti. I.346.31–347.1), highlights what he considers the summit of the Timaeus.
Second case: metaphysics and the commentary on the Parmenides
For Proclus, the Parmenides is Plato’s only entirely theological or metaphysical dialogue, and as a consequence the only completely dialectical one (Theol. Plat. I.7–10, cf. I.4.18.20–24). When we run into demonstrations, divisions, analyses or definitions, we need not think of geometry, as these are primarily dialectical methods.59 Any references to the geometrical method may be no more than metaphors for logical certainty. For example, Proclus first introduces geometry at a crucial point, namely in the description of the method of the Parmenides: “The method of the arguments proceeds through the most precise powers of reason, … firmly establishing each of the matters at hand ‘by geometrical necessity’ (geōmetrikais anankais, in Prm. 645.11–16). Perhaps Proclus here merely mentions geometry, rather than logic, to avoid giving the impression that the Parmenides concerns logic: instead, he takes it to be the metaphysical dialogue which derives all of reality from a first principle, the One.60
Other references to geometry, however, show that there is more to it. The second mention does not occur until the second half of the dialogue, in Proclus’ comments on what he considers the final stage of analysis, after the maieutic first half of the dialogue: the first hypothesis. From Parmenides’ One-Being his audience ascends to the One itself, using a common notion (the One is not many, functioning as what we called a “cognitive pilot”), as in geometry.61 Later references concern the presence of self-evident starting points and the subsequent synthetic order of reasoning in Parmenides’ investigation of the hypotheses.62
Furthermore, the expression “by geometrical necessity” is used on two more occasions, to emphasize that the taxis of demonstration; that is, posterior following from prior, imitates the ontological hierarchy (in Prm. 1132.22–5, 1162.22). Interestingly, the very same point is made in the one and only reference to the geometrical method in the Platonic Theology, concerning the method of the Parmenides: by his mode of demonstration (ho tropos tōn apodeixeōn), says Proclus, that is, using the least possible number of simplest and best-known starting points, or common notions, to derive an increasingly complex plurality of conclusions, Parmenides provides an intellectual paradigm (paradeigma noeron)63 of the order of geometry or the other mathēmata.64 This order of reasoning is “necessary” (anankē), because of a necessary (anankē) correspondence between the order of demonstration and the causal order of reality: logoi carry an image “of the things which they interpret” (Ti. 29b4–5).65 To conclude, the main two purposes of the geometrical method in this commentary seem to be (a) to emphasize the analysis leading to the first principle and (b) to emphasize the subsequent metaphysical aim of the dialogue; that is, deriving lower levels of reality from the One, synthetically presented according to the order of reality.
Third case: ethics and the commentary on Alcibiades I
The commentaries in which the geometrical method plays no part are those on Republic and Alcibiades. Although Proclus analyses parts of both dialogues almost ad nauseam in terms of syllogistic reasoning – using reductions, syllogisms in Barbara, definitions, axioms or common notions, conversions, necessary conclusions, which are then used as premises for further deductions, corollaries and expressions such as a fortiori – he either never (in R.) or hardly (in Alc.) mentions geometry.66 In R. does not contain any reference to the geometrical method. This absence is surprising, considering the importance of geometry and its methods in the Republic. Likewise, it is interesting that in in Alc. we find only some references in passing. For our purposes, some brief comments on in Alc. suffice.
For Proclus, the primary aim of the Alcibiades, self-knowledge, is ethical, and the dialogue concerns the refutation of false opinions. It does not contain an orderly presentation of reality – with one exception (see below). The method suitable for the ethical aim, Socratic dialectic (in Alc. 169.12–171.5), is of course similar to that of geometry – it involves analysis and synthesis (in Alc. 179.14–180.2), as well as common notions and something like postulates (premises agreed upon with the interlocutor) (in Alc. 175.24–176.6) – and elsewhere, as we saw, Proclus explicitly compares lower dialectic, using hypothetical starting points and scrutinizing their consequences, to geometry (see note 62). Not, however, in the Alc. Perhaps this is because for Proclus the Alcibiades is not only an ethical dialogue, but also an introduction to all of philosophy: ethics, (meta)physics and logic (in Alc. 11.3–14). So naturally, Proclus uses the comments on methodology to emphasize its contribution to logic. For example, when presenting syllogisms in Barbara (which, incidentally, contribute to the metaphysical content of the dialogue), for example, “everything just is beautiful, everything beautiful is good, therefore everything just is good”, he explains what first-figure syllogisms are (in Alc. 318.16–319.1).67
Nonetheless, there are two interesting references to geometry as a methodological paradigm. The first, quoting Aristotle, emphasizes the paradigmatic necessity of geometry (cf. in Ti. and in Prm.), but as opposed to ethics. Necessity is here related not to the method of demonstration itself, but to the knowability of the subject matter:68 Socrates’oimai, “I think” (Alc. 103a), does not express ignorance, but awareness of the lack of necessity concerning knowledge of human affairs (or more specifically “other minds”). This confirms our suspicion that the ethical aim of the dialogue in part explains the absence of the geometrical method: ethics is not a demonstrative science.
Elsewhere in the commentary, we find the remark “a fortiori, as the geometers would say”.69 The expression a fortiori refers to the power of analogy, allowing us to draw conclusions concerning higher levels of reality on the basis of knowledge of the lower. Proclus uses “a fortiori” quite frequently, but only in in Alc. is it explicitly called geometrical. Perhaps that is simply due to the introductory role of the Alcibiades in the Neoplatonic curriculum: Proclus may not expect his readers to make the connection. On the other hand, this remark comes at a crucial point: it is attached to what is the metaphysical core of the dialogue for Proclus, and the only presentation of the order of reality it contains, namely the proportional relation “base lover : divine lover = nature : soul”.
CONCLUSION
We can conclude from the above that there is not one “geometrical method” in Proclus. Different sciences or contexts ask for different varieties of a method that derives conclusions from the relevant kinds of starting points. Moreover, there is far more to the method than starting points and derivations. The main aim of the method, according to Proclus, is didactic. As a consequence, it contains not only synthetic, but also analytic elements. And the metaphysical background of the geometrical method requires that the order of presentation of knowledge corresponds to the causal order of reality.
Geometry provides an ideal methodological paradigm, better than either dialectic or empty logic (in Cra. 1.11), because of the geometry’s intermediate position in between the sensible and the intelligible.70 It combines a rigid structure reflecting the order of reality with the power of reversion to higher levels of reality. Rosán rightly sees Proclus’ focus on the geometrical method as motivated by “the sense of beauty that is inherent in the absolute necessity of a rigorous and all-pervading system”.71 But the system itself is not the be-all and end-all of the geometrical method: it is what it represents, and what you can do with it.
NOTES
1. G. E. R. Lloyd (2005: 118), on “the geometrical method” in Galen.
2. A. E. Taylor (1918: 606–7). Cf. Oeing-Hanhoff (1971: 233), who explains the mos geometricus as “mathematisch-synthetische Methode”.
3. This section owes a lot to D. O’Meara (1989: 177–9, 196–8).
4. In the following, “analysis” does not refer to geometrical analysis. On Proclus’ ignoring of Euclid’s application of that method in structuring a proof, see Netz (1999: 293–4). On the notion of “analysis” relevant in this chapter, see below, note 26.
5. This is also clear from O’Meara’s excellent analysis of the role of geometry in Proclus’ work (D. O’Meara 1989), to which this chapter owes a lot.
6. Note that Hathaway (1982) quite unconvincingly maintains that Proclus unconsciously adopted Euclid’s method. Cleary (2000a: 88) takes the ET to have Aristotle, not Euclid, as its model.
7. I use “science” here in the sense of “grounded knowledge” or “scientific understanding” (i.e. episteme), not in the narrow sense of one of the “natural sciences”. On the classical model of science in Proclus and in general see Martijn (2010c), De Jong & Betti (2010), Betti et al. (2011).
8. On Iamblichus as a possible source of Proclus and the difference between the two in the relation between the methods of geometry and those of dialectic, see D. O’Meara (1989: 170, 172–3). On Syrianus, see Ierodiakonou (2009b).
9. In Eucl., second prologue, chapter 7. On Proclus’ commentary on Euclid, see further Szabó (1965), Breton (1969), D. O’Meara (1988), Bechtle & O’Meara (2000), Helbig (2000), Nikulin (2002), Bechtle (2006), Lernould (2010). I thank Jan Opsomer for his valuable suggestions concerning components of the geometrical method.
10. At the end of the commentary on book I, Proclus seems to have reduced the aim of the book to the former (in Eucl. 431.15–432.19, esp. 432.5–9). Possibly, at the end of the book, Proclus is less interested in the method used and more in the subject matter, or perhaps he changed his mind with regard to the primary aim of the Elements, but that does not affect his views on the nature of the method used.
11. On this passage, see also Nikulin (2003).
12. Dianoia has to refer to discursive intellect, rather than “understanding” (Morrow).
13. Morrow’s translation here seems a bit sloppy. The focus of Proclus’ analysis slides from the order of presentation of knowledge (foundation vs. derived knowledge) to the order of ontological composition (elements vs. composite), possibly because of an influence of Aristotle’s definition of stoicheion in Met.1014a.26–1014b.15. For other references to Aristotle, see Heath (1956: 116).
14. On the relation between these two in in Eucl., see Martijn (2010a: 91).
15. For reasons of space this chapter does not contain a discussion of the role of the axioms or common notions, i.e. the self-evident starting points of geometry, in the geometrical method. For examples see below §§ “First case: physics and the commentary on the Timaeus” and “Second case: metaphysics and the commentary on the Parmenides”.
16. This impression is reinforced by the remark further on that Euclid had to distinguish (ἔδει διαστείλασθαι, in Eucl. 75.27) starting points from consequences, and that he does just that, and “thereafter also” (επειτα καί, in Eucl. 76.4) distinguishes between axioms, hypotheses/definitions and postulates – but not that he had to.
17. Metaphysics does not have any (discursive) starting points, cf. in Prm. 702.9–11.
18. This seems to me a case of variatio for purposes of emphasis, rather than a meaningful distinction of different forms of cognition resulting from different aspects of the method.
19. Proclus criticizes Apollonius several times because the proofs he presents involve later theorems, which makes them unsuitable for an introduction (ἀλλοτρίαν … τῆς στοιχειώσεως), in Eucl. 335.16–336.8, 280.4–11 & 282.24–283.3. Note that Proclus often refers to Euclid as “the author of the Elements” when he discusses the proper order of the work. An interesting case in point is in Eucl.193.1–9 (with in Eucl. 339.11–344.8): Proclus rejects the indemonstrability of the fifth postulate (and hence its status as a postulate), but postpones talk about its demonstration until the time is right (Morrow refers to in Eucl. 364.13 & 371.10). Cf. in Eucl.162.8, 248.4, 310.17, 321.9–20, 326.12, 334.3.
20. αἰτίων τῶν δεδομένων, i.e. probably principles, rather than Morrow’s “from traditional premises”. The disjunction seems to be non-exclusive, as the three examples apparently overlap.
21. In Eucl. 75.10–14. We find these elements also in other writers using the term stoicheiōsis, albeit not all at once. Suda mentions introductory teaching, order and conciseness (Σ 1241 = Photius, Lexicon 540 [Porson]); Porphyry also mentions introductory teaching (in Cat. 134.28 [Busse]); and Pseudo-Galen focuses on the order “from elements to end”, that is, from the principles to what is sought (Definitiones Medicae 356.4–15 [Kühn]). Cf. Epicurus, Her. 37 [Arrighetti]. Thanks to Jan Opsomer for pointing me to these passages.
22. Cf. De Jong & Betti (2010: 193–4); Betti et al. (2011: 2). Alternatively, justification may be related to coherence and consistency of the subject, cf. Nikulin (2003: 193–5).
23. Cf. D. O’Meara (1989: 171), who speaks of “pedagogical virtues”. Note that these are properties which Proclus even tries to give to his own commentary: see in Eucl. 200.11–18. Cf. in Eucl. 429.9–15.
24. On this and a related issue in Proclus, that texts reflect reality, see Martijn (2010a: chapter V).
25. The same parallel is present in Proclus’ reading of Platonic dialogues. See Martijn (2010a: 276–80).
26. Analysis is too complex a notion to do it justice here. Let me merely point out that although Proclus’ concept of analysis in in Eucl. is far from precise, context allows us to choose the most relevant notion from the many ancient varieties. Beaney (2003) divides analysis into three main varieties (with many subspecies): decomposition of a complex into simples, regression or working back to first principles, and transformation/interpretation of a statement to a correct logical form; Oeing-Hanhoff (1971) distinguishes, among the ancient varieties, mathematical analysis, practical analysis of an aim to its means of realization, logical analysis into figure and mode, analysis of conclusions to their premises, conceptual analysis, material analysis, grammatical analysis, erotic analysis and astronomical analysis. Proclus clearly has in mind regression, or analysis of conclusions to their premises, which for him also means analysis of effects to their causes. For further references on analysis in Proclus, see Martijn (2010a: 133–4). Note that the above oversimplifies the relation between reality and knowledge in Proclus. See also Martijn (2010b: 159).
27. See, for example, the treatment of Post. I–III at 185.1–187.28, and the description of geometrical procedures at 57.9–58.3.
28. Constructions are needed in the “problems”, which are inferior to the theorems. See in Eucl. 209.10, 243.13–25, 79.3–11.
29. In Eucl. 57.18–26. On the dialectical methods in mathematics in general, see in Eucl. 42–3; on analysis and synthesis see moreover in Eucl. 8, 255; on division and collection, in Eucl. 12–15, 19, 318; on definitions, in Eucl. 131, 178; on demonstration, 33–4. See also D. O’Meara (1989: 171–3).
30. Cf. in Eucl. 211.19–21, 242–3, 255.18. See also Beierwaltes (1979: 250 and note 20, 262). Analysis is also one of the methods for finding lemmata (propositions which are used to establish other propositions, but themselves need to be demonstrated), next to division and something resembling anginoia (cf. Aristotle, An. Post. I, 34) – intuition, which is not really a method, but a “talent” (ἐπιτηδειότης), or “nature” (οὐ μεθόδοις … τῇ φύσει, in Eucl. 211), and consists in “seeing” on the basis of as few principles as possible. This is the primary task, not of dianoia, discursive thinking, but of its summit, logos or a kind of nous (e.g. in Ti. I.246.20–248.6).
31. For Proclus’ explanation of the method, see in Eucl. 254.21–256.8. Analogia does not occur in the in Eucl. often, as it is not introduced by Euclid until book V (and applied in plane geometry in book VI). Cf. in Eucl. 9.4 for the common notion and in Eucl. 405.1–406.9 for an application.
32. Note that the method can go both up and down the ontological ladder. On analogia in Proclus, see the insightful discussions in Gersh (1973: 83–90) and Beierwaltes (1979: 65, 153–7).
33. Heath (1956: 139). On related issues, see further de Haas (2011) and Harari (2008).
34. On this feature see Netz (1999), who shows that Proclus imposes this structure on Euclid.
35. On enunciation and conclusion, see in Eucl. 203.1–210.25, esp. 210.5–16.
36. For example, (ek/pros-)keisthō, as at in Eucl. 190.6, 274.10, etc.
37. Different from porisma in another sense, namely of a problem whose solution requires not only construction or theory but also discovery: see in Eucl. 212.12–17, 301.22–305.16.
38. On logical variables as a sign of universality, invented by Aristotle, see Lukasiewicz (1951: 7–8).
39. Cf. Cleary (2000a: 77). Damascius uses the expression in this manner quite often, e.g. Pr. I.58.15 [Westerink & Combès].
40. Another formulaic element, which we will not go into, is the use of the second person. A practice of geometry that we do not find in any of Proclus’ texts is the use of diagrams.
41. Cf. in Eucl. 26.10–27.10. Cf. in Alc., see below, § “Third case: ethics and the commentary on Alcibiades I”.
42. On the Elements of Theology as an exposition more geometrico see also D. O’Meara (2000).
43. In the manuscripts, the propositions are distinguished, but not called propositions. The division into subsections is Dodds’, not Proclus’, but it is justified by the content.
44. What Dodds calls the “supplementary theorems on causality” (esp. props. 97, 108 & 110) describe the metaphysical basis for the method. Many other propositions use it (explicitly, e.g. props. 113 & 164, cf. 100, 139, or more often implicitly, e.g. 105, 155, 197). Note that in general the analogies do not move up, but down the ontological ladder, fitting the overall direction of the ET.
45. As such a non-geometrical feature of ET for D. O’Meara (2000: 286), but not for Proclus.
46. More precisely, he uses formulations which elsewhere signify corollaries (e.g. “From these things it is also clear that …”, props. 14, 22, etc., cf. prop. 26, where he merely adds a further conclusion).
47. Cleary (2000a: 68). He is referring to logical, not ontological, dependency.
48. Another seeming violation of the “proper order” is noted by Lowry (1980: 94): Neoplatonic metaphysics is not just a hierarchy of hypostases, but it also involves metaphysical processes responsible for the relations existing within the hierarchy. These processes are described in props. 7–112. Against Lowry I would say that since these processes are in a sense caused by the first principle(s), the proper order is largely maintained.
49. For a similar case of such “cognitive pilots”, cf. Martijn (2010a: chapter III, esp. 106).
50. D. O’Meara (1989: 177–8). The formula ὅπερ ἔδει δεῖξαι occurs ten times, nine in book I (cf. ὅπερ ἔδει ποιῆσαι, Inst. Phys. I.9 [Ritzenfeld]), and the tenth as the very last sentence of the work, to emphasize the grand conclusion that there is an unmoved mover.
51. Opsomer (2009: 193–203). Nikulin (2003: 195) mentions that, as in the ET, Proclus does not distinguish different kinds of starting points (definitions, axioms, postulates). Such a distinction, however, is not among Proclus’ criteria of stoicheiōsis.
52. Thus, pace Martijn (2010a: 216) and D. O’Meara (1989: 177–9), the mos geometricus here does not only serve “to reinforce Aristotle’s argumentation by imposing a syllogistic rigour”.
53. On this topic, see also Lernould (2001, 2011), Martijn (2010a: ch. 3).
54. Proclus’ book III on body and soul of the world respectively. On these passages, see also Martijn (2010a: 166–204). Proclus’ comments on Ti. 53b–55c (on the regular polyhedra) are not extant.
55. On this last passage, see Martijn (2010a: 255–61). Cf. in Ti. II.13.8.
56. On references to “geometrical necessity”, see below.
57. On the syllogistic structure of this passage, see Martijn (2010a: 115–32); Lernould (2001, 2011). On the analytic aspect, see Martijn (2010a: 118, 123, 131–2).
58. in Ti. I.331.29–332.9. The reference is to R. 458d5, where we find geometrical vs. erotic necessity.
59. Theol. Plat. I.9.40.1–18. See also Proclus’ explanation of the dialectical methods (in the context of an argument for the necessity of the Forms) at in Prm. 980–83 [Steel].
60. On the polemics concerning the aim of the Parmenides (logical exercise or theology), see Theol. Plat. I.9–10 (34–46) and in Prm. 630.11–645.6.
61. In Prm. 1092.21–1094.13 with 1079.4–20. See also D. O’Meara (2000: 282–3).
62. In Prm. 1099.29–1100.8, 1140.16–18, 1151.28–1152.11, cf. 1195.21–1960.1, 1206.1–25. The description of lower dialectic contains a rather different reference to geometry: like geometry, Zeno’s lower dialectic has hypothetical starting points (in Prm. 701.21–702.11), as opposed, of course, to the non-hypothetical nature of Parmenides’ higher dialectic (in Prm. 1033.30–1034.29, re. R. 533b6c).
63. “Intellectual paradigm” probably refers to the dialectical order as discursive yet rooted in intellect (nous), owing to the intuition of principles. Cf. in Prm. 701.24–6, see also above.
64. Theol. Plat. I.10 45.19–46.2. On this passage, see also A. C. Lloyd (1990: 16–17) and Cleary (2000a).
65. Theol. Plat. I.10 46.2–22, cf. above, § “The elements of Elements”.
66. See in Alc. 11.18–18.10, esp. 15.1–18.10, in R. beginnings of essays 3, 4, 13, 15, 17 and considering its title probably also 2, which is not extant.
67. In the preceding pages, Proclus emphasized that the dialogue contains ethics, theology “and all of philosophy, so to speak” (in Alc. 315.1–318.15).
68. In Alc. 23.8, quoting Aristotle, EN 1094b26 and Plato, Ti. 29b3–d3.
69. In Alc. 134.15. He also identifies a “geometrical corollary” (in Alc. 217.8). Cf. the formula at in Alc. 216.21, similar to that introducing corollaries in ET.
70. Partially explained by Nikulin (2003: 195). See further Cleary (2000b).
71. Rosán (1949: 227). On the beauty of mathematics, see Martijn (2010b). On Proclus as a systematic philosopher, see Beierwaltes (2007), Gerson & Martijn (forthcoming).