La Philosophie des Mathématiques de l’Antiquité Tardive

No student of Platonism would deny the central position which mathematics holds in Platonic philosophy. Whether we look to the Meno, the Phaedo, or the Republic, it is clear that the method of Platonic dialectic was, in some very important ways, based upon the analogy of mathematics.¹ And, if even today we debate the precise details of how mathematics informs Platonism, it is no surprise that some of Plato’s earlier readers, members of the Neoplatonic schools of Late Antiquity, were very much concerned with similar questions. The essays collected in this volume exhibit many of the reasons why Neoplatonism is useful not only as a branch of study in its own right but also as a tool for aiding readers of Plato, Aristotle and modern thinkers about mathematics.

The work originated in a conference held at the University of Fribourg (September 24-26, 1998), which took as its basis two texts, Iamblichus’ De communi mathematica scientia and the first prologue of Proclus’ In primum Euclidis Elementorum librum Comentarii.² As the revised acta of that colloquium, the editors have assembled ten contributions, arranged chronologically by topic, beginning with studies of Iamblichus and culminating with an examination of Proclus’ relationship to modern mathematical thought. Thus the volume offers a selective treatment of Late Antiquity’s engagement with both mathematics and philosophy. It includes attempts to locate the (intermediate) ontological status of mathematical objects in Neoplatonic metaphysics (as found in the works of Iamblichus, Syrianus, Proclus and John Philoponus) as well as a description of the sixteenth century reception of Proclus’ commentary on the first book of Euclid’s Elements.

The first essay by Francesco Romano takes the earlier work of D. J. O’Meara, Pythagoras Revived, as a springboard for discussing the role of mathematics within Iamblichus’ larger project of unifying all the sciences ( ἐπιστῆμαι).³ Romano suggests that — beyond the decisive Pythagoreanization of Platonism which it (according to O’Meara) accomplished — Iamblichus’ work attempts to establish his Pythagoreanizing Platonism as a unifying basis for all the sciences and, moreover, that mathematics (particularly arithmetic) occupies a central role in this movement. To prove his case, Romano considers primarily the De communi mathematica scientia, though he also examines, to some degree, both Iamblichus’ educational history and O’ Meara’s reconstruction of the lost Iamblichean work On Pythagoreanism. On the whole, the essay — if unable to persuade indubitably — is highly suggestive in its implications for an understanding of what a “liberal education” meant to Iamblichus (and perhaps others in Late Antiquity): if Romano is correct, mathematics was the foundation of an Iamblichean “liberal education.” Particularly interesting in this regard is Romano’s treatment of the titles of the lost chapters (V, VI, and VII) of On Pythagoreanism. He understands these titles to indicate a substantial theoretical application (sérieuse théorisation) of the fact that numbers, for Iamblichus, constitute the essence of things in an ontological sense; thus, as ontologically prior entities, numbers (and their study) claim a very real disciplinary priority. As we do not possess these chapters of On Pythagoreanism, it may be objected that Romano has entered a realm of pure speculation, but his proposal seems very fruitful for trying to gain a broader and more detailed understanding of Pythagoreanizing Platonism.

Gerald Bechtle’s work, the second essay in the volume, examines the fate of the dialectical methods (division, definition, demonstration, and analysis) in the Iamblichean corpus, with specific attention to the De communi mathematica scientia. Starting from the treatment of these dialectical methods in Alcinous’ Didascalicus, Bechtle constructs a compelling account of how, historically speaking, they attained the form which they have in Iamblichus. According to Bechtle, the Iamblichean dialectical methods stand in some manner of analogy (Übertragung) to their counterparts, the mathematical methods, though, significantly, these two sets of reasoning tools remain independent (unabhängig) of one another (much more clearly than is the case with Proclus, for example). Important in this connection is the manner in which the mathematical methods are focused upon mathematical (i.e. dianoetic) forms as ontological intermediaries between forms-in-matter (enhyletisch) and intellectual forms (the objects of dialectic). Though there is far more to Bechtle’s treatment than may be adequately addressed in this review, his use of Alcinous deserves special note because it has potentially broad-ranging implications for future studies of Neoplatonic thought. In describing the historical background upon which Iamblichus’ dialectical methods are constructed, Bechtle offers an insightful way of understanding how the Neoplatonists read Plato: he suggests that we should consider the way Alcinous reads Plato in conjunction with Aristotelian ideas as similar to the manner in which a modern literary critic produces a structuralist reading of a given work. In other words, there is a basis of Platonic thought which is always assumed for the Neoplatonists, but, in order to reach the underlying significance of a given passage of Plato, it may be useful for them to employ other theoretical approaches. This explains why many Neoplatonic authors frequently appear (falsely) “eclectic” in their use of philosophical ideas and terminology. It would be interesting, then, to see whether Bechtle (or others) could extend this same sort of interpretation to further Neoplatonic texts (such as Enneads IV.8.1 [6]: what is the reading of the Presocratics implied by this series of ἔνδοξα ?) and whether one could develop more specifically (even if only for a given author) the ways in which a given theoretical approach may be employed (and whether there are identifiable interpretative trends at given points in the Neoplatonic tradition).

Third in the collection is an essay by Linda M. Napolitano Valditara which considers the eighth chapter of the De communi mathematica scientia as a piece in a hypothetical history of readings of the so-called divided line passage of Plato’s Republic (509 D 6-511 E 5). Napolitano suggests that mathematical objects may be independent, ontological entities (i.e. objects of διάνοια not related to the forms) in the divided line passage. This reading, as Napolitano convincingly shows, is what Iamblichus advocates, what a tradition descending from Aristotle supports and perhaps even what Plato himself intended. Napolitano’s methodology and her treatment of two little understood fragments of (pseudo-) Brotinus and (pseudo-) Architas offer a way of breathing new life into a Platonic question which has been discussed to near exhaustion.

Taking as its point of departure a few of the same texts which Napolitano considers (e.g. the divided line passage and Aristotle Metaphysics 987b14ff.), Ian Mueller’s contribution, the fourth in the volume, treats the convoluted question of the concept of mathematical number in Syrianus (with some help from Proclean texts). Mueller’s discussion indicates that the later Platonists (in this case, particularly Syrianus) understood mathematical (arithmetical) numbers to depart from the Euclidean conception of number as a multiplicity of monads, instead believing them, in the first instance, to be forms ( λόγοι). Although Mueller’s discussion is compelling as far as it goes, it leaves many difficult questions relatively untouched: Mueller, for example, points to the fact that, whereas for Euclid geometrical objects and mathematical number are essentially (in an ontological sense) the same, the later Platonists consider mathematical number ontologically superior. Thus, we find this puzzling distinction geometry-imagination versus arithmetic- δόξα to which Mueller refers (p.73), but we find no thoroughly satisfying treatment in the extant scholarly literature. The point here is not to criticize Mueller but to note how painfully apparent the need is for further study in the area: before we can achieve a more comprehensive understanding of the concept of mathematical number in the later Neoplatonists we need a fuller understanding of their ontology in general.

John J. Cleary, in the fifth essay of the volume, offers a discussion which in many ways covers similar ground to Mueller’s preceding piece: both are interested in understanding the ontological status of mathematical objects in later Neoplatonism. Cleary, however, as he indicates at the outset of his work, has a somewhat broader project in mind: he wants to situate the study of mathematics both within its historical (inherited problem) context and within Proclus’ larger philosophical project of the systematization of theology. Obviously, such ambitious aims can only be met partially in a piece spanning seventeen pages. Cleary’s essay, nonetheless, is quite rewarding for its ability to clearly link diverse elements in Proclus’ philosophy: by its conclusion we arrive at a very lucid, large-scale account of the role of geometry for Proclus, which suggests that geometry may function as a tool to allow us to consider spatial figures in a non-spatial fashion (a manner thus closer to the forms). Hence, on Cleary’s reading, by using the understanding ( διάνοια) to project images from the form in understanding ( τὸ διανοητὸν εἶδος) into the imagination ( παθητικὸς νοῦς), we can glimpse elements latent in the forms if not the forms themselves. Moreover, while providing this very specific description of the function of geometry, Cleary continues to remind us of the way in which this conception fits into a larger Proclean schema: we thus understand that mathematics (and specifically the treatment of Euclid’s Elements) in Proclus serves as a double for the complex (and perhaps still more elusive) question of being (including the interplay between the One and the indefinite Many) and that, consequently, it retains a goal higher than its own study.

Sixth in the collection we find an essay which departs sharply from the earlier works: abandoning ancient philosophy for modern mathematics (and its philosophy), Markus Schmitz advances the very controversial thesis that Euclid provided us with the only correct form of plane geometry. He argues not that non-Euclidean geometries are logically inconsistent but that they are best considered as (Euclidean) geometries of curved surfaces (e.g. manifolds) and not of planes. The key point in Schmitz’ argument is his treatment of the Tarskian classical correspondence theory of truth in modern mathematical philosophy: he contends that this conception of truth (i.e. “The truth of a sentence consists in its agreement with (or correspondence to) reality.”⁴) may essentially be reduced to the Stoic καταληπτικὴ φαντασία insofar as it is a theory based upon the identification of an internal representation (Vorstellung) with an external object (Gegenstand). He argues, moreover, that Euclidean geometry is incomprehensible on this understanding of truth. To remedy this dilemma, he suggests an alternative definition of truth (drawn from a statement of Thomas Aquinas) which does not rely upon correspondence between some proposition and an external reality but instead upon the internal understanding ( intellectus) of the essence ( quidditas), or being, of something ( res). This theory of truth, in turn, dictates a preference for the oft-disparaged (ontological) manner of definition offered in Euclid’s Elements over the more “rigorous” modern style of mathematical definition. On the whole, Schmitz’ treatment seems to point out important shortcomings in the current philosophical standard of mathematical truth but not to provide sufficient rationale for his own alternative definition of truth: it seems quite plausible that there could be difficulties in trying to develop more fully how the intellectus recognizes the quidditas of a given res, and, in any case, to address a century of argumentation to the contrary, it would be a worthwhile future endeavor to expand the treatment of truth offered in the current piece.

The seventh contribution to the volume is a piece by Alain Lernould devoted to mathematics and physics in Proclus’ Procli Diadochi in Platonis Timaeum Commentaria, particularly as regards Proclus’ comments on Timaeus 31b-32c. In his discussion Lernould is largely concerned with understanding whether, for Proclus, the physical bond which links the first bodies in this passage is to be understood as a mathematical bond and, if so, in what sense. Proceeding by a close and well-informed reading of both Proclus and Plato, Lernould brings us to the conclusion that we began from something of a false question: the physical bond linking the first bodies is a mathematical one but only because mathematics is identified with theology for Proclus. Consequently, the original distinctions imposed by the terms “mathematics” and “physics” become less meaningful. Lernould’s treatment is, on the whole, very admirable for its attention to the subtleties, both mathematical and philosophical, of the Platonic passage, while maintaining a sufficiently Proclean lens: he manages to explain clearly the geometrical figures described by the original passage without losing sight of the way in which Proclus understands them within his larger philosophical project. The essay thus exhibits quite well how reading a Neoplatonic commentary alongside Plato can — even without providing a solution to modern Platonic debates — provide a deeper understanding of both the original text and the commentary.

Giovanna R. Giardina offers the eighth contribution to the volume, a discussion of the concept of number found in John Philoponus’ Ad Nichomachi Introductionem Arithmeticam scholia. Put briefly, according to Giardina Philoponus distinguishes at least three different types of numbers (which correspond to the three ontological levels of reality): physical number (corresponding to the sensible realm), mathematical (dianoetic) number (corresponding to the discursively rational realm of the soul) and noetic number (corresponding to the intelligible realm). These three types of number are all also found in Syrianus. Where Philoponus differs, however, is in his understanding of the διάνοια of the craftsman — which he takes to be part of the intelligible realm not the rational. This distinction is significant in that it results in Philoponus’ notion of mathematical number being assimilated to the Aristotelian concept of the ἔνυλον εἶδος. Giardina’s treatment seems, on the whole, to be lucid and well-constructed. It extends our knowledge of Neoplatonic treatments of mathematical number (as found, to some degree, in earlier essays in the volume) to a later date than many similar treatments and shows how lively and important the debate about the ontological status of number continued to be throughout centuries.

In the ninth contribution to the volume Mario Otto Helbing shows the importance of Neoplatonic discussions of mathematics from another angle — their reception in the sixteenth and seventeenth centuries. Helbing shows that Proclus’ commentary on the first book of Euclid’s Elements was not only known during this time period but also that it was widely read. He claims that Kepler, Scheiner, Torricelli and many more of the researchers of the time used Proclus’ work to aid their thinking and discoveries. The piece serves only to indicate the existence of this reception and to cite a few of the relevant texts, but it is a wonderful starting point for future research. We are often taught to think of early modern thinkers returning to classical roots (which traditionally include Plato and Aristotle) but, insofar as Helbing is correct, we must now reckon with their returning to Late Antiquity’s reading of these roots as well.

The volume concludes, appropriately enough, with Neoplatonism arriving at the modern day. In a piece which attempts to bridge many temporal and conceptual gaps Giovanni Sommaruga considers the nature of mathematical objects both in Proclus and in modern thought. Selecting a modern-day physicalist Platonist, Penelope Maddy, Sommaruga engages in a discussion of those points on which Proclus’ and Maddy’s philosophies seem (in any sense) comparable. Sommaruga arrives at seven points of comparison between the two thinkers (which I will not cite here), noting in conclusion that Proclus could perhaps be used to revise the modern debate. The modern debate on mathematical objects, according to Sommaruga, has been heavily influenced by oversimplified notions of Platonism; a closer reading of Proclus (with the appropriate construction of modern parallels) could yield new perspectives on what it is that Platonism actually claims about mathematical objects. Sommaruga’s discussion is to be admired for the sheer difficulty of its task: on many ethical issues those who attempt to compare modern and ancient philosophy are frustrated by the incongruence of terminology and societal concerns, but, when studying the exact sciences, the comparison is perhaps still more difficult. As I am not a specialist in modern mathematical philosophy, I do not feel able to comment well on the accuracy of Sommaruga’s comparison; I do, however, think that his treatment serves as a reminder that even the less-studied realms of ancient philosophy can prove fruitful for modern students.

Readers intrigued by any of the above commentary are encouraged to turn to the volume itself. Though the work is primarily aimed at a scholarly audience (and one interested in Neoplatonism), it is well-organized with useful indices and should bear fruit for any willing to consider its ideas. These ideas, while they may be difficult and in some ways foreign, are always clearly expressed and accessible to the diligent and patient student.

Certainly the volume has limitations: as is frequently the case with essays arising from an academic conference, the works in the volume are diverse and difficult to place into a single academic category, be it historical, mathematical or philosophical.⁵ Nevertheless, despite this diversity, the papers are dense and well-edited, presenting in a single, affordable paperback volume important — if not ground-breaking — contributions to much-ignored areas of ancient philosophy and, more generally, ancient society.⁶

Notes

1. I have in mind Meno 86e1ff., Phaedo 95e7ff, and Republic, Books VI and VII, though other texts may be added.

2. The editions used are as follows: Iamblichus. 1975/[1891]. De communi mathematica scientia liber. Edited by N. Festa. Second edition by U. Klein. Stuttgart. (Teubner) and Proclus. 1967/[1873]. In primum Euclidis Elementorum librum Commentarii. Edited by G. Friedlein. Leipzig. (Teubner).

1.3] D. J. O’Meara. 1989. Pythagoras Revived. Mathematics and Philosophy in Late Antiquity. Oxford.

4. A. Tarski. 1940. The Semantic Conception of Truth and the Foundations of Semantics. Philosophical and Phenomenolgical Research, Vol. 4 (1944), 343.

5. This diversity also has consequences for the reviewer: though I consider myself reasonably well acquainted with traditional topics of discussion in Neoplatonism, at least three of the essays in the current collection extend far beyond my expertise. Consequently, I apologize to the authors under review if I misconstrue or misunderstand the importance of some of the work in the volume.

6. I noted only the following “errors”: p.3, paragraph 1, line 11, “Principe” seems to be unnecessarily capitalized; pp.16-44, throughout his article Bechtle uses the anglicized names “Plato” and “Proclus” instead of the more common “Platon” and “Proklos” (I doubt this is an error but remain unclear as to its logic); p.49, fourth paragraph, line 8, the rough breathing mark should be moved to the following line to unite it with the Greek phrase; p.132, first paragraph after first citation, line 2, insert “.” between “i.e” and “,”; p.144, translation of in Tim. II, 41.9-14, line 3, the footnote marker “48” should perhaps be moved so as not to be construed as an exponent; p.196, paragraph 2, line 4, insert “do” between “less simple and” and “not conform”; p.202, first full paragraph, lines 5-6, “can…not” should be “cannot”.