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Ancient Philosophy

A Translation of Arthur Ahlvers’ Zahl Und Klang Bei Platon/Number and Sound in Plato by Arthur Ahlvers, translation by John Black (Studies in the History of Philosophy, 67: Edwin Mellen Press) Excerpt from Translator’s introduction: The motivation for producing this translation arose while I was writing a short monograph on Plato's Timaeus. So many of the topical articles in the literature made reference to the Arthur Ahlvers work that I began to feel that it might be of service to present-day classicists to make it available in English, it being less true nowadays than it was fifty years ago that a reading knowledge of German is an unconditional pre-requisite for serious classical scholarship. My casual entertaining of this idea was transmuted into firm resolve by the value of Zahl und Klang bei Platon for my own research. Despite the comparative obscurity of its Swiss author, and both despite and as a result of the unorthodoxy of his interpretations of Timaeus, it threw light on many questions, reinforced a few speculations as to their answers, and thus led me to the no doubt overblown conceit that in my own work I had something to offer the academic public. Translating Ahlvers' book for English-speaking classicists is thus repaying in a small way the multiple favours, presumably posthumous and certainly inadvertent, he has done me.

Zahl und Klang bei Platon was published by Verlag Paul Haupt, of Bern, in 1952, as part of the romantically-titled series Noctes Romanae: Forschungen über die Kultur der Antike, under the general editorship of Walter Wili. It deals with a number of interpretative issues surrounding Plato's mathematically-based accounts, derived from Pythagoras and the Pythagoreans, of reproduction among the ruling class of the Republic, of terrestrial and celestial music, and of atomic stereometry.

More than this, it indicates surprising ways, to be explained later, in which these accounts are essentially connected.

Thus, in Chapter I, Ahlvers offers a re-analysis of Plato's derivation of the nuptial number in the Republic, while Chapters II, III and IV are devoted mainly to Timaeus. Chapter II examines the issue of whether or not the Greeks recognized the musical intervals of a third, both major and minor, as consonant. Chapter III proposes a way to explain, by drawing upon a musical interpretation of the geometry of the triangles which compose their faces, Plato's choice of the "fairest," half-equilateral, right-angled triangle as the basic unit of this composition, as well as some of the relationships between the atomic shapes of the four elements. In Chapter IV, which in many ways constitutes a postscript, Ahlvers argues that Plato's work in these areas is best understood as Pythagorean rather than Democritean in inspiration.

The significant contribution made by Ahlvers to the discussion of the nuptial number concerns not its identification (as 12,960,000) – in which he agrees with most other commentators – but its precise derivation, which occurs in an exceedingly obscure passage of the Republic (VIII 546b3-d3). Ahlvers argues convincingly that Plato wishes the reader to conceive of this number as essentially constituted in terms of the factors 3, 4 and 5, multiplied together, the result being raised to the fourth power. Other commentators, as Ahlvers shows, have reached the same number via differing interpretations of Plato's intentions as to its derivation. To be fair to all, it must be admitted that Plato gives a number of compatible derivations all reaching the same figure of 12,960,000: it is only one part of the relevant passage whose interpretation Ahlvers disputes.

In Chapter II the topic switches to the apparently unrelated one of whether Greek music acknowledged the third as a consonance. Contrary to the extant works of early commentators,' Ahlvers believes that the practice of Greek music did involvetreating the thirds as consonant, and, what is more, that Plato's theoretical contributions were fully cognizant of and fully in accordance with this practice. How plausible are his arguments on this score is difficult to assess: yet, while Ido not find them decisive, I think they are well worth examining in detail. The freshness of approach must commend them, even if they offend against orthodoxy. Of particular interest is Ahlvers' suggestion as to Plato's meaning in the passage (Timaeus 35b4-36a5) about inserting arithmetic and harmonic means between the notes defined by the musical tetractys in order to produce the series of consonant intervals, prior to the insertion of the whole tones and semitones required to generate the entire diatonic scale. This suggestion differs markedly, yet plausibly, from the received interpretation, with the result that both major and minor thirds are admitted to the realm of the consonant.

Ahlvers turns in Chapter III to the composition out of triangles of the four regular polyhedra which, in Timaeus, form the shapes of the atoms of the four elements.' His distinctive contribution is in the explanation of why Plato uses the right-angled half-equilateral triangle with sides 1, 3 and 2, instead of the equilateral triangle itself, as the basic building block of those polyhedra which have equilateral triangular sides. Briefly put, Ahlvers' explanation is that the ratios of the sides of this triangle are, within appropriate limits of approximation, those required for specifying the proportions' involved in the intervals of the enharmonic division of the fourth. Given certain ideas, here attributed to Plato, about the aesthetics of enharmonicism and thus of this "fairest triangle," Ahlvers draws a connection between the beauty of the style of music, the beauty of the triangle and the beauty of the universe constructed, he claims, on their basis. Plato's view then emerges as even more thorough than previously thought in its undisputed application of musical harmony to the composition of the elements of the world.

In an interesting digression, Ahlvers builds on the above to offer a fascinating explanation of the uneasy relationship between the equilateral-triangle-faced polyhedra associated with fue, air and water and`the square-faced cube associated with earth. The faces of the latter are composed of right-angled isosceles triangles, as opposed to half-equilaterals, and thus do not allow earth to participate in the normal processes of transmutation occurring between the elements, which operate by the dissolution of the atoms into their faces and the reconstitution of these into the other relevant polyhedra.' Plato seems, however, to be uncomfortable with this consequence of his geometric stoichiometry, for elsewhere in Timaeus he appears almost to contradict himself by including earth in the mutual transmutations. By referring to a method of exhaustion attributed to Theaetetus, Ahlvers shows how the square face of the cube can also be regarded as composed, within limits of approximation themselves defined by musical analogy, of the half-equilateral triangles so crucial to the other elements. He thus offers Plato a way to include earth in the processes of transformation, which gives rise, if one wants to explain the apparent self-contradiction, to the suggestion that Plato may have been drawing upon a tradition whose detail he had not fully assimilated.

The final Chapter looks more closely at the question of what are the sources for the entire discussion in Timaeus, and in other dialogues which call upon the interrelated subjects of arithmetic, geometry, stereometry, cosmology and music. Ahlvers' conclusion, that Pythagoras looms much larger in the background to Plato than Democritus, seems unassailable on grounds independent of the analyses andinterpretations he offers in the earlier Chapters. Yet these also add to its support, and reaffirm the crucial place of both Plato and Pythagoras in the history of atomism…

My goal above all has been to preserve the technical clarity of the original, and to present it to a modem reader in readily accessible form. Ahlvers typifies an approach to academic writing which is less common now than it used to be: he oscillates between steady periods of closely-argued exposition and analysis, on the one hand, and occasional polemical outbursts against competing commentaries, and commentators, on the other. Despite its value as identifying areas where the author is perhaps on slippery ground, I have found it difficult to avoid toning down the vituperation; for one thing, it seems foreign to contemporary philosophical discourse, and thus unnatural; for another, the high degree of invective can only appear unwarranted to a reader who is unfamiliar, as now we all are, with the context of scholarly debate in which Ahlvers was a participant; in each case the danger is that a reader will be distracted from the interpretative point by the tone in which it is expressed.

Similar considerations have influenced my attitude to the complex syntax of the German, and in many cases I have resorted to simplification in the interests of clarity, while leaving some instances untouched for the sake of flavor. Ahlvers is fond of the inclusive "we," drawing the reader into collusion with him as a co-developer of his account: I have eschewed this attitude in the English, preferring the use of the first-person singular to identify a view or an intention as the author's, but have allowed some innocent "we"s to remain, when, for example, they are used to invite the reader to accompany the writer in a change of topic.

Matter, Imagination and Geometry-Ontology, Natural Philosophy and Mathematics in Plotinus, Proclus and Descartes by Dmitri Nikulin (Ashgate New Critical Thinking in Philosophy: Ashgate) For ancient scientists and philosophers who followed the Platonic-Pythagorean program, argues Nikulin (New School for Social Research), it was not at all evident, and in fact impossible, that mathematics and its methods could be used to describe the natural world. His central concern here is how and why it became possible for early modern science to do so. He discusses the notion of matter, which Plato and Plotinus considered barely possible; various aspects of the relation of mathematics to physics; and the role and structure of cognitive faculties involved in considering and constructing material and geometrical objects.
Excerpt: Much of the difference in considerations of the relation of geometrical entities to physical things (when the possibility of the application of the former to the cognition and constitution of the latter is either implied or denied) appears to be grounded in a difference in understanding of matter and being. As the first part of the book attempted to show, in his approach to matter Plotinus mostly follows Plato's interpretation of matter as pure receptacle and the seat of forms, yet he also embraces Aristotle's conception of matter as the ultimate substrate, utter potentiality and indefiniteness. In construing the notion of matter (if it has a notion), Plotinus presents matter as non‑being, as radical otherness to being, stressing its unlimitedness and paradoxality. Descartes, however‑and such an approach might be considered exemplary for modern philosophy and sciencetakes matter as substance; that is, as primarily and adequately represented in being and thought through its clearly conceivable main attribute, extension. A body is then a shaped or formed part of that extension, defined solely in terms of geometrical characteristics. A physical body is therefore presented in such a way as to be already subject to mathematical considerations.

The Cartesian ontological approach, which places being as a primary phenomenon in the center of consideration, is further contrasted to Plotinus' position, which takes being as a synthesis of sameness and otherness, or of oneness and multitude (dyad), themselves engendered by the first principle, the One. If being is not itself primary, then the One has to be postulated as the definitive cause of being, prior to being (i.e., as properly not being). If, however, matter is equally represented as non‑being, the question arises of whether it is possible, and how it is at all possible, to make a distinction between the One (the ultimate source of being) and matter (the complete lack of being). As it was argued, Plotinus has resources to establish such a distinction and thus to characterize matter in definite terms, although such a characterization eventually involves a paradoxical description, since it appeals to that which never is and never will be, that which is inescapably missed by any rational discourse.

Substance is thus introduced by Descartes as that which is, that which exists due only to itself and is conceived univocally in its essence, characterized by a (necessary) essential attribute. Still, his attempt to portray matter as an independent substance involves ambiguity, for on the one hand, matter‑as characterized by its essential attribute, extension‑has to be considered substance, but on the other hand, matter does not exist due solely to itself and therefore is not properly a substance.

In Plotinus' approach, being has to do not only with a synthetic unity, but with a limit as well, which is also thoroughly stressed by Plato and Aristotle. In the Cartesian ontological approach, being, which is God, is conceived as perfection, tautologically expressed as reality, self causation, inexhaustible power and, moreover, as infinity: it is infinity and not limit that has to constitute perfection. However, Descartes is reluctant to bring actual infinity into the material world and into the realm of mathematical entities, because, as he argues throughout his writings, in regard to the infinite, we are only able to know that it is, and not what it is (hence the distinction between cognitive procedures of "knowing" and "grasping"). This further leads him to recognize that the physical and the mathematical has to be indefinite as a whole, rather than infinite, and only as such can both be known. Still, admitting actual infinity inevitably involves paradox, which for Plotinus has to be either avoided or logically resolved, whereas for Descartes paradox rather serves an indicator of a potential growth in philosophical and scientific knowledge, and as such can be fruitful.

The very possibility of the application of mathematics to the study of physical phenomena, as discussed in the second part of the book, is the cornerstone of modem science and is consequently denied in ancient science. Thus, for Plato and for the later Neoplatonic thinkers (in particular, for Plotinus and Proclus), mathematics can give knowledge about those things that cannot be otherwise and therefore has nothing to do with the ever‑fluent physical things, about which there can only be a (possibly right) opinion. Aristotle develops a different approach to physics, which he considers scientia, but this science is not mathematical (Aristotle's physics remains the only science about the world until its radical mathematics‑oriented revision in the late middle ages and early modernity).

Furthermore, ancient Neoplatonic thinkers carefully distinguish between arithmetic and geometry within mathematics itself. A reconstruction of Plotinus' theory of number, which embraces the late Plato's division of numbers into substantial (ideal) and quantitative (monadic, or properly mathematical), shows that numbers are structured and conceived in opposition to geometrical entities. In particular, numbers are constituted as a synthetic unity of indivisible, discrete units, whereas geometrical objects are continuous (except for the point) and do not consist of indivisible parts. This mutual irreducibility of number and geometrical magnitude is overcome in modern science, being canceled in Descartes, who, as it has been argued, considers number not primarily in relation to the finite (a limit), but rather to the infinite. Moreover, a decisive step undertaken by Descartes is the non‑discrimination of number and (extended and thus continuous) magnitude: even if they are still formally, distinguished, both can be used interchangeably in scientific considerations and are taken to represent each other univocally. Besides, the Cartesian project of the universal scientific method, the very way it is constructed and introduced, presupposes that the method should be equally suitable for both mathematical (geometrical) and physical entities. Geometrical objects are applicable to physical bodies, insofar as both are supposed to be structured according to order and measure. This order and measure is then to be discernible in both geometrical entities (themselves expressible through numbers) and physical things, allowing for the possibility of the application of the former to the latter.

The notion of the intermediary further plays an important role in the development of the argument. In the Neoplatonic reading of Plato, as found in lamblichus and Proclus, mathematical objects are considered intermediate entities between physical things (bodies) and noetic, merely thinkable, entities (notions). As the previous analysis of Plotinus' teaching on number intended to show, arithmeticals (numbers) should be placed in the same ontological category with ideal forms, or noetic objects. Being distinct from numbers, geometrical figures are to be considered intermediate, insofar as they are in a certain respect similar to both thinkable and physical things and, in another sense, are different from both. As Proclus shows, geometricals on the one hand are divisible and in a certain sense extended, as bodies. On the other hand, like noetic objects, geometricals are precise and do not change their properties over time. Therefore, mathematical entities‑numbers, as well as geometricals‑are to be conceived differently from physical bodies. Descartes, on the contrary, tends to abolish all the intermediate structures in ontology, epistemology and cosmology, and thus simplifies the picture of being and of the world, to ensure the possibility of the consideration of geometrical figures and physical bodies in similar terms­that is, as merely extended, as subjects to order and measure and by the same cognitive procedures.

The notion of intelligible matter becomes of central importance at this point: introduced by Aristotle as a matter of mathematical objects, it is thoroughly elaborated by Plotinus and Proclus. In Plotinus, intelligible matter is conceived as a universal substrate of multiple thinkable forms. It is further associated with an ineradicable otherness within being (a potentiality for being), and is thus present to all ideal entities and to mathematical objects, including numbers and geometrical figures. Intelligible matter is further interpreted as indefinite thinking, (not yet informed by the objects of thinking) when thinking tends to think its own cause, which is not being and thus cannot be properly conceived or thought, since it is not a particular object of thinking. The indefinite thinking, as dyad, thus inevitably "misses" its origin and is informed, as intelligible matter, only when it comes to think itself in definite terms as, and through, noetic objects. As it was argued, Proclus, in his elaboration of the notion of intelligible matter, takes it to be a specific geometrical matter (i.e., primarily a matter of geometrical objects). Such matter can then be consistently interpreted as imagination, as a sui generis extension where geometrical objects can be conceived as divisible, as having parts, as extended and as constructible by the movement of another geometrical object (e.g., the point). It is then intelligible matter that separates the mathematical from the physical and makes the two ontologically and epistemologically incommensurable.

In Cartesian ontology, on the contrary, there is no way to distinguish between different kinds of matter, since there is only one matter (which is extension and substance) that is present as the common matter of all extended objects, in particular, of both geometrical and physical things. This inability to distinguish between specifically geometrical and specifically physical matters allows for the possibility of putting both geometrical entities and physical bodies in one and the same extension, and of applying the same set of rules and procedures to the consideration of both. Because of this, Descartes is capable of expressing bodily properties in the precise language of mathematics. Moreover, since he intends to present arithmetical number and geometrical magnitude as mutually expressible through one another, he can build physics as a mathematical enterprise, which further leads to the substitution of the properly physical by the geometrical. In a sense, the whole physical world is then omitted (or, rather, bracketed) and the laws of geometrical, imaginary construction are imposed onto the physical and eventually expel it as concrete and imprecise from scientific considerations.

The third part of the book portrays the relation of matter (specifically, of intelligible matter) and of geometrical objects to cognitive faculties and to the imagination in particular. In the Platonic tradition, as represented in Plotinus, the intellect, seen through the category of life, is capable of conceiving the first principles. Construed as being and pure actuality, the intellect is further presented through a distinction (which cannot be taken as a real one) between thought as thinking and thought as thinkable, as the objects of thought that exist in an uninterrupted communication. On the contrary, discursive thinking, essentially involved in mathematical and logical argumentation, is incomplete and only partial. Discursive reason carries out its activity in a number of consecutively performed steps, because, unlike the intellect, it is not capable of representing an object of thought in its entirety and unique complexity and thus has to comprehend the object part by part, in a certain (correct) order.

The Cartesian cogitating mind is quite different from the intellect in its structure: the mind is a thinking substance, discursive par excellence. Mind is also self transparent and essentially reflective. Reflectivity is considered proper to the mind alone, whereas its ontologically complementary counterpart, matter, is not reflective. In comparison with the Platonic intellect, the Cartesian mind, which conceives of ideas as any content of thinking, appears to be rather simplistic and deliberately simplified. Such simplification of the mind in its structure, constitution and functioning, undertaken for the sake of clarity of understanding, appears to involve difficulties and ambiguities. In a sense, the simplification of ontological and cosmological structures (e.g., the non‑distinction of a specific difference between the geometrical and the physical) can be taken to represent a distinctive feature of modern philosophy and science, which allows modern thinking to become particularly efficient in reconstructing and reconsidering the physical cosmos as knowable in a strict and precise way.

Imagination appears further to play a crucial role in the constitution and understanding of the sphere of the geometrical. Plotinus, Plutarch of Athens, Syrianus, Proclus and Porphyry, who have their predecessors in Plato and Aristotle, present imagination in its capacity to produce mental images different both from thinkable objects and from sense‑data. Imagination is portrayed as distinct from the intellect and discursive thinking, on the one hand, and from senseperception, on the other. Put otherwise, imagination is intermediate; it is as if "in between" sense‑perception and discursive thinking, both separating and uniting them. Plotinus compares imagination to a double‑sided mirror, reflecting both the sensual and the intelligible, sharing certain features with both, but being neither of them. Furthermore, imagination is intimately connected with some kind of extension and movement, insofar as geometrical objects exist and can be constructed in imagination as geometrical in the proper sense (i.e., as extended, divisible and visualizable), thus exemplifying irrationality and otherness (which brings forth a multiplicity of geometrical figures of the same kind). Moreover, since imagination and intelligible matter appear to share exactly the same features, in Syrianus and Proclus the two are identified with one another. Intelligible matter thus can be taken as geometrical matter‑that matter in which geometricals not only exist, but also can be retrieved by kinematic construction (i.e., construction by movement) according to their ideal notion and formative principle.

In Descartes we do not find a consistent "theory" of imagination; his treatment of imagination involves a number of difficulties and ambiguities, particularly due to his hesitation about its ontological status. Imagination, split further into the mental and the corporeal, marks a connection of the finite mind to the body. Unlike the mind‑reason, imagination is unable to represent and access the reality or essence of a thing, because reason can think that which imagination cannot represent, namely, the infinite. Imagination (which is not an intermediate faculty for Descartes) submits sense data to the interpretation of the mind, which interprets the objects of imagination as extended things, that is, as physical material things and geometrical objects, insofar as both are extended. Through imagination, geometrically extended figures are brought and constructed into physical things, so that the former appear to become not only semblances, but much more substitutes for the former. Geometrical objects are therefore to be recognized as unchanging patterns (themselves constructed after thinkable essences) of physical things, so that both are to be considered (or even imagined into) the one and same matter and extension.

Modern philosophy and science, as it was argued, accept the verum factum principle, according to which an object can be known and does not exist to the extent that, and insofar as, it is and can be produced or constructed. This principle is traced then in Descartes, for whom the physical world in its substantial materiality is still a divine construction. Because of that, Descartes has to confine his efforts to presenting a consistent account of the (imaginary) reconstructed world as an object of scientific consideration. However, in this imaginary recreation of the world, the finite mind easily assumes the role of the demiurg, who knows the world insofar as he himself has created that very world. Only that which is put into the object of cognition by the cognizing subject can be admitted then as knowable in the proper sense, which eventually cancels and destroys all objective teleology.

The discussion of the construction principle was mostly focused on construction in geometry, which is already prominent in antiquity (e.g., in Archimedes and Apollonius), although construction is used mostly in problems and not in theorems (in Euclid), and does not appear to play a central role in theoretical considerations within the Platonic approach to mathematics‑where, in general, the (already) existing has a higher ontological status than the produced. The method of the kinematic generation of a geometrical figure by uniform movement becomes especially important in early modem science as the model for consideration of the physical. As Descartes intends to show, the geometrical and the physical are both movable and reproducible in the same way, insofar as everything extended can be considered constructible under a geometrical pattern, precisely describable and easily imaginable. This allows for the establishment of physical material things as represented, constructed and studied in their motion and change by means of mathematics exactly in the same way as geometrical figures are represented, constructed and studied.

Engendering a geometrical figure by movement is found already in Proclus. Imagination, considered a constructive and creative capacity, appears to play a crucial role in producing geometrical entities, intermediate between physical things and noetic objects. Moreover, geometrical figures turn out to be inescapably dual, for they can be considered both as already existing and as originated by movement. For Proclus, imagination represents geometrical intelligible matter and is itself intermediate between discursive thinking and sense‑perception. Furthermore, imagination is taken by Syrianus and Proclus to be capable of embodying geometrical figures by tracing them as generated or produced by the movement of another geometrical entity (e.g., a point). The geometrical figure appears to be divisible, formed, multiple (since an indefinite number of figures of the same kind can be constructed) and extended only within the imagination, because the physical representation of the figure is distorted, whereas the formative principle of the figure within discursive reason is unextended and indivisible. The purpose of the construction of a geometrical figure in the imagination for Proclus is then a sui generis alienation of the figure, when it is objectified as a constructible image, according to the figure's immanent formative principle, and then studied in all its properties. In this way, the properties are rendered visualizable as imaginable, as if being projected onto the screen of the imagination and into geometrical matter, for better and clearer consideration of and by discursive reason.

The kinematic construction of a geometrical object thus does not properly create the object anew, but rather reproduces it under a non‑geometrical and nonphysical ideal pattern of the intellect‑the noetic form‑itself represented through the formative principle in discursive reason. And the ideal form and formative principle are not themselves constructible, in particular, within the imagination. A geometrical object can then be considered existing as four‑fold: in its intelligible notion form in the intellect; in its formative principle in discursive thinking; as the geometrical figure properly, conceivable and constructible in the imagination; and as a physical imitation, accessible to sense‑perception. All these levels of representation of a geometrical entity are never confused within the Platonic account of geometry, whereas the Cartesian acceptance of the extended as substantial opens the possibility of conceiving the geometrical and the physical in similar terms, and thus of applying mathematical entities and precision to the description of physical things. Once the principle of construction is expanded to all "external" reality (taken as extended substance or matter), and once geometrical entities as intermediate are expelled, then every extended object can be considered imaginable and constructible under a geometrical pattern. Hence the physical becomes reducible to the geometrical, since physical bodies and geometrical figures are considered already (re)constructed in one and the same material extension, being subjects of the imagination. The physical world of modern science and philosophy is thus constructed as (and eventually substituted by) a strictly thinkable, and properly knowable, geometrical world.

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