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Proclus on Mathematics and
the Ascent of the Soul

Clemency Williams

Proclus has been heralded as “the last great representative . . . of Neoplatonism” (Morrow 1970: 160) and thus stands in a most complete and nuanced relationship to Plato and his Neoplatonic critics. An examination of the evolution of the Neoplatonic conception of mathematics and its relation to the ascent of the soul is nicely illustrative of the ways in which Proclus adopted, developed and refined the philosophy of his Neoplatonic predecessors. In order to gain some insight into this relationship, let us first compare two interpretations of a Pythagorean maxim, first by Iamblichus and then by Proclus. The difference in the interpretations is subtle but significant, highlighting a key distinction in the characterization of this relationship by the two Neoplatonists.

At the end of his Protrepticus, Iamblichus gives us a list of thirty-nine Pythagorean symbols (sumbola) with a brief interpretation of each one. The thirty-sixth Pythagorean symbol[1] is:

Honor a figure and a step rather than a figure and three obols.

Iamblichus says, “this advises one to philosophize and to pursue the mathematical sciences not cursorily, and by these as if they were ladders, to ascend to the proposed end” (Iamblichus 1967: 108, 11-12 and 125, 1-8). Proclus, on the other hand, in his Commentary on the First Book of Euclid’s Elements, says, “by this they meant that it is necessary to pursue the science of geometry, which, with each theorem, makes a step upwards and draws the soul to the heights” (Proclus 1977: 84, 16-24). Thus, while Iamblichus asserts that one must not cursorily pursue “the mathematical sciences,” Proclus specifically prescribes geometry. Further reading of the works of Iamblichus, however, shows that in terms of the upwards progression of the soul, he believes the study of arithmetic best assists such an ascent.

The notion that mathematics assists the soul in its ascent was by no means unique to Iamblichus and Proclus. It was the outcome of a Platonic prescription that was first adopted by Plotinus and then later expressed and modified by his followers. In order to understand and appreciate the geometrical emphasis of Proclus and the critical ways in which Proclus developed and altered these accounts of his predecessors, I will proceed by investigating earlier accounts of the relationship as seen in Plotinus, Numenius of Apamea, Nicomachus of Gerasa, Iamblichus and Syrianus.

Although it is mentioned, mathematics does not play a central role in Plotinus’ account of the ascent of the soul. In his exposition of this ascent in Ennead I,3, he distinguishes three types of travelers on the road—the philosopher, the lover and the musician—and explains how each of the three proceed on the upward journey.[2] The musician is drawn by the tone, the rhythm and the design of objects of sense and is lead upwards by the beauty that manifests itself in these forms. The lover, bound by visible beauty, is lead to recognize the beauty in the arts, sciences and virtues and then to reduce them to unity and to be shown their origin. Thirdly, the philosopher, needing only a guide, as he is by nature “winged” and in no need of separation from the sense realm like the others, should use mathematics both to train himself in abstract thought and to gain confidence in the existence of the immaterial. Having completed this, the philosopher is ready to proceed to the study of dialectic, the method that “brings with it the power of pronouncing the final truth upon nature and the relation of things” (Plotinus 1951: I, 3, 1-3).

Plotinus does not say anything further about mathematics in this context. He does not elaborate on the nature of these mathematical studies nor does he specify how exactly the study of mathematics will give the philosopher “confidence in the existence of the immaterial.” He does, however, create a relationship between the study of the mathematical sciences and the study of dialectic, with the former being a precursor to the latter, and further on in the work states: “Philosophy…contemplates the physical world with the aid of dialectic, as the other skills use arithmetic” (Plotinus 1951: I, 3, 6). Although he gives a detailed description of dialectic, he does not explore its connection with mathematics any further.

Numenius of Apamea, too, does little more than mention the relation of the mathematical sciences to the ascent of the soul. In his extracts from his dialogue, On the Good (fr. 2, 17-23), he has adopted Plato’s prescription in the Republic (527 ff.) that the mind’s eye is directed away from the material world by the mathematical sciences which assist it in the contemplation of pure immaterial being:

[I]n reality you need a method that is not easy but more than human. The best is to neglect the sensible, to embrace the mathematical sciences with a youthful enthusiasm, to consider the numbers, to learn, after repeated attention, the object of the supreme science, what is the One.

Again, although Numenius suggests a connection between the mathematical sciences, numbers, the supreme science and its object, he neither describes these mathematical sciences nor explains their relation to the “supreme science” and its object.

Nicomachus of Gerasa not only mentions the nature of the mathematical sciences and their relation to the ascent of the soul, but he examines them in some detail. In his Introduction to Arithmetic, he develops Plato’s position in the Republic that mathematical sciences prepare one for a higher pursuit, acting as “ladders” and “bridges” to pure being, purifying the mind’s eye, turning it away from material reality, and facilitating its access to true immaterial being (7, 21-8,7). He recounts (8, 8-9, 4) the dialogue between Socrates and Glaucon in the Republic (527 d ff.), proposing that the mathematical sciences, when used in everyday activities¾such as arithmetic for calculation, distributions, contributions, exchanges and partnerships; geometry for sieges, the founding of cities and sanctuaries, and the partition of land; and astronomy for farming and navigation¾arouse the soul to turn upwards and thus come closer to the higher world of being.

Nicomachus then examines these mathematical sciences in more detail (9, 5-11, 23), adding a fourth to the above categories and arranging them into the following hierachical structure: Arithmetic, Geometry, Music and Astronomy. Arithmetic is first, since it is concerned with numbers and numerical relations which are fundamental to the other sciences and because it is logically prior to them. Geometry is second because it relies upon arithmetic, music third, as it is relies upon both arithmetic and geometry, and astronomy last, being reliant upon all three. The primacy of arithmetic is further asserted by Nicomachus in his Theologoumena Arithmeticae,[3] in which he relates various divinities to each number and discusses each of the first ten numbers in relation both to their specific mathematical properties and to various non-mathematical subjects, such as physics, ethics and theology. His intention is to reduce all forms to formal properties of number, thereby so organizing the universe.

Nicomachus, however, does not mention dialectic, but asserts that these mathematical sciences are an indispensable precursor to philosophy; as Plato says,[4] “if anyone attempts philosophy in any other way, he must call on Fortune to aid him. For there is never any path without these” (Nichomcus 1926: 15-16). In fact, Nicomachus’ characterization of arithmetic as “existing before all other sciences in the mind of the creating God, like some universal or exemplary plan” (9, 9-9, 12), conveys the concept that numbers are the highest type of form and from them other forms are classified and created. This is an apparent inconsistency in the scheme of Nicomachus. For, on the one hand the mathematical sciences prepare one to comprehend a higher being; however, the highest mathematical science is in fact arithmetic and the highest form of being is number. Thus, if Nicomachus wants to sustain that mathematics leads us to a higher science of being, but the mathematical study of number is supreme, then that science must be subordinate to the mathematical study of number. While he asserts that the mathematical sciences are intermediary (following Plato’s Republic), he simultaneously espouses number as the highest reality; as a result, there hardly seems space in his conception for the Platonic “dialectic.”

Iamblichus, Porphyry’s[5] pupil, also elaborates in depth about the ascent of the soul, and, like Nicomachus, his work has a notable Pythagorean emphasis. In his work, Protrepticus, he describes how the soul must proceed in its upwards path: “the soul progresses little by little from the lesser to the greater things, passing through all beautiful things, and finally discovers the most perfect goods; the turning [of the soul] must progress on a course beginning from general things” (7, 9-12). Next he poses that it is theoretical philosophy that will assist an individual to ascend: “we are taught by this theoretic science according to what law the divine principle or entity which is in each of us ascends to the higher sphere” (11, 12-26). Given his explanation of the Pythagorean symbol, it is clear that Iamblichus believes mathematics to have a role in the ascent of the soul. Furthermore, given his emphasis on arithmetic and numbers, like Nicomachus, the type of mathematics which best assists the soul’s ascent is arithmetic.

Iamblichus characterizes the mathematical sciences in his work, On General Mathematical Science, arguing that mathematics is concerned with realities that are intermediary between intelligibles (which are immaterial and indivisible) and sensibles (material and divisible), and that these mathematical objects are higher than sensibles and lower than intelligibles, functioning as a bridge which joins causes to effects (10, 8-11, 7). In his On Nicomachus’ Arithmetical Introduction, he praises Nicomachus and claims that he is simply going to reproduce Nicomachus’ work because of the excellent way in which it introduces the reader to arithmetic and Pythagorean mathematics (5, 15-25). However, Iamblichus does add his own embellishments. Notably, on several occasions (20, 10-14 & 19-21, 4; 23, 18-24) he criticizes Euclid and his geometrical approach. This further confirms Iamblichus’ arithmetical emphasis and highlights a contrast between the Nicomachean numerical approach and the Euclidean geometrical approach to mathematics. Iamblichus explores the relation between mathematics and philosophy. He argues (55,8-22), with Plotinus, that mathematics prepares the soul for the study of the intelligibles by accustoming it to the immaterial, purifying and strengthening it for the transition to pure being (cf. PlotinusI, 3, 3). Therefore, according to Iamblichus the mathematical sciences serve as a preparation for the soul, so that it may comprehend pure immaterial being.

In his Commentary on the Metaphysics, Syrianus, borrowing the language of Plato (Rep. 527d, 8-e 2), affirms the cleansing and regenerative qualities of mathematics. He states that mathematics can help to lead us to higher things (191, 10-12), functioning as a bridge (96, 28-30). Syrianus is quite explicit about the relationship between mathematics and the soul. Mathematics is intermediate between the sensible and intelligible realms, and mathematical objects form a part of the soul (4, 5-11). The mathematicals in the soul exist at different levels. In terms of arithmetic, Syrianus describes the soul’s generation of mathematical numbers from the two principles that it contains within itself. From these two principles, defined as a monad and a dyad (132, 14-20), all mathematical numbers are generated and acquire their features. The generation of geometrical figures, however, is different (91, 25-92). It is the partless and “essential” principles in the soul that produce geometrical figures. Furthermore, he characterizes geometry as “weak.” Because of its weakness, geometry unfolds these geometrical objects on the imagination so that it can contemplate them more easily. Syrianus, then, characterizes arithmetic and geometry quite differently.

Such, then, was the legacy Proclus inherited. His description of the relationship between mathematics and the ascent of the soul reveals the selective way in which he borrowed certain elements from the work of his predecessors and the critical ways in which he developed and altered them. While retaining the key concept that “mathematics makes ready our understanding and our mental vision for turning towards that upper world” (20, 14-17), he prescribes a method of ascent that is quite different from the previous Neoplatonic accounts. Nicomachus and Iamblichus, both noted for their Pythagoreanizing tendencies, put forth that it is arithmetic which best facilitates the upwards motion of the soul. In contrast, however, Proclus emphatically asserts that it is the study of geometry which best achieves this goal. Placing geometry as prime, he proposes a definite return to Plato in the midst of a tradition that was becoming increasingly Pythagorean.

In the opening sentences of the second part of his prologue, Proclus is quite explicit about the relative positions of arithmetic and geometry: “That geometry is a part of a general mathematics and occupies a place second to arithmetic, which perfects and defines it…has been asserted by the ancients and needs no lengthy argument here” (48, 9-15). However, assuming that Proclus’ ultimate goal is to ascend as “close” as possible to the upper realm of the nous, how are we to understand his emphasis on geometry as a means to achieve this over arithmetic, when it occupies a lower position in his established hierarchy? How is his choice compatible with the situation?

Proclus posits geometry as the mediator, by means of images, between material reality and the Forms. Even though arithmetic too deals with images, these images are realized at a higher level. It is thus precisely because geometry occupies a lower position than arithmetic that Proclus thinks it worthy for study. Because the images of geometry are realized at a lower level, they are more readily perceptible; hence, the soul can better comprehend its innate principles because it can recognize them more easily through the manifest geometrical forms. Thus geometry is better suited to the needs and capacities of the ascending soul: “the soul, exercising her capacity to know, projects on the imagination, as on a mirror, the ideas of the figures; and the imagination, receiving in pictorial form these impressions the ideas within the soul, by their means affords the soul an opportunity to turn inward from the pictures and attend to herself” (141, 2-9).

Therefore, despite the acknowledged primacy of arithmetic, the explanation for Proclus’ particular interest in geometry is to be found in his assertion that mathematicals are projections of innate principles by the soul from the upper realm of the nous and that these principles manifest themselves at a perceptible level. Because these geometrical manifestations are expressed at a more perceptible level than arithmetic, the soul is better able to perceive them, and in these imperfect manifestations recognize its innate intelligible principles. Furthermore, although Proclus still upholds the enthusiasm and primacy of the Pythagoreans with respect to number and geometry and their functions in the ascent of the soul, Proclus prefers to bring Plato’s views of mathematics back into prominence.

In his depiction of the mediatory role of mathematics in the Republic, Plato favors geometry (510b-511b). Furthermore, Plato creates a strong link between geometry and dialectic, the “perfect” and “unhypothetical” science. Proclus examines this connection in detail (42, 9 ff.), concluding, in accordance with Plato, that “dialectic is the capstone of the mathematical sciences” (43, 10-11). He asserts that geometry mirrors dialectic better than arithmetic because of its greater discursive and demonstrative character. In the second prologue, Proclus characterizes geometry by four elements¾demonstration, definition, division and analysis¾and concludes that these four components comprise all Platonic dialectical methods. Furthermore he asserts that Euclid’s work “contains all the dialectical methods” (69, 13-19). Therefore, in this respect also geometry is ideally suited to assist the soul’s ascent because of its close association to the method of dialectic. Geometry calls attention to a science higher than mathematics, which is expressed at a higher, unhypothetical level and “with each theorem lays the base for the step upward and draws the soul to the higher world” (84, 13-23).

The comparison of Proclus’ Commentary to other earlier Neoplationic works reveals not only his debt to his predecessors but also his innovation. Adopting their notions of the importance of mathematics in the ascent of the soul and the hierarchical characterization of the mathematical sciences, in order to assist the soul on its ascent to the noetic realm, Proclus sustains that it is in fact geometry that can best facilitate this upward movement. For, the soul can easily recognize, in the perceptible manifestations of ideal geometrical forms, its innate principles, “to behold the secret and indescribable figures in the inaccessible places and shrines of the Gods, to uncover the unadorned divine beauty and see the circle more partless than any center, the triangle without extension, and every other object of knowledge that has regained unity” (141, 22-142, 2).

References

Iamblichus. 1975. De Vita Pythagorica. Ed. L. Deubner, 1937 (repr. Stuttgart).

————. 1967. Protrepticus. Ed. H. Pistelli, 1888 (repr. Stuttgart: B.G.Teubner, Stuttgart).

————. 1975a. De Communi Mathematica Scientia. Ed. N. Festa, 1891 (repr. Stuttgart).

————. 1975b In Nicomachi Arithmeticam Introductionem, Ed. H Pistelli, 1894 (repr. Stuttgart).

Merlan, P. 1953. From Platonism to Neoplatonism. The Hague.

Morrow, G. R. 1970. Dictionary of Scientific Biography. Oxford.

Nicomachus of Gerasa. 1926. Introductionis Arithmeticae Liber II. Ed. Richard Hoche, Leipzig, 1866 (trans. D’Ooge, 1926).

Numenius. 1973. Fragments. Ed. and trans. E. des Places, Paris.

O’Meara, D. J. 1989. Pythagoras: Revisited Mathematics and Philosophy in Late Antiquity. Oxford.

Plotinus. 1951. Opera. Ed. P. Henry and H-R. Schwyzer. Paris.

Proclus. 1977. In Primum Euclidis Elementorum Librum Commentarii. Ed. G. Friedlein, Leipzig, 1873 (repr. 1967) (trans. Morrow, 1977).

————. Theologie Platonicienne. Ed. and trans. H.D. Saffrey and L. G. Westerink. Paris.

Pistorius, P. V. 1952. Plotinus and Neoplatonism. Cambridge.

Steel, C.G. 1978. The Changing Self A Study on the Soul in Later Neoplatonism: Iamblichus, Damascius and Priscianus. Brussels, 1978.

[1]. T. Taylor (The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid’s Elements 1788, 1789) mentions (I.113), “I do not find this aenigma among the Pythagoric symbols that are extant; so that it is probably no where mentioned but in the present work” (cited in Morrow 1970: 69n81). However, the symbol is mentioned elsewhere. As seen from the above, Iamblichus mentions and explains it in his Protrepticus.

[2]. Here the analogy is borrowed from Plato’s Phaedrus (248d ff.).

[3]. This survives only in a summary by the Byzantine Patriach Photius from the ninth century and in excerpts preserved in an anonymous compilation also entitled Theologoumena Arithmeticae. See O’Meara 1989: 15n24.

[4]. Nicomachus here refers to a passage in the Epinomis (991 d). The attribution to Plato of this work has been long debated. The quote Nicomachus uses is cited imperfectly. See Nichomachus 1926: 185n3.

[5]. Very little of the work of Porphyry, the pupil of Plotinus, remains. However, what survives of his corpus suggests that he was more concerned with ethical and religious than with mathematical means of ascending to the immaterial. In his Life of Pythagoras, he refers to the mediatory role that can be played by the mathematical sciences, bridging the gap between the material and the immaterial (Ch. 47, 58, 12-19). However he concerns himself little with arithmetic or geometry.