Imagination and Mathematics

Chapter 8 considers the role of the imagination as it appears in Proclus’ commentary on Euclid, where mathematical or geometrical objects are taken to mediate, both ontologically and cognitively, between thinkable and physical things. With the former, mathematical things share the permanence and consistency of their properties; with the latter, they share divisibility and the possibility of being multiplied. Hence, a geometrical figure exists simultaneously on four different levels: as a noetic concept in the intellect; as a logical definition, or logos, in discursive reasoning; as an imaginary perfect figure in the imagination; and as a physical imitation or representation in sense-perception. Imagination, then, can be equated with the intelligible or geometrical matter that constitutes the medium in which a geometrical object can be constructed, represented, and studied.

Matter and Evil

The problem of evil, one of the central ethical and ontological problems in Neoplatonism, is discussed in Chapter 11. In Ennead I.8, Plotinus presents matter as utter deficiency, as the absence, lack, or privation of the good, and thus as radical evil. In De malorum subsistentia, Proclus explicitly responds to Plotinus and argues that evil does not exist since non-being does not exist. This means that evil is not opposite to the good because the good has no opposite, and opposites belong to the same genus. Therefore, in its elusive and indefinite nature, evil should be characterized by the rethought and redefined concepts of privation, subcontrariety, and parypostasis. In its inescapable deficiency, then, evil is the privation and subcontrary of the good that exists parypostatically; that is, as elusively present in its absence as the misplacement of being and the displacement of the good.

The Many and the One

In response to Plotinus’ treatment of the one and the many, Chapter 7 is devoted to the discussion of the same problem in the opening sections of Proclus’ Theologia Platonica II 1–3. Beginning the deduction of his grand philosophical synthesis with only two elements in reference to the first two hypotheses of Plato’s Parmenides, Proclus demonstrates that both the one and the many are necessary for the generation of the plurality of henadic, intelligible, psychic, and physical phenomena. While the many is dependent on the one and the one is not dependent on anything, the interaction between the one and the many is essential for the constitution of being. Through a number of precise and subtle arguments, Proclus further establishes the structure of participation as defined by the triad of the unparticipated–participated–participating, which is then applied to the clarification of a number of difficulties and implicit presuppositions in the text.

Unity and Individuation of the Soul

Chapter 4 shows that the understanding of the soul in the Enneads is marked by Plotinus’ attempt to establish a Platonic account of the soul, which he does in constant polemics against other theories. On the one hand, such an account takes into consideration the exegetic and hermeneutic tasks of reading and interpreting Plato. Yet, on the other hand, it also establishes the soul as both uniting and separating, and hence as mediating between the intelligible and the sensible. In his psychology, Plotinus provides explanations for the unity of the soul and for its individuation, which he understands from the perspective of the synthetic unity of the one and the many. Against the material and formal accounts, it is argued that it is the rational principle or logos that is the source and principle of both unity and individuation of the soul in Plotinus.

Number and Being

In the treatise “On Numbers,” as well as in Ennead V.5, Plotinus considers the place and the ontological status of number within being and thinking, taking number to be a mediation between the original unity of the one and the multiplicity of the infinite and making an important distinction between essential and quantitative number. Yet he does not explicitly address the question of the constitution of number, leaving the problem without a definitive answer. The chapter reconstructs the constitution, derivation, and construction of number in its various representations, doing so with reference to the hints and arguments provided in the texts of the Enneads.

The System of Physics

Chapter 10 considers the structure of Proclus’ rarely discussed Elements of Physics and its original contribution to the understanding of physics in antiquity. It is argued that the purpose of the treatise is not only a systematic arrangement of the arguments scattered throughout Aristotle’s works on natural philosophy, using the structure of Euclid’s Elements as a model. Proclus also aims to develop a universal theory of motion or physical change that establishes the first principles as definitions, formulates and demonstrates a number of mutually related propositions about natural objects, and culminates in establishing the existence and properties of the prime mover. Unlike modern physics, which presupposes the applicability of mathematics to physics, Proclus shows that the study of natural phenomena in the more geometrico way can be a systematic rational science arranged by means of logic rather than mathematics.

Eternity and Time

This chapter discusses the notion of eternity in Ennead III.7, where, against the Pythagorean interpretation of eternity as intelligible substance, Plotinus argues that it primarily characterizes being as the intellect, which is the paradigm of all things. The intellect is marked by simplicity, which, however, presupposes a differentiation between the thinkable and thinking. Yet the intellect is also life, which is the life of thinking that turns toward the one, the source of the intellect—of its being, thinking, and life. This allows the intellect to think its objects as determinate in an act that surpasses discursive thought, thus defying and suspending temporality. Against the Pythagoreans, then, eternity is the everlasting non-discursive and self-identical life of the intellect.

The One and the Many

Chapter 1 examines the relation of the one to the many in Plotinus, which is fundamental for his thought. It establishes a system of axiomatic claims about the one, such as that there is a principle of the whole, which is also the principle of being that transcends being; that the act of producing is ontologically prior to and more perfect than what is produced; and that everything perfect produces of necessity. It further argues that both the one and otherness transpire in the constitution of three different representations of the many, which are the ideal numbers, the intellect that thinks the plurality of the noetic objects, and matter. Otherness, then, inevitably appears as dual and ambiguous, as both the rationally conceivable principle of negativity and also as pure indefiniteness, in which respect it is similar to matter.

Beauty, Truth, and Being

Chapter 9 discusses Proclus’ concepts of truth and beauty in mathematics, science, and ontology. He accepts both Plato’s claim in the Philebus that the good is present in the thinkable in beauty, truth, and symmetry, as well as Aristotle’s assertion that beauty is exemplified in mathematics through order, symmetry, and definiteness. Proclus then demonstrates that in mathematical objects beauty transpires in four ways: in the beautiful shape of a sensible reproduction of a geometrical figure; in the beauty of the perfect geometrical figure existing in the intelligible matter of the imagination; in the mathematical object’s definition, with all its properties becoming explicit in constructions and demonstrations by discursive reasoning; and in the beauty of the form in the intellect. On this account, science brings the knowledge, beauty, and goodness of its objects to completion within the unity of the intellect.

Memory and Recollection

Chapter 5 establishes the role of memory and recollection in their mutual relation in Plotinus. This requires a careful reading of the relevant texts in Plato and Aristotle and also a meticulous reconstruction of the arguments in the Enneads. Memory for Plotinus is not the Platonic storage of images or imprints coming from the sensible or the intelligible. Rather, memory is a power of imagination used by the soul for the reconstruction and reproduction of the currently absent, which the soul performs starting with the initial sensible impact that becomes the occasion to form a particular memory. Recollection, on the contrary, takes the form of a rational discursive rethinking that reproduces the soul’s experience of the intelligible objects. Recollection, then, is constituted by a triple motion of the descent, staying, and return of the soul to the intelligible.