Friday, April 8, 2016

Schelling on Sexual Difference

Friedrich Wilhelm Joseph Schelling (fl. 1775-1854) describes, in The First Outline of a System of the Philosophy of Nature, how Nature configures its specie in qualitative opposition for the further propulsion of each species: the species must be divided within and opposed to itself in order for its opposed centrifugal forces to conjointly animate the species into the production of generations in time. Since the purpose of sexual difference and reproduction is thus for the universal species rather than any of its individual members, sexual differentiation and reproduction require the individual to be sacrificed as an instrumental medium for the purposes of the species. The self-organization of Nature thus proceeds, from one spherical self-opposition to the next, through an infinite succession of species, in successive unsuccessful attempts to represent, in and for themselves, the absolute synthesis of transcendental and natural philosophy.


The development of the absolute product in which the activity of Nature would exhaust itself is nothing other than an infinite progress of formation. The process of formation is nothing other than a configuration... Nature contests the Individual; it longs for the Absolute and continually endeavours to represent it. It seeks the most universal proportion in which all actants, without prejudice to their individuality, can be unified.

Throughout the whole of Nature absolute sexlessness is nowhere demonstrable, and an a priori principle requires that sexual difference is taken as point of departure everywhere in organic nature. Nature has either unified the opposing sexes in one and the same product and has developed simultaneously in different directions (as in many species of worms, where mating is always doubled, as well as in most plants), or Nature has distributed the opposed sexes into different stocks (individuals). Here the one-sided sexual division is again distinguished only at different stages of development.

The universal separation into opposed sexes must occur according to a determinate law, and indeed neither sex should be able to originate without the other simultaneously originating with it. We see that where both sexes are unified in one individual, they originate through one and the same formation. Therefore, the law which is observed in the latter must be extended over Nature as a whole.

Thus, if according to our principles, the production of various genera and species in Nature is only one production captured at different stages, then the formation of the opposite sexes in the same genus and species must be only one formation, one natural operation, such that the different individuals of the same genus amount to only one individual, but developed in opposite directions. It must be demonstrated that the separation into different sexes is just the separation which we have furnished as the ground of inhibition in the production of Nature. That is, it must be shown that Nature is actually inhibited in its production by means of this separation, without on that account ceasing to be active.

In other words, Nature will drive the individualization of the product to the extreme in both directions. Therefore, the most acute moment of individualization in each organism is also the most intense moment of natural activity in it... The opposing natural activities that are operative in the product toward opposite directions always become more independent from one another; the more independent from one another they become, the more the equilibrium is disturbed within the determinate sphere of Nature that they describe. If they arrive at the maximum point of mutual independence, then they greatest moment of disturbed equilibrium is also reached.

However, in Nature, the highest point of disturbed equilibrium is one and the same with the moment of the reestablishment of equilibrium. Between the two no time elapses. Those antithetical activities must, therefore, combine themselves according to a necessary and universal law of nature. The product will be a mutual one, constructed from both of the opposed directions (of the formative drive); Nature will in this way return by a circular course to that point from which it had departed; the product will, as it were, turn back on itself and will have adopted once more the general character of its stage of development... From this moment forward, since the joint product is secured, Nature will abandon the individual, will cease to be active in it, or rather it will then begin to exercise an antithetical effect upon it; from now on the individual will be a limit to its activity, which Nature labours to destroy.... The genus must appear as an end of Nature, the individual as a means – the individual would expire and genus remain – if it is true that the individual products in Nature ought to be seen as unsuccessful attempts to represent the Absolute.

This unimpugnable law of nature is most conspicuous, again, in the organisms which succeed to sexual development through perceptible metamorphoses. Flowers wilt, the transformed insect dies, just as soon as the genus is secured. The individual seems here almost to serve merely as a medium, only as a conduit, through which the organic vibration, the formative force (the spark of life) propagates itself. – But is this law of nature not also just as operative in the higher organisms, and does no the individual here too deceive us, seeming as if it were Nature’s end and not merely a means? We do not perceive as strongly in higher creatures that demise of organisms, after the point at which the peak of opposition is achieved, partly because it happens with very attenuated speed, and because the product that was a longer task for formative Nature is also a longer task for destructive Nature; partly because here the sexes are much more separated than at the lower stages. 

If one makes a general comparison of the proximity and distance between the sexes of various organisms, one finds that for the most long-lived organisms the sexes are the most separated, and on the other hand, the more ephemeral the product the closer the sexes are to each other. Where Nature seems to want to preserve the individual longer in one speciese, it breaks the sexes further asunder from one another, as it were, makes them flee from one another. How separated the sexes are in the higher animal species, how near in the flowering plants, where they are gathered in a single calyx (as in a bridal bed)! In conclusion we may suggest that the separation of the sexes happens against the will of Nature, as it were, and that because individual products originate only by means of this separation, these products are abortive experiments of Nature.

The individual passes away, only the species remains, but Nature never ceases to be active. However, since Nature is infinitely active, and since this infinite activity must present itself by means of finite products, Nature must return into itself through an endless circulation. We cannot leave our last proposition without mentioning the consequences that flow from it. The most important conclusion that proceeds from it is this: the variety of organisms is finally reducible just to the variety of the stages at which they separate themselves into opposed sexes.

Apparently paradoxical – but necessary. Nature is only one activity – therefore its product only one as well. Through the individual products it seeks to present just one – the absolute product. Thus its products can be distinguished only through the variety of stages. But many are already inhibited at the lowest level. The ones that stand at the higher stages must have had to pass necessarily through the lower, in order to succeed to the higher.

The formation of each organism will occur completely in step with the formation of all meaning organisms, up to the stage at which that separation occurs in it; the individual formation of every organism  first begins with the development of the sex. Now the joint entity that no single individual completely expresses, but all together express, is called the species. In organic natural products both species and individual are necessary... But Nature organizes to infinity, i.e., each sphere to which Nature is limited must again contain an infinity in itself. Within every sphere other spheres are again formed, and in these spheres other, and so on to infinity.

[Excerpts from the First Outline of a System of the Philosophy of Nature, Keith R. Peterson Translation, SUNY Press, 2004, pp.35-44]

Friday, March 4, 2016

Aristotle's Abstraction Epistemology: Proclus' Neoplatonic Critique

In his Commentary on the Parmenides, Proclus criticized Aristotle’s epistemological theory of abstraction. Aristotle had, in On the Soul (De Anima), briefly described how all knowledge must be derived from the abstraction of intelligible forms from sensible objects. He writes:

“Since it seems that there is nothing outside and separate in existence from sensible spatial magnitudes, the objects of thought are in the sensible forms, viz. both the abstract objects and all the states and affections of sensible things. Hence no one can learn or understand anything in the absence of sense, and when the mind is actively aware of anything it is necessarily aware of it along with an image; for images are like sensuous contents except in that they contain no matter.” (De Anima 432a3)

Abstraction is thus rendered as a mental process whereby the active intellect ‘abstracts’ or separates off a representative form from some sensible object by subtracting the individuating matter from the more common intelligible form. The critical function of subtracting individuating matter into a more universal form was previously described as a dialectical ascent towards knowledge (noesis) the Ideas in Plato’s Phaedo (78b-79d). But where Plato’s dialectical ascent, from material individuals to immaterial Ideas, had involved a discursive process of hypotheses and refutations, Aristotle’s abstraction seems to have naturalized this discursive dialectic into an innate and non-discursive intellective faculty: it is no longer a process of hypothetical thought but an innate power of the agent intellect. Once abstraction has been naturalized, then it is possible to subordinate the old Platonic dialectic of Ideas to the new knowledge of perception, induction, and understanding. Aristotle thus makes abstraction the foundation of his theory of induction in the Prior Analytics:

[I]t is impossible to consider universals except through induction (since even in the case of what are called abstractions one will be able to make familiar through induction that some things belong to each genus even if they are not separable, in so far as each thing is such and such), and it is impossible to get an induction without perception – for of particulars there is perception; for it is not possible to get understanding of them; for it is neither from universals without induction nor through induction without perception. (Prior Analytics 18, 81b1-10)

How, then, are the representative images meant to be abstracted from sensible objects and transmitted to the active intellect? Aristotle left to posterity the task of elaborating how the representative form is meant to be abstracted from sensible objects and transmitted to the active intellect. His most authoritative commentator, Alexander of Aphrodisias, described how universal forms (i.e. genera) are “constructed [syntithenai] by a separation in thought [te-iepinoiai cho-rismos] of the other things which exist along with them” (Quaestio 79, 17–18, cf. Riin Sirkel 2011) Hence, the intelligible forms are meant to be transmitted from substances and intellect, by first subtracting the matter from the form; then transmitting the form from the substance to the intellect; and finally by duplicating the form in the intellect. It seems that, in the absence of discursive dialectic, both the mechanics of subtraction and transmission remain opaque, even in Alexander’s more developed theory of abstraction.

In response to the Forms as Thoughts argument of Plato’s Parmenides (132b-c), Proclus launches into a fiery criticism of the entire Peripatetic tradition of abstraction epistemology. In order to reassert the primacy of the One over the many, souls over perception, and Forms over matter, Proclus seeks to reduce to absurdity the possibility that the intelligible forms, or Ideas, can be abstracted, constructed, and – in any sense – derived from sensible substances alone. He first distinguishes the originary and paradigmatic Platonic Forms, or Ideas, from the ‘later-born’ forms that Alexander had purported to construct through abstraction:

“This ‘later-born’ entity, then, which is called a “thought” is obviously different from the reason-principle in the real sense. For the “later-born” is a dimmer entity that the many, inasmuch as it arises from them and is not prior to them, whereas the real reason-principle is more perfect than they. Whence the former is less substantial than the many particulars, whereas the latter is more substantial, and inexpressibly more perfect than the objects of sense.” (253/892, All page citations are taken from the Murrow and Dillon trans.)

Proclus then proceeds to argue that the soul may not “derive these common properties from the objects of sense themselves” because the objects of sense, or impressions are distinct from the objects of opinion that are ‘derived from sense-object’: since impressions are in sensible objects, opinions are in the intellect, and sensible objects are distinct from the intellect, the impressions of sensible objects must remain distinct from the opinions of the intellect. Abstraction theory stipulates, however, that the abstract intelligible forms may be transmitted over and between this distinction between sensible impressions and intellective opinions. What motivates this transmission of intelligible forms? If the sensible objects do not innately possess forms, then Proclus can conclude that the power to motivate the transmission of intelligible forms cannot arise from “any other source than the soul itself”:

“And it does not derive these common properties from the objects of sense themselves; for that which is derived from sense-objects is an impression (phantasma) and not an object of opinion, and must remain the same, when taken within, as when it was originally apprehended, in order that I may not become false or “non-existent”, but it may not become anything more perfect or noble; nor is it produced from any other source than the soul itself.” (253-2540/893)

Proclus then asks how souls may “produce these general notions” without the Forms, or “reason-principle of things”. He denies that the soul may “derive these common properties from the objects of sense themselves” because, as he has previously distinguished, the ‘general notions’ that are meant to be constructed from impressions are not opinions but must “remain the same” as nothing more than abstract perceptions, impressions, or phantasms, of the perceptible:

“Further, if Matter possesses the common element in the many individual entities in its essential state, and is thus essence more truly than individuals (for it is eternal, while each of them is such to destruction, and takes its individuality from it; for it is through the form in it that each thing participates in essence), while the Soul possesses only the ‘later-born’ common properties, how can we avoid making the Soul of lesser account than Matter, if the form residing in Matter is more perfect and more of an essence than that in the Soul? For this latter is what is most properly termed ‘later-born’, while the former is eternal; and the latter arises from the many particulars, while the former is the principle of coherence for the many, and so the latter is an offspring of the former.” (254/893)

Proclus then offers an indirect argument against abstraction: if matter were an element common to all individuals, and the individua could be destroyed while their common element of matter endured, then matter would be construed as the absolute essence of all things while all individual determinations would be construed as perishable accidents. Since matter would then be essentially prior to the all of the individual determinations that are abstracted into the ‘later-born’ properties of by the soul, matter would be rendered superior to souls. But Proclus protests: “how can we avoid making the Soul of lesser account than Matter, if the form residing in Matter is more perfect and more of an essence than that in the Soul?” Proclus considers this to be an absurd conclusion because he believes that matter impotent to generate and transmit intelligible forms to souls. Hence, even if it were - per impossibile – supposed that matter were essentially prior to souls, then matter could not be known by souls through the transmission of any intelligible forms.

 “Furthermore, the general concept in the many is narrower in range than each of them; for each of the individual entities is amplified by additions and accidental accretions, and the ‘later-born’ concept comprehends each of the many; for which reason it serves as a predicate for each of these, and the individual is in any one of the general concepts as a whole; for this common quality is predicated not only of the general concept mentioned above, but of every individual subject as a whole. How, then, could it be composed from that source, and out of the common quality in the many?” (254/894)

Proclus’ most decisive argument seems to be that, since the abstract intelligible forms are always ‘narrower in range’ merely one among many particular predicates, and no collection of predicates is sufficient to compose a whole individual, then any collection of abstract intelligible forms of predicates requires some ‘additional accretion’ to constitute an individual. This seems to be an early example of the problem of individuation: how are many predicate-properties united in one individual subject? Aristotle had proposed matter as the individuating ground of all predicate-properties. But once Proclus has, in the previous argument, rejected matter as an ultimately individuating ground, the Aristotelians are compelled to find some further source of individuation (e.g. Scotus’ Haecceity) Each of the general concepts abstracted is a predicate of the individual entities. But since each abstracted predicate is ‘narrower’ and does not contain all of the added accidental accretions of the individual entity, it seems that no individual entity could ever be known by knowing all of its predicates. However, the theory of abstraction purports to uncover the common concepts that are abstracted from the individual entities. But since the subject term of the individual entities cannot be known, it seems that neither can the abstraction of the common concepts from the individual entities ever be known.

“For if on the one hand it arises out of the many themselves, where are we to see that infinite number of men, to all of which we apply the same predicate? Or if it arises out of the common quality in the many, how can this be more comprehensive than its own cause? It must therefore take its origin from somewhere else, and receive from some other source this power of comprehending each form. Of this source, indeed, it is an image, coming into existence in a way contrary to what one would expect, by virtue of reminiscence, on the basis of sense-objects, of the causal principle aroused within us.” (254/894)

Proclus then asks how it is possible for one general concept ‘arise’ from many individuals or many qualities if its cause is no ‘more comprehensive’ than the indeterminacy of matter. Since all of these alternatives involves contradictions of one from many, Proclus concludes that the general concepts of abstraction must originate from “some other source” which is the “power of comprehending each form” through the Intellect.

“I would go so far as to add this, that every proof is made on the basis of concepts prior and more august and more universal than itself… how then can the universal concept be worthy of honour, if it is ‘later-born’? For in ‘later-born’ entities the more universal a concept is, the less substance it has; which leads to the species having more substance than the genus. Our rules about the truest type of proof would then have to be abandoned, if we lay down that only ‘later-born’ universals reside in soul; for these are certainly not more powerful than, nor causative of, nor prior in nature to, more particular concepts.” (255/ 894)

Proclus concludes that “every proof is made on the basis of concepts prior and more august and more universal than itself”, i.e. than the concepts abstracted premises and supposits of the proof. Hence, the subordination of soul to matter must compel us to abandon the ‘truest type of proof’, i.e. dialectic through division, definition, and demonstration. The absurdity of abstraction from matter implies that we must accept that essential reason-principles are prior to ‘later-born’ concepts that are abstracted by the soul from material individuals.

“If these results are absurd, it must then follow that prior to the ‘later-born’ concepts there exist essential reason-principles, which are always present and active in the divine souls and those of the classes of being superior to us, but which in us are sometimes obscured and sometimes operative, and that sometimes on the theoretical level only and sometimes on the providential level, when we, in union with the gods, take a hand in administering the whole world (cf. Phaedr. 246c2).” (255/ 894)

For a further criticism of Aristotle's abstraction of numbers, see my commentary on Metaphysics XIII 6-8:

Sunday, February 21, 2016

Aquinas and Aristotle against Plato

Aquinas seated between Aristotle and Plato

The medieval synthesis of pagan philosophy and Christian theology in Albert and Aquinas may have drawn from five principal sources: (i) biblical scripture; (ii) Aristotle; (iii) Augustine; (iv) Pseudo-Dionysius; and (v) Avicenna. Although Platonic elements may be found in all of these sources, Aquinas consistently labours to distinguish his thought from its Platonic inheritance.  (See especially his Commentary on Pseusdo-Dionysius’ the Divine Names (17)) And although Aquinas has no full-length work on Plato and the Platonists, he consistently follows Aristotle in attacking the via Platonica for separating, duplicating, and reifying the logical intentions of the mind as universal forms in separated substances. R. J. Henle summarizes Saint Thomas’ objections:

“The Platonic theory [of Ideas] requires that we maintain that the Idea (1) is truly separated in being, (2) is truly one, (3) is the real formal cause of the particulars, (4) is truly related to them as a cause, a principle, a justification of knowledge, of predication and of being. If the separation is stressed, the theory tends towards pure extrinsecism; if the invasion of the particulars is stressed, the unity of the form drives towards entitative union and pantheism. These ambiguities and tensions are thus inherent in the pure Theory of Ideas. As we have seen, Saint Thomas himself recognizes these different pressures and their logical conclusions.” (Saint Thomas and Platonism,1970, p.378)

Aquinas’ central criticism is that Plato has externally reflected the ideas of the mind into separated substances that are meant to cause all particular phenomena. The result, Henle alleges, is either to cast the ideas of all possible cognition into extrinsic separation from phenomenal world, or to colonize the world with these concepts, until the creator-creation distinction is reduced to a pantheistic system of univocal concepts. Henle concludes:

“The Platonic argument thus doubles the ontological correlates for natural knowledge, but, by insisting that the separated form is what we truly know, it casts a shadow on the metaphysical structure of the material entity, obscuring the intrinsic form and rendering its status in being and knowledge extremely ambiguous.” (375)

Henle acknowledges (341) that Aquinas’ central criticism of separated substance is derived directly from Aristotle’s Metaphysics (Bk. I ch.4), where Aristotle summarizes:

“Plato accepted [Socrates’] teaching, but held that the problem applied not to sensible things but to entities of another kind-for this reason, that the common definition could not be a definition of any sensible thing, as they were always changing. Things of this other sort, then, he called Ideas, and sensible things, he said, were all named after these, and in virtue of a relation to these; for the many existed by participation in the Ideas that have the same name as they. Only the name 'participation' was new; for the Pythagoreans say that things exist by 'imitation' of numbers, and Plato says they exist by participation, changing the name. But what the participation or the imitation of the Forms could be they left an open question.  Yet what happens is the contrary; the theory is not a reasonable one. For they make many things out of the matter, and the form generates only once, but what we observe is that one table is made from one matter.”

Here Aristotle describes how, in response to the Heraclitean Flux of sensible things that ‘were always changing’, Plato recast Socrates’ definitions as unchanging supersensible  entities of reason, which he called Ideas. Later Aristotle represents these Ideas as separate substances:

“For it is evident that they are one in their intelligible expression, for one will express the same notion in speaking of each. Therefore, if there is a man-in-himself, who is a particular thing and is separate, the things of which he is composed, such as animal and two-footed, must also signify particular things and be separable and be substances.”  (Metaphysics 1039a)

Saint Thomas comments:

“[Aristotle] accordingly says, first, that it is evident that the animal present in man and that present in horse are one and the same in their intelligible expression… Hence, if, because of the fact that species are predicated of all individuals according to one intelligible expression, there is a common man, who is man-in-himself, existing by himself, “and who is a particular thing,” i.e., something subsistent which can be pointed to and is separable from sensible things, as the Platonists maintained…  [But] it is not possible for some one thing to be present in many things which exist separately. For you are present only in yourself, since you are not in many things which exist separately, as in flesh and bones, which are your parts. Therefore, if animal is one and the same, it will be incapable of existing in many species, as in man and in horse, since the separate Forms, according to the Platonists, are substances which are distinct from each other.”

Aquinas, following Aristotle, infers that Plato’s argument from science (Rep. 477b) implies that the ideas must be necessary, immutable, and separate substances from mutable material world. Since everything in the world consists in matter in motion, the immateriality and immobility of the Ideas seems to place them beyond the world. Since Ideas are supersensible, they must also be super-substantial separated substances. Plato’s separate substances are thus alleged to have illegitimately hypostatized logical intentions into a duplicate parallel reality. This duplicated reality of Ideas is considered to be illegitimate because, contrary to its scientific purpose, it cannot begin to explain the sensible world. Henle thus summarizes (343) six arguments that Aquinas made against Plato:

“1. The Ideas cannot be causes of motion or transmutation precisely because they were set up to explain the immobility of science and are therefore principles of immobility rather than of change.

2. They cannot serve to explain knowledge of the sensibilia because they are separated from them.

3. They cannot be exemplar principles because – aside from the metaphorical character of this assertion – (a) as separated exemplars they would render the obvious agency of immediate natural causes superfluous.

4. The Ideas cannot be the formal intrinsic causes of material individuals since they are separated.

5. They cannot explain becoming.

6. Moreover, the Platonists are inconsistent, for they do not posit Ideas of artefacts.”

Henle’s six arguments, from beginning to end, each depend upon the contention that Platonic Ideas are separated substances: for Ideas may only be considered immobile, supersensible, non-natural, extrinsic, and natural beings once they have been separated from the mobile, sensible, natural, and intrinsic productivity of the world that is becoming. It may come as a surprise, then, that, not only did Plato never describe Ideas as separate substances, but he even insisted that the Ideas can never be so separated from the world: Plato criticizes the spurious separation of the Ideas from the world in his criticism of the idealist ‘friends of the gods’ (Sophist 246a-249d); the separation of the universals forms from particulars instances in the Greatest Difficulty Argument (Parmenides 133a-134a); and even represents the forms as constituted of the elemental building blocks of the world in the Timaeus. (28a-31b) Although Thomas Aquinas could have had access to Calcidus’ early Latin translation of Plato’s Timaeus, Henle acknowledges that he overlooked this important later development. (325) 

Aristotle, who was in the best position to know Plato’s doctrines, seems to have represented Platonic Ideas as separated forms for the purpose of distinguishing himself from and reformulating Plato’s forms. He indicates that because the sensible things of the world are “always changing”, and Plato’s Ideas are supersensible entities of reason, the Ideas must be separated from the sensible things of the world. Yet Plato’s insistence on the supersensibility is not meant to imply an ontic separation of forms from substances, but rather an even more intimate ontological bond between the Ideas and the word: for since the perfect paradigms of all predicate-properties (e.g. largeness) are the onto-noetic condition for the intelligibility and being of every imperfect instance (e.g. something large), there is an immediate ontological continuity between the paradigms and their instances that is only logically distinguished by their universal and particular scope. The impartation of the intelligibility and being of perfect paradigms into imperfect instances thus constitutes a kind of hyper-dynamic ontological motion beyond the observable dynamics of physical motion. And since these paradigms may only be thought as the universal predicates of sensible properties, even the most abstract paradigms must be thought through a kind of hyper-intuition beyond sensible intuition.

Aquinas’ criticisms of Plato are thus directly derived from Aristotle’s misrepresentation of Plato’s ontology for the purpose of advancing his own competing substance-metaphysics. In fact, it is Aristotle’s substance-metaphysics rather than Plato’s ontology that is most responsible for introducing so many sharp philosophic divisions between form and matter, intention and existence, logic and nature. Once it is recognized that Plato’s Ideas were never conceived as separate substances, it becomes possible to respond to Aquinas’ criticisms of Plato:

1. Ideas can impart motion because each of perfect paradigm imparts a hyper-dynamic motion into the imperfect instances.

2. Ideas can explain sensibilia because each of the paradigms is the hyper-intuited exemplar of sensible exemplifications.

3. Since the Principles, Ideas, and forms are the originary onto-noetic source of all intelligibility and being, there is no originary division of the mythical, metaphoric, linguistic or logical from the real entities of nature. Hence, natural causes should be explained, along with all other orders of causation, through the formal relations of Ideas emanating from the supreme Principle of the Good.

4. Since there is an immediate ontological continuity between paradigms and exemplifications, the Ideas are never extrinsic to material substances.

5. The Ideas are, not only not static, but are indeed the originary hyper-dynamic source of ‘becoming’ in logic, nature, and society.

6. The hyper-dynamic ‘becoming’ of the Ideas in human society implies that the Ideas can be continually reconstructed into artefacts, such as mathematical objects and even an Idea of a bed. (Rep. 596b)

Aquinas seems to have followed Aristotle in making two critical mistakes: first, he adopts Aristotle’s distinction of form and matter, in which the material world is distinguished from its forms; and then, on the basis of this first distinction, he concludes that the necessity, immutability, and separation of the Platonic forms are incompatible with materiality, mobility, and interpenetration. These mistakes did not originate with Plato, nor even with Aquinas who did not possess all of Plato's texts, but rather in Aristotle’s own substance metaphysics, which he afterwards interpolated into his interpretation of Plato when he characterized the Ideas as ‘separated substances’. For Plato, to the contrary, the Ideas are not separated but connected to all ‘substances’ through an unbreakable ontological continuum that hyper-dynamically imparts the intelligibility and being of perfect paradigms to imperfect exemplifications; and the ‘matter’ (from ‘hyle’ meaning ‘wood’) is merely a metonym for the passive matrix into which all forms are plastically received to take shape. Aristotle’s suggestion that Plato’s supersensible Ideas must be separate substances, in which universal forms are cast beyond the mobile and material world, can thus be nothing less than a deliberate misrepresentation of Plato’s authentic doctrines.

See also my lecture Plato, Logic, and Ontology:

Friday, February 19, 2016

Contra Aristotle on Contradiction

Aristotle Contemplating Homer, by Rembrandt
Aristotle advertises the Principle of Non-Contradiction, in the Metaphysics (Bk. IV, ch.3-6), as the first, most certain, and unhypothetical principle upon which all demonstration depends. He writes: “[T]he most certain principle of all is that regarding which it is impossible to be mistaken; for such a principle must be both the best known… and non-hypothetical.” In contrast to hypotheses, unhypothetical principles are not conjectured as merely possible, but are rather meant to be known immediately and necessarily through some direct noetic intuition: Aristotle’s allusion to unhypothetical principles at Metaphysics 1005b14 plausibly alludes to Plato’s unhypothetical principles of knowledge prescinding from the supreme principle of the Good at Republic 510b7-511b7. Where hypotheses are only possibly true and must be discursively demonstrated from some higher and prior conditions, unhypothetical principles are - by definition - fundamental, indemonstrable, and discursively incontrovertible. The incontrovertibility of these unhypothetical principles might suggest that Aristotle, no less than Plato, has quite dogmatically demanded that we accept an indefensible assertion.  Aristotle’s indirect argument for the Principle of Non-Contradiction, however, crucially relies upon classical Platonic principles of dialectic, noesis, and emanation.

Aristotle vehemently repudiates his ‘uneducated’ critics, who ask for the principle of non-contradiction to be demonstrated, for failing to realize that not every principle can be proven, for the simple reason that any attempt to demonstrate everything would produce an infinite regress of demonstrations. (For a critical discussion of Aristotle’s abhorrence of infinity, see my Theological History of Infinity) Although he declines to elaborate what this education is meant to consist in, judging by the introduction of unhypothetical principles in the Republic, we may guess that Aristotle intends it to resemble something of what Plato had described in the Republic, consisting in a dialectical ascent, from less to more certain hypotheses through, up the ladder of discursive reasoning until it can recognize the first unhypothetical principle of non-contradiction. Although Aristotle admits that unhypothetical principles cannot be demonstrated by any direct deduction from premises to conclusion, he nonetheless contends that these first principles of philosophy can only be corroborated by an indirect argument, in which Platonic dialectic is formalized for the purpose of refuting any contrary denial of the principle of non-contradiction. However, under closer scrutiny it appears that without the hypothetical and discursive construction of the cosmos through Platonic dialectics, Aristotle’s indirect argument quickly collapses into ampliative circularity.

Aristotle first argues that the principle of non-contradiction is held to be the most certain of all principles, that “answers to the definition” of a hypothetical principle, because “it is impossible for anyone” to affirm and deny the same thing. He suggests that the Principle of Non-Contradiction is meant to be indirectly proven from the impossibility of affirmative and negative opinions in Heraclitean flux: for if Heraclitean flux were extended to all opinions, and all opinions were simultaneously affirmed and denied, then any and all opinions would be both true and false; true by default; and no determination could ever be made between truth and falsity. To preserve the possibility of determining truth, Aristotle rejects this kind of ‘pan-inconsistency’ of all opinions. But this rejection of paninconsistency, in which all opinions are inconsistently true and false, does not immediately also imply the rejection of paraconsistency, in which the truth and falsity of some opinions does not produce an explosive inconsistency of all opinions, triviality, or trivialism.  Indeed, Paulla Gottlieb has indicated that Aristotle even appears to endorse at least some paraconsistent syllogistic conclusion at Prior Analytics II 15 64a15 when he writes: “Consequently it is possible that opposites may lead to a conclusion, though not always or in every mood, but only if the terms subordinate to the middle are such that they are either identical or related as whole to part.”

Aristotle’s most famous indirect argument is from the impossibility of meaningfully denying the Principle of Non-Contradiction: for if a sceptic of the Principle of Non-Contradiction makes any significant statement, then the critic must have, in the very act of making this statement, already presupposed the principle of non-contradiction. There are two stages to this argument: analytic and hypothetical. First, if every meaningful statement possesses some determinate shape of signification, then, in the very act of making a meaningful statement, even the sceptic must analytically presuppose the possibility of some determination. Second, such a determination is only possible on the hypothesis that some determination is true and its contrary determination is false. But since this truth and falsity of contrary determinations is only possible on the further hypothesis that for any determination that is true, its contrary cannot be true but must be false, it seems that for every act of determinate signification we must presuppose the lawful prohibition of the coincidence of contrary determinations.

The first analytic stage of the argument is undoubtedly true, but the second hypothetical stage of the argument is problematic for three reasons: first, because it surreptitiously deploys discursive and hypothetical arguments to indirectly demonstrate what is advertised as a non-discursive and explicitly unhypothetical principle; second, because it neglects any explanation of the source, scope, and specificity of its determinations; and third because this entire hypothetical inference remains ampliative. An argument is ampliative if its premises and rules are insufficient for the necessary demonstration of its conclusions. Hypothetical inferences are always ampliative, and insufficiently demonstrative, for the simple reason that there could always be alternative hypotheses that remain to be considered. Plato deliberately deploys hypothetical arguments to produce contrary theses and motivate his dialectic, but Aristotle’s didactic demonstrations are meant to be necessary, and which have neglected to consider alternative formulations for the scope of non-contradiction: specifically whether non-contradiction implies the rejection all inconsistencies (i.e. paninconsistency) or merely some instances of inconcistency that do not entail triviality (i.e. paraconsistency).

The greatest difficulty for Aristotle is that his entire indirect argument seems to have circuitously presupposed the very possibility of a determinate disjunction that it has been developed demonstrate. For every indirect argument involves a disjunctive syllogism (i.e. A v B; ¬ B, TF: A); the function of disjunction (i.e. A v B) already involves at least two determinate disjuncta (i.e. A & B); and these disjuncta are each, in turn, meant to be determined as distinct disjuncts by the Principle of Excluded Middle (i.e. Ex(Fx v ¬Fx)). But if Aristotle maintains that the Principle of Excluded middle presupposes determination, and determination presupposes the Principle of Non-Contradiction, then he has circuitously presupposed this very principle for the purposes of demonstrating it. Where Plato’s supreme unhypothetical principle of the Good is meant to be virtuously circular, because it is the emanative beginning and assimilative end of all discursive reasoning, the circularity of Aristotle’s indirect demonstration for the fundamental unhypothetical Principle of Non-Contradiction surely compromises its purely formal status as a first principle of reason.

For the purpose of distancing himself from any dialectical ascent towards the Good, Aristotle transposes the question of contradiction from polysemous words to putatively unambiguous and punctiliar facts. Any finite number of meanings can be reduced to individual meanings, and individual meanings can be self-identically represented as an abstract variable according to a formal notation convention. Where Plato had formalized Socratic dialectic into a written dialogue, Aristotle formalized Plato’s dialogue into didactic instruction. Since, however, even the words used in didactic instruction can be ambiguous, and liable to contradiction, Aristotle adopts a formalized language with symbols that appear simply identical to themselves. This formal notation is meant to prevent contradictions between terms, but inadvertently engenders an even broader conflict between formality and physicality: for once all terms have been formulated into a consistent system, then this formal system remains distinct and indifferent to the physical world.

Aristotle points to the true purpose of his indirect argument when he contends that the ultimate implication of rejecting the Principle of Non-Contradiction must be to reject all determination, individuation, and definition, and thus to entirely “do away with substance and essence.” His purpose, in defending the principle of non-contradiction, is thus, not merely to preserve the possibility of any determinatively significant statements, but rather, and more metaphysically, to preserve the real definition of essences and the very determinate shape of all substances. For if significant statements did not presuppose determination, and determination did not presuppose non-contradiction, then there could be neither determinations, nor essences, nor substances, and all essences would be reduced to a swamp of accidental attributes, wherein nothing could ever endure and never be known.

Aristotle’s Principle of Non-Contradiction is thus meant to preserve the determinacy of signs, substances and essences. The indirect argument declines to specify whether the principle applies to all instances of contradiction, i.e. paninconsistency, or only some instances of contradiction, i.e. paraconsistency. Aristotle can only prohibit any possibility of contradiction by making the further assumption that a prohibition on the totality of all inconsistencies also implies a prohibition on any particular inconsistency. However, this assumption requires the principles that obtain in every part to be identified with those of the whole and vice versa, so that the whole cosmos may be lawfully regulated, or nomologically constrained, by one self-identical and supreme principle.

The self-identity of forms is, moreover, only possible by the imposition of some superior paradigm of identity, unicity, and individuality. The Principle of Non-Contradiction thus requires that we hypothesize a supreme principle of identity, unicity, and individuality to nomologically constrain the parts by the whole. However, Aristotle cannot admit this hypothetical presupposition without compromising its status as an unhypothetical first principle of reason. Once Aristotle has formalized dialectics into didactic instruction, and further formalized this didactic instruction into various formulae, he can no longer ascend the dialectical ladder from hypotheses, to refutation, to the supreme principles, and his indirect argument for the Principle of Non-Contradiction ineluctably collapses into ampliative circularity. Were the Principle of Non-Contradiction to have, more Platonically emanated from the first principle of the Good through the second principle of Intellect (Nous), then Aristotle might more plausibly preserve all determinations in the virtuous circle of Platonic emanation and assimilation.

See also my lecture Plato, Logic, and Ontology:

Thursday, January 21, 2016

Plato and Aristotle on Names and Propositions

Plato's Cratylus, by Nancy Rourke

Aristotle is generally regarded as the founder of logic: he boasts that, before his formalization of logic, absolutely “nothing existed at all.”[1] Yet a minority tradition has always recognized Plato as its founding genius.[2] He can be read to have anticipated a response to Aristotle’s interchangeability thesis of subject and predicate terms when, in the Cratylus, Hermogenes claimed that “any name which you give, in my opinion, is the right one, and if you change that and give another, the new name is as correct as the old… For there is no name given to anything by nature; all is convention and habit of the users.”[3] Since names are not given by nature, they can be freely substituted with any artificial linguistic convention.[4] Socrates argues to the contrary that if we admit that there are true and false propositions; whole propositions are composed of particular words; and what is true of the whole is true of the parts; then we must also admit that there can be true and false words.[5] The truth of words, which is prior to any composition in sentences, statements, and propositions, must then depend upon some correspondence between the appearance of the name and the essence of the thing named. Since the denial of this essential correspondence threatens to dissolve the true, fixed, enduring meaning of all names, Socrates claims that names must be “independent and maintain to their own essence the relation prescribed by nature.”[6] Names are, thus, not merely artificially contrived, but are rather given by the most skilled poets and statesmen, who are said to have, through the art of dialectics, learnt how to distinguish and determine their intrinsic and essential relations.[7] Before Aristotle had made propositions the basic truth evaluable units of logical analysis, Plato had already made names into the images of language that individually imitate the essential differences in and between the things named.[8]

Plato once again examined this possibility of recognizing the falsity of language, difference, and negation in the Sophist.[9] After dividing reality into five major kinds, or Arch-Ideas, including those of Existence, Identity and Difference, he argues that negative prefixes, such as ‘not-tall’, “do not mean something contrary to what exists but only something different.”[10] Since the primary Arch-Idea of Existence is prior to those of Identity and Difference, the affirmation of some existing entity must always precedes its denial, negation, and differentiation.[11] And since the names of terms are prior to their composition in propositions, term negation is also prior to propositional negation. The term negations that are meant to distinguish the meaning of names must thus correspond to the real structural differences in the essences of the things named.[12] It is this essential correspondence of the difference of names to their named essences that ultimately allows Plato to affirm that the putative non-existence of negations and differences may be relationally present in and among things by participating in an identity at higher levels of existence.[13] The decisive juncture in the development of ancient term logic thus occurs - at precisely this moment - when Plato describes how these real and essential differences uniquely admit for the possibility of the “blending of any one form with another.”[14] Where previously all names had been individually affirmed to be indifferently and bivalently true or false, once the reality of these differences has been acknowledged, Plato can divide and combine the terms of speech into internally distinct but intrinsically related propositions.[15] Plato even further elaborated the conditions for their truth and reference when he wrote: “Wherever there is a statement, it must be about something… And the true one states about you the things that are [or the facts] as they are… Whereas the false statements states about you things different from the things that are.”[16] Plato rather than Aristotle may thus be recognized to have invented the first theory of how subject and predicate terms may be combined into propositions.[17] Where Aristotle would later evacuate this intrinsic relation of names to essences between subject and predicate terms, Plato had already at the dawn of logic insisted on preserving it through a logic of propositions that imitated the fundamental and essential relations of metaphysics.[18]

Aristotle afterwards developed a new logic of terms by widening this difference between the names and the essences of the things named until names could be isolated and interchangeably conjoined in the propositions of syllogisms. This widening difference allowed Aristotle to develop what was arguably the first formal logic of deductive inferences, in which rules of valid inferences could be defined prior to any consideration of the meaning of the words, as well as their various essential and metaphysical relations.[19] This formalization was accomplished by reverting to a sophisticated adaptation of Hermogenes’ opinion, first proposed in the Cratylus, that the names of terms are conventional, artificial, and interchangeable.[20] Aristotle thus distinguishes, in the Categories, between predicates that are ‘said of’ but not present and predicates that are present in the subject. He defines the presence of a predicate, not by the Platonic participation “as parts are present in a whole”, but by the separate subsistence of “being incapable of existence apart from the subject.” As an example of a predicate that is merely ‘said of’ but not ‘present in’ the subject, Aristotle writes: “‘man’ is predicable of an individual man, and is never present in the subject.”[21] Where formerly Plato may have held the predicate ‘man’ to correspond to the Idea of man that could be instantiated in the inner essence of any individual man, Aristotle now suggests that it may be predicated independently from any further participative or essential relation.[22] This relative indifference of ‘said of’ predicates crucially allows Aristotle to affirm an asymmetrical transitivity of predicates, in a cascading movement from predicate to subject. He writes: “When one thing is predicated of another, all that which is predicated of the predicate will be predicable also of the subject.”[23] The meaning of the predicate can thus only be truthfully transferred from predicate to subject because the truth of a predicate is grounded in the subject, just as separable accidents are grounded in primary substances.

In On Interpretation, Aristotle further extends this difference between names and essences to radically re-assert the ontological dependence of both subject and predicate terms upon propositions. He writes: “Nouns and verbs, provided nothing is added, are like thoughts without combination or separation; ‘man’ and ‘white’, as isolated terms, are not yet either true or false.”[24] Where Plato had preserved the truth value of names in correspondence to the essence of the things named, Aristotle claims that “there is no truth or falsity about [names], unless ‘is’ and ‘is not’ is added.”[25] Once their truth is made entirely dependent upon the addition of the copula ‘is’ in the proposition, then the intrinsic and essential relation between subject and predicates can be made into an extrinsic and accidental relation of conjoining any two terms by a third copula. Aristotle’s extrinsic relation of two terms united by one copula may thus seem a fitting substitute for Plato’s intrinsic relation of many particular predicates united in one universal form. Yet Aristotle seems, in the Prior Analytics, to have recognized how any such substitution would produce paradoxes of relations: for if, according to the Third Man Argument, he is warranted in concluding that an infinite regress of universal forms must frustrate any hypothesis of one separate universal form of ‘man’ over many particular men, then he must similarly conclude that a no less vicious infinite regress of extrinsic relations would frustrate any hypothesis of one separate copula between two terms. And since the copula is essential to any proposition, such an infinite regress of extrinsic relations between terms would likely threaten to dissolve the essential bonds in all propositions. Thus, by the time of the Prior Analytics, Aristotle appears to have shifted from his old theory of two terms united by one copula to a new theory of two terms that are interchangeably applied to one another: where the propositions of On Interpretation had been formulated as ‘A is B’, the premises of the Prior Analytics are formulated as ‘B applies (hyparchei) to A.’[26] Although this immediate application of predicates seems to have been meant to obviate any requirement for the relations to be signified by the copula, it has only succeeded in delaying the paradox by virtually representing these copula relations in a sophisticated system of formalized syntax.

For the influence of Plato and Aristotle's logic on Trinitarian theology, see my essay:

[1] Aristotle. Prior Analytics, 185b34. Joseph Maria Bochenski expressed this general consensus when he wrote that it is “no exaggeration to say that Aristotle... was the first formal logician.” Cf. Bochenski, Joseph Maria. A History of Formal Logic, 1961: 40.
[2] Lutoslawski expressed the dissenting minority party opinion when he writes: “The first man whom we meet in the history of human thought as a logician… is Plato.” Cf. Lutoslawski, The Origin and Growth of Plato’s Logic, 1897: 3. Alcinous and some of the Middle Platonists seem to have regarded Plato’s dialectic as somehow methodologically superior to Aristotle’s syllogisms. Cf. Alcinous, Handbook of Platonism, 156.25-35. And Plethon even accused Aristotle of sophistry for not acknowledging his debt to Plato. Cf. Lutoslawski, The Origin and Growth of Plato’s Logic, 1897: 8-11.
[3] Plato. Cratylus, 384d.
[4] Plato. Cratylus, 385a. Socrates objects to the absurdity of absolute conventionalism when he comments: “Suppose I call a man a horse or a horse a man. You mean to say that a man will be rightly called a horse… and a horse again would be rightly called a man…?
[5] Plato. Cratylus, 385b-c. “[I]f propositions may be true and false, names may be true and false.” For modern logic, only propositions can be elevated as true or false. However, this restriction has resulted from the rejection of any Platonic participation of names with the paradigm of the things named.
[6] Plato. Cratylus, 386d-e. Plato clearly intends to associate Hermogenes’ linguistic conventionalism with sophistry when he contrasts the true dialectical way of distinguishing names with the untrue artifice of the Sophists, Callias, and Protagoras. Cf. Plato. Cratylus, 391b-c.
[7] Plato. Cratylus, 388c-391b. Before entering into a discussion about the Homeric origin of divine names, Socrates concludes that “the work of the legislator is to give names, and the dialectician must be his director if the names are to be rightly given… Cratylus is right in saying that things have names by nature, and that not every man is an artificer of names, but he only who looks to the name which each thing by nature has, and is able to express the true forms of things in letters and syllables.” Cf. Plato. Cratylus, 390d-e.
[8] Plato, Cratylus, 431a-b. Socrates comments: “For the name, like the picture, is an imitation… But if I can assign names, as well as pictures to objects, the right assignment of them we may call truth, and the wrong assignment of them falsehood.”
[9] Lutoslawski explains that the “question of error was left unsettled in the Cratylus (429d), and in the Theaetetus (187d, cf. 200d). It is only here [in the Sophist] that Plato explains error as a judgment about Not-Being, while in all earlier works the possibility of thinking and judging Not-Being was denied in agreement with Plato’s philosophical predecessors.” Cf. Lutoslawski, The Origin and Growth of Plato’s Logic, 1897: 429.
[10] Plato. Sophist, 254d-257b.
[11] The Eleatic Stranger affirms the asymmetrical priority of positive existence to negative difference when he comments: “[W]e say that two of the three will not blend with one another… Whereas existence can be blended with both, for they both exist.” Cf. Plato. Sophist, 254d; Horn, Lawrence. A Natural History of Negation, 2001: 19.
[12] The Eleatic Stranger highlights this isomorphic correspondence when he comments that “the nature of the different appears to be parcelled out, in the same way as knowledge.” Cf. Plato. Sophist, 257c.
[13] Plato. Sophist, 258b-d. The Eleatic stranger concludes “‘that which is not’ unquestionably is a thing that has a nature of its own.” Cf. Bäck, Allan. Aristotle’s Theory of Predication,2000: 40-41.
[14] Plato. Sophist, 260a-c. Plato more emphatically suggests that the atomic division all names into interchangeable parts threatens to annihilate the possibility of discourse, dialectic, and philosophy itself when the Eleatic Stranger describes how “the attempt to separate every thing from every other thing not only strikes a discordant note but amounts to a crude defiance of the philosophical muse… This isolation of everything from everything else means a complete abolition of all discourse, for any discourse we can have owes its existence to the weaving together of forms.” Cf. Plato. Sophist, 259e-260a; Parmenides 135c.
[15] Plato sketches of how noun and verb terms may be mixed to form complex propositions when he concludes: “Neither in this example nor in the other do the sounds uttered signify any action performed or not performed or nature of anything that exists or does not exist until you combine verbs with names… Because now it gives information about facts or events in the present or past or future; it does not merely name something but gets you somewhere by weaving together verbs with names. Hence we say it ‘states’ something, not merely ‘names’ something, and in fact it is this complex that we mean by the word ‘statement’ [i.e. proposition].” Cf. Plato. Sophist, 262b-d.
[16] Plato. Sophist, 262d-263b. For a contemporary examination and defence of Plato’s theory of truth and reference, see Berman, Scott. “A Platonic Theory of Truthmaking,” Metaphysica 14 (2013): 109-25.
[17] Lutoslawski, The Origin and Growth of Plato’s Logic, 1897: 430.
[18] Plato suggests the metaphysic al subordination of linguistic discourse, including all names, terms, and propositions, to dialectical thinking when he writes: “thinking and discourse are the same thing, except that what we call thinking is, precisely, the inward dialogue carried on by the mind with itself without spoken sound…. Whereas the stream which flows from the mind through the lips with sound is called discourse.” Cf. Plato. Sophist, 263e. Plato also seems to reaffirm the divine origin of the possibility of thinking and discourse when he writes: “Must we not attribute the coming-into-being of these things out of not-being to divine craftsmanship and nothing else… from a cause which, working with reason and art, is divine and proceeds from divinity?” Cf. Plato. Sophist, 265c. This reference to the creative cause of mixed forms seems to be a clear reference to the Demiurge, or divine craftsman, of the Timaeus (28a) which Plato elsewhere describes as the formal cause of order in the World-Soul (26e, 27b, 273b, 530a, 265c, 28a, 28c) and the king (basileus) of the world (274e, 28c, 28d, 272e), who gives order to heaven and earth (28d, 30c, 97c, 98a, 30c, 97c, 28e, 966e, 967b) for the highest good (30a, 37d). Cf. Menn, Stephen. Plato on God As Nous, 1995: 4-7. 
[19] Bochenski, Joseph Maria. A History of Formal Logic, 1961: 40.
[20] Plato. Cratylus, 384d.
[21] Aristotle. Categories, 2b-c, 20-25.
[22] Gerson notes that this ‘said of’ and ‘present in’ predicate distinction seems like the “smoking gun of antiharmonization” between Plato and Aristotle, but he responds that this distinction is alone insufficient to warrant the conclusion that ‘said of’ predicates may not also remain essentially related to the ‘present in’ predicates of Plato. Cf. Gerson, Lloyd. Aristotle and Other Platonists, 2005: 83. However, Aristotle’s example of the predicate ‘man’ seems to have been deliberately chosen to emphasize this precise discontinuity and disharmony, for ‘man’ is also the predicate that Aristotle deploys against Plato’s theory of the Ideas in the famous Third Man Argument. Cf. Fine, Gail. On Ideas, 1993: 203.
[23] Aristotle. Categories, 3a, 10-15.
[24] Aristotle. On Interpretation, 16a, 1.10-15.
[25] Aristotle. On Interpretation, 16a, 1.15-20.
[26] Geach, Peter. “History of the Corruption of Logic”, 1968: 48-49. Cf. Aristotle. Prior Analytics, 24a15,25.

Saturday, January 9, 2016

A Theological History of Infinity

"Infinity" by Joe MacGown

A. The Agony of Ancient Infinities

The ancient Greeks elevated finite proportions above this primeval chaos in their founding Chaoskamf of Olympian gods and chthonic titans to initiate a gradual emancipation of the proportionate finitude of Greek humanity from the vast universality of Asian animality.[1] Anaximander first theoretically envisioned this cosmogony of finitude when he described the primordial chaos as not only an unlimited (apeiron) flux, but also as an inexhaustible reserve of generation and destruction; the “the first principle of existing things is the unlimited”, “eternal and ageless”; “deathless and indestructible.”[2] The Pythagoreans were, however, the first to fold the appearances into ‘arithmological’ limit-forms by distinguishing the limited quantitative forms (peras) from unlimited qualitative content (apeiron).[3] Each of the limit forms was, not merely a token of some natural number, but a structural partition of the cosmos originating from the eidetic light of its central hearth.[4] The very possibility of knowledge by counting was thus insuperably rooted in the finite structure of any countable content.[5] Parmenides made this ontological ground explicit by conjoining the possibility of thought and being into one eternal and immutable onto-noetic sphere. He writes: “For it is the same thing that can be thought and that can be.”[6] By conjoining the form of limit to the ground of being, the unlimited chaos of non-being was immediately cast into the nullifying void.[7] But since limited-being could only be conceived as the negative opposite of some unlimited-nonbeing, which could not be conceived at all, the unintelligibility of non-being also implied the unintelligibility of any and all limited-beings. After only the briefest flash of divine infinity, the titanic forces of the dark crept back from behind the doors of night.

Aristotle is ordinarily credited with introducing the first philosophical account of infinity. Yet it is rather Plato to whom we owe the first hint of an eidetic infinity in-and-beyond all predicated properties. Plato’s theory of universal Ideas (eide) re-cast all of the predicated properties that constituted Anaxagorean cosmic-mind (Nous) as intelligible but super-sensuous Pythagorean limit-forms, in which any and all predicates could signify a singular universal paradigm (e.g. largeness) that formally possesses an indefinite multiplicity of possible instances (e.g. large things).[8] It thus implied a two-fold eidetic infinity, in which infinitely many predicates could be positively combined to signify an equivalent infinity of perfect paradigms, and each perfect paradigm could be negatively divided, particularized, and instantiated in infinitely many properties.[9] Plato’s divine method of the Philebus has, however, often been interpreted as a method of stamping the limited paradigms of Ideas upon the unlimited manifold of phenomena to construct a medley of limit-unlimited mixtures.[10] These mixtures (mikton) of the limit (peras) and the unlimited (apeiron) have consequently been read to re-impress the firm mark of finitude that had previously been inscribed by the Pythagorean limit-forms.[11] Yet Plato conspicuously declines to make these mixtures into merely another finite limit-unlimited compound for the simple reason that the very effort to completely circumscribe the unlimited in the limit would also inadvertently allow the unlimited to escape at the very moment of its capture, because the unlimited would be mis-identified as merely another limit-form, and could thus never be genuinely circumscribed within any limit.[12] Hence, when Plato proceeds to explain the nature of the mixture, he suddenly turns to describe it in more spirited terms as a “progeny” and a “coming-into being”, which is made by the “king of heaven and earth.”[13]

Plato’s students applied his eidetic infinite to resolve previously unsolved problems in arithmetic and geometry. For example, Plato playfully alludes to Theaetetus’ discovery of infinite continued fractions (e.g. 1/√2) that had been rejected by the Pythagoreans.[14] His student Eudoxus further devised a method of exhaustion that even anticipated infinitesimal calculus by circumscribing polygons with increasing sides inside and outside a circle, and dividing the area of the polygon with the square of the circle’s radius to produce evermore exact approximations of pi (π). Archimedes later applied this method of exhaustion to calculate the area of parabolas, ellipses, cylinders and spheres.[15] Leucippus and Democritus then applied a negative rendition of Plato’s eidetic infinite to infinitesimally divide every conceivable substance into aggregates of homogenous atoms: where Plato had united infinitely many but identical predicates into one perfect paradigm (e.g. largeness), the Atomists divided any one individual whole substance into infinitely many atomic parts.[16] But by infinitesimally dividing all beings, the Atomists not only recapitulated Parmenides’ paradox of limit-beings opposed to unlimited-nonbeing in every atom, but also introduced a new mereological paradox: for since each and all particularly delimited atoms can only be conceived once they have been divided; and the gap of division is an instance of the void of non-being between atoms; the unintelligibility of non-being intrudes into and dissolves all atoms. And since no part can be conceived that is not either divided from or combined into a whole, the elimination of genuine composition of parts into wholes inadvertently eliminates any genuine part. Therefore once the Atomists had rejected Platonic participation and infinitesimally divided every limit-form, any genuine composition of parts-into-wholes became impossible simply because every putative composite ‘whole’ amounted to nothing more than the adventitious aggregation of some further divisible parts.[17]

Once the Atomists had separated qualities from quantities, and made Plato’s eidetic infinity of predicated qualities into a strictly quantitative and infinitesimal division, Aristotle could reject both infinity and infinitesimal division for the “many contradictions [that] result whether we suppose it to exist or not to exist.”[18] Any conception of the infinite thus seems negatively opposed to the finite, as the non-finite that universally exceeds the finite, just as any conception of the finite seems negatively opposed to the infinite, as some partition within the infinite. Yet if the infinite could genuinely exceed the finite, then it would also escape from its negative opposition to the finite, so that it would no longer negatively oppose so as to exceed the finite. Hence as soon as the infinite exceeds the finite, it paradoxically becomes no longer infinite. Against both Plato’s infinite principle of the World-Soul and the Atomists infinitesimal divisibility of atoms, Aristotle thus argued that any quantitative conception of the infinite must produce paradoxes, in which the infinite is distinguished from, opposed to, and limited by the finite. He wrote: “Everything is either a source or derived from a source. But there cannot be a source of the infinite or limitless, for that would be a limit of it... [Thus] there is no principle of this, but it is this which is held to be the principle of other things.”[19] To preserve the stable mixture of substances together with the infinite divisibility of mathematics, he instead distinguished actual from potential infinity: actual infinity is the totality of an interminable series of finite quantities; while potential infinity is the possibility of a continuous numerical series.[20] He affirmed the possibility of counting a potentially infinite sequence of magnitudes (e.g. 1, 2, 3,…, ∞), but rejected the hypostatization of mathematical infinity as anything actual.

After Aristotle’s criticisms the positive principle of actual infinity was re-introduced by Speucippus in the extreme transcendence of the One, and by Xenocrates as an emanation of the World-Soul from the One and the Dyad. Already in the Republic, Plato had hinted that the supreme Idea of the Good “is not essence but still transcends essence in dignity and surpassing power.”[21] In the first hypothesis of the Parmenides, he had also quietly hinted at divine infinity when he described the One as neither whole nor with parts; with no beginning or end; and without shape or limits.[22] Iamblichus likewise reports that Speucippus, the second scholararch of the Platonic Academy, had accepted Plato’s Pythagorean principles of One and the Dyad but, by overly accentuating its positive transcendence, had inadvertently severed their causal reciprocity. He argued for a radical separation of the paradigmatic cause from its instantial effects when he wrote: “what is itself the cause of some quality in other things cannot have that quality in the same way.”[23] This extreme transcendence produced a radically disconnected ontology in which the One could be serially reproduced at innumerable eidetic levels, just as surely as atoms had been negatively divided and multiplied in a kind of eidetic atomism that anticipated Liebniz’s monadology.[24] Yet Xenocrates, the third scholararch of the Platonic Academy, further extended Plato’s eidetic infinity to the second principle of the indefinite Dyad in a triadic model of supreme principles, which clearly anticipated, not only the pagan triads of Numenius and Plotinus, but also the later Patristic formulation of the Trinity.[25] He thus initiated the gradual transformation of Plato’s four-tier ontology into a cascading series of nested triads, in which the supreme principle of the One intellectually comprehends and actively generates the indefinite Dyad, which are altogether sexually conjoined to produce the Mixture of the World-Soul.[26] Once he had re-conceived the One as the positive and generative intellect, and the Dyad as the negative and effusive fountain of unlimited emanation, then each of the essential ingredients had been prepared for Plotinus to radicalize Plato’s eidetic infinity as a pure infinity beyond being that only becomes intelligible through its negatively diffusive emanation.

Although God is nowhere explicitly described as ‘infinite’ in the Jewish scriptures, numerous attributions of ‘eternity’ have contributed to casting a halo of transcendence and infinity around the Godhead.[27] Philo of Alexandria seized upon this fortuitous alignment to re-interpret the Timaeus as a cosmogony of divine creation in which Plato “affirms that the Creator of the gods is also the father and creator and maker of everything.”[28] He similarly described the passage in which “God breathed into his face the breath of life” as an allegory of the emanation of the spirit of God into the natural cosmos and human soul.[29] Consequently, there should be little surprise that he would also incorporate the Middle Platonist notion of infinite emanation. Henri Guyot has admitted that Philo did not, in fact, use the word ‘infinite’ (apeiron) to describe God, but nonetheless argued that Philo may have been the first to attribute infinity to God.[30] Since divine transcendence implies that no determinate, limited, and creaturely qualities can be adequately attributed to God, Philo must have conceived of God as indeterminate, unlimited, and ipso facto infinite.[31] Philo’s characterization of the true “God of all gods” as “transcending all visible essence”, “invisible Being”, and “appreciated by the mind alone” may have thus echoed a shared Middle Platonist belief in an intellectual intuition of the transcendent principle of the One.[32]

Socrates similarly gestured towards the supreme Idea of the Good that transcends, not only all particular good things, but even all other universal Ideas.[33] By transcending all particular finite beings the Good was rendered infinite, but by transcending each universal Idea it also re-entered into the finite. This awesome entrance of an infinite power into the finite world is highlighted in Dionysius Longinus’ treatise, On the Sublime (Peri Hypsous), in which the aesthetic of the sublime is first described as an “irresistible force” that “illumines an entire subject with the vividness of a lightning flash, and exhibits the whole power of the orator in a moment of time.”[34] Longinus adoringly characterizes Plato as a ‘demi-god of literature’ and the ‘supreme master of style’, and repeatedly characterizes the sublime, in terms reminiscent of Plato’s eidetic infinite, as the majesty of nature that exceeds the limits of human perception.[35] And his singular reference to Jewish scripture, in which God said “let there be light, and there was light” (Gen. 1:3), also recalls Philo’s allegorical re-interpretation of Genesis as a philosophic cosmogony.

The pure nova of negative infinity finally burst forth, like the all-comprehending Oculus of the Roman Pantheon, from the Enneads of Plotinus. There the One is no longer represented as merely the extreme transcendent cause of Speucippus, nor the active intellect of Xenocrates, nor even the indeterminate creator of Philo, but rather as the infinite and plenitudinous fountain of all intelligible beings. Aristotle had rejected actual infinity for the paradoxes resulting from the infinite related in opposition to the finite, but Plotinus sought to escape from the site of these paradoxes into the totally transcendent One, beyond the finite and the infinite alike. He writes: “[I]ts Being is not limited… Nor, on the other hand, is it infinite in the sense of magnitude… All its infinitude resides in its power…. Absolutely One, it has never known measure and stands outside of number… And having no constituent parts it accepts no pattern, forms no shape.”[36] The infinite power of the One is, thus, neither a quantitative, nor numerical, nor an extended infinity, but rather an infinite reserve of potential power from which every composition of quality and quantity ceaselessly flow. And since even mathematical quantities must emanate from this singular source, Plotinus may also assimilate Aristotle’s potential infinity into the One. But because the One is placed in an extreme transcendence beyond being, and every being was suspended from what is not-being, it enters the scene as a negative infinity that threatens to dissolve all particular beings.[37] Plotinus’ celebrated mystagogical ascent to the One is, consequently, an agonistic approach towards an asymptotic null-point that annihilates the particularity of all positive beings.[38] 

The intractable opposition of Plotinus’ pure and negative infinity to all finite and particular being could only be resolved in and through the mysterious union of the infinite in the finite and the finite in the infinite, in which the negative infinity of the indeterminate but all-creative one was positively represented in an individual. Such an individual union required the Church fathers to reject any superficial identification of the Platonic One with the Jewish God in a Gnostic drama sacrifice and atonement.[39] If God could become a man, then the One cannot plausibly remain infinitely removed from the finite world, but must also transcend its very transcendence to become positively manifest to the finite. At this singular moment, the sublime aesthetic of infinity was made fully present within the proportions of the beautiful.[40] Once, however, the indeterminate infinitude of God was re-finitized in man, it also seemed inadmissible to render God as immaterial, unlimited, and infinite. Consequently, the double-transcendence of God in Christ suggested to Tertullian that the Christian God should be cast in a kind of Stoic mould as the finest of finite matter.[41] Origin argued, on a similar basis, that since “whatever is infinite will also be incomprehensible” so “number will be correctly applied to rational creatures or understandings.”[42] And on the rare occasion that Augustine speaks of infinity, he also echoes this Platonic tradition, but attributes to God perfect knowledge of infinitely many numbers in a potentially infinite sequence, like the naves of the Latin Basilica, processing towards a semi-circular altar that opens its all-comprehending circumference upon the congregation just as God humbled himself to enter the world.[43]

The apogee of ancient agony over the question of divine infinity thus arose during the Arian controversy concerning whether Christ, who had been begotten by God, could be recognized as equally divine, and ipso facto, equally infinite with God the Father. Arius and his supporters argued that since Christ was begotten he must also have begun to exist at some previous time, before which he had not existed.[44] The resulting diminution of the divine infinity of Christ thus combined elements of both Tertullian’s re-finitized infinite and Plotinus’ negative infinity: for Christ’s beginning portrayed the Son as temporally limited like the finitized infinite of Tertullian’s materially bounded God; while the God without a beginning portrayed the Father as the pure supra-temporal power of Plotinean infinity. And since Christ’s finitinitized infinitude was distinct and subordinate to God’s pure infinitude, Arian also suggested, like Speucippus, an extreme transcendence of disconnected superior and subordinate principles. Worst of all, once the infinite power of God had been disconnected from the only begotten Son, Arius had also unwittingly diminished God: for once the finitized infinitude of Christ was subordinated to the infinitude of God, then the divine infinity of God excluded the finitized infinity of Christ; but since exclusion may only result from finitude, Arian’s subordinationist theology also implied that, not only Christ, but even God the Father must be bounded in a kind of finitized infinity. Athanasius loudly protested against these implications, and championed the orthodox doctrine that Christ could never be subordinated, even as a finitized mediating principle between God and man, without also abandoning his co-equal divine infinity, the very possibility of substitutionary atonement, and the central doctrines of the Christian faith.[45]

Gregory of Nyssa has been honoured as the first to attribute a positive conception of infinity to God for the purpose of refuting the subordinationist implications of Arianism.[46] In response to Eunomius, Gregory wrote: “the Divine Nature is without extension, and, being without extension, it has no limit; and that which is limitless is infinite.”[47] God is thus no longer described as ‘not finite’, in the merely negative terms of Philo’s indeterminate infinity. The divinity of Christ rather implies a distinct second person who, after Tertullian, is united in one God. But where Plotinus’ pure infinity had produced an infinite negativity that dissolved all particular beings, Gregory’s Trinitarian theology preserves finitude in a positive infinity by incorporating the finitude of Jesus’ humanity into the intra-Trinitarian procession of the divine essence. Where Tertullian, like Philo, rendered each person finite, Gregory recognized, in response to the Arian heresy, that the full divinity of Christ could only be maintained if each person were distinct yet fully infinite. Hence, God is consequently rendered so effusively infinite in goodness, power, and love as to beget a second divine person, which is related back from the second to the first by a third person.

The Christological controversies are generally regarded as purely doctrinal disputes in the formative years of Christian theology. But once it is recognized all of the classical formulation of divine transcendence may have been shaped by the Platonic eidetic infinity, then it becomes possible to re-read these controversies on the nature of Christ as conflicts that were, in many respects, centred upon the nature of infinity: for Philo God was indeterminately infinite; for Plotinus, the One was a pure negative infinity; but for Gregory, God must be attributed a positive infinity precisely because of the positive incarnation of divine infinity in the human finitude of Christ. Since, moreover, any notion of infinity that exceeds and evacuates finitude produces paradoxes, it is possible to argue that it is only the Christian Trinity which preserves the finite in the infinite and the infinite in the finite; which pacifies its ancient agony; and which plumbs the idea of infinity to its inmost depths. In the Trinity alone may each distinct person be distinguished, and, from this ‘second difference’ between the divine persons, it may likewise exceed but remain interrelated by an analogous relation to a third.[48] Gregory’s divine infinity may thus integrate all of the differences of the divine persons precisely because it transcends and intermediates each of their distinctions in the Trinitarian economy of differences within the identity of the divine essence. And since its infinity is shared between the divine persons in the Trinity, it may also succeed in realizing the hints of the positive and negative double-transcendence of Plato, in a triune infinity that had only been dimly anticipated but placed beyond the world in Xenocrates’ triad of supreme principles. This Trinitarian infinity was modelled into the classical Greek Basilica, in which many semi-circular domes surround one sovereign dome, which often depicts Christ Pantocrator as the solemn judge of humanity in the towering majesty of his divine infinity.

B. The Medieval Dispute Over Divine Infinity

Pseudo-Dionysius of Areopagite made Gregory of Nyssa’s divine infinity into the paradoxical pillar of theology, in which God is both dizzyingly absent and dazzlingly present. He characterized God as unutterable, ineffable; beyond Mind, beyond Life, beyond Being” and who “transcends all the opposition between the Finite and the Infinite” [49]; as “boundless” and “infinitely powerful” [50]; and whose greatness is “Irrepressible, Infinite, Unlimited, and, while comprehending all things, is Itself Incomprehensible.”[51] But where, for Gregory, divine infinity had acted as a foil for the finitized infinity of Arian subordinationism, Pseudo-Dionysius described it, in terms more reminiscent of the Pythagorean limit-forms, as “the Form producing Form in the formless, as a Fount of every form; and it is Formless in the Forms, as being beyond all form.”[52] He also insisted that this divine infinity is the supra-numerical fountain of the numerical unity in creatures, since God “exceed[s] all Number, penetrating beyond all Infinity.”[53] For Pseudo-Dionysius the effusive power of divine infinity may act to numerically unite all finite creatures from a super-celestial height beyond ever difference, and even the contradictory mysteries of faith.[54] The truest knowledge of theology may thus only be conferred by his singular “spiritual light”, which illuminates every mind by “renewing all their spiritual powers.”[55] This spiritual illumination of the contradictory mysteries of faith became the signature of the Gothic aesthetic of divine infinity. Thus when Abbot Suger of St. Denis began to rebuild Saint-Denis as the first Gothic church, he had this verse by its pseudonymous patron inscribed on the doors: “The noble work is bright, but, being nobly bright, the work/ Should brighten the minds, allowing them to travel through/ the lights/ To the true light, where Christ is the true door.”[56]

Once divine infinity had been elevated to a spiritual clerestory to shine its light upon every definite proposition, early medieval theologians largely followed Augustine in maintaining totally silent on the subject. [57] Yet Augustine’s spirit could not rest quietly until it too had reached these divine heights. Anselm of Canterbury, who had imparted so much impetus to the beginning of Latin Scholasticism, had all-but named divine infinity when he characterized God as “a being than which nothing greater can be conceived.”[58] This background Dionysian belief is also evident in the fourth of the Five Ways for proving the existence of God, in which Thomas Aquinas describes how a gradation of goodness in beings always presupposes an utmost paradigm of goodness.[59] He argues that divine simplicity entails divine infinity because what is simple is not finitely bounded or partitioned.[60] And he clearly reaffirms an infinite divine power when he answers that God “himself, his essence, his wisdom, his power, his goodness are all without limit, wherefore in him all is infinite.”[61] But against Anselm’s ontological argument, Aquinas adopted Aristotle’s caution and rejected the possibility that finite creaturely intellects could know actual infinities.[62] He argues that since any idea of number must be abstracted from counting many things, and it is impossible to count an infinite multitude, then “no set of things can actually be inherently unlimited, nor can it happen to be unlimited.”[63] Hence, Aquinas attributed infinity to the divine essence, but denied that it could be known by any created intellect. Yet once Aquinas rejected the possibility of creaturely knowledge of infinity, he may also have inadvertently suggested that even his own attribution of divine infinity could not be truly known. This unwieldy combination of Dionysian divine infinity and Aristotelian potential infinity thus created a crack in the shell of Thomistic theology from which the new virtual infinity could be borne.

The Franciscan Order heralded this new notion of infinity for the purpose of emulating the infinite charity of Christ by exceeding every finite limit. Their anarchic impetus to re-enact the more authentic faith, poverty, and charity of the apostles beyond every institutional limit set by the diocese and the monastery compelled them to search for some intelligible sign of infinity. St. Bonaventure first charted this course when he described how Being itself “is most highly one and yet all inclusive”, “last because it is first”, the “greatest in power” and closest “to the infinite.”[64] Henry of Ghent further argued that even Aristotle’s potential infinite may signify the positive and pure act of divine infinity.[65] He rejected Avicenna’s view that the essences of things must exist simply because they are eternally apprehended by the divine intellect, and re-interpreted existence as a relation of any possible essence to the cause of divine creation. Once existence had been folded back into the causal interrelations of essence, Henry could render the enumeration of a potential infinity of existents as a sign of actual infinity in the divine essence. But since even this signification must be recognized as an intellective being, Henry’s sign of infinity inadvertently reintroduced the sign of potential-to-actual infinity as an infinite signifier on the representational and semantic plane of knowledge. Yet where Plotinus’ pure infinity had been hung like a super-celestial star beyond the firmament of all Being, Henry’s infinite signifier rested instead in a virtual matrix of possible knowledge. By thus rendering Aristotle’s potentially infinite enumeration into the sign of infinity, Henry of Ghent began to conversely represent divine infinity as little more than that which can be signified by a virtual infinity.

John Duns Scotus completed this Franciscan virtualization of divine infinity by inverting the Aristotelian-Thomistic ascent from creatures to God into a pure Plotinean descent from virtually signified divine infinity to every finite determination.[66] Where Aquinas had proceeded from divine simplicity to divine infinity, Scotus reversed this procedure to argue from virtual infinity to divine simplicity.[67] He contended, against Aquinas, that it was spurious to infer from Boethius’ principle of individuation that “not being limited by matter entails being infinite.”[68] The consistently Aristotelian consequence of any negation of the material, the limited, and the finite should rather be nothing more than a potential infinity of finite enumerations, which might only be elevated to the heights of divine infinity if divine infinity had already been implicitly presupposed. Thus Scotus makes this hidden presupposition explicit when he invokes Plotinus’ pure infinity to represent God as an “infinite power” of “infinite motion.”[69] But because all knowledge of infinity is meant to be derived from Henry of Ghent’s signification of a potential infinity, and any potential infinity is little more than a quantitative enumeration, Scotus must also transform divine infinity into an infinitely extended quantity. He writes: “We might imagine that all the parts… remained in existence simultaneously. If this could be done we would have an infinite quantity, because it would be as great in actuality as it was potentially.”[70] Every finite partition of this quantitatively extended virtual infinity then becomes yet another virtual representation within a thoroughly semanticized ontology, which can be viewed in the structurally superfluous decorations on the flattened screens of late Gothic Rayonnant facades.[71]

Nicholas of Cusa may have been the first medieval thinker to enter and exceed this Franciscan virtualization of divine infinity by suspending every quantitatively extended and virtual infinity from the singular union of quality and quantity in the divine infinity of Christ in the Trinity. He followed Duns Scotus in re-imagining divine infinity as an extended virtual space, but followed Pseudo-Dionysius in radically re-affirming the indivisibility of the Godhead beyond all contradictions in a way that challenged its virtual representability with a series of aesthetic paradoxes. For example, a century before Copernicus he famously advanced the opinion that since, in an infinite sphere every point is equidistant from the periphery, the centre must be located everywhere and nowhere, and every point both is and is not located in the centre of the universe.[72] Aristotle had anticipated these paradoxes but rejected any actually infinite extension in space. Yet after Duns Scotus had virtually represented Plotinian infinity in a quantitatively extended space, Nicholas of Cusa returned to Plato and accepted these paradoxes as ineliminable fixtures of the imaginative intellect, that were suspended from an analogical source of spiritual light.[73] He believed this doctrine to have been mystically revealed to him on a return voyage from Constantinople, but it may also have been shaped by the influence of Dionysian Neo-Platonism emanating from Chartres.[74] Once the quantitative multiplicity had been represented as a virtual infinity, Nicholas of Cusa could similarly invoke this current of Dionysian Platonism to suspend every aesthetic paradox from the central paradox of Christ in the Trinity.[75] By constructing a complex aesthetic topography of paradoxical spaces that converged upon the theological centre of Christ, Nicholas of Cusa could represent an inconsistent plurality of perspectives converging towards the kind of mystagogical infinity that is displayed in Northern Renaissance triptychs, such as Van Eyk’s Ghent Alterpiece.[76]

C. The Sublime Aesthetic of the Modern Virtual Infinity

The modern notion of infinity developed from the further partitioning, separation, and enfolding of the late Franciscan virtual infinity in the conceptual void that had been broken open by its fall from the divine infinity. William of Ockham dropped the axe that finally severed the virtual from the divine infinity when, contrary to Duns Scotus, he added a further distinction to Henry of Ghent’s signification of divine infinity: where Ghent had made potential infinity a sign of actual infinity, Ockham recommended that it should, more modestly, be understood as merely a sign of some further but merely metaphorical potential infinity.[77] And he further argued that even if God were signified by some potential infinity, then God would not be perfect (ens eminentissimum) because, for any degree of perfection a potential infinity may always produce some greater degree of perfection. Ockham thus concluded that, since God is maximally perfect, God could not be signified by any potential infinity, which could only signify an iterative enumeration of imperfectly finite things.[78] The Godhead was thus decapitated from any place beyond the Franciscan virtual infinity. And once this virtual infinity was made to signify nothing beyond itself, it became possible for it to be folded into a self-enclosed, a-theological, and secular infinity. This further finite enclosure of infinity had been previously foreshadowed by the successive stages in the abstract symbolization of natural numbers, geometry, and algebra.[79] But it was Ockham’s self-signifying virtual infinity that finally suggested, to Raymon Llull, Peter Ramus, and others, the project of constructing a completely virtual representation that might, in principle, replace all reality by exhaustively quantifying and calculating each and every part.[80]

Once the virtual infinity had been made into a totalizing virtual representation, modern physicists, like the ancient atomists, could begin to infinitesimally divide and chart the motion of each part within the whole. Giotto di Bondone, and Lorenzo Ghiberti and Filippo Brunelleschi began to illustrate how it could be surveyed to draft the earliest linear perspective paintings of the Italian Renaissance. Galileo Galilei subsequently applied Eudoxus’ eidetic infinitesimal division of infinite continued fractions to calculate the movement of projectiles and falling bodies. He referred back to Nicholas of Cusa’s On Conjectures when he described the many paradoxes of infinity (e.g. Aristotle’s Wheel) that result in a coincidence of contrary quantifiable (quantum) and non-quantifiable (non-quantum) magnitudes.[81] The theoretical conditions for René Descartes’ coordinate system for charting geometric magnitudes had similarly emerged within the representational matrix created from amidst this late-Franciscan sundering of a virtually infinite realm of representation from the mysterious depths of divine infinity.[82] Once divine infinity had been transformed into a self-signifying virtual infinity, and all its physical movements had been exhaustively represented with no sign of divinity betwixt its infinite magnitudes, Blaise Pascal could bemoan the solemn terror of a sublime void that devours all finite understanding: “I feel engulfed in the infinite immensity of spaces whereof I know nothing, and which know nothing of me.”[83]

This early modern project of replacing reality with the virtual reality of a Mathesis Universalis was nearly carried to completion in the infinite physical cosmology of Isaac Newton and the infinitesimal monadology of Gottfried Wilhelm Leibniz. In response to Descartes’ Principles of Philosophy, Newton transformed the bounded heliocentric universe into an unbounded Euclidean space. He wrote that space “extends infinitely in all directions” and time "is eternal in duration.”[84] By extending the dimensions of the Cartesian coordinate system across an immeasurable physical plane, he re-constructed a new Stoic cosmology, in which the finite material universe hangs vertiginously amidst an infinite void.[85] Leibniz likewise radicalized the infinitesimal intension of all composite substances into an actual infinity of atomic monads. His study of Galileo’s paradoxes of infinity had persuaded him that infinity could not be understood as either a sequence of whole numbers (numerus omnium unitatum), the number of all numbers (numerusn umerorumo mnium), but only as the greatest number (numerums maximus), which alone abided by the logical axiom that the whole is greater than its parts.[86] He wrote: “I believe that there is no part of matter which is not, I do not say divisible, but actually divided; and consequently the least particle ought to be considered as a world full of an infinity of different creatures.”[87] But Leibniz’ actual infinity of numerums maximus, like his highest being of ens realissimum, did not so much return to the Dionysian divine infinity as affect an infinitesimal division of the post-Cartesian virtual realm of representation. The influence of this virtualization of space, light, and motion may be seen in the punctuated rhythm of the innumerable ornaments, optical effects, and curvilinear spaces, that are celebrated in the great Baroque churches, such as Gian Lorenzo Bernini’s St. Peter’s Basilica and Christopher Wren’s St. Paul Cathedral.[88]

The aesthetic of the sublime was borne at this time from the Pascalian dread of a lonely consciousness suspended between the Newtonian infinite cosmos and the Leibnizian infinitesimal monad. Plato and Aristotle never appear to have distinguished the sublime from the beautiful, which rather tends to be described in terms that prefigure Kant’s notion of the moral sublime.[89] And while Aquinas had elevated the beautiful to a transcendental attribute of the divine essence he found no reason to distinguish it from divine infinity.[90] The sublime instead emerges only after the uniquely modern sundering and supplanting of divine infinity by the virtual infinity, and its terrible aesthetic of all-enveloping but entirely empty spaces. Longinus’s treatise on the sublime effects of rhetoric was thus predictably re-introduced in the Baroque age by Nicolas Boileau to illustrate these dizzying heights reflected from an “absolutely unknowable void.”[91] But elite disillusionment with its suffocating artificiality prompted a Rococco return to the sublime aesthetic of virtual infinity in the varied simulations of nature, such as Marie Antoinette’s rustic retreat, le Hameau de la Reine à Versailles.[92] Edmund Burke thus distinguished the sublime, in contrast to the pleasure of the beautiful, as a terrific excitement of pain, danger, and death.[93] Immanuel Kant, likewise, initially described the sublime in these allegorical terms.[94] But in the in the Analytic of the Sublime, he transformed it into a pure judgment of the imagination when he describes how “sublimity is not contained in anything in nature, but only in our mind.[95]  This transformation of the aesthetic of the sublime into a pure judgment of virtual infinity transposed it from the realm of Newtonian-Leibnizian representation to an ever more rarified plateau of transcendental judgments. And since Cusa and Galileo had already shown that the virtual infinity had never been any less paradoxical than Aristotle’s actual infinity, Kant’s last great critique collapsed into a captivating but intractable morass of aesthetic paradoxes.[96]

Immanuel Kant’s critiques had seemed to destroy, together with every objectified proportion, any possibility of the mediation of the sublime within the beautiful. [97] But he had also inadvertently carved out a Dionysian door for a sublime escape from the virtual realm of representation beyond the limits of reason alone. The tightening knot of solipsistic systematicity, especially in the hands of Johann Gottlieb Fichte, prompted a new Romantic search for the sublime in some semblance of infinitude beyond the pure judgments of the imagination, into the irreducible spheres of feeling (Herder), morality (Schiller), and poetry (Schlegel). Johann Gottfried Herder quickly recognized that, for all its pretensions of finitistic sobriety, Kant’s “supersensible reason aiming at absolute totality” amounted to “an unbounded fantasy that steps into the infinite.”[98] Friedrich Schiller took one step further beyond its bounds when he recast the sublime as an irresistible vital power of sensuous infinite “before which those of ours vanish into nothing.”[99] Where the Kantian sublime had merely pointed to an infinite ideal, Schiller re-envisioned it as a captivating ‘genii’ that could tear the “independent spirit away from the net, wherewith the refined sensuousness ensnared him, and which binds so much the more tightly, the more transparently it is spun.”[100] Schiller’s moral rapture thus summoned the moral imagination beyond every net of virtual representation.[101] But it was Karl Wilhelm Friedrich Schlegel who first found the key for the re-divinization of the virtual in the poetics of the sublime when he wrote: “Romantic poetry alone is infinite.”[102] And he pointed even further beyond when he wrote: “The true object of the art should be, instead of resting in externals, to lead the mind upwards into a more exalted region and a spiritual world… by means of a system of Christian philosophy founded on religion.”[103] For the purpose of restoring the sublime aesthetic of the divine infinity, Schlegel advocated the restoration of “the grand, the boundless, and infinite, concentrated in the idea of an entire Gothic fabric”, which he celebrated as “a rare and truly beautiful combination of contrasting elements, conceived by the power of human intellect, and aiming at faultless perfection in the minutest details, as well as in the lofty grandeur and comprehensiveness of the general design.”[104]

This Romantic restoration of the Gothic aesthetic required the aesthetics of the infinite to be re-imagined as well, beyond the counterfeit model of virtual representation, in a genuine theology of divine infinity. Kant had presented the virtual infinity as little more than the empty postulation of a regulative ideal, consisting in a Euclidean matrix of pure intuition in terms reminiscent of the infinite Newtonian cosmology, and the thing-in-itself in terms reminiscent of the infinitesimal Leibnizian monadology.[105] But his antinomies of cosmological and aesthetic judgment had also suggested that its insuperable contradictions might only be resolved by an aestheticized dialectic that would inspire the romantic genius to create ever new forms of art. Thus the Oldest System Programme of German Idealism describes how poetry “obtains a higher dignity” when it “becomes again in the end what it was in the beginning – teacher of the human race.”[106] Fichte had initially conceived the infinite as an asymptotic approximation to some infinite goal that merely ‘ought-to-be’.[107] But Schelling had fully naturalized Fichte’s infinite ‘ought’ into the “infinite activity of nature” at “every stage of becoming.”[108] This infinite spirit of nature had, for Schelling, remained bounded within the finite parameters of physics until it was liberated by Hegel from the virtual net of representation.[109] In the Philosophy of Nature, he recognized, for the first time since Nicholas of Cusa, how “space is a contradiction” whose “truth is the self-transcendence of its moments” in time.[110] And in the Science of Logic, Hegel argued that Fichte’s infinite ‘ought-to-be’ must produce a finitized infinity of determinate limits; engender an intrinsic self-contradiction between every finite limit and its own infinite ‘ought’ to exceed the finite; and, by this contradiction, propel a ceaseless oscillation between the finite and the finitized infinite until even this determinate oscillation may be dialectically sublated into the “genuine infinite.”[111] Once every virtual representation of such potentially infinite but ultimately finite spaces had been exposed as contradictory, and the contradictions of the virtual infinity had been folded back into the Idea, then its negative infinity could be made positive, and its aesthetic of the sublime could once more usher hungry souls from the matrix of representation back into the mysterious Gothic space of divine infinity.

[1] Hesiod. Theogony. ll. 116-138. After the Greeks had vanquished the Persian Empire on sea and land, Pheidias chose to sculpt the image of the centauromachy into the pediment of the Parthenon as a triumphant celebration of the victory of the finite human proportions.
[2] Anaximander Fragments 1-3. Recent scholarship has emphasized the productive generation of limited beings from Anaximander’s unlimited Apeiron, which abysmally foreshadows every subsequent positive infinity. Cf. Finkelberg, Aryeh. “Anaximander’s Conception of the “Apeiron””, Phronesis, 38, 3, 1993: 231; Sweeney, Leo. Divine Infinity in Greek and Medieval Thought, 1998: 542-546.
[3] Philolaus writes: “Nature (physis) in the world-order (cosmos) was fitted together out of things which are unlimited and out of things which are limiting, both the world-order as a whole and everything in it.” Fr. 1. Porphyry reports that Pythagoras “learned the mathematical sciences from the Egyptians, Chaldeans and Phoenicians.” Cf. Porphyry. The Life of Pythagoras, trans. Kenneth Sylvan Guthrie: §6, p.82.
[4] Philolaus writes: “The first thing fitted together, the one in the centre of the sphere, is called the hearth.” Fragment 7. Cf. Philolaus of Croton: Pythagorean and Presocratic. Huffman trans., 1993.
[5] Klein, Joseph. Greek Mathematical Thought and the Origin of Algebra. 1968: 64-66.
[6] Philolaus. Fragment 2. Cf. Fragment 3: “For you cannot know what is not - that is impossible - nor utter it.” Cf. Philolaus of Croton: Pythagorean and Presocratic. Huffman trans., 1993.
[7] Fragment 7: “For this shall never be proved, that the things that are not are…” Cf. Coxon, A. H. The Fragments of Parmenides: A critical text with introduction, translation, the ancient testimonia and a commentary, 2009. 
[8] Plato. Phaedo 100b; Republic, 596a. This principle of the unlimited plenitude of Ideas was extended to any Idea that could be signified by any name (e.g. the Ideas of couches and tables). When Socrates admits to the unseemliness of Ideas of “hair or mud or dirt”, Parmenides reassures Socrates that his doubts have arisen because he still gives too much “attention to what the world will think” and “philosophy has not yet taken hold” of him. Cf. Plato. Parmenides, 130d-e.
[9] Rosen, Stanley. “Ideas”, The Review of Metaphysics, 16, 3, 1963: 416-418. Plato also entertains an anti-foundational circle of infinite reason at the conclusion of the Theaetetus, where he characterizes the search for apodictic certainty in finite judgments as “the most vicious of circles” leading towards the “most absolute darkness.” Cf. Plato. Theaetetus, 209e; Cf. Findlay, John Niemeyer. Plato: The Written and Unwritten Doctrines, 1974: 228. And, in his most enigmatic dialogue, the Parmenides, Plato casts out upon the “vast and hazardous sea” of dialectic to signal the inconsistency of Parmenidean ontology in the first hypothesis, and Pythagorean arithmology in the second hypothesis. Cf. Plato. Parmenides, 137a, 137c-142a, 142b-155e; Cf. Sayre, Kenneth. Plato’s Late Ontology, 1983: 49-60.
[10] Plato. Philebus, 16c-17a; 26d-e.
[11] Ibid. 26e; Cf. Achtner, Wolfgang. “Infinity as a Transformative Concept in Science and Theology”, In Infinity: New Research Frontiers, Heller, Michał & Woodin, W. H. eds., Cambridge University Press, 19, 2011.
[12] Plato suggests this problem in the Philebus when, after signalling that pleasure is a metonym for the unlimited, he describes a paradox of pleasure, in which whatever is sought for pleasure becomes no longer pleasurable in the very instant in which  satisfied. Cf. Plato. Philebus, 31b-32d.
[13] Ibid. Philebus, 26d, 27a, 28c; Timaeus, 30b. The unlimited may thus be incorporated into the infinite self-motion, goodness, and power of the World-Soul. Cf. Plato. Phaedrus, 248e; Laws, 256b.
[14] Plato. Theaetetus, 147d; Euclid, The Elements, Bk. X. The Pythagoreans responded with a more murderous move when they purportedly drowned Hippasus at sea after he had discovered that the square root of the number two is an irrational number that could never be reduced to a finite fraction. Cf. Iamblichus, Vita Pythagorica, 18: 88.
[15] In the Method of Mechanical Theorems, he anticipated by Cavalieri’s method of indivisibles and Newton-Leibniz’ infinitesimal calculus when he attempted to calculate the surface of a sphere with an infinity of right triangles. Plato may also be read to have suggested the possibility of calculating the surface and composition of a sphere using right triangles when in the Timaeus he describes the onto-noetic construction of every complex structure within the cosmos from basic right triangles. Cf. Plato. Timaeus, 54a.
[16] Aristotle. Physics, Bk. III, ch.4; Plato. Republic, 596a-b; Phaedo, 100c; Paul Tannery hypothesized that Zeno’s arguments for the absurdity of infinitesimal divisibility were a response to non-extant Pythagorean writers, whose monadic limit-forms suggested infinitesimal atoms. Cf. Tannéry, Paul. L'Histoire de la science héllène, 1887; Berryman, “Ancient Atomism”, Stanford Encyclopedia of Philosophy, 2011. <>
[17] This mereological paradox of composition applies mutatis mutandis to every later reiteration of materialism from the Epicureans to the French materialists.
[18] Aristotle, Physics, Bk. III Ch. 4. 203b30.
[19] Ibid. 202b-208a.
[20] Aristotle may have alternatively attributed a non-quantitative actual infinity to the power of the First Mover. Cf. Mondolfo, Rodolfo. L’infinito ne pensiero dell’ antichita classica, 1956. For a dissenting opinion, see Sweeney, Leo. Divine Infinity in Greek and Medieval Thought, 1998: 143-167.
[21] Plato. Republic, 509b.
[22] Plato. Parmenides, 137d.
[23] Dillon, John. Heirs of the Old Academy: A Study of the Old Academy (347-274 BC): 42; Cf. Iamblichus, De Communi Mathematica Scientia.
[24] Ibid., 54-56. John Dillon argues that only five ontological levels deserve to be counted, but a consistent application of Speucippus’ causal equivocity may have implied a potentially infinite series of levels. Aristotle seems to corroborate this criticism when he characterized Speucippus’ as a “bad tragedy.”; Cf. Aristotle, Metaphysics, Bk. 14, 1090b20.
[25] Many of the distinctive Middle Platonist doctrines such as the Ideas in the mind of God, the correlation of the Greek pantheon with the supreme Ideas (e.g. One : Zeus :: Dyad : Poseidon :: Mixture : Athena), and even the emanation of the world from the One can be traced back to Xenocrates. The doctrine of emanationism, in which the Godhead continuously pours forth creation, is generally attributed to Plotinus, but Varro suggests that Xenocrates may have been the first describe the Ideas as actively generated from the mind of God, and Aetius reports that he had already described the Dyad in distinctly emanationist terms as the ‘Everflowing’. This, together with allusions by Plato (Republic 508e, Phaedrus 245c), hints that Plotinus had quite correctly recovered and radicalized an authentic doctrine of Plato and his earliest students. Cf. Dillon, John. Heirs of the Old Academy: A Study of the Old Academy (347-274 BC): 100, 154. Cf. Dillon, John. The Middle Platonists, 80 B.C. to A.D. 220: 22-39.
[26] Dillon, John. Heirs of the Old Academy: A Study of the Old Academy (347-274 BC): 100-107. Plato often indicated that he favoured a four-tier ontology, which may have been modelled on the Pythagorean Tetractys: for example, in the Republic (533e-534a), he describes four epistemic levels of noesis, dianoia, pistis, and eikasia; in the Philebus (23c-26e), he describes four principles of limit, unlimited, mixture, and cause; in the Timaeus (31b, 39e) he describes the four elements and four creatures. Aristotle also hints that Plato may even have intended to theoretically reconstruct the cosmos on these Pythagorean principles when he reports (Metaphysics 1090b) that “[t]here are some who think that, because the point is the limit and extreme of the line, and the line of the plane, and the plane of the solid, there must be entities of this kind.” In light of the evidence of Plato’s unwritten lecture On the Good, this report may be read as a veiled reference to Plato, or Plato’s students. Cf. Aristoxenus, Elementa Harmonica II 30-31; Cf. Gaiser, Konrad. “Lecture on the Good.” Phronesis, 25, 1, 1980: 25-27. Cf. Findlay, John. Plato and Platonism: An Introduction, 1978: 40-48.
[27] Cf. Gn. 17:1, 21.33; Dt. 4:39, 32:40, 33:27; Ps. 8:27, 32:9, 90:2, 139:7, 134:6, 143:3, 144:3, 145:5, 147:5; Jb. 11:4, 38:1; Is. 46.9; Rm. 11:33; Ep. 3:8; Jn. 1:3.
[28] Philo of Alexandria. The Works of Philo, Complete and Unabridged, C.D. Yonge trans.; Cf. Philo, On the Eternity of the World: 15; Plato, Timaeus, 29e.
[29] Ibid. Cf. On the Creation, 134; Questions and Answers on Genesis I, 5.
[30] Guyot, Henri. L'Infinite Divine: Depuis Philon Le Juif Jusqu'a Plotin, 1906. Guyot’s thesis has been contested by H.S. Wolfson for spuriously inferring divine infinity from the absence of determinate attributes. However, once determinate attribution is recognized as a Pythagorean limit-form, then the absence of any determinate attributes may be correlated with the current of Middle Platonist emanationism. Cf. H.A. Wolfson, Philo, Foundations of Religious Philosophy in Judaism, Christianity and Islam, Cambridge, 1947: 2.126-138
[31] Guyot, Henri. L'Infinite Divine: Depuis Philon Le Juif Jusqu'a Plotin, 1906, 50-55; Cf. Geljon, Albert-Kees. “Divine Infinity in Gregory of Nyssa and Philo of Alexandria”, Vigiliae Christianae, 59, 2, 2005: 169.
[32] Philo of Alexandria. The Works of Philo, Complete and Unabridged, C.D. Yonge trans.; Cf. The Special Laws, I. III. 20.
[33] Plato. Republic, 509b.
[34] Longinus. On the Sublime, H.L. Havell trans., 1980: I.4.
[35] Ibid.: IV.4. Cf. Shaw, Phillip. The Sublime, 2007:4-5. Plato does not make the modern distinction between the sublime and the beautiful because he conceived of the beautiful (kalon) as the fine, the temperate, and the good. Plato thus tended to assimilate beauty to the virtue, which is exemplarily portrayed in the sublime life of Socrates. Cf. Hippias Major, 285e; Charmides, 154b-e. Longinus may have been a friend of the founder of Neo-Platonism, Ammonius Saccas, and the teacher of Plotinus’ biographer Porphyry. Cf. Longinus. On the Sublime, H.L. Havell trans., 1980, Introduction by Andrew Lang, 1980: xv-xviii.
[36] Plotinus, Enneads. V.5. 10-11, in MacKenna, Stephen. Plotinus: The Enneads. Second Ed.
[37] Cunningham, Conor. Genealogy of Nihilism, 2002: 5: “Plotinus develops a meontological philosophy in which non-being is the highest principle. The One is beyond or otherwise than being.”
[38] Hart, David Bentley. “Notes on the Concept of the Infinite in the History of Western Metaphysics”, in Infinity: New Research Frontiers, ed. Michael Heller, 2014: 262-3: “[I]f the truth of all things is a principle in which they are grounded and by which they are simultaneously negated, then one can draw near to the fullness of truth only through a certain annihilation of particularity.”
[39] The Gnostics expressed this agony when they asked why, if the One is perfectly Good, and every emanation is less good and more evil, should evil emanate from the goodness of the One? The Gnostics proposed to circumvent, but could never resolve this problem, by producing a phantasmagoria of new myths, such as the stray passions of the Aeon Enthymesis, that promised to emancipate the initiate by successively infinitizing the finite. Cf. Irenaeus of Lyon. Against the Heresies. BK.I.21.4: “deficiency and suffering had their origin in ignorance, the entire system originating in ignorance is dissolved by knowledge (gnosis).” Cf. Jonas, Hans. The Gnostic Religion, 2001: 188-194.
[40] The nativity, crucifixion, and the pieta are among the most deliberate attempts to represent this singular coincidence of the sublime and the beautiful.
[41] Tertullian. De Carne Christi V, 4
[42] Origen. De Principiis, Bk.II.Ch.9.
[43] Augustine of Hippo. City of God, IV, Bk.XII,18. Centuries later, Georg Cantor would cite this passage as a Patristic precedent for his no less Platonic doctrine of transfinite sets. Cf. Cantor, Georg. “Letter from Cantor to Hermite”, Nov. 30, 1895, in Meschkowski (1967), 262.
[44] Williams, Rowan. Arius: Heresy and Tradition, 2001, 48-66.
[45] Augustine also argued that when Christ had said “I and the Father are one” (Jn. 10:30) he had also affirmed co-equality of Christ and God. Cf. Augustine of Hippo. Tractates on the Gospel of John, tr.36.9.
[46] Mühlenberg, Ekkehard. Die Unendlichkeit Gottes bei Gregor von Nyssa Gregors Kritik am Gottesbegriff der Klassischen Metaphysik. Göttingen, 1966; cf. Sweeney 1992: 473–504.
[47] Gregory of Nyssa. Against Eunomius, IX.3.
[48] This analogous relation of any of the two divine persons to a third is tantamount, not only to Augustine’s subsistent relations in the immanent Trinity, but also to the principle paradigm of the analogy of being in the vestigial trinitatis of grammar and creaturely relations. 
[49] Pseudo-Dionysius of Areopagite. The Divine Names, II.1:77; V.10: 142.
[50] Ibid. The Divine Names, VIII.1:155.
[51] Ibid. The Divine Names, IX.3:163.
[52] Ibid. The Divine Names, II.1:77.
[53] Ibid. The Divine Names, IX.2: 162.
[54] Ibid. The Divine Names, IX.7: 167: “[T]he same things are both like unto God and unlike Him: like Him in so far as they can imitate Him that is beyond imitation, unlike Him in so far as the effects fall short of the Cause and are infinitely and incomparably inferior.”
[55] Ibid. The Divine Names, IV.4: 91: “From the Good comes the light which is an image of Goodness; wherefore the Good is described by the name of “Light,” being the archetype thereof which is revealed in that image.”
[56] Abbot Suger. On What was Done in His Administration, XXVII. Concerning the Cast and Gilded Doors.
[57] John Scottus Eriugena, who himself had translated and commented upon Pseudo-Dionysius, had described God as the “infinity of infinities.” Cf. The Division of Nature, I.517b; II.525a. No mention of it is found, for example, in Peter the Lombard’s Sentences. Cf. Burns, Robert. “Divine Infinity in Thomas Aquinas: 1. Philosophical-Theological Background”: 58.
[58] Anselm of Canterbury. Proslogion, Bk.I, ch.5.
[59] Thomas Aquinas. Summa Theologica. IA. Q.2, A.3. Etienne Gilson describes how “the very foundation of this doctrine [for Aquinas is] the universally accepted doctrine in medieval theology… that God is infinitely above anything we can think and say about him.” Gilson, Etienne. The Unity of Philosophical Experience, 1937: 108.
[60] Thomas Aquinas. Summa Theologica. IA. Q.3 & Q.7.
[61] Thomas Aquinas. On the Power of God, Q.1, A.2.
[62] Aquinas argues against Anselm that even if the word ‘God’ signifies that of which nothing can be thought “it does not therefore follow that he understands that what the word signifies exists actually, but only that it exists mentally.” Anselm had assumed that the divine essence implied divine existence, but once Aquinas, following Aristotle and anticipating Schelling, had divided essence from existence, the former could not imply the latter. Cf. Thomas Aquinas. Summa Theologica. IA, Q2, A.1. Obj. 2.
[63] Thomas Aquinas. Summa Theologica. IA. Q.7. A.4. Where Plato had robustly conceived of eidetic numbers as Ideas generated by the supreme Principles and multiply instantiated in numerically distinct sensible objects, Aristotle rejects eidetic numbers to thinly re-conceive them as little more than abstract concepts generalized by the intellect from quantities of numerical distinct sensible substances. Cf. Plato, Parmenides, 143c; Aristotle, Metaphysics, 987b. For Aristotle’s criticisms of Plato’s eidetic numbers, see Metaphysics XIII, ch.6-8.
[64] Bonaventure. The Mind’s Road to God. Ch.5, 7.
[65] Henry of Ghent. Summa II, A.11, Q.2. Gilson describes this as a “complete reversal of the Greek idea of infinity conceived as the condition of that which, being left unfinished, lacks the determinations required for its perfection.” But this may only be construed as a reversal if ‘infinity’ is rendered as the Pythagorean-Platonic ‘unlimited’ (apeiron) opposed to the proportionate limit-forms. Once, however, every limit-form is elevated, in the Platonic tradition,  to an infinitely reproducible perfect paradigm, then it becomes evident that Henry of Ghent had not so much reversed as reiterated the Greek idea of infinity on the quasi-transcendental and epistemic plane of intellective being. Cf. Gilson, Etienne. History of Christian Philosophy in the Middle Ages, 1955: 448-9.
[66] Leclerc, Ivor. The Nature of Physical Existence, 2002: 68.
[67] Cross, Richard. Duns Scotus: Great Medieval Thinkers, 1999: 26. Cf. Thomas Aquinas. Summa Theologica, I.II.3c.1.1.49b.
[68] John Duns Scotus. Ordinatio.,n.143; Cit. Cross, Richard. Duns Scotus: Great Medieval Thinkers, 1999: 40.
[69] Ibid. D.2, Q.2.7.I 7: 89. Scotus argues, for instance, that since “the totality of essentially ordered causes is from some cause that is not any part of the totality”, and an infinite number of essentially ordered causes could not exist at once, there must be an infinitely superior cause of the essentially ordered causal series. Cf. John Duns Scotus. Ordinatio. D.2, Q.2.7.III 39-53: 104-109.
[70] Ibid. Quod.5,m.2; Cf. Cross, Richard. Duns Scotus: Great Medieval Thinkers, 1999: 40; Richard Cross notes that this virtual actualization of potential infinity anticipates post-Cantor mathematics.
[71] Pickstock, Catherine. “Duns Scotus: His Historical and Contemporary Significance”, Modern Theology, 21:4, 2005: 547: “[U]nivocity is for Scotus a semantic thesis regarding the constancy of meaning through diverse predications, all the same he tends to semanticise the field of ontology itself, through his thesis of essential and virtual inclusion.”
[72] Nicholas of Cusa. On Learned Ignorance II, c.11; Cf. Hoff, Johannes, Kontingenz, Berührung, Überschreitung: Zur philosophischen Propädeutik christlicher Mystik nach Nikolaus von Kues, 2007: 309-324.
[73] Borsche, Tilman. “Das Bild von Licht und Farbe in den philosophischen Meditationen des Nikolaus von Kues”, in Viderere et videri coincident: 163-182. Cf. Hoff, Johannes. The Analogical Turn: Rethinking Modernity with Nicholas of Cusa. 2013: 40.
[74] Bond, H. Lawrence. Nicholas of Cusa: Selected Spiritual Writings. Introduction, 1997: 20. Thierry of Chartres had earlier responded to the nominalist tritheism of Roscelin of Compiègne by recollecting Pseudo-Dionysius’ Neo-Platonic doctrine that, just as every multiplicity is derived from some prior unity, so might every contradiction be resolved by the complex participation of all differences in the One. Cf. Pseudo-Dionysius of Areopagite. The Divine Names, IX.7: 167: “[T]he same things are both like unto God and unlike Him: like Him in so far as they can imitate Him that is beyond imitation, unlike Him in so far as the effects fall short of the Cause and are infinitely and incomparably inferior”; Cf. Nicholas of Cusa, On Learned Ignorance, I.19.57; Proclus. Elements of Theology, 1099: 32–35; Plato. Sophist, 256e-257c.
[75] Milbank, John. Mathesis and Methexis: the Post-Nominalist Realism of Nicholas of Cusa, Unpublished draft: 31: “Cusa notably affirms the latter by saying that everything paradoxically participates in the very God who cannot be participated.”
[76] Hoff, Johannes. The Analogical Turn: Rethinking Modernity with Nicholas of Cusa. 2013: 70.
[77] William of Ockham. Quodlibetal Questions. II Q. 2; 1980: 112–16; Cf. Gracia, Jorge ed. A Companion to the
Philosophy of the Middle Ages, William of Ockham, 2008: 708-714.
[78] Ibid. VII, Q.21. Cf. Tweedale, Martin. “Scotus and Ockham: On the Infinity of the Most Eminent Being”, Franciscan Studies, 23, 1963: 265.
[79] The Babylonian place-value number system made subsequent zero signs into a multiplier of natural numbers (i.e. 1, 10, 1000,…, etc.); the Egyptians had likewise employed place-holder symbols for unknown magnitudes in quadratic equations; Aristotle had separated abstract numbers from Pythagorean arithmology and Platonic eidetic numbers; and Diophantus, Al-Khorwarizmi, and François Vieta had gradually substituted natural numbers for abstract variables. Cf. Klein, Joseph. Greek Mathematical Thought and the Origin of Algebra, 1992: 104: “Accordingly, [to Aristotle's theory of abstraction] the methematika have their being 'by abstraction', that is, their separate mode of being arises from their being 'lifted off', 'drawn off', 'abstracted'. This is why the 'dependence' of mathematical formations works no detriment to their noetic character.”
[80] Pickstock, Catherine. After Writing: On the Liturgical Consummation of Philosophy, 1998: 49-70.
[81] Knobloch, Eberhard. “Galileo and Leibniz: Different Approaches to Infinity”, Arch. Hist. Exact Sci., 54, 1999: 91; Cf. Nicholas of Cusa. De coniecturis. In Werke. Ed. by P. Wilpert, v.1, 1967: 147.
[82] Descartes indicated his indebtedness to this tradition when he famously argued, in his Third Meditation, from the innate idea of an infinity being to the existence of God. His innate idea of an infinite being is neither Augustinian divine illumination, nor even Platonic recollection (anamnesis), but rather a distinctly Ockhamist repetition of Duns Scotus’ argument from virtual infinity to divine simplicity. Cf. Descrates, René. The Third Meditation.
[83] Pascal, Blaise. Pensées, §205.
[84] Harrison, Edward. “Newton and the Infinite Universe”. Physics Today, 39, 2, 24, 1986: 24.
[85] Harrison, Edward. “Newton and the Infinite Universe”. Physics Today, 39, 2, 24, 1986: 24. Newton wrote “there is nothing in space, yet we cannot think that space does not exist, just as we cannot think that there is no duration, even though it would be possible to suppose that nothing whatever endures. This is manifest from the spaces beyond the world, which we must suppose to exist (since we imagine the world to be finite).”
[86] Rescher, Nicholas.Leibniz' Conception of Quantity, Number, and Infinity”. The Philosophical Review, 64, 1, 1955: 111. The paradox of an infinite set of all numbers that is not greater than its parts had already been recognized by Galileo, but would also later emerge as one of the decisive problem for naïve set theory, for example, in Cantor’s Paradox and Russell’s Paradox.
[87] Leibniz, Gottfried Wilhelm. “Letter to Foucher”, Journal de Sçavans, March 16, 1693, G I 416.
[88] Cf. Leibniz, Gottfried Wilhelm. The Principles of Philosophy Known as Monadology. §§36-41
[89] Plato. Hippias Major, 285e; Charmides, 154b-e; Aristotle. Metaphysics, 1078b.
[90] Thomas Aquinas. Summa Theologica. I, Q.39, A.8. Cf. Caygill, Howard. A Kant Dictionary, 2000: 91.
[91] Milbank, John. “Sublimity: The Modern Transcendent”, in Transcendence: Philosophy, Literature, and Theology Approach the Beyond, Regina Schwartz ed. 2004: 211.
[92] This naturalization of the sublime produced an early dispute between the advocates of an infinitely objective account of the beautiful, which Wolf and Baumgarten described as a formal proportion, and an infinitesimally subjective account of the beautiful, which Shaftesbury and Hutcheson characterized it as inner sense. Cf. Caygill, Howard. A Kant Dictionary, 2000: 91.
[93] Burke, Edmund. On the Origin of the Sublime and the Beautiful. Of the Sublime. 1757: §VIII, p. 148. John Milbank seeks to save Burke’s aesthetic of the sublime from Kant’s ‘metaphysics of the sublime’ when he writes: “Burke it is established in a distinctively different fashion which rather than placing the Sublime over the Beautiful by removing the role of eros, instead achieves the same thing by dividing the erotic itself.” Cf. Milbank, John. in Transcendence: Philosophy, Literature, and Theology Approach the Beyond. Ed. Regina Schwartz, 2004: 222.
[94] Kant, Immanuel. Observations on the Sublime and the Beautiful. Kant often describes the sublime as the infinite that exceeds all proportions and the beautiful as the finite teleological relations within proportions: Lofty oaks and lonely shadows in sacred groves are sublime, flowerbeds, low hedges, and trees trimmed into figures are beautiful. The night is sublime, the day is beautiful… the sublime must be simple, the beautiful can be decorated and ornamented… the virtue of the woman is a beautiful virtue. That of the male sex ought to be a noble virtue.” He unambiguously repudiates his former psychological description of the sublime when he writes: “The transcendental exposition of aesthetic judgments that has now been completed can be compared with the physiological exposition, as it has been elaborated by a Burke and many acute men among us, in order to see whither a merely empirical exposition of the sublime and the beautiful would lead.” Cf. Kant, Immanuel. The Critique of Judgment, Guyer and Matthews trans., 2000: §29, 5: 277, p. 158.
[95] Kant, Immanuel. The Critique of Judgment, Guyer and Matthews trans., 2000: §§28-29, 5: 263-277, pp. 147, 158: The transcendental exposition of aesthetic judgments that has now been completed can be compared with the physiological exposition, as it has been elaborated by a Burke and many acute men among us, in order to see whither a merely empirical exposition of the sublime and the beautiful would lead.
[96] Kant himself describes the principle antinomy of aesthetic judgment, in which the purity of the concepts of beauty and sublimity result a dilemma between the pure aesthetic judgment that is thought but not felt, or felt but not thought: for if there were some universal concept that could demonstrate an aesthetic judgment, then there would be no freedom either for artistic taste or creativity; but if, to the contrary, there were no such universal concept, then neither could there ever be any expectation of universal agreement. Cf. Kant, Immanuel. The Critique of Judgment. Guyer and Matthews trans., 2000: §8, §46.
[97] John Milbank describes how "the sublime is only sublime as a rupture in this or that context, or of this or that beautiful proportion" Cf. Milbank, John. “Sublimity: The Modern Transcendent”, in Transcendence: Philosophy, Literature, and Theology Approach the Beyond, Regina Schwartz ed. 2004: 221. Heine also colourfully characterized Kant as the “arch-destroyer in the realm of thought”. Cf. Heine, Heinrich. Religion and Philosophy in Germany. Snodgrass trans., 1959: 109.
[98] Herder, Johann Gottfried. Kalligone, 1800; Cf. Guyer, Paul. “Free Play and True Well-Being: Herder’s Critique of Kant’s Aesthetics”, 2006.
[99] Schiller, Friedrich. “On the Sublime”, In Naive and Sentimental Poetry, and On the Sublime: Two Essays, Julius A. Elias trans., 1967. Schiller emerges at this decisive juncture as, not merely the first to ethicize the sublime aesthetic of the infinite, but also the first to aestheticize the post-Kantian political order.
[100] Hegel writes of Schiller: “It is Schiller who must be given great credit for breaking through the Kantian subjectivity and abstraction of thinking and for venturing on an, attempt to get beyond this by intellectually grasping the unity and reconciliation as the truth and by actualizing them in artistic production.” Cf. Hegel, Georg Wilhelm Friedrich. Lectures on the Philosophy of Aesthetics, Bosanquet trans., 1886: 116.
[101] Schiller highlights the escape from the virtual realm of phenomenal representation when he writes: “We gladly allow the imagination to find its master in the realm of phenomena, for it is ultimately, however, only one sensuous force, which triumphs over another sensuous one, but nature in all of its limitlessness can not attain to the absolute greatness in us ourselves… Man is in its hand, but the will of man is in his own.” Cf. Schiller, Friedrich. “On the Sublime”, In Naive and Sentimental Poetry, and On the Sublime: Two Essays, Julius A. Elias trans., 1967.
[102] Schlegel, Karl Wilhelm Friedrich. Athenaeum Fragments, 1798, Source of original German text: Friedrich Schlegel, Kritische Schriften, ed., Wolfdietrich Rasch. Munich: Carl Hanser Verlag, 1958. 
[103] Schlegel, Karl Wilhelm Friedrich. Appeal to Painters of the Present Day, 1804, Source of original German text: Friedrich Schlegel, Kritische Schriften, ed., Wolfdietrich Rasch. Munich: Carl Hanser Verlag, 1958.
[104] Schlegel, Karl Wilhelm Friedrich. From The Fundamentals of Gothic Architecture, 1803, Source of original German text: Friedrich Schlegel, Kritische Schriften, ed., Wolfdietrich Rasch. Munich: Carl Hanser Verlag, 1958.
[105] Kant, Immanuel. The Critique of Pure Reason, Paul Guyer trans., 1998: A25/B40; A209/B25. See especially the cosmological antinomies: A485/B513.
[106] This anonymous text, which was discovered by Franz Rosenweig in Hegel’s notebooks and Hegel’s handwriting, unabashedly confesses: “I am convinced that the highest act of reason, which, in that it comprises all ideas, is an aesthetic act, and that truth and goodness are united like sisters only in beauty-- The philosopher must possess just as much aesthetic power as the poet. The people without aesthetic sense are our philosophers of the letter. The philosophy of the spirit is an aesthetic philosophy.” Cf. Classic and Romantic German Aesthetics, J.M. Bernstein ed., 2002: 185-187.
[107] Fichte, Gottlieb. Introductions to the Science of Knowledge, Heath and Lachs trans., 1991: I, §3, 115-118.
[108] Schelling writes: “Nature is limited must again contain an infinity in itself. Within every sphere other spheres are again formed, and in these spheres others, and so on to infinity… Nature organizes, where it organizes, to infinity.” Cf. Schelling, Joseph Wilhelm Friedrich. First Outline of a System of a Philosophy of Nature, Keith Peterson trans., 2004: 34, 44.
[109] Hegel remarks that the infinite and the infinitesimal “have their true notions in philosophy itself; it is wrong headed to think that they should be borrowed and adapted from mathematics, where they are not employed in conformity with the Notion, and where they are often taken up at random.” Cf. Hegel, Georg Wilhelm Friedrich. Hegel’s Philosophy of Nature, M.J. Petry trans., 1970: §259, I. 234.
[110] Hegel, Georg Wilhelm Friedrich. Hegel’s Philosophy of Nature, M.J. Petry trans., 1970: §257, I.229. Hegel recognized that Kant’s antinomies of aesthetic judgment, like Nicholas of Cusa’s paradoxes of virtual infinity, produced contradictions for any intuitive conception of space and time. He writes: Time, like space, is a pure form of sensibility or intuition; it is the insensible factor in sensibility… It is limited, and the other involved in this negation is outside it. Consequently, the determinateness is implicitly external to itself, and is therefore the contradiction of its being. Time itself consists of the abstraction and contradiction of this externality and of the restlessness of this contradiction.” Cf. Hegel, Georg Wilhelm Friedrich. Hegel’s Philosophy of Nature, M.J. Petry trans., 1970: §258, I.230.
[111] Hegel, Georg Wilhelm Friedrich. Hegel’s Logic, A.V. Miller trans., 1969: §§269-272, pp. 136-137.