THINKING BEYOND IDENTITY: NUMBERS AND THE IDENTITY OF

INDISCERNIBLES IN PLATO AND PROCLUS

John V. Garner

Abstract: In his Euclid commentary, Proclus states that mathematical objects have

a status in between Platonic forms and sensible things. Proclus uses geometrical

examples liberally to illustrate his theory but says little about arithmetic. However,

by examining Proclus’s scattered statements on number and the traditional sources

that influenced him (esp. the Philebus), I argue that he maintains an analogy

between geometry and arithmetic such that the arithmetical thinker projects a “field

of units” to serve as the bearers of number forms. I argue that this conception of a

“multitude,” wherein each unit differs in no way from the others, implies that Plato

and Proclus do not recognize unqualifiedly what would become the principle of the

identity of indiscernibles. I argue that Cratylus 432c in particular provides support

for a reading of Plato as consistently thinking beyond the principle of identity. I

conclude by drawing out an important epistemological and ethical lesson from this

reading.

Introduction

In part one of the Prologue to A Commentary on the First Book of Euclid’s Elements,

Proclus outlines his general theory of the soul’s projective understanding of mathematical forms.1

Part two of the commentary then applies this theory specifically to geometry.2 What is missing in

his commentary is an in-depth consideration of the general theory’s implications for arithmetic.

Euclid himself treats arithmetic in Book VII of the Elements; but we have little of what Proclus

may have thought about that text.3 Despite his focus on geometrical examples, I will argue here

that Proclus’s commentary does have some important hints about his views on number.

Part I briefly reviews the main features of Proclus’s general theory of “mathematicals.” It

then pieces together an account of the place of number in this theory. Part II argues that the Platonic

precedents for this view of number, especially in the Philebus and Cratylus, shed light on the

importance attributed by both Plato and Proclus to the practice of what I call “thinking beyond

identity,” or more specifically to thinking beyond what would later be called the principle of the

identity of indiscernibles.4 Above all, the practice of thinking beyond identity, as I argue, helps

philosophers to distinguish the rigorous demands of thinking numbers properly from the more

forgiving practice of making or understanding images well. While this thesis has evident

implications for Platonic epistemology and the relation of the knower to the sensible world, I also

wish to emphasize the ethical implications. For without this basic distinction, I suggest, the

philosopher’s very attitude to, or way of life within, the sensible world risks becoming excessively

austere and uncharitable.

Part I – Mathematical Projections and Unit Groups

Two features of Proclus’s general theory of mathematics are particularly salient.5 First, he

argues that what he calls “mathemeticals” are entities different from both Platonic forms and from

sensible things. Mathematical objects, he writes, “have the status neither of what is partless and

exempt from all division and diversity nor of what is apprehended by perception and is highly

changeable and in every way divisible.”6 The triangle itself or the circle itself is partless, and yet

any figure that is indeed triangular or circular, even a hypothetically perfect figure that we might

envision mentally, would have the added complexities of containing lines and points. Even a

perfect, extended triangle, therefore, is not what it is to be a triangle per se but rather simply the

perfection of triangle-in-instantiation. 7 This “intermediate” status of mathematicals will be

important for us later for grasping Proclus’s alternative to Aristotelian induction. But it suffices

here to note that things classed as intermediates or mathematicals will “go beyond the objects of

intellect” because of their plurality; but they “surpass sensible things in being devoid of [sensible]

matter.” 8 They are inferior to forms but superior to sensible things. Following loosely the

parallelism of the Republic’s “divided line,” Proclus argues that distinct cognitive faculties are

coordinated to these distinct strata of objects. The perfectly simple awareness of the perfectly

simple forms is carried out by intellect (nous); the imprecise perception of sensible things belongs

generally to the domain of opinion (doxa).9 According to the tripartite ontological distinction

forged before, therefore, the faculty of grasping the mathematicals will be an intermediary between

these powers. This faculty he calls understanding (dianoia).10

Second, and most importantly for our purposes, Proclus considers the proper activity of

dianoia to be not simply the passive reception of information either from an external object or

from the higher intellect. Rather, when attempting to understand mathematical ideas, dianoia is an

activity, indeed an activity that uses what Proclus calls the self-motion of the imagination

(phantasia).11 When the understanding tries to grasp the precise nature of the partless circle or the

square itself, for example, it inevitably spreads the concentrated insight of nous into an “expressed”

format accessible to dianoia. The imagination, writes Proclus,

is moved by itself to put forth [proballei] what it knows, but because it is not outside the

body, when it draws its objects out of the undivided center of its life [i.e. intellect], it

expresses them in the medium of division, extension and figure. For this reason, everything

that it thinks is a picture or a shape of its thought [noēmatos]. It thinks the circle as

extended, and although this circle is free of external matter, it possesses and intelligible

matter [noētēn … hulēn] provided by the imagination itself.”12

Notice that Proclus says not that the imagination produces images of things that are previously

seen with the senses. Rather, he claims that the fundamental role of the dianoetic imagination is to

produce an expression of what is previously intellected. This projection (probolē) occurs precisely

because the soul cannot in its present condition comprehend the full significance of its higher,

strictly intellectual grasp of forms. 13 Because of this weakness, the soul must express the

concentrated truths by projecting them into what Proclus, modifying Aristotle, here calls

“intelligible matter.”14 This intelligible matter provides the triangle in intellect with the lines and

points that render it, precisely, a humanly understandable entity, i.e. something extended and able

to be studied by the soul through the use of its more familiar, extra-intellectual resources.15

Importantly, this projective activity of understanding is not a pointless self-expressive act.

Rather the understanding’s projection is accompanied by and enables what Proclus calls a

subsequent psychical “regathering” (sunagein au palin) into concentrated form of the expressed

form’s contents.16 In short, the soul uses its imaginary projections as self-created stepping stones

to bring itself up to the heights of the intellect. This self-ascent is possible because soul’s dianoetic

projections are purer and more accurate to the form in intellect than is anything in the sensible

world. Thus, in becoming themselves subjects of study by the understanding, these projections

guide the soul as a whole to trace them back to their origins.17 Since, as Proclus argues, the soul

“draws her concepts both from herself and from Nous” it follows that the soul’s study of the source

of projections can lead the soul nowhere but back to itself, on the one hand, and back to a source

higher than its own understanding, i.e. to intellection per se, on the other hand.18 Thus, the ultimate

goal of the projective understanding is in fact, according to Proclus, to allow the soul to recover

itself into intellectual activity, to draw itself up to intellectual concentration. Because the activity

of understanding is double in just this way—creative imagining and reflective re-concentration—

Proclus describes the understanding as “always writing itself and being written on by Nous.”19

That is, it “writes itself” insofar as it projects for itself an intelligible matter that can bear

expressions of forms; and it is “written” by intellect insofar as these projections are discovered to

contain more than the soul could have given itself from its own resources alone (while still at the

level of understanding).

As has been evident through my exclusive use of Proclus’s geometrical examples, what is

missing in Proclus’ commentary on Euclid is his views on arithmetic and Euclid Book VII. Yet

Proclus does provide us with some clues which, when combined with a close attention to Proclus’s

own sources, can yield some informed hypotheses about the direction a Platonist of his ilk should

go.

First, Proclus’s own clues are in fact fairly prevalent. He claims that geometry “occupies a

place second to arithmetic, which completes and defines it (for everything that is expressible and

knowable in geometry is determined by arithmetical ratios).” 20 Proclus probably means that

arithmetic deals strictly with what we call rational numbers; and knowledge of irrationals only

enters when we deal with the indefinite divisibility of geometric extension, which can be

approximated only through an endless process of creating more and more precise ratios

approximation.21 By saying that arithmetic “completes” geometry, he means that “arithmetic is

more precise than geometry, for its principles are simpler.”22 Proclus soon explains what he means

by “principle” here, which really accords with his definition of an “element” given in the second

prologue, namely: what proves is an element of what is proved by it; and more specifically, an

element is that into which something complex can be resolved.23 Geometric principles “resolve”

into arithmetical ones in the sense that geometry as a practice will presuppose the ability to think

number. One cannot intuit and project for oneself an extended understanding of triangles unless

one can already implicitly grasp the three.

Having laid out arithmetic’s priority, Proclus even identifies for us the most basic element

of arithmetic. “A unit,” he says, “has no position, but a [geometrical] point has; and geometry

includes among its principles the point with position, while arithmetic posits the unit.”24 The

fundamental element of arithmetic is thus the unit (hē monas). It has an ontological priority for

Proclus because of its pure, non-composite, and indivisible nature. Of course, this identification of

the basic element of arithmetic with “the unit” accords precisely with the rank Euclid gives the

unit in Book VII: “A unit [monas] is that by virtue of which each of the things that exist is called

one [hen],” says the first definition. “And a number [arithmos],” in turn, “is a multitude [phēthos]

composed of units [monadōn].”25 For Proclus, since it is the basic element of arithmetic, the unit

is the principle of all proofs involving number and the basic element of resolution of any group.

When a man is thus “counting a group of men,” says Proclus, “one man is his unit.”26 We shall see

in a moment, however, that the purest count never takes the man as the unit but rather, more

primordially, posits simply the units themselves.

Now, it is important to remember that Proclus speaks of geometry and arithmetic

analogously, such that magnitudes are analogous to numbers. The element in geometry analogous

to the unit in arithmetic is thus Proclus’s “point with position.”27 But Proclus is very careful to

argue that the point must always be conceived as divisible, i.e. “a magnitude consists of parts

infinitely divisible.”28 This divisibility guarantees that the point will never be truly independent of

the line or the interval between points, of which the point is a mere division.29 For this reason,

Proclus can point out an essential disanalogy between the point and the unit, and between geometry

and arithmetic generally. The geometer, he says, “does not assert, as does the arithmetician, that

something is least.”30 The unit, that is, is indivisible. Thus, it is not a point, though it serves the

same role as an element or principle that points serve in geometry.

Even though the point and the unit differ essentially, what is most important for our

purposes is the continued use Proclus makes of the analogy between geometric points and

arithmetic units. For just as he carefully argues that geometric ideas grasped by intellect have no

extension but are expressed as extended, so too can we see that numbers, conceived as consisting

of elemental units, may also be understood through Proclus’s model of creative understanding.31

This suggestion on my part is not entirely speculative, as the analogy between projected points and

units is made clear by Proclus himself in several places. “We must not suppose number in [the

soul] to be a plurality of monads [plēthos monadōn], nor [should we] understand the idea of

interval as bodily extension, but must conceive of all the forms as living and intelligible paradigms

[paradeigmata] of visible numbers [tōn phaimomenōn arithmōn] figures, ratios, and motions.”32

Just as geometric “extension” is very different than the “idea of the interval” itself, argues Proclus,

so too is number “in the soul” very different from a “plurality of monads.” The geometric form in

itself is “without motion or genesis, indivisible and free of all underlying matter,” say Proclus,

even though the things “latent [kruphiōs] in the form are produced distinctly and individually on

the screen of the imagination.”33 Maintaining Proclus’s analogy, therefore, we can infer that

numbers also admit of such a projection into what we have just seen are units; these units are thus

likewise “projected” entities. A group of three units is the product of a creative understanding, as

it tries to make sense to itself of the nature of the three, which it grasps intellectually in a more

concentrated form. Proclus indeed specifies, “All mathematicals are thus present in the soul from

the first. Before the numbers [pro tōn arithmōn] the self-moving numbers [hoi autokinētoi] […].”34

In this passage, I read the first use of “numbers” to refer to the projected unit groups Proclus

mentions just afterwards, while the second reference, to the “self-moving numbers,” glosses the

forms, as is made clear in the same kind of expression that Proclus uses later in reference the forms

for figures: “Prior to sense objects […] are the self-moving intelligible and divine ideas of the

figures.”35 Thus, the eidetic numbers themselves must be taken to be prior to the plurality of units

generated by dianoetic projection.36

Since, based on the preceding passage, the term “numbers” can refer in Proclus to

something other than the forms themselves—i.e. also to the projected groups of units—the

following passage is rendered particularly salient for my argument. Proclus writes, “[Let] us say

that it is by virtue of [the soul’s] otherness, that is, the plurality and diversity of the ratios in her,

that the understanding, when she has been constituted and has noted that she is both one and many,

projects numbers [tous arithmous proballei] and the knowledge of numbers, which is arithmetic

[…].”37 Above all, this passage must not be interpreted as saying that numbers themselves, i.e. the

forms, are dependent products of the soul’s projection. Rather, it must be understood that the soul,

which is here intellectually engaged in intuiting absolutely self-definite numbers, must use the

understanding to grasp the nature of those numbers.38 Thus, from out of its own resources as a

“lower” faculty than intellect and from out of the resources of the intellect itself which are gifted

to it, the projected interpretation of number is generated. The understanding’s own deficient and

indefinite nature (relative to intellect) infects the projection, and this infection explains why a

number would be conceived in a non-concentrated form, as a potentially boundless “plurality of

monads.”39

Therefore, without any direct, surviving evidence of Proclus’s comments on Book VII of

Euclid, we can plausibly reconstruct an outline of the Proclean application of the theory of creative

dianoia to arithmetic. Unit group numbers are not numbers themselves. Rather, just as in geometry,

“we must grant that [the geometer] is investigating the universal [e.g. circle], only this universal

is obviously the universal present in the imagined circles,” so too should we grant that the

theoretical arithmetician is investigating the universal (e.g. three), only this universal is obviously

the universal as present in a projected unit-group.40

Part II – The Tradition: Pure Units, Perfect Images, and the Identity of Indiscernibles

We must recognize that Proclus is working deeply within a tradition in his text. We have

already seen how he follows the general tendency of the Euclidean tradition in his conception of

the unit as the primary arithmetical element.41 What makes Proclus special, I would argue, is that

his acceptance of Euclid’s terms is suitably qualified to fit more unequivocally (than it might

otherwise be interpreted) within, for example, the Phaedo’s schema, which clearly rejects the idea

that a number itself—or what Proclus calls the intelligible number—is dependent in any sense on

units.42 According to neither Plato nor Proclus may the two itself, for example, be sufficiently

understood simply as the “emergent property” of a unit that is brought together with another unit,

nor of a single item divided.43 Proclus thus interprets Euclid with Plato or Plato with Euclid, in

combination with his Plotinian heritage, to yield a compelling, albeit probably not unique, reading

of the place of unit group numbers. This interpretation of Proclus not only solves certain problems

stemming from the Aristotelian critique in Metaphysics Mu and Nu, but it also supports, more

generally, Proclus’s efforts to establish an anti-abstractionist epistemology, a point I will return to

later.44

While Proclus is indeed part of a long tradition, one could plausibly argue that Plato’s

Philebus is the core source of the theory of perfectly identical units as intermediates between forms

and sensible groupings.45 There we find Plato’s most revealing passage specifying the nature of

these unit group numbers. The passage arises in the context of Socrates’s attempt to classify

knowledge into more precise and less precise kinds.46 He begins by looking at the knowledge

offered by various arts:

Socrates: Let us, then, divide the arts [technas], as they are called, into two kinds, those

which resemble music, and have less accuracy [akribeias] in their works, and those

which, like building, are more exact. […] And of these the most exact are the arts which

I just now mentioned first.

Protarchus: I think you mean arithmetic [arithmētikēn] and the other arts you mentioned

with it just now.

Socrates: Certainly […] (56c-d).

In context it is clear that Socrates thinks a rudimentary building activity lacking the use of

measurements is less precise than a similar practice that would be governed by the discipline of

using measures.47 Yet a bigger point emerges in the passage above when Socrates adds that any

applied science using measures will, itself, have only as much “precision” (hē akribeia) as it

employs arithmetic. Notably, Proclus himself uses this same term akribēs repeatedly in the Euclid

commentary to describe the superiority of mathematicals to sensibles. Indeed, just as Proclus later

repeats, Socrates says not only that arithmetic makes all practical precision possible but also that

any arithmetic-in-application is itself made possible by a study that is even more precise than it.48

Thus, Socrates continues:

Socrates: But, Protarchus, ought not these [arithmetical arts] to be divided into two kinds?

What do you say? […] Are there not two kinds of arithmetic, that of the people [tōn

pollōn] and that of philosophers [tōn philosophountōn]?

Protarchus: How can one kind of arithmetic be distinguished from the other?

Socrates: The distinction is no small one, Protarchus. For some arithmeticians reckon

unequal units [monadas anisous], for instance, two armies and two oxen and two very

small or incomparably large units [stratopeda duo kai bous duo kai duo ta smikrotata ē

kai ta pantōn megista]; whereas others refuse to agree with them unless each of countless

units is posited [thēsei] to differ not at all from each and every other unit [mēdemian allēn

allēs diapherousan] (56d-e, translation slightly modified).49

Socrates’s wants to show that the many, if they do indeed use measures, always count unequal

things in their counting. They use imprecise units as measures to count imprecise things. Socrates

is not disparaging this activity of applied counting or measuring. Indeed he just praised it as

superior to guesswork.50 However, he is saying that the impure precision found there has a prior

source that is more purely precise in itself, and this source needs to be acknowledged. If there were

no such perfectly precise study of pure units, then there would be no possibility of an applied

counting of impure units which, while impure, is still more precise than guesswork. In other words,

there can only be an impure precision in applied counting if there is a pure precision in a purely

precise count. But a purely precise count is more precise only if its object also admits of more

purity. Philosophical arithmetic thus thinks only of units that “differ not at all from each and every

other unit.”51

Some interpretive problems arise with this passage, and I will turn to them momentarily.

But first it is important to see that Socrates does not say in the Philebus that these pure unit numbers

are themselves the absolute measures of all precision tout court. They are the measures of purity

of objects counted in a count. Indeed he ultimately claims, in agreement with Republic VII, that

dialectic is superior because it harbors and makes possible the intellection, or noēsis, of that which

is “eternal and self-same.”52 This passage thus agrees in essence with both the Republic and the

Phaedo in allowing that a share in awareness of the forms is prerequisite to the study of unit groups,

and that among the forms are found “the two” and “the three” and other numbers.53

Now, given this long tradition of speaking of unit group numbers as superior in precision

to applied measures but inferior to the objects of dialectic and intellection, it is important to see

that Socrates’s very conception of such units presents certain logical difficulties. To highlight this

point, I would like to return first to an interpretive difficulty in the Philebus’s passage on the

numbers consisting of units that “differ not at all from each and every other unit.” For one might

be tempted to argue that Socrates cannot really mean that these units differ in no way from one

another.54 Rather, Socrates really must have meant, one might argue, only that the things counted

must belong to the same type.55 The motivation for such a reading would be quite clear: the

principle of the identity of indiscernibles (henceforth “principle of identity”), if one accepts it as

an absolute principle, demands that items can be “two” only if they are “unequal” in some way.56

Or, as Leibniz says, there cannot be “two individuals entirely similar or differing only in

number.”57 Indeed, some readers have insisted that Socrates’s examples suggesting that there can

multiple entities that differ in no way are intrinsically absurd examples.58

Here I want to suggest a different standard for reading this Philebus passage and this issue

of identical units in general. If we take it for granted that the principle of identity holds absolutely,

or that Plato adhered to it, or that if he did not, then he was confused, then we will certainly claim

that Socrates is confused if he thinks these units (in the plural) “differ in no way.” On that

assumption, we will argue that Socrates’s pure units are artificial, fictions, abstractions, and in

general not a truth-preserving presentation of a number.59 However, if we do not take the principle

of identity for granted, the passages can be read more directly; and the prospect is raised that

Socrates is not confusing anything at all. Rather, he can be seen as proposing that we conceive of

a field lying beyond any identity principle. If we can “posit” such a field of pure units, we can

thereby use this pure plurality as a mediating measure of the sensible world. This measure, as

sharing with forms an internal purity, nevertheless shares in common with the sensible world its

plurality. In the pure presentation of a pure plurality, the thinker can enable herself to measure

multitudes. By contrast, the simple and self-same forms of the “two” or “three,” while they must

inform this positing, cannot serve as direct measures for sensible items or groups.60

In short, Platonists can respond strategically to the accusation of absurdity and defend the

very practice of thinking beyond identity. Only if one first thinks a field of units can one in turn

have a relatively accurate conception of sensible sets of armies or items. The Platonist can argue

that it is precisely an immovable adherence to the principle of identity—found classically in the

empiricist traditions but also in Leibniz—that would prevent one from positing this field.61 Thus,

per the Philebus’s argument, the genuinely precise measures of sensible sets would be thought

improperly. Contingent, sensible measures of things would then risk being taken as the ultimate

measures of groups rather than as things measured by a prior measure. Platonists can argue that

strict adherence to the identity principle entails a failure to think number.63

For Socrates, I want to argue, two units can in principle be counted as two, even if they are

not “unequal” in any way. What Socrates means, I would argue, is not that we must envision units,

with a material imagination, as if they are like points in a group. Rather, he simply means that we

must creatively project a domain of difference and multiplicity that violates any strict adherence

to the principle of identity. Only this self-enabling of oneself to think non-identity will in turn

enable the soul to properly frame sensible nature and to grasp the way its multiplicity and diversity

images the forms.

Indeed, I want now to defend this last claim and argue that a proper understanding of the

sensible world as imaging the forms is enabled specifically by “thinking beyond identity.” Indeed,

this ability to understand images is precisely what is at stake in this debate about whether or not

the principle of identity holds necessarily. This point is made remarkably clearly in Plato’s

Cratylus. For Socrates there not only has us think of a case that, if it is possible, violates the

necessity of the principle of identity; it also shows us why thinking beyond identity is vital for

thinking the nature of the truth of an image qua image.

The example arises at Cratylus 431e ff. There, Socrates is in the midst of his effort to refute

a strict and absolute naturalism about names. His larger argument will be that if naturalism is true,

then conventionalism is required to some extent. Part of what Socrates needs in his argument is a

conception of the way that a name can be correct—i.e. it can function as a name of the thing it

names—even if it contains some errors in its formulation. But Cratylus denies this possibility.

(Names, we should recall, have already been classified generally by the interlocutors as a species

of image (eikōn).)

Cratylus: [If] we add, subtract, or transpose a letter [stoicheiōn], we don’t simply write the

name [onoma] incorrectly, we don’t write it at all, for it immediately becomes a different

name, if any of those things happens.

In response, Socrates marks a distinction between numbers (arithmoi), on the on hand, and names

(onomata), on the other hand.

Socrates: What you say [about missing letters] may well be true of numbers, which have

to be a certain number or not be at all. For example, if you add anything to the number

ten or subtract anything from it, it immediately becomes a different number [allos

arithmos], and the same is true of any other number you choose. But this isn’t the sort of

correctness [orthotēs] that belongs to things with sensory qualities [poiou], such as

images [eikonos] in general. Indeed, the opposite is true of them—an image cannot

remain an image if it presents all the details of what it represents.

Pausing here, we should note that Socrates’s conception of number is the typically Greek one

whereby they are conceived strictly as the rational numbers. The subtraction or addition Socrates

speaks of here is the subtraction or addition of one, or of a unit, from a number conceived as a

group of units.

Socrates (continues): See if I’m right. Would there be two things—Cratylus and an image

of Cratylus—in the following circumstances? Suppose some god didn’t just represent

your color and shape the way painters do, but made all the inner parts [ta entos panta]

like yours, with the same warmth and softness, and put motion, soul, and wisdom like

yours into them—in a word, suppose he made a duplicate of everything you have and put

it beside you. Would there then be two Cratyluses [duo Kratuloi] or Cratylus and an

image of Cratylus [Kratulos kai eikōn Ktatulos]?

Cratylus: It seems to me, Socrates, that there would be two Cratyluses [duo … Kratuloi].

Socrates: So don’t you see that we must look for some other kind of correctness [orthotēta]

in images and in the names we’ve been discussing, and not insist that if a detail is added

to an image or omitted from it, it’s no longer an image at all. Or haven’t you noticed how

far images are from having the same features as the things of which they are images?

(Cratylus, 431e-432d).

Socrates is saying that the case of numbers is not like the case of names or images. In the case of

conceiving numbers, an “error” in one’s presentation of the three means that one presents not three

units but rather four units or two units. One has thus presented a different number than what one

set out to present.64

By contrast, Socrates argues that in the case of names or images, extra features that do not

perfectly present the original can be presented in one’s presentation, or certain features can be

lacking in it, and yet the presentation—i.e. the image or the name—can remain an image or name

of the original. The image or name of X does not, simply due to an addition or subtraction of a

feature, become a name or image of some non-X or some Y.65 Nor does it even become a bad or

false image.66 Indeed, here Socrates not only says that such imperfections in names or images are

possible; he says that these divergences are necessary, if something is a name or image.

We can draw two important conclusions from this passage. First, Socrates’s thought

experiment clearly violates the principle of identity. Since the two Cratylus’s properties differ in

no way, just as the Philebus insists units differ in no way, it follows that thinking this thought of

two Cratyluses can be understood as an example of thinking the Two “into” a multitude of

identicals. Cratylus is thus presented as subjugated to the thought of a number form. The thought

of “two Cratyluses” thus, in truth, allows the form of the Two—not of Cratylus proper—to carry

out its proper function in the understanding’s presentation. (This is precisely to treat Cratylus as

what Proclus would call the “matter” of the presentation of the two.) But for just this reason,

Socrates argues, the result is that this kind of thinking is no longer a thinking of the original (i.e.

Cratylus) and its image or name (i.e. the image or name of Cratylus). We are now thinking the

Two; and thus a kind of injustice has been done in the task we set out to do, if indeed we did set

out to think-and-image Cratylus. If we are to grasp the image-original relation, therefore, a

different kind of thinking will be needed. We must think “beyond” the demand to attain perfect

duplication.67

Second, what differentiates images from numberings appears, in the two Cratyluses

example, to be precisely a question of the “elements” in terms of which the presentation must be

forged. To think a number is to think that number actively “into” the elements necessary to its

contents, namely units. By contrast, to think Cratylus properly “into” an image is to present

Cratylus into a medium wherein necessary differences from the original are always preserved.68 If

we imagine that something definite—such as Cratylus—appears in this “element” of imaging, then

we must imagine Cratylus’s definiteness as becoming mixed with an indefiniteness to generate a

mixture that can only ever approximate Cratylus’s definiteness. In this sense, any imaging must

present something unequal to the original; it presents the original into a medium of mixture where

nothing can be exactly identical either to itself or to anything else.69 For if it were identical to itself,

it would be the original itself, and thus not an image. And if it were identical to anything else, it

would be a duplication, and thus a presentation of number, not an image. Thus, the proper

presentability of Cratylus—or indeed any form—achieves its truth, goodness, and fulfillment as

an image only if and insofar as it presents something—the image or name—that remains always

unequal to—and yet still ordered to—the original. If we think of the element of imaging as the

Philebus’s “more-and-less,” we can see that truth and goodness in the imaging of originals cannot

be, and ought not to be, a matter of feature-by-feature exactitude.

In short: To properly think of the nature of images at all is to think of that which necessarily

cannot be an absolute and perfect duplication of an original. For then the original and image would

be thought as equals; and the thought would render the original, so to speak, as the matter of the

number two. The schematization would not, therefore, present a true image of an original. It would

try to present either two identical originals or two identical images; and either of these thoughts

sufficiently leaves out the concept of an image-of-an-original, which is what it is to be an image

per se. Thus, the schematization would in fact be the presentation of the number two of which

Cratylus is taken up as the “matter,” not a presentation of an image of Cratylus.

Conclusion – Thinking Beyond Identity in Epistemology and Ethics

From the Cratylus we have arrived at three key claims. First, Plato is comfortable, in more

places than just in his discussion of monads in the Philebus and Republic, with implying that

numbers are properly presented as consisting of units that differ in no way. Second, he is

comfortable with thinking experimentally the implication of these multiple identicals. The

implication is that one thereby thinks a number “into” intelligible units or “into” a multitude of

identicals. This presentation of number lacks perfect simplicity (as sensible groups do as well), but

it is a well-defined presentation, informed by the number form itself. Whereas the principle of

identity would have required that “two Cratyluses”—or any supposed multitude of identicals—be

just one and the same substance (or else they would be a mere abstraction or fiction), Socrates

concludes, precisely, that they would be two, i.e. a precise instantiation of the Two.

Third, and most importantly, the thought of the two identicals is a “generative” thought.

That is, as soon as one thinks it, one no longer thinks the image-original relation but generates

another medium for thinking number. If one were attempting to discern what makes a name or

image true or good qua image—as the Cratylus was doing—then one can be assured that, if one

ends up thinking the set of identicals, one has missed out on one’s aim.70 But, for this same reason,

the power to generate this field of identicals likewise shows us that, if we aim to image Cratylus

in a presentation rather than to present a number, then we are required (but also thereby permitted)

to not perform a perfect duplication. Imaging is thus revealed to us as being true or good not by

the standard of perfect duplication but by a different measure, i.e. whether or not one captures the

pattern, style, or type of the original (tupos).71 It is imperative, demandingly, that we not think we

can perfectly duplicate through an image; and it is permitted, graciously, that we do not have to do

so.

Thus, we arrive finally at the grander, epistemological lesson I think we should draw.

Proclus also insisted on this lesson, which is that any abstractionist epistemology is doomed to

misunderstand the nature of the image-original relationship.73 For an abstractionist epistemology

pretends that the soul can draw from out of what has less “precision”—such as the imprecise

activities of measuring and counting things in the imprecise world of the senses—an awareness

that has more truth and precision than that very starting point. As Proclus argues decisively, this

is impossible, unless the soul were already endowed with access to that precision and truth and

providing it to its experience.74 For no soul that lacks access to the original could grasp the image

as an image, i.e. as something lacking in being the original while also serving to present it.75 Thus,

likewise, no soul could even begin to abstract from sensibles without being previously informed

by the original of the original’s content, which is above and beyond what this sensible example

sets before sensibility. This recognition is required if one is to grasp the sensible reality as not the

original to which it is ordered; and thus, it is required if one is even to be inspired to try to

“abstract.” This lesson, I think, is the general basis for why Proclus insists that unit numbers must

be creatively projected by the soul a priori. For without a positing of a pure multitude, the soul

cannot even recognize that the multitude of sensible objects are not the original measures of

number. Thus, Proclus insists that these intermediate numbers do not come from sensible things

but “come from the soul, which adds perfection to the imperfect sensibles and accuracy to their

impreciseness.”76 While it lacks the developed theory of projection or imagination, Socrates’s

argument in the Philebus is conducted in the same spirit, since it is the pure precision of the

philosopher’s “posited” units that makes possible any precision in any empirical sense.

In short, without thinking beyond identity, one might hold that the goodness and truth of

an image is found in its “differing in no way” from the original. Thinking a multitude of identicals,

by contrast, enables us to grasp that even the fulfillment of such a rigorous mimetic demand would

not yield the desired unification with the original. It would yield only a perfect duplication, or an

instance of the two. Thinking beyond identity thus frees us from the law of perfection in this sense:

an image-qua-image—and thus, by implication, the nature of sensible things at large—can attain

its good and true version only by remaining necessarily distinct from the original. Only if we can

understand the nature of images in this way—as not being required to attain perfection—are we

freed to leave perfection aside and, instead, to seek simply the goodness proper to the image. Only

then can the images be interpreted properly as images of originals that must remain distinct. Thus,

the realizations enabled by thinking beyond identity are, I would argue, the necessary

epistemological pre-condition for distinguishing the originals (or forms) from images (or

sensibles); and this distinction is what is needed in order to grasp the import of the originals in

their own self-determining distinctness. All of these realizations must be enabled before any so-

called “abstraction” could begin.77

If indeed a proper understanding of sensibles or forms is not possible without the

epistemological pre-conditions outlined above, then Proclus can be understood when he claims

that sense-experience has no importance for epistemology other than to awaken the soul to the

gifted source of its own essential projections. Those projections, which serve as “paradigms” for

sensible things, are themselves merely “likenesses” of the forms grasped intellectually. 78 In

geometry, these paradigms are the “perfect figures,” and they must be prior for soul to the

“imperfect figures” of the senses. In arithmetic, as I have argued, the same epistemology holds

true. The units we attain in thinking numbers fulfill perfectly the demands of the law for presenting

number; and this fulfillment frees us to treat images, and thus sensible things, in light of the more

charitable idea that they can never, and ought not even be asked to, present perfection to us.

In just this sense, the epistemological lesson here is also a wholly ethical lesson. For the

true measure of something is whatever it is to be a truly good version of that something. This

lesson applies to images—and to sensible things—as much as to anything else. And thus the

measure of the proper truth and goodness of an image qua image is not to be a double-original.

Rather, the measure of the truth and goodness of an image qua image is goodness-as-an-image.

This means that goodness-as-an-image is achieved only in remaining an image, i.e. in remaining

distinct from and ordered to the original. The same holds true of the sensible world of becoming

as a whole, and thus to treat the sensible world fairly, justly, or in the broadest sense well, we must

not ask it to be perfect. Thus, we learn an ethical lesson here as much as anything else.

In the end, then, thinking beyond identity, which occurs when we properly present number

as a multitude of identicals, is just the thought we need in order to be able to think—by contrast—

images as images. The prior thought of number makes possible the posterior thought of sensible

things. At the same time, the positing of number reveals to us our prior, but previously merely

enfolded and unexpressed, intuitive grasp of the originals. For this reason, I would argue, Socrates

says in the Republic that it is the study of numbers that every soul needs if it is to begin its path of

understanding. Only then are we empowered to discern originals and their images in light of them.

By positing units that violate the identity-demand, we can become empowered to discern forms

and their images charitably, and in their own proper lights.

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1 Proclus, Euclid, 3-38. I would like to thank Andrew Gregory and the participants at London Ancient Science

Conference 2017 for commenting on a presentation of an earlier draft of this project. 2 Proclus, Euclid, 39-69. 3 For speculations regarding an explanation for this fact, see for example O’Meara, Pythagoras, 167. 4 This paper owes a great deal to the work of Leonard J. Eslick, especially his essay “Two Cratyluses,” which draws

out some of the implications of Cratylus 431e ff. for the principle of identity. 5 The following paragraphs owe enormously to Nikulin, “Imagination.” 6 Proclus, Euclid, 9 [11]. 7 See Garner, Emerging Good, chapter 4 for an elaboration on this claim as it pertains to the Philebus. 8 Proclus, Euclid, 4 [4]. See the discussion of “intelligible matter” in Proclus just below. 9 Proclus, Euclid, 3 [4]. On doxa in Proclus, see Helmig, Forms and Concepts, chapter VI. 10 Proclus, Euclid, 3 [4]. On dianoia in Proclus, see Helmig, Forms and Concepts, chapter VII. 11 Proclus, Euclid, 41-46 [51-57]. See also Nikulin, “Imagination.” 12 Proclus, Euclid, 42 [52-53]. 13 Proclus, Euclid, 23 [28]. Here, Proclus clarifies that the value of mathematics is for its own sake, even though it

nevertheless would not occur without human weakness. “We must therefore posit mathematical knowledge and the

usefulness that results from it as being worthy of choice for their own sakes, and not because they satisfy human needs.

And if we must relate their usefulness to something outside them, it is to intellectual insight that they must be said to

be contributory.” 14 See especially Nikulin, “Intelligible Matter.” 15 “For the understanding contains the ideas but, being unable to see them when they are wrapped up, unfolds and

exposes them and presents them to the imagination […]” (Proclus, Euclid, 44 [54-55]). 16 Proclus, Euclid, 3 [4]. 17 For an example of this projection, we can think of the way the understanding grasps a triangle. By examining

mentally the projected, extended triangle, made up of divisible parts, the soul “in turning back to itself […] would

obtain a superior vision of the partless, unextended, and essential geometrical ideas that constitute its equipment”; and

this self-examination leads “to more perfect intellectual insight” and “[emancipates] it from the pictures projected in

imagination” (Proclus, Euclid, 44-45 [55]). 18 Proclus, Euclid, 14 [16], my italics. Proclus argues at length that the origin of the dianoetic projections must come

from two sources, i.e. from the soul’s own less-than-intellectual power of dianoia, on the one hand, and the power of

intellect “gifted” to it from above, on the other hand. See Proclus, Euclid, 30 [36-37]. 19 Proclus, Euclid, 14 [16], my italics. On the broader Proclean conception of self-constitution, see MacIsaac, “Origin.” 20 Proclus, Euclid, 39 [48]. 21 See Taylor, “Forms and Numbers,” which outlines the method for the approximation of the irrational. See also

Fowler, Mathematics. 22 Proclus, Euclid, 48 [59]. 23 Proclus, Euclid, 60 [72]. 24 Proclus, Euclid, 48 [59]. 25 Euclid, Elements, 277. 26 Proclus, Euclid, 33 [40]. 27 Proclus, Euclid, 48 [59]. 28 Proclus, Euclid, 216 [278]. 29 Proclus, Euclid, 216 [278]. 30 Proclus, Euclid, 33 [40]. 31 See Nikulin, “Imagination,” 158: “the difference between (‘quantity’ and ‘magnitude’) matches that between

numbers and geometrical figures.” 32 Proclus, Euclid, 15 [17]. Importantly, Proclus contrasts the groups of units with the pre-expressed forms of numbers,

wherein “each number, such as five or seven, appears to every mind as one and not many, and as free of extraneous

figure or form” (Proclus, Euclid, 78 [96]). 33 Proclus, Euclid, 45 [56]. 34 Proclus, Euclid, 14-15 [16-17]. 35 Proclus, Euclid, 112 [140]. 36 See Plato, Phaedo, 96a ff on forms for numbers. See also Klein, Greek, passim on “eidetic numbers.” 37 Proclus, Euclid, 30 [36], my emphasis. 38 The indefiniteness and plurality of the soul is presumably the source of the soul’s need to express the concentrated

number form as a field of identical units with indefinite or endless possible units. 39 In the same passage, Proclus clarifies that the soul is gifted these resources by the Demiurge who “took in hand the

unity and diversity of the universe, and the mixture of sameness and otherness […], and constructed [the soul] out of

these kinds, together with rest and motion […]” (Proclus, Euclid, 30 [36]). Following this construction of the soul,

Proclus adds that it is only when the soul self-reflects and “has noted that she is both one and many” that the soul

“projects numbers” (Proclus, Euclid, 30 [36]). 40 Proclus, Euclid, 44 [54]. 41 We need not see Proclus’s theory of the creative understanding as original in order to see the theory as important

for what it offers. Proclus’s commentary may simply be the most complete document that survived. For example, as

the editors of Syrianus, Metaphysics explain, Syrianus also “makes [Euclid’s] units or ‘monads’ merely the matter or

substratum of number, on which we have to impose, as form, the triad, pentad, heptad, ennead, etc., that we carry in

our souls […]. Only so can the units compose three, five, seven or nine; they are not number otherwise […]. There is

only one triad, just as we nowadays think there is only one number, three” (3). Proclus is perhaps clearer that the units

are posterior to the attempt to understand the fully unified, intellectual form. While the units are not unreal, they

nevertheless cannot exist unless a prior intellection of number forms is ongoing. Importantly, Proclus’s very notion of

“projection” itself has a long history. As the editors of Syrianus, Metaphysics explain, “So although Porphyry already

uses the term ‘projection’, it may have been introduced into geometry by Iamblichus, who sought to integrate

Pythagorean philosophy with Platonic. Syrianus […] ascribes to Plato the simpler idea that the objects of geometry

reside in the imagination, but not the idea of projection” (5). On this history, see also Helmig, Forms and Concepts,

chapter VII, 2.4. 42 Certainly, we must be cautious about assuming that the Phaedo and Proclus mean the same thing by “intelligible

number.” As we have seen, Proclus’s statements would be consistent with a conception of forms as contained “in” a

correlated intellect, while the Phaedo may certainly be interpreted as offering a stronger sense of the mind-

independence of forms. On this point see Siorvanes, Proclus, 50. 43 See Phaedo, 96a ff. See also Tait, “Second Best Method.” For Proclus the thesis, as I have argued, is that the

understanding’s active projection of units really flows from the soul’s share in intellection of numbers themselves.

We need not see the five units in the group as existing apart from the soul’s actual intellection of the form five. In this

sense, when the soul grasps a sum such as 5+7=12, then we might speculate that Proclus’s theory is more in line with

the Kantian thesis that an a priori “construction” is demanded if the conception is to be really possible (cf. Kant,

Prolegomena, Part One). See also Morrow, “Introduction” to Proclus, Euclid, lix. 44 For an overview of the Aristotelian critique of the supposed Platonic doctrine of units, see especially the introduction

to Annas, Metaphysics and Helmig, Forms and Concepts. For the response already foreseen in Plato, see Garner,

Emerging Good, chapter 4. 45 For the view that the Philebus does not imply a conception of “intermediates,” see the summary offered in Arsen,

“Utility.” For a general defense of intermediates in Plato, including in the Philebus, see Wedberg, Mathematics. 46 Proclus was certainly familiar with this passage: “In general, as Socrates says in the Philebus, all the arts require

the aid of counting, measuring, and weighing, of one or all of them; and these arts are all included in mathematical

reasonings and are made definite by them […]” (Euclid, 21-22 [25]). 47 Philebus, 56a: “Take music first; it is full of this; it attains harmony by guesswork based on practice, not by

measurement; and flute music throughout tries to find the pitch of each note as it is produced by guess, so that the

amount of uncertainty mixed up in it is great, and the amount of certainty small.” 48 Proclus, Euclid, 10-11 [12-13]. 49 The passage continues:

Protarchus: You are certainly quite right in saying that there is a great difference between the devotees of

arithmetic, so it is reasonable to assume that it is of two kinds.

Socrates: And how about the arts of reckoning and measuring as they are used in building and in trade when

compared with philosophical geometry and elaborate computations—shall we speak of each of these as

one or as two?

Protarchus: On the analogy of the previous example, I should say that each of them was two (Philebus, 56e-

57a). 50 Philebus, 56a, quotation in note 47. 51 The passage thus echoes Republic VII, when Socrates says that arithmetic as studied by experts

leads the soul forcibly upward and compels it to discuss the numbers themselves [autōn tōn arithmōn], never

permitting anyone to propose for discussion numbers attached to visible or tangible bodies. […] Then what

do you think would happen, Glaucon, if someone were to ask them: “What kind of numbers are you talking

about, in which the one [to hēn] is as you assume it to be, each one equal [ison] to every other, without the

least difference [diapheron] and containing no internal parts [morion]?”

I think they’d answer that they are talking about those numbers that can be grasped only in thought

[dianoēthēnai] and can’t be dealt with in any other way (525d-526a).

The shift between the term to hen here and hē monas in the Philebus need not be taken as a technical shift. The

immediate qualifications that follow make it clear that the basic sense is the same. Even so, it is true that this to hen

passage is vague enough to allow that Socrates might be read to mean that the units must belong to the same type. If

that is all he means, then the Philebus goes further, positing its pure units as the same in a stronger sense that merely

“same in type,” for they contain absolutely no differences one from another. On this point, see Pritchard, Mathematics,

Ch. 7. See also Wedberg, Mathematics. 52 Philebus, 61e. See also 58e-59c. 53 See Republic VII, 525d cited above as well as Phaedo, 96a ff. See also Republic VI, 510d, which mentions

mathematicians speaking and drawing not for the sake of the images but for “the square itself [tou tetragōnou autou]”

and the “diagonal itself [diametrou autēs].” 54 See, for example, the direction Pritchard goes with this passage: “the units dealt with by mathematicians are sensible

objects taken as instances of unity. Now, the Philebus passage is quite explicit as to this. The units are not said to be

identical; it is required only that they should be posited as identical. The closeness of this view to Aristotle’s is plain

[…]” (Mathematics, 124). In other words, these posited units take the sensible things and interpret those objects, taking

them as unities. Contra this reading, nothing about Plato’s use of the term thēsei here suggests that pure units are

interpretations of sensible things or groups. On the contrary, Socrates’s whole conception in this passage of the pure

precision of pure arithmetic as prior in reality and in knowledge to the applied precision of applied arithmetic suggests

that the very ability to grasp sensible objects as unities would depend on a prior real ability to originally posit pure

units and take these units themselves as the more real unities. 55 Wedberg, Mathematics, 117 considers and rejects this interpretation. 56 For a discussion of this point, see for example Mueller, “Academic Theories,” 116. He also argues that Aristotle’s

critique of Plato is motivated by the conception that no true substance—only an abstraction that ignores features of

the truth—could be conceived as identical to another: “For, on one reading, his doctrine of abstraction only entails

that one can ignore differences between units, not that one can genuinely conceive undifferentiated units” (116). See

also Eslick, “Aristotle,” which explores the Aristotelian version of the identity problem. 57 Leibniz, “Arnauld,” 336. See also Eslick, “Two Cratyluses.” 58 See, for one example, Ademollo’s reading of the Cratylus example to which we will turn shortly: “Strictly speaking

the idea of there being two Cratyluses is intrinsically absurd. For there cannot be two tokens, as it were, of the same

particular; and a perfect duplicate of Cratylus would not be another Cratylus, but a wholly distinct individual, who

after his coming to be would have a different history from Cratylus’. Yet it is rather natural to speak loosely of

Cratylus’ perfect duplicate as ‘another Cratylus’” (Ademollo, Cratylus, 365). 59 See note 56. A similar intuition may lie behind Gadamer’s reading of Philebus 56c-d as an abstractionist thesis,

along Aristotelian lines (see Gadamer, Ethics, 201). 60 “The only objects that are capable of perfectly exemplifying numerical Forms are objects that have the exact number

of parts corresponding to the numerical Forms. Because intermediate numbers are composed of wholly simple units

that determine their arithmetic properties, they are the only objects that have an exact number of parts.” Arsen,

“Utility,” 208 note 28 here summarizes Wedberg’s core argument succinctly. For the full argument that the

intermediates are alone suited to serve the functions of arithmetic in Plato, see Arsen, “Utility” and Wedberg,

Mathematics. 61 On the principle of identity in the Stoics see, for example, Barnes, “Retour éternel.” 63 Otherwise, one could argue, as does Mueller following Syrianus, that either there are such units, or arithmetic is

false or a fiction. See Mueller, “Academic Theories,” 116. 64 See the excellent summary offered by Barney, Names, 128-129: “[The] individuation of numbers is fine-grained

and exact: if x and y are numbers, and y differs from x even in the slightest possible degree, then x and y are different

numbers.” Later Barney clarifies the way that, nevertheless, there can be names of numbers and, furthermore, per

Socrates’s later argument, there must be such conventional names accompanying numbers. “[Given] the Two

Cratyluses argument, any name of x will differ from x to some degree, which means to at least as great a degree as y

does; so, depending on the nature of those differences, we may have no way of telling, on the basis of its content alone,

whether the name names x or y (or some other number differing from x in a similar way) more correctly. No matter

how well designed, the name of a number—like, say, a photograph of a particular snowflake—will not have content

which is sufficiently precise to disclose its object to the exclusion of all others. And the upshot is that convention is

in at least some cases a necessary adjunct to mimesis (435c1–2).” 65 In the following pages of the Cratylus, Socrates clarifies that images, in order to retain their status as images of

originals, only need to capture the general pattern, style, or type (tupos) of the original; number-presentations, by

contrast, need to attain perfection in their elements. See Cratylus, 432e. 66 See Barney, Names, 117, note 14. Helpfully, Barney argues that the Cratylus’s freeing of images from the burden

of having to be perfect duplicates of the original does not mean that the original is no longer “the standard.” For the

standard or criterion of goodness, F, can function as the appropriate measure or norm of X, even if doing or being X

at all implies that one cannot actually be perfectly identical to the F that measures any X. In her example, running a

race in zero time is the original measure; and even though running a race cannot be done in zero time, the standard

has not changed. In general agreement, I would only suggest the possibility of distinguishing perfection (or zero time)

from the good itself. Both the perfect instance (zero time) and the imperfect instances (e.g. the sub four minute mile)

share in the good, and the latter, while imperfect, still counts as a good example. On the good versus perfection in

Plato, see Garner, Emerging Good, chapter 3. 67 We might distinguish two senses of the importance of “thinking beyond identity” suggested by this study. First, the

thought of multitude of identicals thinks beyond the logical or ontological necessity of the principle by thinking the

mere possibility of perfect duplicates. Second, if the number-presentations can be contrasted with images in the way

Socrates does, then this proves that not all things must be measured, in a normative sense, by the measure of perfect

duplication. Thus, necessarily, an image must not be asked to live up to the standard of numbers, since that is not the

norm for a good image. 68 We might refer to the Philebus, for the element of imaging here is probably related to what the dialogue calls the

apeiron or the “more-and-less.” Socrates there attempts to think this principle as a kind of pure relativity. See Garner,

Emerging Good, chapter 2. 69 As Eslick puts it, “If we are to have many particular instances of the same Form, it can only mean that these

particulars are separated from their own essence, and that they are image-beings, deficient in reality” (“Two

Cratyluses,” 84). 70 Like the Cratylus, the Philebus is also invested in specifying the conditions of possibility of the goodness of

something that is always ontologically an emergent, or a non-original. It does this in the way it tries to think the

goodness of pleasure qua pleasure. See Garner, Emerging Good, chapter 3.

71 Cratylus, 432e. 73 Proclus, Euclid, 10-15 [12-18]. For Proclus’s critique of abstraction, see Helmig, “Proclus’ Criticism” and Helmig,

Forms and Concepts. By contrast, Klein suggests that there is a kind of practical abstraction model in Plato whereby

pure units are units of “theoretical logistic,” which arises from out of a “practical logistic” (or everyday counting of

things) precisely “when its practical applications are neglected and its presuppositions are pursued for their own sake”

(Klein, Greek, 23). We may worry that Klein’s explanation commits to the same kind of Aristotelian position Proclus

finds wanting (see note 56 above). For how could the “neglect” of some features of practical applications—which

applications are imprecise in themselves per the Philebus—yield the result of a science with greater precision, namely

pure unit arithmetic? As Proclus puts it, “And how can we get the exactness of our precise and irrefutable concepts

from things that are not precise?” (Proclus, Euclid, 11 [13]). In support of my reading against abstraction, see also

Tait, “Noēsis,” 182. See also Syrianus, Metaphysics, 90.9-15 for a precedent to the Proclean position. 74 Proclus, Euclid, 10-11 [12] and 112 [140]. For a more in-depth articulation of the role of projection in grasping the

sensible world, see especially part 3 of MacIsaac, “Non enim.” See also Helmig, “Proclus’ Criticism.” 75 “For in saying that matter receives from nature the substantial things that are more truly existent and clearer, while

soul out of them fabricates in herself secondary images and later-born likenesses—likenesses inferior in being to their

originals, since the soul has abstracted from matter things that are by nature inseparable from it—they do not thereby

declare the soul to be less important than matter and inferior to it?” (Proclus, Euclid, 12-14 [14-15]). Evidently, Ibn

Sīnā and Aquinas, among others, try to respond with grades of abstraction, different kinds of matter, and a system of

abstraction aided by quasi-independent intellectual input. See Rahman, Avicenna’s Psychology, Ch. V as well as

Aquinas, “On Being,” 2.82-84. 76 Proclus, Euclid, 10-11 [12]. 77 Thus, Proclus forcefully insists that if the soul “weaves this enormous immaterial fabric and gives birth to such an

imposing science without knowing or having previously known these ideas, how can she judge whether the offspring

she bears are fertile wind eggs, whether they are not phantoms instead of truth? […] If, therefore, mathematical forms

are products of the soul and the ideas of the things that the soul produces are not derived from sense objects, [then]

mathematicals are their projections, and the soul’s travail and her offspring are manifestations of eternal forms abiding

in her” (Proclus, Euclid, 11-12 [13]). 78 Proclus, Euclid, 112 [140]. See Proclus, Theology, 113-127 on henads.