Inventing Intermediates: Mathematical Discourse and Its Objects in Republic VII

Inventing Intermediates: Mathematical Discourse and Its Objectsin Republic VIILee Franklin

Journal of the History of Philosophy, Volume 50, Number 4, October2012, pp. 483-506 (Article)

Published by The Johns Hopkins University PressDOI: 10.1353/hph.2012.0080

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Inventing Intermediates: Mathematical Discourse and Its

Objects in Republic VIIL e e F r a n k L I n *

plato is commonly taken to be committed to the existence of intermediates, ideal mathematical particulars distinct from Forms.1 This is despite the fact that he avoids direct pronouncements on the issue in Republic VIVII, the longest sus-tained discussion of mathematics in his corpus.2 nevertheless, interpreters argue for the commitment by reconstructing Platos reasoning about what is necessary for the theorems of mathematics to be true. In addition to realist tenets about the truth of mathematics, however, such reconstructions assign to Plato a problematic assumption about mathematical discourse. and because Plato has much to say in these passages about mathematical discourse, this assumption, once articulated, offers a new strategy by which to consider his views on the objects of mathematics in the Republic. By examining the discussion of mathematics in Book VII, I shall show first that Plato in fact does not endorse the main argument by which he is thought to arrive at a commitment to intermediates. I then offer a new account, according to which Plato regards the objects of mathematics as theoretical fictions, objects invented by mathematicians to promote their theoretical aims.

Consider the following line of thought, called argument P by Myles Burnyeat:

(1) The theorems of mathematics (arithmetic, geometry, etc.) are true; math-ematicals exist.(2) They are not true of physical objects in the sensible world; mathematicals are not physical objects in the sensible world. Therefore,

* Lee Franklin is associate Professor of Philosophy at Franklin & Marshall College.

Journal of the History of Philosophy, vol. 50, no. 4 (2012) 483506

[483]

1 Versions of this paper were presented to the Philadelphia area Colloquium on ancient Philosophy and at the Franklin & Marshall College Colloquium on Mathematics and the Good in Platos Republic. I am greatly indebted to andrew Payne, allan Silverman, Mary Beth Willard, Hugh Benson, nicholas Smith, rachel Singpurwalla, and two anonymous referees of this journal for many helpful comments.

2 See Burnyeat, Platonism and Mathematics, 21720; Burnyeat, Plato on Why Mathematics is Good for the Soul, 3335; and annas, On The Intermediates.

484 journal of the history of philosophy 50:4 october 2012

(3) They are true of ideal objects distinct from sensible things; mathematicals are ideal objects distinct from sensible things.3

In this argument, Plato is thought to infer the existence of intermediates from his commitment to mathematical truth. The core intuition, implicitly linking the clauses of premise (1), is that mathematical entities of some sort are necessary as the truth-makers of mathematical theorems. But as becomes clear in premise (2), argument P restricts the candidates for truth-makers to items of the sort de-scribed by mathematical proofs and theorems, objects of which these theorems are true. The argument does not consider the possibility that universal mathematical properties, or Forms, suffice to ground mathematical theorems. The basis of this restriction, ostensibly, is the fact that ancient geometers and number theorists spoke about ideal mathematical particulars overtly in their proofs.4 as Burnyeat writes, [T]he entities referred to in a given science are entities whose existence is neces-sary for the theorems of the science to be true.5 But to restrict the candidates for mathematical truth-makers in this way is to assume that the type of entity overtly spoken about in mathematical reasoning directly reveals the type of entity that grounds mathematical theorems. It is to assume that mathematical discourse is transparent with respect to its own foundations.

as I shall show here, however, Plato distinguishes the truth-makers of mathemati-cal theoremsFormsfrom the mathematical particulars of which such theorems are, or would be, true. and since mathematicians speak and reason overtly only about the latter, mathematical discourse is opaque. These points emerge in the first section of this paper, where I examine the pedagogical role of mathematical dis-course in our coming to apprehend Forms as such. In Socratess arguments in Book VII for including arithmetic and geometry in the education of philosopher rulers, Plato highlights the juxtaposition of reasoning about particularsfor example the figures depicted in a geometric constructionwith the universal theorems such reasoning purports to prove. He frames this contrast as an opposition of the sort characteristic of summoners, the category of items that rouse the soul to inquiry (523a10524d5). In line with the way summoners engender learning, this juxtapo-sition provokes a puzzle about the very grounds of mathematical truth, reflection on which leads us to recognize Forms as the entities necessary and apt to ground mathematical theorems. To see this, we must reconsider Platos concern with mathematical truth. For Plato, it is not just the exact or unqualified character of mathematical theorems that demands explanation. More importantly, an account of mathematical truth must explain its universality. This requires universal math-ematical properties, or Forms; ideal particulars do not suffice, however perfectly they instantiate mathematical theorems. Consequently, for Plato, mathematical

3 Here I combine two arguments that Burnyeat presents as corollaries to each other. each premise combines the correlated premises from the two arguments, separating them by a semicolon. See Burn-yeat, Platonism and Mathematics, 22122. Similar reasoning is offered by Wedberg, Platos Philosophy of Mathematics, 5354; and in the introduction to annas, Aristotles Metaphysics: Books M and N, 23.

4 Burnyeat, Platonism and Mathematics, 221, 22930; and Wedberg, Platos Philosophy of Math-ematics, 56.

5 Burnyeat, Platonism and Mathematics, 221.

485mathematic al d i sco u rse i n r e p u b l i c v i i

discourse is opaque: the items that mathematicians speak and reason about overtly are distinct in kind from those that ground mathematical theorems. In Platos view, we cannot reason from the objects spoken of in mathematical proofs to a conclusion about what must exist for the truth of mathematical theorems. Plato would not endorse argument P.

While this result does not entail that Plato denies the independent existence of intermediates, it opens the door to an alternative account of their ontologi-cal status. In the second section, I argue that Plato regards the ideal particulars spoken of in mathematical proofs as theoretical fictions. In prescribing a new method for astronomy, Plato identifies the epistemological factors that compel discourse about ideal particulars in general. Because mathematicians are not yet acquainted with Forms as such, the universal truths they seek to apprehend must be framed as truths about particulars. But the items of the sensible world fail to instantiate these truths. Moreover, this failing is not due primarily, or in an un-qualified way, to the imprecise character of sensible particulars. rather, the states and configurations that must be studied for the mathematicians theoretical aims are nowhere instantiated, even in approximation. as a result, the astronomer can-not arrive at the proper subject matter by idealizing from sensible phenomena, as is sometimes thought.6 rather, he must simply devise the figures needed for his inquiry. Subtly likening the astronomer to a craftsman or demiurge, Plato characterizes mathematical discourse as a kind of theoretical do-it-yourself, and intermediates as objects invented, or imagined, by mathematicians to promote their theoretical aims.

1 . m e t h o d o l o g i c a l s u m m o n i n g

Mathematics has a complex pedagogical role for Plato (Republic 532b6d1),7 a central aspect of which is based in a peculiar feature of mathematical discourse: reasoning about particulars to prove universal theorems. In this section, I show that the pedagogical function located in this feature entails that mathematical discourse is opaque. arguing for the inclusion of arithmetic and geometry in the education of philosopher rulers, Socrates points to the way the practitioners of these disciplines speak about their objects (525d56, 527a34). arithmeticians insist on speaking of units free of sortal designation or material parts (525d7e3). Similarly, he highlights the geometric use of constructions, whereby geometers depict ideal bearers of geometric properties (527a610; cf. 510d5511a2). as I shall argue, these remarks about mathematical discourse serve to illustrate its role in our coming to apprehend Forms as such. Specifically, the juxtaposition of discourse about ideal particulars with the universal theorems it purports to prove raises a puzzle about the foundations of mathematical truth. reflection on this puzzle leads us to countenance Forms as the truth-makers of mathematical theorems, that is as universal essences necessary and apt to ground such universal truths. In so doing, however, this discovery reveals that mathematical discourse

6 For instance, see Heath, History of Greek Mathematics, 285; and Mourelatos, Platos real as-tronomy.

7 all citations of the Republic are from Slings, Platonis Rempublican.


obscures its own foundations: the ideal particulars mathematicians speak and reason about are not the grounds of the truths they discover. Plato presents this account by framing mathematical discourse as a summoner.

I begin, therefore, with some preliminary remarks about summoners. after de-scribing the qualifications for ruling in the , Socrates raises the question of how the city might bring about qualified philosopher-rulers (521c13). Describ-ing education as the turning () of a soul, he asks which subject draws the soul from the realm of becoming to the realm of what is (521d45).8 Before answering this question, though, Socrates digresses to add that the desired subject matter must also be practically useful for warriors, since philosophers-to-be come to the educational program from military service (521d5522e4; cf. 537b15). after establishing the propriety of teaching arithmetic to soldiers, Socrates returns to examining how the very same subject, used properly, is really fitted in every way to draw one towards being (523a13). Having thus contrasted the practical and theoretical benefits of mathematics, Socrates introduces summoners to explain its usefulness for the latter purpose specifically.

Summoners are initially described as items that call () or com-mand () the intellect to inquire, since in their case perception of-fers no sound result (523a10b4). To illustrate, Socrates discusses three fingers that vary in size. They do not summon the intellect with respect to fingers, since none of them appears to be both a finger and not a finger at the same time. But they do summon the intellect with respect to size (and hardness and thickness), since one and the same finger may appear both long and short at the same time (523c10524a3). Summarizing, Socrates says, Those that strike the relevant sense at the same time as their opposites I call summoners (524d14). The contradic-tion reported by perception compels the intellect (, 523d3, and 524a5, 524c7, and 527e4) to take up the matter, and this endeavor brings the intellect to a better understanding. a summoner draws the soul toward being, then, by provoking a rational compulsion to inquire into and resolve contradiction. This is Platos analysis of the mechanism, if you will, by which the soul is reoriented and educated.

By repeating the terms of this account, Socratess later remarks about arith-metic and geometry indicate that we are to view these disciplines as summoners. In comments that present the disciplines as closely parallel, Socrates states that each of them compels the soul to employ intellect or thought (526b13, 526e3), and emphasizes their ability to turn the soul from becoming to being (525c56, 526e35), contrasting each disciplines psychotropic powers, and relegating their practical benefits to the status of by-products (, 527c3).9 In each case, moreover, the disciplines theoretical benefit centers on the way its practitio-ners speak about their objects (, 525d58, . . . , 527a14). altogether, these indications are most explicit in the case of geometry,

8 Unless otherwise specified, translations are from Grube and reeve, Plato: republic.9 There is a clear parallel between the practical benefits assigned to arithmetic and geometry. Both

are useful for military purposes (522c1e2, 525b12, 526c11d5), and both sharpen the intellect of the student generally (526b59, 527c38).


where the discussion of an opposition in geometric discourse concludes in the remark that geometry draws the soul towards truth ( . . . , 527b8). nevertheless, the figure of a summoner has typically not been invoked to explain the pedagogical function of theoretical mathematics.10 This is for two reasons, it seems. First, the example Socrates uses to show that number is a summoner apparently involves no sophisticated arithmetical or geometric reasoning, but simply a conflict in counting: We see the same thing to be both one and an unlimited number at the same time (525a46).11 Second, because it originates in perception (, 523c2, 524a6, and 524d3, 9), it is hard to see how summoning might occur in nonempirical mathematical inquiry. as a result, the discussion of summoners has not been taken to provide a model for the way sophisticated mathematical inquiry promotes learning.

To counter these points, let us compare the experience by which Socrates introduces summoners with that described later to show that number, specifi-cally, provokes summoning. The former arises from perceptual conflict involving properties readily accessible to the senseslarge and small, thick and thin, and so onand results in thought that aims to distinguish the conflicting proper-ties. It is not clear exactly what sort of reflection Plato has in mind here, in part because Socratess terminology varies. Summoners are said, alternately, to rouse the intellect ( , 523d4, 523d8, 524b4, c1, d6), thought ( , 524d2), and calculation (, 524b4). Similarly, the resulting reflection is described by reference to three questions that are not obviously equivalent. What does the sense mean or signify (, 524a6, 523d5) by its reports? are the properties reported one or two (524b5)? and finally, concerning each of the properties reported, What is it (523d4; cf. 525a1)? nevertheless, two details indicate that Plato intends a type of reflection that is rudimentary, at least from a philosophical standpoint. First, the resulting questions focus on clarifying the properties reported, in an effort to understand how two opposite properties can be compresent: [T]he soul is puzzled as to what this sense means by the hard, if it indicates that the same thing is also soft . . . (524a67; see also 523d4).12 Most likely, then, Plato here describes the inquiry by which we realize that these properties hold in a qualified way, so that one and the same thing can be long in relation to one item and short in relation to another, or hard in one part and soft in another.13 Crucially, though, such reflections are metaphysically innocent:

10 See annas, An Introduction to Platos republic, 27374. I. robins (Mathematics and the Con-version of the Mind, 361) does not give an account of summoners, beginning his account from the example to show that number is a summoner, but failing to discuss its relation to the broader model. Likewise, reeve (Philosopher Kings, 7273) describes summoning as the first step toward knowledge.

11 robins (Mathematics and the Conversion of the Mind, 361) connects this example with the kinds of units arithmeticians speak of in their proofs. even so, his account presents summoning as a precondition for theoretical arithmetic, not a type of learning that is provoked by theoretical arithmetic itself.

12 Socratess ease in shifting from considerations of length to those of hardness undermines a reading by which the consideration of the fingers is already an instance of mathematical reasoning. For this reading, see Fowler, The Mathematics of Platos Academy, 113; and Smith, Platos Divided Line, 39.

13 But see reeve (Philosopher Kings, 7374), who reads the example as parallel to the case described at 602c4603a7, so as to involve the invocation of measurement in assessing these properties. One problem with this reading is that in the case of summoning, Socrates stresses that the opposites are


to engage in them neither depends on, nor directly attains, insight into Forms as such.14 This accords with the second detail: Socratess emphasis that his example represents one of the first times the intellect summons reasoning ( . . . , 524b24; see also 524c1011). The reflection provoked by this instance of sum-moning occurs, then, at the very beginning of our engagement in philosophy.

By contrast, the summoning provoked by number is more sophisticated. Glaucon says, We see the same thing to be both one and an unlimited number [ ] at the same time (525a35). The crucial word here is , unlimited or infinite, for in no ordinary sense of the term do we perceive that a sensible particular is infinite. To apprehend this requires a theory asserting, for instance, the infinite divisibility of matter, or the indefinite characterization of sensible particulars. That Plato has the latter in mind, specifically, is indicated by a subtle progression in the appearance of number before it is designated as a summoner. number is first invoked ordinarily, to count sensible particulars: three fingers (523c4). a moment later, though, number is invoked to count properties themselves, in the souls effort to determine whether the opposites reported by the senses are one or two (524b45). Thus before it is discussed explicitly as a summoner, number is predicated of two distinct kinds of subject: sensible particu-lars and properties. This progression prepares us to see the puzzle concerning number as a conflict between these predications, that is between counting the particular as one thing (among others like it) and counting its properties.15 This puzzle will not be apprehensible to all, however, but only to those already famil-iar with properties as countable subjects in their own right.16 accordingly, the resulting question What is the One itself? calls for no ordinary clarification, but a sophisticated inquiry into the nature of unity, to determine whether it is more genuinely exemplified by particulars or properties.

In this way, Socratess examples indicate that the category of summoners repre-sents a template of learning that can be satisfied at different levels of sophistication.17

contemporaneous, whereas in the Book X passage this is explicitly not the case. I am indebted to andrew Payne for bringing this difference to my attention.

14 This characterization may be contested on the grounds that Socrates goes on to infer that the properties distinguished through summoning are distinct in kind from sensibles (524b10c13). But, as evidenced by Socratess reference to a previous discussion with Glauconwe called [] the one intelligible and the other visible (524c13)this is Socratess argument, given as a philosopher reflecting on the metaphysical conditions underlying summoning, not an argument performed by all who experience it. For a reading that collapses summoning with the results of Socratess reflec-tion on it, see Byrd, The Summoner approach, 377. In my view, Socratess reasoning here models the reflection by which we move from the rudimentary stage of comprehension involved in the first example of summoning to the more sophisticated comprehension required to experience the second example, concerning number.

15 But see robins (Mathematics and the Conversion of the Mind, 361), who takes the conflict to stand between counting the sensible particular under two different sortal properties, and so to pave the way for the ideal units that populate arithmetic reasoning. Given that robins thinks this conflict arises exclusively from perceptual judgments of an ordinary kind, however, it is hard to see how we apprehend the infinity of the sensible particular in his view.

16 The brief argument by which Socrates concludes that properties and sensibles are distinct in kind thus plays an important role in preparing Glaucon, and us as readers, to understand the puzzle presented here. See n. 14.

17 For a similar reading, see Byrd, The Summoner approach, 37779.


against this, one might assert that the role of perception in summoning restricts it to the very beginning of our philosophical education. But there is considerable breadth in the way Plato employs perceptual terms in this passage. For instance, Socratess remark that an item can be perceived as a finger (523d56) allows per-ception to include ordinary conceptual attributions. Moreover, as we have seen, perception varies even more widely in the two examples of summoning, from or-dinary sense perception to a type of intellectual apprehension informed by prior theorizing. Indeed, Plato employs perceptual terms repeatedly in this passage to describe sophisticated intellectual apprehension: in Socratess assertion that intellect is compelled to see large and small as distinct, contrasting this intellectual seeing explicitly with ordinary eyesight ( . . . , 524c68; cf. 511a12); in his reference to the vision of Being ( , 525a2; my translation);18 and finally, in his description of the instrument of the soul by which alone can the truth be seen ( , 527d8e3). This usage suggests that the perception involved in summoning may range over several different types of apprehension, some of which occur in theoretical mathematics, or reflection on it.

I shall expand on these remarks after examining how arithmetic and geometric discourse satisfy the template of a summoner. For this, let us start with the opposi-tion ascribed to geometric discourse. Socrates says, [n]o one with even a little experience of geometry will dispute that the science is entirely the opposite of what is said about it in the accounts of its practitioners (527a24). He proceeds to describe the dynamic, practical language geometers employ in developing their constructions, although the discipline is pursued for the sake of knowledge (527a610). For our purposes, there are two main interpretive questions here. The first concerns the opposition at issue. Socrates cites two contrasts: one between practical and theoretical aims, and a second between the subject matter appropri-ate to each of those aims. now, the contrast between practical and theoretical aims presumably does not rise to the level of strong opposition described by Socratess words ( , 527a23). For if it did, we should expect Socrates to have noted this earlier when he asserted that arithmetic can be studied for both practi-cal and theoretical ends (522b7c9, 525b14). rather, this contrast points to the stronger one captured in Glaucons remark that geometric accounts are for the sake of knowing what always is [ ], not of what comes into being and passes away on any given occasion (527a6b5).19

The second question, then, concerns how we are to understand the phrase that which always issince it is one side of the opposition at stake. Prima facie, this wording suggests that geometers are thinking about eter-nal geometrical figures, in contrast to those that come to be and pass away.20 But there is a problem with this reading. Because, according to Socrates, it is part of

18 Grube and reeve translate this as study. While this is one meaning of , this translation omits the connotation of sight, which is repeatedly employed in this passage as a metaphor for intel-lectual apprehension.

19 I have amended Grube and reeve to capture the force of the words , which suggest that it is not becoming in general, but what comes to be at a specific time.

20 So, perhaps, annas in her introduction to Aristotles Metaphysics: Book M and N, 23, but certainly Wedberg, Platos Philosophy of Mathematics, 1079.


an opposition familiar to anyone with the least experience in geometry, the sense in which geometry apprehends that which always is must itself be familiar in this way.21 In this reading, then, anyone familiar with geometry would know that geometers are reasoning about eternal geometric figures. But as Plato acknowl-edges, the ontological status of the objects of geometry is a question of great and lasting philosophical difficulty (534a58), not something obvious to the lay geometer. accordingly, it is more reasonable to think that that which always is describes not eternal geometric figures, but invariant geometric theorems. For what would be familiar about geometry is that its proofs hold universally, that is, in all relevantly similar cases. a geometric theorem asserts what is always the case for figures characterized thus-and-so. The proper contrast with this is not the perish-ability of geometric figures, but their particularity, a point Plato brings home by including the phrase to convey that geometry does not concern what comes about on any given occasion.22 That geometers speak of their figures as if they were constructing them suggests that they are studying the features of some one circumstanceand thus a fact that may come and go with itwhen they are really investigating what holds in all such circumstances.

The contrast between universal and particular may be located very precisely in the structure of geometric proof, in the unmediated transition from what came to be called the , where a conclusion is reached about the figures depicted in a construction, to what came to be called the , where a general thesis is asserted to have been demonstrated. It is unclear when these terms were first applied to the formal steps of euclidean proof. Similarly, it is unclear precisely when the regimented proof structure found in euclid came to be conventional for Greek geometers.23 nevertheless, there is good reason to believe that this structure, or something very much like it, was well established by Platos day. evi-dence for this is based in eudemuss account of the development of geometry, as reported by Proclus, which attributes numerous results that appear in the Elements to members of Platos academy.24 Similarly, this structure is manifest in aristotles writing about mathematics.25 Finally, Platos own depictions of mathematics in the Meno and Theaetetus imply a structure of this sort, insofar as they emphasize the use of geometric constructions, which depict particular geometric circumstances.26 Thus it is safe to assume that anyone familiar with geometric proof in Platos day would know that geometers sought to prove universal theorems by investigating particular cases, depicted in a construction.

21 See also robins, Mathematics and the Conversion of the Mind, 365.22 See n. 19. 23 For a discussion of this development, see netz, Shaping of Deduction, ch. 7.24 Morrow, Proclus: A Commentary on the First Book of Euclids elements, 5457.25 See, for instance, Meteorology 373a811, and 375b20376b22.26 Meno 82b985b7, and 87a37; and Theaetetus 147d4148b3, especially 147d4: . One

might contend that the first Meno passage does not emphasize a transition from particular to universal, insofar as the boy does not clearly grasp the conclusion regarding the diagonal as a general truth. This may be part of what Socrates wants to highlight in his qualification of the boys understanding at 85c9d1. nevertheless, this can be explained as a feature of the slaves youth and inexperience in geometry, and thereby not taken to be representative of the geometry of Platos day. On the way Plato emphasizes the use of construction in the latter Meno passage, concerning hypothesis, see Franklin, Investigation from Hypothesis, 9698.


now, because they lacked any rule or procedure for universal generalization, ancient geometers said nothing to justify the transition from reasoning about particulars to a universal conclusion. Consequently, scholars of ancient geometry have puzzled over the generalization at the heart of geometric proofs.27 Why did ancient geometers feel warranted in drawing a general conclusion from the facts of a single case? reflection on this question brings to light a key point about the type of particulars geometers spoke about. Following r. netz, I take the geom-eters generalization to be based in the repeatability of their reasoning; geometers are licensed to conclude that the same must hold in every similar case, because the exact same inferences could be drawn.28 In order for this to obtain, though, geometers must restrict their discourse to just those characteristics from which the conclu-sion follows in the most general sense. In the first instance, the geometers must speak of perfect bearers of the relevant properties. Moreover, they must leave out of consideration those properties irrelevant to the regularity at issue, even where this means speaking of figures characterized by determinable geometric properties without specifying how they are determined. For instance, a proof showing that a triangle has angles equal to two right angles should not specify that the triangle depicted by the construction is a right triangle; for if the inference involves this specification, the conclusion will be shown to hold only of right triangles. a for-tiori, no proof should speak of material composition for the figures, since such factors are irrelevant to what is true of them geometrically.29 Thus, in order to draw conclusions with the greatest universality, it is crucial that geometers reason about ideal figures: perfect bearers of geometric properties, free of both material designation and, in many cases, fully determinate specification. In pointing to the opposition of universal and particular in geometric discourse, then, Plato is pointing to geometric discourse about intermediates, or their like.

One advantage of this reading is that it enables us to see the same concerns at work in Socratess earlier remarks about arithmetic, and so to maintain the close parallel between the two disciplines. Socrates says that arithmeticians prohibit anyone from discussing numbers attached to visible or tangible bodies. . . . If, in the course of argument, someone tries to divide the one itself, they laugh and wont permit it . . . taking care that the one never be found to be many parts rather than the one (525d7e3). Here we find the same combination of discourse about particulars of special character leading to general results. The units on which the arithmeticians operate lack the sortal designations by reference to which num-

27 See Mueller, Philosophy of Mathematics, 1113; netz, Shaping of Deduction, ch. 6; and Burnyeat, Platonism and Mathematics, 23031. Fowler (The Mathematics of Platos Academy, 1013) leaves aside philosophical questions about the generalization at the heart of geometric proof. nevertheless, he confirms the characterization of mathematical subject matter given here.

28 netz, Shaping of Deduction, 24546. For a contemporary account of the cognitive and visual archi-tecture that enables visual constructions to aid in the discovery of universal theorems, see Giaquinto, Visual Thinking in Mathematics, chs. 45.

29 By the same token, there would be no reason for a geometer to assert that the triangle in his construction is eternal, because this would be irrelevant to the sum of its angles; what holds of a figure because it is a triangle holds irrespective of whether it is eternal or perishable. For this reason, the eternality of geometric figures would not be familiar to anyone experienced with geometry.


bers are typically predicated of particulars: there are no fingers to count here (cf. 523c4), just pluralitiesof idealized units.30 as in geometry, the absence of such designations enables the arithmeticians reasoning about a particular collection of units to apply to any similarly numbered collection. To wit, Socrates emphasizes that the units under investigation are such that each one [is] equal to every other, without the least difference . . . (526a34). The truths of arith-metic hold universally because they hold by virtue of the quantity of a collection of things, and independently of what is quantified. This is made clear by proofs about units that are not units of anything at all.31 Thus, common to arithmetic and geometry is an opposition between the particularity of their discourse and the universality of their results.

Before we consider the pedagogical role of this opposition, it is worth noting how very basic and general is the feature of mathematical discourse that Plato highlights in these passages. To appreciate the juxtaposition of particular and uni-versal does not require that one be a mathematical initiate, possessed of detailed knowledge of the most sophisticated problems that concerned the mathematicians of his day. This is emphasized by the fact that the opposition is apprehensible to the lay observer of geometry.32 Likewise, it is a feature about which there is agree-ment among scholars of ancient mathematics, despite their differences on the finer points of mathematical method and progress.33 Indeed, Socratess comments about arithmetic and geometry are devoid of details that might enable us to iden-tify specific mathematical problems or methods in his remarks.34 Plato can make such details clear when he chooses, as in the interrogation of Menos slave (Meno 82e1385c9), or the description of Theodoruss and Theaetetuss reasoning in the Theaetetus (146d147e).35 He withholds such detail because he is concerned

30 Thus I dissent from attempts to interpret Socratess wording here to refer to the Form of one, e.g. Cherniss, Aristotles Criticism of Plato and the Academy, 518. On this point, see Burnyeat, Platonism and Mathematics, 226n34. Burnyeat characterizes the arithmeticians discourse about these units as the result of abstraction. But I see nothing in the passage to support this construal. Indeed, Socratess wording seems to undermine this account. The arithmeticians do not allow others to propose numbers that have material parts for discussion. Were the abstractionist account correct, Socrates should say that they disallow others from discussing the material parts of the units they discuss. The arithmeticians simply speak of units without material composition, with no suggestion that they arrive at these units by stripping familiar particulars of their material features.

31 On the characterization of units in theoretical arithmetic, see Philebus 56d4e3. See also annas, Aristotles Metaphysics: Books M and N, 721; Burnyeat, Platonism and Mathematics, 2627; knorr, The Evolution of the Euclidean Elements, 13842; and Fowler, Mathematics of Platos Academy, 11112.

32 To wit, there is no indication that Socratess interlocutors, Glaucon and adeimantus, are expert mathematicians. By contrast, consider the characterization of Simmias and Cebes as having studied with Philolaus in Phaedo, 61d67.

33 See Mueller, Philosophy of Mathematics, 1113, 5859; netz, Shaping of Deduction, 187; knorr, Evolution of the Euclidean elements, 13843; Fowler, Mathematics of Platos Academy, 1011, 1416; klein, Greek Mathematical Thought and the Origin of Algebra, 1819; and Burnyeat, Platonism and Mathemat-ics, 22930.

34 See Fowler (Mathematics of Platos Academy, 117), who for this reason treats Socratess remarks on geometry only briefly.

35 This is not to say that the mathematical details of these passages are beyond dispute. In particular, the Theaetetus passage has generated great controversy. See Szab, The Beginnings of Greek Mathemat-ics, ch. 1; knorr, The Evolution of the Euclidean Elements, chs. 34; Burnyeat, The Philosophical Sense of Theaetetus Mathematics; and knorr and Burnyeat, Methodology, Philology, and Philosophy.


here with a broad and fundamental feature of mathematical discourse: reasoning about ideal particulars to prove universal theorems.36

Let us now consider how, in Platos view, the juxtaposition of particular and universal in mathematical discourse promotes philosophical learning. More than a superficial contradiction, this opposition poses a puzzle about the very foundations of the truths arithmeticians and geometers discover. To see this puzzle, and the way it fits into the template of summoning, two preliminary points are in order. The first concerns what we may call the metaphysical orientation of mathematicians. In brief, I take it that mathematicians are not yet acquainted with Forms as such, that is, as unitary essences common to and responsible for the character of a plurality of like particulars. This is suggested by the depiction of those at the penultimate level of the Divided Line as apprehending only shadows and reflections outside the Cave (516a67; see also 532c2),37 and in the description of mathematicians as dreaming, insofar as this metaphor imputes a failure to recognize that the items dreamed are merely images of others (533c13; cf. 476c15). That mathemati-cians proceed with cognizance of Forms is sometimes inferred, however, from their reliance on hypotheses (510b5, 510c3d2, 533c23). On the supposition that these hypotheses are definitions, or other axiomatic propositions, scholars take Plato to assert that geometrical proofs are conducted with awareness of their dependence on universal principles, that is Forms.38 In the first instance, though, it is not obvious that invoking a definition entails the awareness that the definition signifies a unitary essence, responsible for the truths that follow from it. One can be aware of the logical structure of proof without reflecting on its metaphysical underpinnings. In any case, I align myself with an alternate reading, by which to hypothesize the odd and the even . . . and other things akin to these is not to offer definitions of these properties, but to predicate them unreflectively in the course of geometric reasoning.39 Such usage occurs when geometers designate the figures they investigate, as in the of construction-based proofs: Let aBC be a right triangle. On this account, geometers do engage with Forms, insofar as Forms are the properties predicated of the figures they reason about, but they do so unknowingly, as in the predications of ordinary discourse.40 That they invoke Forms in this way manifests a prephilosophical orientation toward particulars, the bearers of properties, as the exclusive objects of speech, thought, and inquiry.

36 So Mueller, Philosophy of Mathematics, 13: So far as I know, the basic method of proof in every historical form of mathematics in which proof has played an explicit role has involved the setting out of an apparently particular case and arguing on the basis of it.

37 Pace Burnyeat (Platonism and Mathematics, 22728), who takes this passage as evidence that mathematicians are aware of Forms (517ab). I see nothing to support such a reading.

38 See robinson, Platos Earlier Dialectic, 15253, 197; Taylor examination of Hare; and annas, An Introduction to Platos republic, 251. Benson (Dialectic in the Republic) argues that both dianoetic and dialectical investigation proceed from hypothesis. In accord with this view, he seems to deny a metaphysical difference between the uppermost levels of the Divided Line. against the view that hypoth-esizing mathematicians directly countenance Forms, see Miller Beginning the Longer Way, 32526.

39 a reading of this kind is found in Hare, Plato and the Mathematicians; and in Franklin, Particular and Universal. See also Smith, Platos Divided Line, 3334, who agrees that Forms are the objects of hypothesis.

40 See Miller, Beginning the Longer Way, 326.


a second piece of evidence for the view that mathematicians are aware of Forms is Socratess remark that although geometers make their statements about physical constructions, they are reasoning about something else, for the sake of the square itself and the diagonal itself [ . . . ] (510d78). Since Plato frequently uses the phrase the F itself to refer to the Form, this comment seems to say explicitly that geometers know they are reasoning about, or for the sake of, Forms. This reading is opposed, however, by others who argue that the phrase does not always refer to Forms, and that within Platos descriptions of mathematical practice it refers instead to the ideal particulars about which mathematicians make their arguments.41 I shall soon suggest that we do not need to choose between these competing interpretations. Plato wants both meanings to be in play, so as to juxtapose the conception mathematicians have of their subject matter with that of philosophers.

To see this, we must make the second preliminary point, concerning truth. In contrast with our own philosophical usage, where truth is primarily a predicate of propositions, possessed or lacked discretely, Plato often speaks of truth as a feature that objects display in greater or lesser degrees, for example to distinguish the occupying different levels of the Divided Line (511e24). In addition, the sense in which a given object is true is predicate-specific; different subjects may be more or less truly F depending on the manner in which the predicate F belongs to them, that is, the sense in which they are F. This comes out most clearly in the discussion of imitation in Book X, where the painting of a bed is said to be less truly a bed than one constructed by a carpenter, and this, in turn, less truly than the Form of Bed itself (597a5d2).42 Crucially, the Form of Bed is not a bedone cannot sleep in itbut the being of Bed, what it is to be a bed (597a45). Thus a particular bed and the Form are true in different senses, because they bear the predicate bed in different ways: the first as a mere exemplification, and the second as the very essence itselfthat which causes any particular bed to be a bed and, more broadly, whatever else it is in virtue of being a bed.43

Putting these preliminary points together, we may redescribe the mathemati-cians orientation toward particulars as a blinkered stance toward truth. Unaware of Forms and their distinctive manner of being, the mathematician believes that the only way to be F is to be an instance of it. accordingly, he will also believe that

41 The controversy is an old one. See annas, An Introduction to Platos republic, 251. For readings in which the phrase does not refer to Forms, see adam, The republic of Plato, 68; Burnyeat, Platonism and Mathematics, 21920n19; and Denyer, Sun and Line: The role of the Good, 3045. Denyers discussion of the issue is particularly helpful.

42 In my view, the distinction between whole categories of with respect to truth may be derived from this predicate-specific sense. Forms are preeminently true because each Form is the essence of just one property, even if it participates in and is characterized by other Forms. Conversely, sensible particulars occupy a lower ontological level because they are whatever they are by dependence on Forms; no sensible particular is an essence. all of the predicates that apply to it do so in a way inferior to the way a Form is the single essence it is.

43 See Socratess discussion of sophisticated causes in Phaedo 103c10105c7. More broadly, this causal role is also implicit in the aims of the What is F? question, insofar as the answer to this ques-tion should say what is common to all and only F things, in such a way as to ground the explanation of their regular character, i.e. what F is like.


the truest Fs are those that exemplify the property in the most perfect, unqualified sense. Thus, in speaking of units that lack material composition or ideal geometric figures, the arithmetician and the geometer take themselves to be speaking of preeminently true objects. But reflection on their own method of proof reveals that these particulars are not true in a different sense. recall that in order to infer a universal theorem from a particular case, the mathematician must grasp that his inferences about that case are repeatable. But in apprehending the repeatability of his inferences, the mathematician must also have a sense that these inferences are grounded not in the particulars themselves, but in something else.44 In appre-hending that what holds of this triangle could be inferred about any other triangle, the mathematician senses that the inference is based in what it is to be a triangle. Put another way, the repeatability of the mathematicians inferences implies that these inferences, and the theorem discovered through them, are grounded not in the particularor any other particularbut in the predicates, or properties, that make the particulars alike. But if this is right, then in one important sense at least these particulars are not true: they are not the cause of what holds of them and every other like particular. Thus, while the mathematician may ordinarily regard the particulars he reasons about as preeminently true, reflection on the repeat-ability of his inferences reveals that these figures are, in a different sense, not true.

Here two points are important. The first is that the opposition with respect to the truth of mathematical particulars will not appear to all who merely engage in mathematical inquiry, but only those who reflect on mathematical discourse and its peculiar combination of universal and particular. Second, ones apprehension of this conflict will initially fall far short of full understanding. The mathemati-cian does not yet have a full and clear apprehension of Forms, or their distinctive manner of truth. The apprehension that the particulars are not true is vague and inarticulate; it is the dawning awareness that something else is responsible for the theorems the particulars exemplify, though exactly what stands in this role is unclear. In characterizing mathematical discourse as a summoner, Plato asserts that it is by reflection on this very conflict that the mathematician comes clearly to distinguish being truly F as the property itself, from being true as an exempli-fication. Confronted by the appearance of ideal particulars as both true and not true, the mathematician is forced to ask, What is true?, where this question asks, specifically, what sort of item could ground repeatable inferences and universal truths of this sort. This leads one to countenance Forms as the kind of entities necessary and apt to ground theorems of the sort discovered in mathematics.

There is an important difference between what is necessary to explain math-ematical truths, in my interpretation of Plato, and what is necessary for a proponent of intermediates. Burnyeat, for instance, emphasizes the status of mathematical theorems as unqualified truths; others emphasize that mathematical theorems are true precisely, rather than approximately.45 Such concerns are then thought to

44 Cf. Phaedo 102c14.45 See Wedberg, Platos Philosophy of Mathematics, 4852; Burnyeat, Plato on Why Mathematics is

Good for the Soul, 1922; Burnyeat, Platonism and Mathematics, 22527; and annas, Aristotles Metaphysics: Books M and N, 2324.


require a special class of particulars to instantiate the theorems in an unqualified, or perfect, manner. In my reading, by contrast, the universality of mathematical theorems is of central interest to Plato. The existence of a single perfect instan-tiation may suffice to explain the unqualified truth of a theorem, but it does not explain why the theorem holds of all similarly characterized items; nor does it explain why the theorem holds approximately of items characterized in ways ap-proximately like it.46 In Platos view, the universality of mathematical theorems can be explained only by reference to universal mathematical properties, that is Forms. Because of their determinate nature, these suffice to explain the unquali-fied, or exact, character of mathematical truths as well.

The metaphysical reflection provoked by mathematical discourse sheds light on the structure common to all summoning, and the importance of perception to it. In the case of mathematics, a rudimentary grasp of truth results in a conflict in the appearance of ideal particulars vis--vis truth. Subsequent reflection on this conflict leads to an improved grasp of truth, culminating in the apprehension of Forms as such, so as to resolve the apparent contradiction. a similar progression occurs in Socratess initial example of summoning. There the finger is reported by perception as large and small simpliciter, not larger than one finger and smaller than another (524a59). again, the contradiction depends on an unrefined grasp of the properties reported. Subsequent reflection clarifies their relational character, and resolves the contradiction. In its most general sense, then, summoning is a progression from rudimentary notions that generate apparent contradictions, which in turn compel reflection that clarifies and improves our understanding of those very properties. In stressing that this process begins in perception, Plato is isolating the rudimentary grasp of a property that is deficient in such a way as to produce contradictory appearances. The crucial point is not that these appearances are achieved through the physical senses, but that they originate in an unrefined quasi-perceptual apprehension of the relevant property, by virtue of which an item may strike us as F or not-F without our being able to explain why.

Thus, summoning effects a revision in our grasp of the property at stake. In the summoning provoked by mathematical discourse, though, the relevant property concerns the metaphysical status and character of its bearers. Consequently, the summoning provoked by mathematical discourse engenders a radical change in the way we think of mathematics itself, from an investigation of ideal particulars taken to be the truest items there are, to an investigation of universal truths that follow from Forms. These are the perspectives on mathematics that obtain before and after the summoning provoked by mathematical discourse. Plato contrasts these perspectives in his use of the phrase F in descriptions of math-ematical method. The key to seeing this is that both the mathematician and the philosopher will employ this phrase as an honorific, to designate the items that are most truly F in his view. Thus arithmeticians who refuse to divide the one itself are not speaking about the Form, but about each of the units that populate their proofs (525d9; my emphasis). Socrates, by contrast, will use to refer

46 The latter is clearly a concern for Plato, given his interest in the practical applications of mathematics.


to the Form of unity. Socrates explicitly highlights this contrast by asking, What kind of numbers are you talking about, in which the one is as you assume it to be? ( , . . . ; 526a13; my emphasis). The same contrast is at work also in Socratess earlier remark that although geometers employ visible constructions, their thought isnt directed to them, but to those other things that they are like. They make their claims for the sake of the square itself, and the diagonal itself, not the diagonal they draw . . . (510d6e1). Geometers recognize that their proofs are not about their physical drawings, but take them to be about the ideal mathematical particulars these depict; it is to these figures that they apply the phrases the triangle itself, the diagonal itself. But from the philosophers vantage point, geometrical proofs are aimed atthat is undertaken for the sake of knowledge ofthe unitary essences in which mathematical theorems are grounded, the Forms.

Of course, these perspectives on mathematics are not equal in Platos view. Only the philosophers view accurately captures the fact that mathematical theorems are grounded in Forms. It is for this reason that these truths can be known only by one who arrives at them through dialectic, making use only of Forms themselves, moving on from Forms to Forms, and ending in Forms (511c12). Conversely, it is for this reason that mathematical theorems can be discovered, but not known, as a result of mathematical inquiry alone (533d56; cf. 527a9b5).47 Finally, this is the meaning of the repeated descriptions of mathematical inquiry as dealing with images, or as dreaming (516a58, 533c13). Mathematical discourse fails to reveal the grounds or causes of the truths it discovers; it is opaque with respect to its own foundations.

Thus, Plato would not adduce argument P as proof that intermediates exist. Cer-tainly, he would endorse the inference from the objective truth of mathematical theorems to the existence of some independent entities that serve as their truth-makers. But for Plato, the entities that serve in this role are Forms. accordingly, he would reject the assumption that mathematical theorems can be made true only by items of which they are true. Precisely because they follow from Forms, mathemati-cal theorems can be true without being true of anything. That mathematicians arrive at their theorems through discourse about particulars is an indication that they fail to grasp the truth of these matters wholly or directly.

2 . i n v e n t i n g i n t e r m e d i a t e s

That mathematical discourse is opaque for Plato does not show that he denies the existence of intermediates. argument P is not the only argument by which Plato is thought to arrive at a commitment to their existence. But it is the argument on which there is the greatest consensus and, for reasons I will not go into here, the most compelling.48 That Plato does not endorse argument P, though, raises

47 See Burnyeat, Platonism and Mathematics, 21920, and n. 19 on this passage, and the way Glaucon, not Socrates, asserts that geometry is itself knowledge, not for the sake of knowledge.

48 Two other arguments appear in Wedberg, Platos Philosophy of Mathematics, 5556. First, one might argue that the existence of a Form as a one over many requires that a plurality of perfect instances for each Form exist. alternatively, one might argue that inasmuch as Forms self-predicate, each Form is itself a perfect bearer of the property it is. I shall here note only two problems of these strategies.


the following question: What exactly is going on when mathematicians speak of ideal particulars, if they are not referring directly to the conditions that render their theorems true? In this section, I show that Plato answers this question in the discussion of astronomy. In prescribing a new method for astronomy, akin to that employed in geometry and arithmetic, Socrates identifies the factors that compel mathematical discourse about ideal particulars. In broad strokes, the theoretical aims of mathematicians drive them to investigate a rather specific set of invariant truths. Because of their orientation toward particulars, however, they can discern these truths only as instantiated by some particulars or other. But the sensible world fails to offer particulars that do this. This failing is typically understood by reference to the imprecise, or approximate, character of sensible particulars, so that the recommended astronomical subject matter may be apprehended through idealization of the observable heavens. as I shall argue, however, Plato places less emphasis on approximation than is typically thought, and qualifies it by reference to other considerations, with the result that the observable heavens enter into the true astronomers purview in no way at all. rather, the astronomer simply creates his own subject matter. In this way, Plato characterizes mathematical discourse as a kind of theoretical do-it-yourself, and intermediates as theoretical fictions: objects that mathematicians invent to promote their theoretical aims.

This view emerges from the role of demiurges in Socratess remarks on as-tronomy. Let us first outline, then, how the figure of a demiurge is introduced to the discussion. In response to Glaucons assertion that the study of the perceptible heavens compels the soul to look upward, Socrates insists that astronomy, as it is currently practiced, has the opposite effect because it relies on the senses (529b3c2). To explain his preferred alternative, Socrates first highlights the deficiency of the objects and motions in the perceptible heavens in comparison to the true ones [ ]. . . . and these, of course, must be grasped by reason and thought, not by the sight (529c6d5). By characterizing these celestial items as objects of , Socrates implicitly likens them to the ideal particulars studied in geometry and arithmetic (see also 511a12, 526a67). accordingly, the remark that they are true need not speak to their independent existence, but only to their characterization as perfect exemplars of the relevant properties, items that will be regarded as preeminently true by the astronomer who is not yet acquainted with Forms. Socrates then asserts, We should use the embroidery in the sky as a model in the study of these other [true astronomical objects] (529d78). To illustrate this modeling, Socrates likens the reaction a true astronomer will have to the observable heavens created by the divine demiurge to that of an experienced

First, each invokes a claim about Forms that has been rejected in recent scholarship. The separation of Forms is typically taken to include the claim that Forms exist independently of whether they are instantiated. See, for instance, Fine, Separation, and for different reasons Silverman, Dialectic of Essence, 13236. Similarly, the self-predication of Forms is, for the most part, no longer understood as a characterizing predication, such that the Form need be a paradigmatic instance of the property it represents. Here see nehamas, Self-Predication and Platos Theory of Forms; and Silverman, Dialectic of Essence, 8795. Second, since each of these strategies requires only the existence of a small number of perfect particulars, neither compels the existence of a presumably infinite realm of abstract particulars corresponding on a one-to-one basis with the panoply of true mathematical theorems.


geometer responding to diagrams drawn by Daedalus or some other craftsman (529e45, 530b34). In both cases, the theorist would think it ridiculous to exam-ine the craftsmans work for the truth of the matters that interest him. Instead, the astronomer must study astronomy by means of problems, as we do geometry, and leave the things in the sky alone (530b6c2). a demiurge is introduced, then, as the fashioner of items that are inappropriate subject matter for geometry and astronomy respectively. and, since the true astronomers modeling is presented as a response to these items, identifying the precise sense in which they are inap-propriate is crucial to understanding the method recommended for astronomy.

To start, we may understand the appearance of demiurges in this passage as part of a broader emphasis on the aesthetic character of the subject matter of astronomy and harmonics. In describing the latter stages of the mathematical program, Plato stresses these subjects role in preparing us to apprehend the ultimate end of the philosophical education, the Good itself, rather than the realm of Forms more generally. Thus, Socrates commends harmonics, a subject akin to astronomy, specifi-cally as a useful [study] in the search for the beautiful and the good (531c67), whereas the earlier mathematical studies were praised primarily for bringing us to behold truth (527e3), or being (526e7). Similarly Socrates stresses, in a way he has not before, that the subject matter of geometry and astronomy is . as alexander Mourelatos has convincingly argued, this term connotes not merely proportion, but commensurable proportions of the sort on which harmonies are based.49 For our purposes, however, the most noteworthy emphasis on beauty comes in Socratess description of the observable heavens as the most beautiful and most exact [ . . . ] of visible things (529c7d1). Since this remark leads directly to the claim that the sensible heavens are deficient, the terms of Socratess praise presumably capture the very respects in which the observable heavens fall short. We might think that Socratess methodological recommendation is motivated by aesthetic considerations, alongside a concern for precision. But as the parallel between the geometer and astronomer unfolds, Socratess concern for precision seems to disappear. The works of Daedalus are praised only as very finely executed [ . . . ]; similarly, the heavens are arranged . . . in the finest way possible [ . . . ] (529e3, 530a5).50 The implication is that the observable heavens are deficient primarily because of a failing with respect to beauty.

That Plato emphasizes beauty over accuracy recasts the deficiency of the observable heavens, and by extension the sense in which the true heavens are modeled on them. To see how, we must focus not on the similarity between the geometer and the astronomer as they respond to the work of a demiurge, but on an important difference: whereas the astronomer responds to the divine craftsper-sons productthe observable heavensthe geometer is responding to diagrams

49 See Mourelatos, Platos real astronomy, 3941; and robins, Mathematics and the Conver-sion of the Mind, 37375.

50 an apparent exception is the remark that the heavens deviatefrom their or-dained motions. For reasons I will present later, I do not think this should be read as the claim that the observable heavens are inaccurate, in the sense of being at best approximations of geometrical shapes.


() carefully drawn and worked out by a human craftsman (529e2). In one reading, these diagrams may be treated as geometric constructions.51 This reading is suggested, perhaps, by the reference to Daedalus, the most sophisticated inventor in Greek myth, but certainly by the fact that Plato uses the term elsewhere exclusively to speak of geometrical constructions.52 In such a reading, the geometers response targets the difference between a geometrical construc-tion, as an imprecise physical rendering, and the idealized figures it is supposed to depict. Similarly, the observable heavens would approximate the idealized celestial system to be studied by the true astronomy. Two details undermine it, however. First, the drawings could have been drawn by any craftsman whatsoever ( , 529d9), and not necessarily a Daedalus. Second, the geometer happens upon () these diagrams.53 Together these details indicate that the diagrams were not created for a geometric purpose, but one proper to a crafts-mans work. The are a craftsmans plansthe sketches or blueprints on which he bases his work.

This explains why the concern for precision vanishes from Socratess remarks. For while a craftsperson will make an effort to draw his plans to scale, he will also designate his plans with the dimensions they are intended to represent. When the are considered as plans, there can be no question of their accuracy: like constructions designated by a geometer, they represent exactly what the crafts-man declares them to represent. Plato calls the craftsmans plans , then, not to identify them as geometrical constructions, but to highlight their kinship with constructions as representational media. But in highlighting this kinship, Plato simultaneously highlights the most important difference between geometric constructions and technical plans. The craftsperson assigns dimensions with the aim of designing a functional product that meets the exigencies of mate-rial construction. The geometer, by contrast, aims at the discovery of universal truths concerning geometric relations, for example the equal, the double, or any other ratio (530a12). Because the craftsmans figures have been designed for practical rather than theoretical purposes, the figures and relations depicted in them yield no insight into important geometric necessities. Compare the circum-stances depicted in the blueprint of an architect with those described in euclid. Though both assign exact dimensions, only the rather baroque circumstances in the latter are apt to reveal a significant mathematical truth. rather than imprecise depictions of the right subject matter, the craftsmans plans are precise depictions of the wrong subject matter.

The astronomer encounters a similar problem, but for different reasons. The astronomer will not examine the motions of the heavens for the truth of the rel-evant because hell consider it strange to believe that theyre always the

51 See robins, Mathematics and the Conversion of the Mind, 37374; and Mourelatos, Platos real astronomy, 3940.

52 There are surprisingly few other uses of this term in the corpus: Cratylus 436d2; Theaetetus 169a3; Euthydemus 290c2; Phaedo 73b1; Hippias Minor 367d89.

53 But see Mourelatos (Platos real astronomy, 4647), who takes this term to suggest that the geometer fortuitously comes upon drawings suitable for his investigation.


same and never deviate anywhere at all . . . since theyre connected to body and visible (530b13). Crucially, Socrates is not here restating the charge of approxi-mation, but another that actually undermines the importance of approximation to his account.54 according to Socrates, one who thinks that the heavens are an appropriate subject matter for astronomy must be thinking that they hold in the same way eternally. The implication is that the demiurge succeeds in giving the heavens, at their original creation, the shapes and velocities that instantiate perfect harmony. The problem is not that the heavens cannot ever obtain this perfect arrangement, but that they cannot maintain it.55 Otherwise, there would be no need to mention the heavens tendency to deviate. Socrates could simply say that the heavens can never perfectly achieve the right consonances because they have body and are observable.56 But since they do deviate, were an astronomer to inspect the observable heavens he would not find the precise dimensions that instantiate perfect, harmonious order. at best, these motions would merely approximate the supremely beautiful arrangement given them initially by the divine demiurge. We can now see why, although beauty and accuracy are mentioned together in Socratess initial praise of the observable heavens, the concern for accuracy drops out thereafter. The observable heavens are not imprecise tout court, but specifi-cally in relation to the arrangements that instantiate perfect celestial harmony.57

The craftsmans plans and the observable heavens are alike in that they fail to present the states and relations conducive to the theoretical aims of the geometer and the astronomer respectively. But they suffer this broad failing for different reasons. Whereas the craftsmans plans lack the appropriate subject matter because they have been created for practical purposes, the divine demiurges product suf-fers from the instability of his materials. Plato uses these differences to provide distinct but complementary pieces of information about the modeling proposed as the method of the true astronomy. The divine demiurges product is the proper system for instantiating the proportions and harmonies astronomy must investigate. Were it not for the inherent instability of matter, the observable heavens might be worth studying for the sake of apprehending the Good. The craftsmans diagrams offer a way to circumvent this problem, for they model a way of depicting figures

54 For a reading of to connote imprecision, see Miller, Beginning the Longer Way, 319.

55 Like all created things, the heavens are subject to degeneration over time. See 546a13.56 The account of creation in the Timaeus confirms this. There, the divine demiurge is said to have

striven to structure the heavens according to the ratios and proportions that are beautiful and good. In so doing, he gave to the physical universe perfect geometrical shapes, and motions that stand in precise consonances, . For instance, consider the shape given the universe as a whole: He gave it a round shape, the form of a sphere, with its center equidistant from its extremes in all direc-tions (Timaeus 33b). See also Mourelatos, Platos real astronomy, 5657.

57 In this way, my reading avoids a problem associated with the emphasis on precision as the focal point of Platos complaint about the sensible world. For if this were Platos concern, it would involve the claim that the items of the physical world fail to instantiate any mathematical properties exactly. But this is surely impossible: at any given moment, the celestial bodies are moving at some precise velocity; similarly, the items of the physical world have some determinate shape, if not one of the simple shapes studied in geometry. For this reason any concern with accuracy must be explicated with respect to some privileged class of mathematical properties. Cf. Wedberg, Platos Philosophy of Mathematics, 50.


in precisely those shapes, dimensions, and relations we wish them to have. Putting these ideas together, we arrive at the following recommendation. The astronomer must employ the representational medium of the craftsman to depict the type of system created by the divine demiurge: celestial systems composed of idealized bodies, moving in idealized orbits, whose dimensions in some way or other stand in perfect to one another.

This, I take it, is the meaning of Socratess recommendation that the true astronomer must inquire by means of problems, as we do geometry, and leave the things in the sky alone (530b67). Since the terms of this conclusion have been the source of controversy, a bit more discussion is in order. I will first con-sider the remark that astronomers must leave the observable heavens alone. Interpreters have been reluctant to take this comment at face value, to mean that astronomers should not consider the phenomena of the observable heavens at all in their researches. This is clearest in the case of those who think that Platos astronomy, like its contemporary counterpart, aims to explain the motions of the observable heavens.58 But even in readings where the recommended astronomy is an a priori study of abstract kinematics, the observable heavens are invoked as a constraint on the astronomers inquiries.59 This is purportedly justified by the fact that the observable heavens have been created by the demiurge to be as good and beautiful as possible. Consequently, the observable heavens may be regarded in the same way a geometer regards his own drawingsas close approximations of the ideal states he wishes to consider. In this reading, the astronomer arrives at the correct subject matter by altering observable celestial phenomena through idealizing stipulations.60 But this comparison overlooks an important difference between the geometers relation to his construction and an astronomers to the heavens: The geometer can overlook the imprecision of his drawings because he devised them with the states and relations they are to depict already in mind.61 But the astronomer does not know the ideal arrangement from which the observable heavens have deviated, and of which they are now an approximation. Since he does not know how to corrector idealizethe celestial phenomena observable to him, the true astronomer cannot look to the observable heavens as a datum for his inquiries at all.

rather, the astronomers task is the more difficult challenge of exploring, in an experimental (but not at all empirical) manner, the many arrangements that an ideal celestial system can occupy, in order to discover how they do, and do not, display and harmony. In such an inquiry, the astronomer generates his own subject matter, depicting for himself, by means of construction, the vari-

58 See Vlastos, Observation in Platos astronomy.59 Mourelatos, Platos real astronomy, 5558; and robins, Mathematics and the Conversion

of the Mind, 37476.60 Mourelatos Platos real astronomy, 56.61 robins, Mathematics and the Conversion of the Mind, 374 notes this in the case of geometry,

but overlooks its relevance to the possibility of an astronomical parallel. Mourelatos (Platos real astronomy, 5556) seems to think that the astronomer already has a firm grasp on what a coherent and perspicuous celestial system looks like, so as to make the adjustments and idealizations needed to proceed from the observable heavens to the true ones.


ous arrangements whose character he wishes to investigate. This is the meaning of the claim that the astronomer, like the geometer, must resort to problems (, 530b6). Most interpreters assume that is a geometrical term of art, and seek its meaning in geometric contexts.62 But since Plato never uses the term in a technical senseeven in passages where he is discussing familiar aspects of geometrical method (e.g. 510b911a2)this strategy produces accounts that strain to read Platos usage through a narrow meaning found elsewhere. But we can read the use of here in line with Platos other uses, which are of two main types: a literal use to describe something built or placed before oneself, as in defensive battlements; and a second, in the context of inquiry, to describe challenges or questions posed to oneself or others.63 a construction may be re-garded as a way of placing before oneself the circumstance whose features one wishes to investigate, or reason out. In this way, the remark that are used by geometers points not outside of Plato, but back to the most recent allusion to geometric method, in the use of the term to highlight the kinship between a craftsmans plans and geometrical constructions. The shift in terminol-ogy, from to , serves to emphasize that constructions, and the circumstances they depict, are products of the geometers own devising.

Thus Plato introduces a demiurge not merely to characterize inappropriate astronomical subject matter, nor to emphasize aesthetic considerations, but as a metaphor for the astronomers own theoretical project. The true astronomer must engage in a kind of theoretical do-it-yourself. The objects and circumstances necessary for his inquiries are nowhere to be found in the sensible world. nor are there approximations from which, through a process of idealization, he can arrive at the proper objects and circumstances. Consequently, he must construct his own subject matter through the use of representations akin to geometric dia-grams. This explains Glaucons choice of words in saying that the recommended astronomy involves a task . . . a lot harder than the empirical geometry already in practice ( . . . , 530c34). Though can be used for any deed or action, it has the narrower meaning of constructive work, or its product, as in Socratess statement that the heavens are arranged . . . in the finest way possible for such works ( , 530a56).64 The depiction of astronomy as a constructive project also explains why such stress is placed on the ordering of the mathematical studies, as in Socratess conspicuous backtracking to include stereometry (528a6d10).

62 For fuller discussion of these alternatives, see Mourelatos, Platos real astronomy, 6062; and Mueller, ascending to Problems: astronomy and Harmonics in Republic 7, 10322. Mourelatos favors a use in Oenopides that refers to the analysis of a construction. Though they resemble each other, the main difference between Mourelatoss reading and mine is, first, that I take to refer to the use of constructions generally, and so to include a broader set of mathematical activity than Mourelatos. Second, there is a difference in emphasis, as the term, in my account, brings out the mathematicians role in creating his own subject matter.

63 For the first use, see Sophist 261a6, Statesman 279d1, Timaeus 74b6. For the second, Theaetetus 180c5, though this use likely has a specifically geometric connotation. For discussion of in aristotle, see Lennox, aristotelian Problems.

64 Translation adapted from Grube and reeve to bring out the force of .


The astronomer studies ideally shaped bodies displaying motions that are really fast or slow as measured in true numbers, that trace out true geometrical shapes (529d23). The components of the astronomers idealized heavensnumber, shape, solidare the subject matter of his previous inquiries in arithmetic, ge-ometry, and stereometry. The ordering of the mathematical studies provides the real astronomer with the raw materials, as it were, for his constructive theoretical project.65

The astronomer is driven to theoretical construction by the absence of ap-propriate subject matter in his environment. By itself, though, this absence is not sufficient to explain the use of construction. For as we have already seen, the result of these investigations is the discovery of universal truths that follow from, and can ultimately be understood by reference to, Forms. So there is, in principle, a second, Form-based route by which these truths may be derived (see 511c12). But the mathematical studies are conducted from hypothesis and the prephilosophical orientation this method manifests. Moreover, insofar as these studies are propaedeutic to the apprehension of Forms, they cannot benefit from awareness of Forms as such. Consequently, the truths at which these studies aim must be framed in a manner hospitable to the mathematicians orientation, that is, as truths concerning particulars. Since his environment fails to offer any instantiations, the mathematician must invent them. It is in this light, I suggest, that we understand Socratess numerous remarks that the discourse of geometers is necessary or compulsory (, 527a6; cf. 510b5, 511a45, 511c7). These remarks are sometimes taken to acknowledge mathematical conventions in place, to the effect that there was no other way to do mathematics in Platos time.66 Be-yond merely noting these conventions, though, Plato seeks to explain them as a result of our metaphysical orientation before philosophical education. We come to inquiry directed toward particulars as the exclusive objects of investigation. If we are to surmount this perspective, we must confront certain truths nowhere realized in our sensory experience. But until we apprehend Forms, we can only confront these truths within the constraints of our prephilosophical orientation toward particulars. Thus if we are ever to apprehend the Forms and the Good, we are compelled to investigate the truths of mathematics via a method that depicts them, constructively, in instantiations we devise. For the rational soul striving to apprehend the Forms, necessity is the mother of invention.

We may describe Platos view as a kind of fictionalism about ideal mathematical particulars, with a few important qualifications. First, fictionalism often attributes, to those who speak of fictional objects, awareness of the make-believe status of their objects. But while nothing prohibits mathematicians from adopting this perspective, Plato tends to portray mathematicians as agnostic, because they are unreflective, about the ontological status of their objects. Thus, when confronted, they can offer no further analysis than to say that they speak of entities that one cannot see except by thought (511a12, 526a67, 529d45).67 In any case, Platos

65 a similar point is made in Miller, Beginning the Longer Way, 32122.66 Burnyeat, Platonism and Mathematics, 219. nevertheless, I agree with the point Burnyeat

makes here, which is that Plato is not criticizing mathematicians for their procedure.67 See Burnyeat, Platonism and Mathematics, 22627.


preferred metaphorconveyed via the parallel between the second and fourth levels of the Divided Lineis not fictional characters, but images: shadows and reflections. This relationship of image to original is employed, most notably, to capture the metaphysical relationship between Forms and sensible particulars. I take it that this feature of the metaphor is satisfied by the fact that the math-ematicals are particulars, if not sensible particulars.68 But another aspect of the metaphor is equally relevant, namely that imagesand especially reflectionsare not mind-independent entities. They exist, rather, just insofar as they appear to someone. Moreover, their appearing requires that a perceiver stand in a specific relation to the originals of which they are images. Whereas spatial relations and optics determine to whom a reflection may appear, in the case of intelligible im-ages, the relevant factors are methodological. Intermediates, we may say, appear to mathematicians through the character of their discourse.

In this light, finally, we may consider anew Platos reluctance to describe di-rectly the ontological status of intermediates. He is disinclined to say that these items simply fail to exist, for much the same reason one might hesitate to say that a reflection does not exist. at the same time, though, they do not fit neatly into either of the two main categories of his metaphysics: they are neither sensible particulars nor Forms, though they bear some resemblance to both categories. In this broad respect, they are like the soulanother entity whose status is some-times murky in the Platonic view.69 But beyond their ambivalent character, the fact that intermediates appear to inquirers by virtue of their method entails that their ontological status can be understood only by inspecting that method.70 ac-cordingly, it is only by appeal to this method, and its pedagogical role, that we can understand the sense in which these items are intermediate. In the view offered here, the ideal particulars spoken about in mathematics are intermediate in that they appear to inquirers who have begun to turn away from sensibles, but are not yet fully turned toward the Forms.

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68 See Smith,Platos Divided Line, 38: My argument requires only that the two levels contain the same sorts of objects, where what sorts the objects is their place within the image-original hier-archies in Platos metaphysics. In this regard, Smith and I are in agreementthe penultimate level of the Divided Line must contain participants in Forms, i.e. bearers of the properties that Forms are.

69 See Silverman, Dialectic of Essence, 6364.70 I take it this is why Socrates introduces the items at the penultimate level of the Divided Line

by reference to the method of mathematicians, in contrast with every other level, where the relevant entities are introduced directly, as objects familiar to Socratess interlocutors. Benson (Dialectic in the Republic, 18889) downplays the uncomplicated reference to Forms at 510b8. That Socrates can expect Glaucon and adeimantus to be familiar with Forms is clear from 507b16.

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Inventing Intermediates: Mathematical Discourse and Its Objects in Republic VII