1 Introduction

In this Element, I will argue that Plato was not a mathematical Platonist.Footnote ¹ My arguments will be based primarily on the evidence found in the Republic’s Divided Line analogy and in Book 7.Footnote ² I will present Plato’s view as it develops in the text, which, while perhaps not as reader-friendly as one might like,Footnote ³ emphasizes the significant changes that Plato intends to be surprising, even shocking, to his reader – changes that are often missed in the interpretative literature. First, I will offer what I take to be an accurate translation of the text,Footnote ⁴ before critically considering which claims remain the same and which change, especially as we transition from the Divided Line to Book 7. Finally, I will bring these claims together into a consistent picture of Plato’s view of mathematics, demonstrating that he was not a mathematical Platonist.

Typically,Footnote ⁵ the mathematical Platonist story is told on the basis of three realist components: (a) that mathematical objects, as Platonic forms, exist independently of us in the metaphysical realm of forms; (b) that the way things are in this metaphysical realm fixes the truth of mathematical statements; and (c) that we come to know such truths by, somehow or other, “recollecting” the way things are in the metaphysical realm.Footnote ⁶ This Platonist story, by confusing the hypothetical method of mathematics with the dialectical method of philosophy, conflates the two types of realism at play in Plato: methodological realism and metaphysical realism. My aim is to show that while Plato is a philosophical Platonist – that is, he adopts metaphysical realism for philosophical inquiry – he is a mathematical as-ifist – that is, he adopts methodological realism for mathematical inquiry. Thus, it is by keeping these methods distinct that we will see that, as regards (c), we come to know mathematical objects by treating our hypotheses as if they were true first principles for the purpose of using these to solve mathematical problems.

As regards (a), I will show that mathematical objects depend on the mathematical problem that we are attempting to solve. It is the problem at hand that gives rise to the needed hypotheses that themselves are taken as if they were true, and it is these hypotheses that give rise to the needed objects of thought that we take as if they exist for the purpose of solving the problem. Mathematical objects, then, exist in a methodological sense but not in a metaphysical sense. Against (b), what fixes the truth of a mathematical statement is its method, not its metaphysics – that is, mathematical truth is fixed by a demonstration that shows that the answer to our problem can be deduced from our hypothesis; it is not fixed by the way things are in the metaphysical realm of forms. As we will see, in mathematics, existence is a consequence of truth – that is, is a consequence of taking our hypotheses as if they were true for the purpose of solving a problem. In philosophy, by contrast, truth is a consequence of existence, that is, is a consequence of our tethering our hypotheses to independently existing forms. It is these considerations, which arise by keeping distinct the mathematician’s and the philosopher’s methods, that allow us to see that Plato was not a mathematical metaphysical realist; rather, he was a mathematical methodological realist.

My aim is to argue that since the method used by the mathematician is distinct from that of the philosopher, then so too must be their objects. From a methodological standpoint, I will show that the mathematician uses the hypothetical method and travels downward from a hypothesis, taken as if it were a true first principle, toward a conclusion. The philosopher, on the other hand, uses the dialectical method to first travel upward from a hypothesis, taken as a hypothesis, toward a first principle, the truth of which is fixed by a form, and they then travels downward from a form-tethered or true first principle to a conclusion. I will further show that, as a result of these methodological differences, the mathematician, in their goal of solving mathematical problems, needs only take their objects as if they exist. This is why, now from an epistemological standpoint, mathematical objects are to be taken as objects of thought, whereas philosophical objects are to be taken as objects of understanding (or, at the end of Book 7, as objects of knowledge). Bringing these two standpoints together, I will argue that mathematical objects, as things that arise from “images,” or from drawn or constructed diagrams, are nonetheless to be taken as distinct from such “images” and so are to be taken as “things themselves.” However, even as “things themselves,” mathematical objects are distinct from “forms themselves”; they are methodologically real – that is, we treat them as if they exist to solve a mathematical problem – but they are not metaphysically real. Indeed, this is why, at the end of Book 7, Plato likens the faculty of thought to that of imagination and, as a consequence, comes to reserve the term “knowledge” for philosophical knowledge only. Thus, taking my evidence primarily from the Divided Line analogy and Book 7, I will argue that Plato was not a mathematical Platonist; mathematical objects are not forms, they do not either exist in some metaphysical realm or fix the truth of mathematical statements, and we do not come to know them via recollection.

2 The Interpretive Lay of the Land

The number of interpretations of Plato’s views of mathematics is vast. Some consider the whole of Plato’s works, others focus on specific dialogues. My interpretation will focus primarily on what Plato says in the Republic’s Divided Line and Book 7. The reason for this is twofold; except for the Meno, these are the only places where Plato presents a sustained account of mathematics, and there seems little debate that this dialogue was written by Plato.Footnote ⁷ In a broad stroke, my interpretation is intended to cut a midpoint between the two prevailing and competing views. The first is the view of Reference CornfordCornford (1932), Reference WhiteWhite (1976), Reference TaitTait (2002), and Reference Benson and BensonBenson (2006; Reference Benson and McPherran2010; Reference Benson2012) that the hypothetical method is part of the dialectical method so that mathematical objects must, in some sense, be part of the realm of forms. The second is the view of Reference BurnyeatBurnyeat (2000) and Reference BroadieBroadie (2020) that the mathematician’s hypothetical method is distinct from the philosopher’s dialectical method, but that Plato adopts a quietist stance on the ontological status of mathematical objects – that is, on the question of whether mathematical objects are to be taken as distinct from forms.

Benson’s part of view has a long history and is well captured by Cornford’s argument that Plato has two types of dialectic at play, each with its own methodology: one mathematical and having as its objects mathematical forms, the other philosophical, or ethical, and having as its objects forms like Justice, Virtue, and Good. Likewise, Reference BensonBenson (2012) sees both types as part of the same method, but further distinguishes between the mathematician’s dianoetic method and the philosopher’s dialectic method, arguing that

Most part of interpreters hold that these unhypothetical first principles are unhypothetical because they are tethered to, or fixed by, a stable metaphysical domain (i.e., by a realm of mathematical objects taken as philosophical forms, or, like Tait, by a realm of foundational mathematical objects taken as geometric forms).

Reference BurnyeatBurnyeat (2000), in contrast, uses his distinct from interpretation to point to two stances that one may adopt as regards the ontological status of mathematical objects: the “internal” stance taken by practicing mathematicians and the “external” metaphysical stance taken by the philosopher of mathematics. He remains oddly silent on what the practicing mathematician’s internal stance comes to but, as regards the latter, holds that Plato “leaves the external question tantalisingly open” (p. 22). Likewise, Reference BroadieBroadie (2020) holds that “Plato shows no interest in this metaphysical question” (p. 15).

Benson (2000) similarly holds that “Plato is less concerned to offer a fourfold ontology associated with the four sections of the Line, than he is to describe the correct method of the greatest mathēma – the knowledge of the Form of the Good” (p. 1). But, as we have noted, Benson, as many other interpreters who believe that the philosopher’s method must be adopted by the mathematician, holds that this external question must be answered. The various, what I will call, metaphysical interpretations that seek to answer this external question agree that one must adopt a mathematical Platonist position but bifurcate over whether this should be answered at a philosophical or at a metamathematical level – that is, whether one must adopt the view that mathematical objects are philosophical forms themselves or are to be founded on an ontologically preferred metamathematical theory of forms, such as foundational theory of geometric forms.

Another option, however, is to argue that mathematical objects are “intermediates” between philosophical forms and sensible objects. Indeed, forgoing his internal/external distinction for the moment, Reference BurnyeatBurnyeat’s (2000) position seems to purposefully leave open the possibility of an interpretation of mathematical objects as intermediates:Footnote ⁸

However, even if he leaves himself open to an “intermediates” interpretation, Burnyeat does forestall those aforementioned Platonist interpretations, like Benson’s, that require that mathematicians or philosophers of mathematics adopt the dialectical method on the basis of a supposed criticism that the mathematicians problematically mistake their hypotheses for unhypothetical first principles and, in so doing, leave their hypotheses unaccounted for. I agree with Burnyeat, and, as I will show, it is this criticism that itself is the mistake of all part of interpretations. As Burnyeat rightly notes, ‘mathematics is not criticised but placed. Its intermediate placing in the larger epistemological and ontological scheme of the Republic will enable it to play a pivotal, and highly positive, role in the education of future rulers’ (p. 42).

Next we must ask: What is this important role? We are told that an education in mathematics will enable the philosopher to grasp the Good, but how does that work? Why does this education take ten years?Footnote ⁹ What is Plato’s criticism of mathematics as currently practiced? Why are the branches of mathematics so ordered? Finally, what is the relationship between mathematical reasoning and philosophical/moral reasoning? With respect to the last question, I begin by noting my agreement with Reference BroadieBroadie’s (2020) claim that ‘the text of the Republic offers virtually no evidence that his [Plato’s] problem lies in meta-mathematical ambition for dialectic or in the theory that ethical reality itself is mathematically structured’ (p. 29; italics added).

I fully agree with the first of these disjuncts but disagree somewhat with the second. As regards the first disjunct, when we focus on what the text says, we will see that Plato’s problem is set at making space for the beneficial role that mathematics plays in preparing the mind for philosophical dialectic, by turning us away from a reliance on beliefs and opinions founded on sense experience. He is also showing us the limits of mathematical inquiry, namely, that it is conjectural and so cannot yield the kind of fixity demanded of philosophical knowledge.Footnote ¹⁰

This conjectural aspect of mathematics, against all part of interpretations, is no criticism of its current practitioners – that is, it is no problem of the method of mathematics that needs fixing by some metaphysical or metamathematical account of its hypothesis as unhypothetical first principles. Again, as Broadie notes:

What I will show is that, to appreciate both the benefits and the limits of the method of mathematics and measure these against benefits of the method of philosophy, these methods must be kept distinct, and, consequently, so too must both their epistemology and their ontology.

So, against all of Cornford, White, Tait, Benson, Burnyeat, and Broadie, I will show that Plato is concerned to offer a fourfold ontology associated with the four sections of the Divided Line. As I will argue, only then can we understand why mathematical inquiry, while distinct from philosophical inquiry, is “good for the soul.” Thus, it is a mistake to claim, as Broadie does, that Plato, in so separating the method of mathematics from that of philosophy, “went well beyond what was needed for making it clear that philosophical thinking, in particular the sort of ethical thinking that would be engaged in by philosophers-rules like those of Plato’s ideal state, is not mathematical in character and is not to be modelled on mathematics” (p. 22).

With Burnyeat,Footnote ¹¹ I will disagree with Broadie; there is certainly textual evidence for the claim that philosophical or moral reasoning is to be modeled on mathematics. By placing the mathematical theory of proportion as the highest theory and the Good as the highest form, Plato is showing us that even though their methods are distinct, moral reasoning is to be taken as “akin to” mathematical reasoning.

Indeed, I will argue that not only does this supposition answer why the mathematical branches are so ordered, again with the metamathematical theory of proportion itself as the highest theory, but it also answers the question of why the study of mathematics is good for the soul, namely, because the “concord” and “harmony,” or the good order, of both the objects of the branches of mathematics and the objects of philosophy are to be accounted for by proportional reasoning. So, while the ethical realm is not structured by mathematics per se, the proportional structure of the realm of forms is to be taken as “akin to” the proportional structure of the realm of mathematical objects in the sense that the good order of the forms themselves is to be accounted for by the mathematical notion of proportion. Thus, Plato’s criticism of mathematics as currently practiced is not, as part of interpretations build their case on, that it makes use of hypotheses. Rather, it is that its arguments are taken to rely on sense experience (e.g., “counted units” in the case of arithmetic, “constructed diagrams and figures” in the case of theories of geometry, “ornaments of the heavens” in the case of theories of astronomy, and “audible concordances” in the case of theories of cosmology).

In our investigation into the order of the branches of mathematics, what we will further see, however, is that the geometrical theory of proportion plays a double role: as Plato’s preferred mathematical theory of cosmology and as the highest, or metamathematical, theory that provides a good ordering of the branches of mathematics. In this metamathematical use, the mathematician qua philosopher-in-training will come to see that the notion of proportion itself is to be taken as a measure of harmony and concord itself. It is this use, when next applied to the philosopher’s forms, that will lend itself to the philosopher’s inquiry into moral matters and, in so doing, get them closer to grasping the Good as the highest form – that is, it will allow the philosopher to see, via the use of proportional reasoning, the sense in which the Good itself provides a good ordering of the forms.

Finally, I will appeal to this account of the metamathematical use of a geometric theory of proportion to conclude that Plato does take a stand on the “external” question, answering clearly that mathematical objects are not forms, either philosophical or foundational. Keeping in mind what Plato shows by placing mathematical objects within the realm of Being, and what the Republic says – namely, that “philosophic natures always love the sort of learning that makes clear to them some feature of being that always is and does not wander around between coming to be and decaying” (485b; italics added) – we will see that mathematical objects must have some “feature of being” but, again, only hypothetically so. Thus, and against both quietest and Platonist interpretations, I will argue that, as regards mathematical objects, Plato is a methodological realist – that is, he is a realist on the basis of what objects of thought are needed to solve both mathematical and metamathematical problems.

More pointedly, as regards the latter problems, what the geometrical theory of proportion brings to the mathematician’s table is an answer to the internal metamathematical question: What branch of mathematics accounts for or good orders the other branches of mathematics? What the geometric theory of proportion brings to both the mathematician’s and the philosopher’s table is talk of harmony and concord itself (i.e., talk of good order itself, as expressed by reasoning in terms of proportions). Indeed, as we will see, this is why the Divided Line is so divided into the geometric ratios that it is! I will show that the use of proportional reasoning itself plays an overarching and essential role in three ways: (a) in the overall argument scheme of the Divided Line and in Book 7, the notion of clarity, as a measure of truth and reality, is accounted for by the proportion of ratios between the lines themselves;Footnote ¹² (b) in his account of the good order of the branches of mathematics; and (c) in his account of the good order of the forms. This last explains why the study of mathematics is needed to grasp the Good.

So, against the quietist interpretations of Burnyeat and Broadie, Plato does answer the “external” questions of the ontological status of mathematical objects and the metamathematical sense in which mathematical objects are “akin.” However, against the part of interpretations of Cornford, White, Benson, and Tait, he does this by reducing both questions to internal questions – that is, to problems that themselves can be answered via the mathematician’s hypothetical method. Simply, then, the philosopher’s dialectical method and its need to appeal to an external philosophical or foundational realm is made mute. Plato’s mathematician is a methodological realist, they are not a metaphysical realist; they take mathematical hypothesis as if they were true first principles for the purpose of solving a problem and, in virtue of this, they take mathematical objects as if they exist. Thus, to require of mathematics that its objects are forms, be these philosophical or foundational forms, is to mistakenly confuse both the method and the epistemology of mathematics with that of philosophy. Finally, and now in hand with Plato, my counsel, as regards current practitioners of philosophy of mathematics, is as follows: We too would do well to keep the methodological requirements for mathematics distinct from those of philosophy – that is, we would do well to place more focus on the mathematician’s method and so on mathematical practice than we do on philosophical metaphysics or mathematical foundations.

3 The Divided Line

In Book 6 of the Republic, in attempting to explain the nature of the Good itself, Socrates first uses the Sun analogy to show the way in which the Sun is an “offspring” (506e) of the Good, and thereby comes to separate the visible and the intelligible realms. Next, Socrates uses the Divided Line analogy to further explain the epistemic and ontic distinctions that result from the distinctions between the visible and intelligible. Following Glaucon’s claim that he has, through Socrates’ use of the Sun analogy, understood “these two kinds” (the visible and the intelligible) (509d), Socrates introduces the Divided Line analogy to further explain his claim that “what the latter (the Good) is in the intelligible realm in relation to understanding and intelligible things, the former (the Sun) is in the visible realm in relation to sight and visible things” (508c). Bringing the two analogies together, Socrates begins with the assumption that the Sun is “sovereign” over the visible realm and the Good is “sovereign” over the intelligible realm (509e).

Socrates then subdivides each realm, according to the clarity of its objects:

It is important to pause here to note that the notion of clarity and the ratios of clarity as set by the various divisions and subdivisions are here intended to do both epistemic and ontological work. As Plato himself remarks,

Given Glaucon’s assent that he has understood both the distinction between the intelligible and the visible realm and the nature of the objects of the first, opaque, subsection of the visible realm, Socrates next considers the objects of the clear subsection, and moves to consider the ontic and epistemic consequences of these distinctions made within this realm:

Thus, physical objects themselves and their images respectively relate, on the basis of the ratio of their clarity or opacity (which is illuminated by the Sun [508b]), ontologically to existence and nonexistence, and epistemically to truth and untruth, and so to knowledge and opinion.

We subsequently come to the subdivisions of the intelligible realm:

In the first subsection of the intelligible realm, then, the soul uses “images”Footnote ¹³ and its method is such that it is forced toFootnote ¹⁴ base its inquiry on hypotheses, reasoning from a hypothesis down to a conclusion.

In the other subsection, the soul reasons from a hypothesis up to an unhypothetical first principle and then down to a conclusion,Footnote ¹⁵ making no use of images but only of forms themselves. Glaucon is here confused, and so Socrates begins anew, now making mention of the mathematicians’ method:

Note, then, that the objects of mathematics are not the previously mentioned “images,” but they are “the odd,” “the even,” “the various figures,” and so on. These objects are treated both as hypothesized and as known, so no account of them is needed. They are taken as if they were “evident to everyone” first principles, but they are not. Indeed, as noted in the last sentence, the purpose of the mathematicians’ as if hypotheses are to solve a given problem, so, unlike the philosophers’ method, the mathematicians’ method is not aimed at arriving at unhypothetical first principles, it is aimed at solving problems. This is another reason why the hypothetical method cannot be part of the dialectical method; mathematical hypotheses are taken as known – that is, as if they were first principles, so no account of them is needed – whereas philosophical hypotheses are taken as hypotheses, so, if they are to be held as known, an account of them in terms of unhypothetical first principles is needed.Footnote ¹⁶

Having so clarified things to Glaucon’s satisfaction, Socrates is now ready to move on:

There is a much missed and important distinction to be made here between “images,” or what a mathematician uses or talks about (e.g., the diagonal they draw) and “things themselves,” or what they think about (e.g., the diagonal itself).Footnote ¹⁸ Drawn diagrams, figures, and so on are “only images” used to aid the mathematician’s thinking about the “things themselves”; their arguments, however, are intended to be about the objects they think about. There is also a much missed and important, but often confused, analogy that Plato will appeal to further on: that just as physical objects are the originals of “shadowy” physical images, so mathematical objects are the originals of “shadowy” mathematical images; just as for the objects of the physical realm wherein “the likeness [the image] is to the thing it is like [the original],” so too for the objects of the mathematical realm. As originals, then, mathematical objects are clearer, and recalling the epistemic and ontic role of the concept of clarity as a marker of truth and existence, statements about them are truer and they are more real than those about mathematical “images.” Thus, when the mathematician uses the faculty of thought they will come to use diagrams and figures “as only images,” and, as a result, will come to see the need to make their arguments about mathematical objects themselves.Footnote ¹⁹ However, these mathematical “kinds of things” (i.e., kinds that arise out of the use of the hypothetical method), while both intelligible and clearer than their “images,” are yet distinct from those that arise out of the use of the dialectical method:

The mathematician, then, has access to objects that are found in the intelligible realm, but that are distinct from the other intelligible kinds of things (forms), both methodologically because is the mathematician is forced to use hypotheses and ontologically because the objects are objects of thought. So, again using the intended proportional reasoning as set out by the ratios of clarity in the divided line, just as the images of physical objects are less clear than physical objects themselves, and mathematical images are less clear than mathematical objects, so too are mathematical objects less clear than the philosophical objects grasped by traveling up to a first principle: the forms. That Plato intends to use these methodological differences to further infer even more epistemic and ontic distinctions is next made clear:

Glaucon is here more than confused; indeed, he is both shocked and surprised by the implications of these methodological differences:

Again, using the ratio of clarity as his cleaver, Plato comes to consider the full epistemological and ontological implications of these differences in method.Footnote ²³ That is, epistemic truth and ontic reality is to be given only to those intelligibles that are reached by the philosophers’ dialectical method. Next employing proportional reasoning as fixed by the ratios of clarity across the realms, we have that the mathematicians’ objects are more clear (true/real) than all of physical objects, physical images, and mathematical images because they are grasped more clearly, by thought and not by the sense perception, but because no first-principled account is given of them they remain less clear (true/real) than those intelligible objects grasped by the understanding. The Divided Line ends with Plato clarifying, again by reasoning proportionally from the ratios of clarity, the epistemic terminology that he intends to employ:

Thus, reasoning proportionally by taking the ratio of clarity to be the mark of both epistemic truth and ontic reality, we have that in the visible realm, what we believe is truer than our opinions because physical objects are more real than images, and in the intelligible realm, what we think is truer than what we believe because mathematical objects are more real than mathematical images, and finally, what we understand is truer than what we think because forms are more real than mathematical objects. We have, then, our four epistemological states and their corresponding ontological objects: understanding/forms, thought/mathematical objects, belief/physical objects, and imagination/images of physical objects.

But is this all the evidence we need to claim that mathematical objects are not forms? Perhaps too these mathematical objects, as Aristotle reports of Plato’s view (Metaphysics, 987b), are yet distinct “intermediates.”Footnote ²⁵ Or, perhaps, as Cornford, White, Tait, and Benson suggest, the mathematician, in light of Plato’s “criticisms,” could or should be now motivated to adopt the dialectical method and so search for those unhypothetical, metaphysical, or metamathematical first principles that would allow them to tether or account for their hypotheses. To speak against these possibilities, and clearly conclude that Plato was not a mathematical Platonist, we need to next consider what Plato says in Book 7, where, in detailing both the educational value of mathematics and the problems with mathematics as currently practiced, Socrates, again using proportional reasoning based on the ratios of clarity, reconsiders the distinctions of the Divided Line. It is to these reconsiderations that I now turn.

4 Book 7

Just after Book 6’s Divided Line analogy, Plato, in Book 7, introduces the Cave analogyFootnote ²⁶ to represent “the effect of [a philosophical] education and the lack of it on our nature” (514a), so that the philosophical journey outside the cave, is to be thought of “as the upward journey of the soul to the intelligible realm” (517b). Interestingly, and telling against any need for an epistemological use of a myth of recollection,Footnote ²⁷ we are simply told that this analogy shows that the “power to learn is present in everyone’s soul” (518c).Footnote ²⁸ Thus, education and learning, “takes for granted that sight [and, by analogy, the capacity of the soul, reason] is there” but it is “not turned in the right way” so that we must “contrive to redirect it appropriately” (518d). The aim of Book 7, then, is to show what subject can be used to “redirect” the soul from its downward journey to its upward one, so that the philosopher may come to “see the Good” (519c).

To this end, Socrates asks: “So what subject is it, Glaucon, that draws the soul from what comes to be to what is?” (521d). Glaucon is next pushed to consider “one of those (subjects) that touches all of them” (522b) – that is arithmetic or “number and calculation” (522c). This subject is claimed as “one of the subjects we were looking for that naturally stimulate the understanding” (522e–523). The problem, however, is that “no one uses it correctly” (523a); for example, they see numbers as “attached to visible or tangible bodies” (525d). In its correct use, this subject must “summon thought” and, in so doing, “wake up the understanding” (514d); for example, numbers must be taken as “accessible only to thought and grasped on no other way” (526a; italics added). So, the philosopher-in-training and the mathematician must both move away from becoming or what comes to be (what is grasped by sense perception) toward being or what is (what is grasped by reason) – that is, they must aim at “knowledge of what is, not of something that comes to be and passes away” (527b). So, in our newly reconsidered divided line, we have an added ontological aspect to the divided line: the physical realm is taken as the realm of becoming and the intelligible as the realm of being. But note that, in light of the Divided Line’s epistemic distinction between mathematical thought and philosophical understanding, these mathematical subjects serve only to redirect the soul; by summoning or using thought, it stirs or wakes up the understanding. So here too, mathematical objects are to be taken as objects of thought, not objects of understanding.

Plato next notes the manner in which the layman and the current mathematical practitioner reason incorrectly; they use their senses and rely on images (both images of physical objects and images as diagrams and figures of mathematical objects) whereas they should use thought and rely only on those mathematical objects themselves firmly located in the intellectual realm. On their upward journey, then, the philosopher-in-training must first take up arithmetic,

Lest one be tempted to make much of the use of ‘understanding itself’ here, note that immediately after we are reminded that such numbers “are accessible only to thought and can be grasped in no other way” (526a; italics added), and as such arithmetic “really does seem to be necessary to us, since it apparently compels the soul to [move upward and] use understanding itself on the truth itself” (526a–b; italics added).Footnote ²⁹ As with arithmetic, likewise too for our account of the subjects of geometry (526c–e); solid geometry (two-dimensional geometry or “whatever shares in depth”); astronomy (three dimensional or “revolving solids”) (528b); and, cosmology (theory of harmony) (530d). That is, for all of the branches of mathematics, we are not to seek an account of any of these subjects on the basis of how they are currently practiced. We consider first geometry:

As for solid geometry, we are somewhat mysteriously told

I believe that Plato is here referring to Theaetetus as his preferred “director”; Theaetetus was both a native of Athens and was known to be developing a theory of solid geometry around the time of the writing of the Republic.Footnote ³¹ As I will show, much is at stake here and there are many suggestions as to just who Plato is referring to when he speaks of “those who investigate it who are too arrogant to obey him” and “investigators who lack any account of their usefulness.” My claim is that both here, and next when he comes to speak of astronomy as it is “handled today” (529a), he is referring to Archytas and his Pythagorean followers. Not only was Archytas a well-known political figure, so talk of directing a city seems apt, but he was also developing a theory of astronomy (or mechanics more generally) based on a Pythagorean or arithmetical theory of proportion, or a theory of proportion build up out of arithmetic ratios. And, more problematically as a Pythagorean, these ratios for the study of astronomy were taken as arising from sense perception (i.e., what was seen “in the ornaments of the heavens” [529b]), and, for theories of harmonics, they were taken as arising from what was heard in “audible concordances” (530c). As Plato sees it, the problem here is twofold: it is both that current practitioners of these mathematical theories rely on sense perception and that they “lacked an account of its usefulness.”

More to the point, what I will show is that these Pythagoreans lacked a geometrical theory of proportionFootnote ³² that could not only provide a mathematical account of both astronomy and cosmology, but that could further provide a metamathematical account of what all mathematical subjects have in common; that is, an account of numbers, figures, motion, and sound in terms of geometric measures or ratios. This claim marks a major point of disagreement with Reference BurnyeatBurnyeat (2000), who assumes that Plato’s theory of proportion is, or is to be based on, Archytas’ arithmetical theory. Indeed, as we will see, this point of departure provides the basis for several other significant differences between us. My appeal to the use of a geometrical theory of proportion also marks an important point of divergence from all part of interpretations, including Reference TaitTait’s (2002). That is, what part of interpretations get right is that, for Plato, mathematics itself must be metamathematically “accounted for”; what they get wrong is the demand that such accounts can only be provided by the mathematician taking up the philosopher’s dialectical method, which would require reference to a fixed domain of mathematical objects as forms, be these metaphysical forms or foundational geometric forms. Thus, while Tait rightly notes that Plato was “concerned with foundations because of ‘the discovery of incommensurable line segments’” (pp. 19–20), I think he, like Burnyeat, is mistaken in his claim that “a geometric theory of proportion has likely still not been discovered by the time of the Republic” (p. 20). It might not have been fully developed, but Plato’s reference, in the Theaetetus, to the mathematical work of both Theodorus and Theaetetus shows that he is aware that it is being developed.Footnote ³³

In line with Benson,Footnote ³⁴ I take Archytas, Philolaus and other Pythagoreans as the referents of those current practitioners who both “rely on images” and “fail to give an account,” and too I take Theodorus and Theaetetus as the “credible mathematicians.” However, I part ways with Benson when he claims that “we would do well to avoid drawing any conclusions concerning the relative flaws and merits of Theodorus’ and Theaetetus’ procedures” (p. 18). Indeed, as I will argue, it is precisely to their, albeit developing, geometric theory of proportion, developed as it was out of the desire to solve mathematical, physical, and metamathematical problems via the use of the mathematician’s hypothetical method that we should look to “ascend to problems” (531c) and give “and account of its usefulness” by showing that, metamathematically, it provides an overarching good-ordered account of mathematics itself. Thus, I will argue against all part of interpretations by showing that Plato has a different, organizational conception of what he intends for a metamathematical foundation.

Having stated and situated my claim and aims, let us now continue onward with what Plato says. We come next to the fourth subject, astronomy “which deals the motion of things having depth [the motion of solids]”:

How then shall we be motivated to proceed in our study of astronomy if not by physical images? Simply, we are to use these physical images as physical diagrams and then use these physical diagrams as images of mathematical objects themselves, this with the aim of solving a physical problem by making it a mathematical one; for instance, by measuring the true speed of motions by true numbers as measured between true geometrical figures.

So, it is by “making use of problems” and, in so doing, attending to the usefulness of our mathematical hypotheses, taken as if they are true first principles, that we are led to those true objects that are needed to solve these physical problems, that we are to undertake our study of astronomy. Turing then to the more general account of the usefulness of mathematics, this schema sums what I call Plato’s methodological as-ifism: we take a mathematical object as if it exists because we take a hypothesis as if it were a true first principle, and we do this with the aim of solving a problem.

Having set our problem-solving strategy, we come next to the fifth and final mathematical subject – that is, the study of the theory of harmony itself. We are first told, as with “astronomical motions” so with “harmonic ones” (530d) – that is, current practitioners, including PythagoreansFootnote ³⁶ (530d) believe that

It is interesting that Plato uses the term “ascend” here; recall that, on my reading, the mathematician, who adopts the hypothetical method, must travel down from hypotheses. So, one might be tempted, as are Cornford, Tait, and Benson (see, for example, Reference BensonBenson [2012, pp. 18–23]), to appeal to this use of ‘ascend’ to argue that the only way the mathematician can ascend is by traveling up from hypotheses toward first principles, so that Plato here intends the directive that the mathematician must adopt the dialectical method. My counter point, which I will argue for in the next section, is that Plato, by using the phrase “ascend to problems” is here indicating that he intends the albeit emerging geometric theory of proportion as itself, allowing us to “ascend” to those metamathematical problems that concern questions of the “kinds of things” of mathematics and the “kinships” among them, and it does this by providing an overarching account of the lower-level mathematical branches.

Returning to Plato, we note that Glaucon here responds that this “ascending to problems” is a “daimonic task”Footnote ³⁷ and Socrates is quick to remind him that it is just this task that is “useful in the search for the beautiful and the good” (531c). As we will see, the mathematician’s metamathematical use of the geometric theory of proportion is useful for their overall aim of providing an overarching good-ordering account of the subjects of mathematics itself, and it is just this use that “contributes something” to the philosopher’s goal of searching for the good.

So, in addition to its being among the five mathematical subjects, the mathematical value and philosophical use of the geometric theory of proportion is that it allows us to give a good-ordering account of what all the other mathematical subjects have in common, and, in so doing, it is the highest subject that allows us to “contribute to our goal [of seeing the Good].”

It is here that, with Burnyeat, I disagree with Broadie’s claim that there is no textual evidence for Plato holding that philosophical or ethical reasoning is, in some sense, mathematical in nature. Indeed, not only does this supposition answer the question of why mathematics is “good for the soul,” but too it answers why the mathematical branches are so ordered, with the mathematical theory of harmony as the highest. This is because “concord” and “harmony,” both mathematical and ethical, are to be accounted for by the metamathematical use of geometric theory of proportion. Thus, we may finally conclude that Plato’s criticism of mathematics as currently practiced is not, as part of interpretations build their case on, that it makes use of hypotheses, rather it is that it is taken to rely on sense perception, (e.g., the counting numbers of “tradesmen and retailers in the case of theories of arithmetic, the “constructed diagrams and figures” in the case of theories of geometry, the “ornaments of the heavens” in the case of theories of astronomy, and the “audible concordances” in the case of theories of cosmology). What we will next see is the way in which the geometrical theory of proportion plays its double role: as a mathematical theory of cosmology and as a metamathematical theory of what measures harmony and concord itself. It is this last use, when applied to the philosopher’s forms, that lends itself to the philosopher’s inquiry into moral matters and, in so doing, gets the philosopher closer to grasping the Good.

In what way, then, does the geometrical theory of proportion allow for the organization of all other mathematical subjects so that we come to see what they have in common, and just how does this relate the philosopher’s goal of grasping the Good? I will take up the details of answering this question in the next section. For now, what I want to point out is that for all of these mathematical subjects it is mathematical practice that summons thought; that is, it is by “making use of” or “ascending to” (531c) mathematical problems, and not concerning ourselves, as current practitioners of mathematics do, with physical or mathematical images, that we are motivated by the use of the faculty of thought to move upward and so “wake up the understanding” (514d). Thus, even when faced with overcoming the errors of current practitioners, Plato does not require that the mathematician be so motivated by these mistakes to move even further upward toward adopting the philosopher’s dialectical method and, in so doing, use the faculty of understanding to move into the realm of the forms. That is why Plato is careful to mention that “all these subjects are merely preludes to the theme [of attaining the good] itself” (531d; italics added); this is why Plato next takes time to remind us that we should

Thus, Plato’s mathematician is neither a dialectician nor can they know the forms: the mathematician’s hypothesis are taken as if they were first principles, and, consequently, the mathematician’s objects are taken as if they exist. To see that this is what Plato fully intends, we turn next to his reconsideration the mathematician’s ontology. That is, now making use of both the Cave analogy’s distinction between being and becoming, and the proportions of the divided line, so that now the realm of being aligns to the intelligible and the realm of becoming aligns to the sensible, Plato tell us that

So, in contrast to the philosopher’s objects (the forms as “the best among the things that are” or the “clearest thing”), the mathematician’s objects, while in the intelligible realm and so in the realm of being, are better than mathematical “images” (which are now taken as akin to physical images or shadows of physical things), but are still less clear than forms, and so themselves remain ontologically “shadowy.” That is, while in the Divided Line mathematical images were taken as akin to physical images and mathematical objects themselves were taken as akin to physical objects themselves, Book 7 ends with us taking mathematical objects as akin to shadowy physical images!!! So reconsidered, the mathematician’s objects are thus even more removed from the philosopher’s forms!

When next pushed by Glaucon to “discuss it [the realm of the forms] in the same way as we did the prelude [as we did with mathematics]” and, in so doing, to further clarify “in what way the power of dialectical discussion works, into what kinds it is divided, and what road it follows” (532d; italics added), Socrates replies:

Regardless, then, of “what kinds” the realm of forms is divided into, Plato insists on two things: that the Good exists and

Having thus situated the necessary role of a mathematical education, Plato next takes the opportunity, again in light of the Cave analogy, to reconsider the mathematician’s method as

Importantly, and a point often missed in the interpretative literature, Plato, as a result of holding the mathematician’s hypothetical method as distinct from the philosopher’s, here reconsiders his use of the term “knowledge,” reserving it for philosophers only.Footnote ³⁸ That is, having reconsidered and further distinguished the mathematician’s method from the philosopher’s, Plato next turns to reconsider his epistemological terms so that the mathematician, using hypothesis as if they were true first principles to solve a problem, has an understanding of their objects, but not knowledge!!

What next follows is the crucial claim for my argument that mathematical objects are not forms: as a result of these reconsidered methodological and epistemological distinctions, Plato will now reconsider the ontological status of mathematical objects. They arise from the use of the hypothetical method and not the dialectical method, and, as a consequence, they are objects of thought and not objects of knowledge; they are “concerned with being” but are also conjectural – that is, they are more akin to the “shadowy” objects of the imagination. Again, by adding the Cave analogy to the divided line, we have that just mathematical objects, as measured against physical images, are as being is to becoming, so too mathematical objects, as measured against forms, are conjectured objectsFootnote ³⁹ – that is, they are objects that we take as if they were real for the purpose of solving a problem. That Plato intends this ontological “akinness” is clear:

Thus, bringing the Sun and the Cave analogies together, and using the ratios of the clarity as set by the Divided Line analogy as markers of existence and truth, we are left to conclude that mathematical objects are in the realm of the intelligible and so are “concerned with being” (534a), but when compared to objects of knowledge (i.e., to forms), mathematical objects, as objects of thought – since thought is akin to imagination – are not forms, just as physical images, as objects of imagination, are not physical objects. This explains the sense in which the mathematician hypothesizes “the odd,” “the even,” “the various figures”; but too once so hypothesized, they can then think about them as if they existed, even if they cannot know they exist because they lack a first-principled account of their being. This, then, is all Plato plans to say of the matter of the distinction between mathematical and philosophical methods, their epistemic faculties, and their objects. And too this is all he is going to say about “the things” that each deals with:

This marks the point on which Reference BurnyeatBurnyeat (2000) builds his interpretation of Plato’s quietist stance of the “external” ontological status on mathematical objects.Footnote ⁴¹ This too is the point where Burnyeat and I fundamentally disagree. Burnyeat holds that “the things these deal with” is a reference to the distinction between mathematical and philosophical as intelligible kinds of things and so he considers the question that Plato intends to “pass by” the question of whether mathematical objects are forms, or intermediate kinds of things. Burnyeat says of this passage: “[t]o refuse to contemplate the result of dividing the objects on the intelligible section of the Line is to refuse to go into the distinction between the objects of mathematical thought and Forms” (p. 34).

It is this reading, moreover, that licenses Burnyeat’s further claim that even if “internally,” from within the context of mathematical practice, we can deny, as Burnyeat and I do, both that mathematical objects “could ultimately be derived from Forms” (p. 34) and “that Plato thinks mathematics is directly about Forms” (p. 35), we cannot, on this “internal” basis, get to the “external” metaphysical claim that mathematical objects are not forms, and so must ultimately rest quiet on the matter. And so, according to Burnyeat, we must conclude that the “external” question “is not discussed in the Republic” (p. 33). I disagree. Not only is this quietest view out of line, as Burnyeat himself notes, with the fact that the “external” question “was certainly debated in the Academy, as we can tell from the last two Books of Aristotle’s Metaphysics” (p. 33), but, as I have shown, in both the Divided Line and Book 7 of the Republic,Footnote ⁴² Plato has shown us, both mathematically and philosophically that mathematical objects are not forms. Moreover, he has further shown us, in the Meno, the Republic, and the Theaetetus, that it does matter whether our mathematical “kinds of things” are proportioned in ratios that are geometric or arithmetic, so that the supposedly “external” question of how mathematical “kinds of things” are ratioed is answered, and too it is answered “internally,” – that is, is answered withing the context of mathematical practice by appealing to a mathematical theory not a metaphysical theory.

On my interpretation, what Plato is here “passing by” is the question of the ratios between the kinds of things in each of the mathematical and the philosophical realm. That is, he is not going to take up the question of how these mathematical kinds of objects should be taken as kinds proportioned by geometric ratios, by, for example, appealing to the specifics of Theodorus’ developing theory, except to say that the geometric theory of proportion is the highest mathematical branch. Likewise, of the kinds of things that philosophy deals with, forms like Temperance, Justice, Good, and so on, he is not going to consider the question of how exactly these kinds of objects should be taken as kinds proportioned by the Good, by, for example, appealing to the Platonic notions of “participation” or “presence” or “imitation,” except to say that the Good is the highest form. Note too that this reading is further in line with Socrates’ dismissive reply (at 533a) to Glaucon’s question of “into what kinds” the objects of understanding (i.e., the forms) are divided. He has already said he is not going to get into the discussion of “kinds of” forms; likewise, he is saying here that he is not going to get into the discussion of either “kinds of” mathematical objects or “kinds of” forms.

The last point that Plato does pause to make even more clear is that the differences in methodology demand differences in both epistemology and ontology. That is, while mathematics can be used to “grasp some image of the Good,” because the mathematicians’ method does not begin with first principles, it cannot offer an account “of the being” of mathematical objects, so they remain as objects of thought as opposed to objects of understanding (i.e., forms). To think otherwise, we are told, is simply irrational:

Plato here tells us of the limits of a mathematical education: the mathematical method, in so far as it reasons down from hypotheses as if they were first principles, as opposed to reasoning up from hypothesis to first principles, cannot grasp “the being of each thing,” whether that thing is a mathematical thing or a philosophical thing, like the Good. But, nonetheless, and importantly, the benefit of a mathematical education on the philosopher-in-training is that mathematics can be used to “grasp some image of the Good.” In the next section, I will consider the question of how this is possible. For now, I pause to remind the reader that Plato intends the geometrical theory of proportion to play a metamathematical role and, in the next section, I will show that Plato intends for this metamathematical role to be akin to the metaphilosophical role played by the Good. What we will see is that the geometrical theory of proportion plays a double mathematical role: as a mathematical theory of cosmology and as the highest, or metamathematical, theory that accounts for the concord and harmony (i.e., for the good order) among all the branches of mathematics. Next, using proportional reasoning to take this role as akin to the role of the Good we come to see the sense in which the Good, as the highest form, accounts for the good order among the forms. It is this use of proportional reasoning, now applied to the philosopher’s forms, that lends itself to the philosopher’s inquiry into moral matters and, in so doing, gets the philosopher closer to grasping the Good. But, while the mathematician’s method can be used to account for mathematical objects (e.g., for irrational numbers) it cannot be used to make mathematicians rational – that is, it cannot be used to account “for the being” of either mathematical or philosophical objects, and as such the philosopher, whose goal is to reach the Good, will need another method.

As I have hinted at, and will now argue, one should see Plato’s entire account of mathematics as an attempt to move past any metamathematical use of the Pythagorean arithmetical theory of proportion so that he can include those “irrationals” that have a logos, or have an account in terms of geometric ratios; for example, those numbers that can be given an account of by a geometric theory of proportion.Footnote ⁴⁴ As we are shown in the Meno, some irrational lengths, which are constructed from the problem of doubling the length of the side of a two-foot square, are to be included as numbers qua objects of thought, because an account of them can be given in terms of geometrically ratioed lengths (e.g., as ratios measured by the diagonal of a two-foot square). To see that Plato further intends this metamathematical role to apply to all the branches of mathematics, we first recall his criticism of Pythagorean arithmetical accounts of astronomy and cosmology and next we recall that both Plato and Theaetetus were students of Theodorus, who was known to be developing a geometric theory of proportion that was aimed to account for all the branches of mathematics. This metamathematical use is further witnessed in the Theaetetus (145c–d) where Theaetetus tells Socrates that he learned all of arithmetic, geometry, astronomy, and harmonics from his teacher Theodorus, who is an expert (145a) in all of these subjects, and who, Socrates further tells us,

And, more importantly for my claim that his geometrical theory was known to Plato, note that just after this, when Theaetetus is asked by Socrates whether he has “found such a class,” Theaetetus replies, “I think we did,” and proceeds to sketch his account of numbers as geometrically proportioned measures, whereby “all numbers can be divided into two classes … all the lines which form the plane figure representing the equilateral number we defined as lengths, while those which form the sides of squares equal in area to the oblongs we called roots, as not being commensurable with the others in length.” Just after this Theaetetus tells us that “there is another distinction of the same sort in the case of solid geometrical figures and harmonics” (Theaetetus, 147e–148b).

So, while Reference TaitTait (2002) rightly notes that Plato was “concerned with foundations because of the discovery of incommensurable line segments” (pp. 19–20), he is wrong to presume that the only “foundation” that can be given is one founded on the philosopher’s dialectical method and an ontology of forms. Simply, the geometric theory of proportion is taken as the theory that answers the metamathematical question of what theory accounts for “the ratios between the things” of the various branches of mathematics. So, against all part of interpretations, no metaphysical move to the philosopher’s dialectical method is needed to answer this metamathematical question. Again, against Cornford, White, Benson, Tait, and Burnyeat, this question is an internal mathematical question, that is, while Plato does not take up the question of exactly how the kinds of objects in the mathematical realm are ratioed, as we are clearly shown in the Theaetetus, it is nonetheless a question that he intends to be settled by mathematicians, like Theaetetus and Theodorus. It is neither a question that requires the use of the philosopher’s dialectical method, and so requires mathematical objects as geometric or metaphysical forms, nor is it an “external” one that is passed over in silence.

5 The Good in Mathematics

How, then, does the learning of mathematics, and, in particular, the metamathematical “investigation of all the subjects we have mentioned,” so that it “arrives at what they share in common with one another and what their affinities are, and draws conclusions about their kinship,” “contribute something” to philosopher’s goal of grasping the Good? To answer this question, we must first answer the question: What is the relationship between mathematical reasoning and philosophical/moral reasoning?. Here my aim will be to show, against Broadie and in agreement with Burnyeat, that Plato does intend for philosophical inquiry to be modeled on mathematical inquiry. But too, against Burnyeat and in agreement with Broadie, this not because the realm of forms is itself mathematically structured, rather it is that our reasoning about both the good in mathematics and the Good in philosophy are to be founded on proportional reasoning. Thus, to answer these questions, I now turn to consider “the good” in mathematics and, in this light, further investigate the philosophical role of the geometric theory of proportion. Recall the discussion that underpins both the Divided Line and Book 7: how the philosopher is to reach the Good. Recall too that Socrates introduces the Divided Line analogy to explain the Sun’s analogical claim that “what the latter (the good) is in the intelligible realm in relation to understanding and intelligible things, the former (the Sun) is in the visible realm in relation to sight and visible things” (508c). Bringing the Sun and Divided Line analogies together, we are then guided to construct the geometrically ratioed divided line under the assumption that the Sun is “sovereign” over the visible and the good is “sovereign” over the intelligible (509e).

Moreover, it is just this assumption that allows us to use the notion of clarity to proportion the ratios of the sub-realms of both the physical and the intelligible realms (i.e., the Sun makes images and physical objects clear to the same degree or proportion that the good makes mathematical objects and forms clear). Moreover, in the visible realm, it is the Sun that ratios the degree of clarity (508b), and, as a result, physical objects and physical images respectively relate, on the basis of the ratios of their clarity or opacity, ontologically to existence and nonexistence, and epistemically to truth and untruth. Analogously, it is the good that ratios the degree of clarity in the intelligible sub-realms to the same degree or proportion as the Sun ratios of the visible sub-realms. As a result, philosophical objects (forms) and mathematical objects respectively relate, on the basis of the ratios of their clarity or opacity, ontologically to existence and nonexistence, and epistemically to truth and untruth. Next, adding the ratios as set by the analogy of the Cave, mathematical objects relate to philosophical objects in the same ratio as physical images relate to physical objects, so that mathematical objects are akin to objects of imagination – that is, they are less real (they are “shadowy” [532b]) than philosophical objects, and the statements about them are less true than philosophical statements (“they are clearer than belief but darker than knowledge” [533d]).

So far, I have merely summed the claims I have considered. Now I will further argue that the geometrical theory of proportion plays an additional role in the mathematical sub-realm, and this role is intended to be taken as akin to the role played by the Good in the philosophical sub-realm. Indeed, this is the reason why the philosopher-in-training must take up the study of mathematics for ten years and why they must proceed from arithmetic, to geometry, to solid geometry, to solids in motions, to the harmony of such motions, and end at the geometrical theory of proportion itself. Only then will they come to see the double role that it plays, both as a mathematical theory of cosmology and as a metamathematical foundation that accounts for the good order of the various branches of mathematics. Finally, it is by appreciating this last role that they are further able to understand the analogous role that the Good itself plays in accounting for the good order of the forms.

Until recently, there has been little discussion in the literature as to why Plato orders his mathematical subjects in the manner he does.Footnote ⁴⁵ To this end, I now turn to consider why it is that the geometrical theory of proportion is the last, and I hold, the highest, mathematical subject. Recall that in his discussion of the theory of harmony, Plato claims that this subject has two uses. The first use is as an “indispensable aid” in the philosophers’ “search for the beautiful and the good” (531c); for this use, Reference BurnyeatBurnyeat (2000, p. 34) is right that the discussion of the “ratios between the things these deal with” (534a), is beside the point of Socrates’ message. However, as noted, he is wrong to assume that this reference is to “the things” that mathematics and philosophy deal with, and, as a result, he is wrong to conclude that Plato adopts a metaphysical quietist position as regards “the distinction between the objects of mathematical thought and Forms” (534a). Against both Burnyeat and Broadie’s quietism, what I have demonstrated thus far is that Plato has considered, in the Divided Line and in Book 7, this metaphysical distinction, and has shown that mathematical objects are not forms.

As I have argued, what Plato has refused to explicitly discuss is the ratios between the things that each of mathematics and philosophy deals with. But, as noted, in the mathematical sub-realm, he, following Theodorus and Theaetetus, for example, will need to do this when he comes to consider the second use of the theory of proportion: the “investigation of all the [mathematical] subjects we have mentioned” with the aim of arriving at “what they share in common with one another and what their affinities are, and drawing conclusions about their kinship” (531d). Thus, in the mathematical sub-realm, what the geometric theory of proportion does is provide a metamathematical good ordering of the branches of mathematics and, in so doing, provides an overarching account of what all the mathematical subjects share in common. However, it does not do this by standing above, or apart from, the other mathematical subjects, rather it is constitutive of what they all have in common – that is, the geometric conception of ratio – and it is in virtue of this, that it provides an overarching account of the all the kinds of mathematical objects in terms of geometric ratios. That is, grasping this metamathematical role allows us to “ascend to problems” by providing an overarching account of the “ratios between the things” that mathematics deals with. For example, as seen from the view provided by the geometric theory of proportion, the “kinds of things” of arithmetic (i.e., numbers) are not to be understood as Pythagorean arithmetical indivisible units;Footnote ⁴⁶ rather, they are to be understood as geometrical measures. More importantly, as measures of geometric ratios, as Plato himself demonstrates in the Meno, their rationality or irrationality is not decided on the basis of whether such measures are arithmetically commensurable or not, but rather by whether they can, in light of a geometric theory of proportion, be given an account in terms of the ratios of their geometrically constructed measures. Again, going back to the Meno, the length of the side that doubles the area of a two-foot longFootnote ⁴⁷ square, while incommensurable, is nonetheless “rational” in so far as an account of it can be given in terms of the ratio of the proportions of the sides (in terms of the length of the diagonal as set by the Pythagorean theorem) when the length of the two-foot side is taken as a measure.Footnote ⁴⁸

This interpretation of the geometric theory of proportion as both account-giving and constitutive of the subject matter of mathematics is not only important for understanding what Plato takes arithmetic numbers and geometric figures to be, but for understanding the account-giving role of the Good as well. That is, the Good, in so far as it serves an analogous role in the philosophical realm as the geometric theory of proportion in the mathematical realm, does not stand above, or apart from, the other forms; rather, it too by being a measure is constitutive of “ratios between the things” that metaphysics deals with – that is, the ratios between the forms themselves. These account-giving and constitutive roles of the good, in both the mathematical and the philosophical sub-realms, are further in agreement with the Greek notion of logos as not only something that merely provides an account or order but also something that provides a good order, that is, an order that establishes both concord and harmony.Footnote ⁴⁹ Finally, we must also note that implicit in the Greek notion of an account as a good order is always a moral, or value-laden, component. It is in all of these senses of “good,” then, that we are to understand why a mathematical education is not only necessary for the philosopher’s education but also good for the philosopher’s soul.Footnote ⁵⁰ That is, it is only when we place our focus on the metamathematical account-giving role of the geometric theory of proportion that the study of the various mathematical subjects can be used to “redirect the soul from its downward journey to its upward one,” so that the philosopher comes to see that, analogously,Footnote ⁵¹ “the last thing to be seen is the form of the Good” (517c; italics added). It is this analogy, then, that allows us to see why the mathematician’s metamathematical investigation into the “ratios between kinds of things” that mathematics deals with “contributes something” to the philosopher’s goal of grasping the Good.

It is now time to consider Reference BurnyeatBurnyeat’s (2000) interpretation of this second, metamathematical, use of theory of proportion and his further claim that it is constitutive of the “kinds of things” that metaphysics deals with, so that philosophical dialectic itself is the metamathematical attempt to account for all hypotheses, this with the more than confusing claim that Plato “leaves us to infer that dialectical debate about the conceptual foundations of mathematics itself … is a debate about values like justice” (p. 46). Thus, in the philosopher’s accounting for their hypotheses in terms of forms, as Burnyeat explains, they will, along the way, need to also account for the mathematician’s hypotheses likewise in terms of forms:

Clearly, I disagree. Where, however, do we part ways? Burnyeat and I agree that this “synoptic view” comes by way of the use of the theory of proportion itself. That is, once the mathematician, after their ten years of study, comes to this view of the account-giving role of the theory of proportion, they then, for another fifteen years, have to apply this role of proportion itself to practical matters (matters concerning the proportioning of both military matters and matters of state administration; see 539e–540a). Only then are they in position to aim, for the next five years, to become a philosopher by applying this account-giving role of proportion to metaphysical matters (matters concerning the proportioning of the forms, ethical matters, and even matters concerning the proportioning of the soul). Were we differ is that for Burnyeat, after the philosopher has grasped the Good as the highest form, they must then turn to give an account of the good order of both mathematical and metaphysical matters in terms of forms, then, as a philosopher king, to matters concerning the good order of the city, its citizens, and themselves (534a–b). Thus, Reference BurnyeatBurnyeat’s (2000) answer to his question – “Is the study of mathematics merely instrumental to knowledge of the Good, in Plato’s view, or is the content of mathematics a constitutive part of ethical understanding?” (p. 6; italics added) – is:

Here I stand with Reference BroadieBroadie (2020) that making the metamathematical debate about the foundations of mathematics a dialectical debate leads to “idiocy of demanding that mathematics should model itself on ethical philosophizing” (p. 23). Ok, perhaps not idiocy, but at least inconsistency: the assumption that the metaphysical realm of forms is itself mathematically structured and the subsequent appeal to an account- giving role of forms for mathematics is clearly inconsistent with Burnyeat’s stated quietism. Burnyeat has it that at a mathematical level the mathematician and the philosopher’s methods are distinct but that at the metamathematical level only the philosopher can account for the good order of mathematics. Simply, Burnyeat has gone too far, both in claiming that mathematics is constitutive of ethical understanding and, consequently, that at least one task of the philosopher is to dialectically account for mathematical hypotheses in terms of forms. Where has Burnyeat gone wrong? As noted, the first error is that he has taken the theory of proportion as constitutive of ethical inquiry, leading to the view that dialectic itself is a metamathematical activity, so that the good order of mathematics is to be accounted for by a philosophical investigation into the good order of the forms. The second is that he takes his metamathematical theory of proportion as arithmetical as opposed to as geometrical.

Again, on my interpretation only the geometrical theory of proportion can contribute to the goal of allowing both the mathematician and the philosopher to grasp the good. As regards Burnyeat first error, mathematics does not need philosophy, either its method of dialectic or its ontology of forms, to account for it’s good order. As I have demonstrated, the theory of geometrical proportion is not just a metamathematical theory, it is the highest mathematical theory (i.e., the theory that good orders all of the branches of mathematics). And too in this role it is to be taken as akin to the Good as the highest form that good orders all of the forms. That is, the metamathematical role of the geometric theory of proportion is to be taken as akin to the metaphilosophical role played by the Good; each is constitutive of “ratios between the things these deal with.” So it is by analogy, and not by constitution, that we are to understand the metaphilosophical use of the proportional reasoning. That Burnyeat has missed this point is a result of his second error. As I have argued, taking the route offered by the Pythagorean arithmetical theory of Archytas will be “labor in vain,” because it simply cannot play this metamathematical account-giving role, and so cannot be used to run the intended analogy between the theory of proportion as the highest theory and the Good as the highest form. According to Reference BurnyeatBurnyeat’s (2000) reading, which uses Archytas’ arithmetical theory of proportion to take “all five mathematical subjects as ‘sister sciences’” (p. 19),Footnote ⁵² “from Plato’s standpoint, Archytas’ fault would be his developing such a mathematics merely in order to explain, from above as it were, the auditory experiences we enjoy” (p. 53; italics added).

In contrast, I hold that, besides Archytas’ reliance “auditory concordances,” both Archytas and Burnyeat’s fault is that they ought to have taken the theory of proportion as both geometrical and as a Philolausean “mother-city.” That is, if this metamathematical theory is to fulfill the mathematician’s goal of good ordering the branches of mathematics and, in so doing, aiding in the philosopher’s grasping of the Good as the highest form, it needs to be taken as the highest theory. Indeed, that Burnyeat takes the theory of proportion as arithmetical, or minimally, that, in nodding to Archytas, he fails to distinguish between the two accounts, is what gets him into the problem of having to explain the order of the mathematical subjects:

My reading avoids this “snag”: the geometric theory of proportion is last in order of education because of the double role it plays: as the theory that accounts for the concord and harmony of the cosmos, and as the highest theory that accounts for the concord and harmony of the various branches of mathematics in terms of ratios. What Burnyeat has failed to consider is that, after their study of the five mathematical subjects, the mathematician must “ascend to problems” and come to metamathematically see that the good order of mathematics will be the one that puts the geometric, as opposed to the arithmetic, theory of proportion last in order of education but also first in order of account.Footnote ⁵³

Thus, I disagree entirely, with Burnyeat’s claim that

As I have demonstrated, Plato is telling both mathematicians and philosophers what they need to do to grasp the good. During their ten years of mathematical study, from within the pedagogical context of discovery, they are to work up from arithmetic, to geometry, to solid geometry, to solid geometry in motion, to the harmony of such motions, and then they are to “ascend” to the geometric theory of proportion as the highest theory. Then, starting from there, now from within the mathematical context of justification, that is, the context that gives the account, they are to work down having grasped the overarching good ordering or account-giving role of the theory of proportion for all the kinds of mathematical objects in terms of geometric ratios. Likewise, during their fifteen years of philosophical study, from within the pedagogical context of discovery, they are to work up from proportional reasoning to an account of “all the kinds” of philosophical objects in terms of their ratios (i.e., from the proportionality of military matters and matters of state administration to an account of the proportionality of forms to an account of the Good as the highest form). Then, starting from there, now from within the philosophical context of justification, for the next five years they are to work down having grasped the overarching good-ordering, account-giving, role of the Good. Thus, in as much as Plato is setting out how one is to become a good philosopher (i.e., a philosopher who aims to grasp the account-giving role of the Good), he is also setting out how one is to become a good mathematician (i.e., a mathematician who aims to grasp the account-giving role of the geometrical theory or proportion). In neither case, however, is there a need for the mathematician to adopt the philosopher’s dialectical method and seek an account of their hypotheses in terms of forms; simply, the account of mathematics is given by a mathematical theory, not by a metaphysical one! So, against Reference BurnyeatBurnyeat’s (2000) claim that Plato’s philosophers in training

I have shown that, in the Republic, the mathematician qua philosopher-in-training can certainly come to see the account-giving role of the geometric theory of proportion and its akinness to the account-giving role of the Good, without having to “make creative contributions” and this is why they can “pass by” the question of the “the ratios between the things these deal with.” The mathematician, however, will have to concern themselves with this question; as Plato’s repeated appeals to mathematicians like Theaetetus and Theodorus show, he does intend for this work to be taken up by mathematicians and not by philosophers. Thus, there is no need for the mathematician to “switch to dialectic and philosophy.”

Recall that even though the mathematician, in their use the geometric theory of proportion, can come to give an account of their mathematical beliefs, they, nonetheless, in contrast to the philosopher, remain “irrational,” or alogos, or without an account, because, as Plato explains, they examine things “in accordance with belief,” whereas the philosopher examines things “in accordance with being.” That is, the geometrical theory of proportion accounts for, or justifies, our beliefs about what we think numbers, squares, and son on are, but it cannot, in so doing, justify that they exist “in accordance with being.” Only those things that are accounted for by the Good itself, as “the best among the things that are” – forms like Truth, Knowledge, Beauty, Virtue, and so on – can properly be said to exist “in accordance with being” – that is, can be accounted for by the Good itself.Footnote ⁵⁴ Thus, the goal of a mathematical education is that, when aimed at philosophers in training, they come to understand that, in a sense analogous to the account-giving role played by the geometrical theory of proportion, the Good, as the highest form, accounts for what exists: it does not stand above, or apart from, the other forms, rather it is constitutive of what they have in common, and in virtue of this, the Good provides an overarching account of the being of those other “kinds of things” that are forms.Footnote ⁵⁵

We are now in position to measure my interpretation against Broadie’s. As I have shown, there is certainly textual evidence that Plato argues for, and, indeed, wants us to conclude that philosophical inquiry is to be modelled on mathematical inquiry. On my reading this is because they are to be taken as akin in the sense that both are to be taken as being founded on proportional reasoning itself. How, then, is this all important proportional reasoning, aimed at grasping the good in mathematics and the Good philosophy, supposed to work? Especially given that the details of “the ratios between the things these deal with” with are “passed by”? In contrast with part of interpretations, it is not making mathematical inquiry part of philosophy inquiry. Yet too, it is neither, as Burnyeat would have it, by making dialectic itself a metamathematical inquiry. In contrast to both views, the answer to this question is found in Plato’s presentation of the divided line itself. Plato has demonstrated the use and value of such proportional reasoning, as built up out of the ratios the divided line, by using it as an argument scheme it on us!Footnote ⁵⁶

The Divided Line and the divided line of Book 7, as reconsidered in light of both the Sun and Cave analogies, shows us the value of the use of proportional reasoning made on the basis of geometric ratios. That is, the divided line itself, is constructed proportionally on the basis of the geometric ratios of the lines,Footnote ⁵⁷ without telling us what those ratios are; it is the proportions, between both the ratioed sections and the ratioed subsections of the line itself, that allows us to follow Plato’s proportional reasoning. And, in so doing, we arrive at what the Sun, the Divided Line, the Cave, and Book 7 “share in common with one another and what their affinities are” and next to use the analogies between these, for example, the akinness between the good-ordering role of the theory of proportion and the good-ordering role of the Good, to “draw conclusions about their kinship.” By placing the mathematical theory of proportion as the highest metamathematical theory and the Good as the highest metaphilosophical form, Plato is showing us that even though their methods are distinct, metaphysical/moral reasoning is to be taken as akin to mathematical reasoning. Not only does this supposition answer of why the mathematical branches are so ordered, again with the metamathematical theory of proportion itself as the highest theory, but too it answers the question of why the study of mathematics is good for the soul, namely, because the “concord” and “harmony,” or the good order, of both the objects of the branches of mathematics and the objects of philosophy are to be accounted for by proportional reasoning. So, while the metaphysical/moral realm is not structured by mathematics per se, the proportional structure of the realm of forms is to be taken as akin to the proportional structure of the realm of mathematical objects in the sense that the good order of the forms themselves is to be accounted for by reasoning based on the mathematical notion of proportion.

Finally, and most importantly for my aim of showing that Plato was not a mathematical Platonist, the appeal to the reconsidered divided line on the basis of what the Sun, the Divided Line, the Cave, and Book 7 “share in common with one another and what their affinities are,” when now aimed at the distinction between the ontology and epistemology of the realms and sub-realms, allows us, by use of proportional reasoning based on the ratios between the realms and sub-realms, to “draw conclusions about their kinship.” Thus, we come to Plato’s final proportional argument that

It is this proportional argument, taken in the context of the reconsidered geometric ratios of the lines, that allow us, as the reader, to reach the conclusion that the objects of mathematics are more akin to the objects of imagination then they are to those of knowledge. What Plato has shown us, through his geometrically structured proportional reasoning, is that mathematical objects, as objects of understanding, are “concerned with being.” But too, as objects of thought, wherein thought is taken akin to imagination, they are more “shadowy” than philosophical objects, as objects of knowledge. Thus, he has shown us that because the method of mathematics is distinct from the method of philosophy, so too must be both the epistemological state of their understanding and ontological status of their objects. So, using proportional reasoning based on the reconsidered geometric ratios of the lines, we are now justified in “drawing a conclusion about their kinship”: mathematical objects are distinct from forms.

6 Mathematics Versus Metaphysics

We now in a position to reconsider, at the level of the practice of mathematics, the typical mathematical Platonist components and to conclude that Plato was not a mathematical Platonist. Recall, then, the first component – that mathematical objects, as Platonic forms, exist independently of us in some metaphysical realm of forms. As I hope I have shown, for Plato, mathematical objects do not exist independently of us; they are conjectured objects and so they depend on the mathematical problem that we are attempting to solve. It is the problem at hand that gives rise to the needed hypotheses that we take as if they were first principles, and it is these together that give rise to the needed objects of thought, that we take as if they were real, and it is both that underwrite the arguments that we think we need to reach a given conclusion. Mathematical objects, then, exist in a methodological sense not in a metaphysical sense. As we have seen, in mathematics, existence is a consequence of truth – that is, is a consequence of taking our hypotheses as if they were true first principles for the purpose of solving a problem. In philosophy, by contrast, truth is a consequence of existence, that is, is a consequence of our tethering the truth of our hypotheses to independently existing forms. We next consider component (b), that the way things are in the metaphysical realm fixes the truth of mathematical statements. What we note, is that what fixes the truth of a mathematical statement is the mathematicians’ method, not any philosophical or mathematical metaphysics; it is the demonstration of the conclusion, given the hypotheses and objects of thought that we begin within the context of a given problem, that fixes the truth of a mathematical statement. Finally, we reconsider component (c), that we come to know such truths by, somehow or other, “recollecting” the way things are in the metaphysical realm. For Plato, at least in the Republic, there is no need for recollection; we simply assume that we have the capacity for such understanding, and we show that we come to understand such truths by our use of the method of hypothesis, which requires only that we can think of the object itself, that is, think of it as if it was independent of any mathematical diagram or figure. Mathematical understanding, then, is neither the result of our discovering the way things are in a metaphysical realm, nor our creating the way things are in our mind or in a community;Footnote ⁵⁸ it is result of what we can demonstrate in the context of a given problem via the use of the hypotheses and objects we begin with. It is these considerations, that arise by keeping distinct the mathematician and the philosopher’s methods, that allow us to see that Plato’s practicing mathematician was a methodological realist not a metaphysical realist.

Next we come to reconsider, now at the level of the philosophy of mathematics, the desire to solve those metamathematical problems concerning “into what kinds of things [mathematics] is divided” and “what road it follows” by providing an overarching good ordering or an organizational account of all of the mathematical subjects. What Plato shows us is that, from a metamathematical standpoint, to “undertake an investigation of all the mathematical subjects we have mentioned” and “arrive at what they share in common with one another so that we can draw conclusions about their kinship” we do not have to turn to philosophy. We do not have to turn to the dialectical method and a metaphysics of philosophical or geometrical forms; rather, we can turn to mathematics itself and use the geometric theory of proportion as the highest metamathematical theory. We can now clearly see why the part of Platonist interpretations, of Cornford, White, Tait, and Benson, fail to hit their mark. Recall that these Platonist perspectives arise in light of the mathematicians’ supposed failure to give a first-principled account of their hypotheses; having seen the supposed error of their ways, the mathematician should now be motivated to adopt the philosophers’ dialectical method and so search for those unhypothetical first principles that would allow them to give an account of these in terms of a domain of stable mathematical objects (i.e., as either geometrical or philosophical forms). But, as I have shown, the mathematician can rest easy in their use of the method of hypothesis and in the metamathematical use of the geometric theory of proportion to give such a foundational account of their objects as objects of thought, without having to adopt the philosopher’s dialectical method.

Moreover, what we require of such a metamathematical foundation, if I may, yet again, use this term in an organizational senseFootnote ⁵⁹ is that it provides an overarching good-ordering account of what constitutes a good order of the “kinds of things” we think about. The geometric theory of proportion, for example, tells us that we should think about numbers as if they were geometrically constructed measures, or measures of ratios, but it does not tell us that numbers are such things – again, this would be to confuse an account of things “in accordance with belief” with those “in accordance with being.” This marks the point of my disagreement with Reference TaitTait (2002); he sees Plato’s foundational goal as one that aims to “make explicit the rational structure we are studying and so to define what is true of the structure,” whereas “[f]or Aristotle, the goal of foundations can only be organizational” (p. 2, n. 2; italics added). Forgoing any interpretation of Aristotle’s organization view, I see Plato’s aim of getting at the “rational structure,” now read as its proportional structure, by organizing all the mathematical subjects in terms of geometric measures or ratios. As already noted, properly speaking, the geometric theory of proportion does not itself have a subject matter; borrowing Reference BurnyeatBurnyeat’s (2000) terminology, it provides an “abstract schema” (p. 73) for organizing the subject matters of the various disciplines, but it itself is not about anything, or in Plato’s terms, it is not “in accordance with being.” As a result, it cannot, as Tait suggests, “define what is true.” But this is no fault. While it was certainly the case that the geometric theory of proportions was not fully developed at the time of the writing of the Republic, it was certainly being developed, and, as I have shown, Plato was well aware both of its geometrical developments and of its arithmetical alternatives. This is yet another reason why, even at the highest level of mathematical investigation, Plato would still hold the principles of the geometric theory of proportion as hypothetical – that is, as if they were true.Footnote ⁶⁰

The geometrical theory of proportion, even if taken as hypothetical, provides us with a foundation as an overarching good-ordering account of the “kinds of things” of mathematics, and in so doing, it allows us to “acquire a systematic and wholly general grasp of what each thing is” by providing us with methodological first-hypotheses (e.g., the hypothesis that numbers are geometrical measures). But it is not a “foundation” in the current philosophy of mathematics sense of the term, that is, it does not provide metaphysical first-principlesFootnote ⁶¹ that would allow us claim that it “defines what is true” and so presume that such things metaphysically exist. To think that it should is, simply, to confuse the method of mathematics with the method of philosophy. Tait’s stronger foundational claim, and his resulting Platonist interpretation, arise from just this confusion. He holds, in line with Reference WhiteWhite (1976), that “Plato was concerned to argue for a proper foundation for them (the so-called mathēmata)” (Reference TaitTait 2002, p. 1), but, “whereas White understands the new foundations to be a new and separate science of dialectics, with its own axioms and theorems, on [Tait’s] account the foundations is to consist in adequate first principles for, say, geometry, itself, to be founded by a process of dialectic” (p. 2, n. 1).

Our reconsiderations, at both the mathematical and metamathematical level, clearly show that the mathematical Platonist story, by confusing the hypothetical method of mathematics with the dialectical method of philosophy, conflates the two types of realism at play in Plato’s Republic: methodological realism and metaphysical realism. I have shown that while Plato is certainly a philosophical Platonist – that is, he adopts metaphysical realism for philosophical inquiry – he is a mathematical as-ifist – that is, he adopts methodological realism for mathematical inquiry. Indeed, if there is one clear message that we should get from Plato, it is that we should never confuse mathematics with metaphysics! They are distinct in method and, so, must be distinct in both epistemology and ontology. Thus, Plato was not a mathematical Platonist!

Why does any of this matter for current practitioners of philosophy of mathematics? Because it shows that we too would do well to keep the methodological requirements for mathematical knowledge distinct from the metaphysical ones of philosophical knowledge. As practitioners of philosophy of mathematics, who, like Plato, undertake answering foundational or metamathematical problems, we would do well to place more focus on the mathematician’s method and so on mathematical practice than we do on mathematical metaphysics. Moreover, if we insist on a metaphysical reading of Plato’s view of mathematics, then, not only do we misread Plato, but we also close the door to understanding the ways in which mathematical practice itself can offer an account of mathematical methodology, mathematical epistemology, and mathematical ontology. Thus, just as Plato was not happy with current practitioners of mathematics, because they confuse mathematical images with mathematical objects, so too am I not happy with current practitioners of philosophy of mathematics, because, by confusing the metaphysical aim of having to reason to first principles with the mathematical aim of having to reason from hypotheses, they confuse mathematical objects with metaphysical forms.