In Mathematics in Plato’s Republic, Sarah Broadie pursues a fresh look at a question about the Republic that has long vexed interpreters: why does Plato require such an intensive, multi-disciplinary education in mathematics as a propaedeutic to dialectic? Broadie answers that Plato’s educational program in mathematics remedies a “vulnerability in reason” that, although inherent to human rationality itself, is compounded by the culture and education of the theoretical city of Callipolis (43). Her account of the origin of the vulnerability of reason rests on the interpretation that dialectic in the Republic refers to Socrates’ familiar elenctic method of testing moral beliefs in the aporetic dialogues (or perhaps even in Republic I). This understanding of dialectic would seem to explain Plato’s worry that dialectic threatens to undermine foundational moral beliefs and encourages a zeal for eristic—a worry that leads Plato to propose delaying instruction in dialectic.[1] In Broadie’s view, the closed nature of Callipolis renders the guardians uniquely susceptible to these dangers because their education largely prevents them from encountering heterodox views, much less entertaining them; thus, a stronger “antidote” for their vulnerability is required. Broadie interprets this postponement as an indication that Plato views the capacity for critical examination as a “vulnerability” in reason that, once activated in the Guardians, “[…] will infect them with nihilism about values and the objectivity of truth” (46). Mathematics, she proposes, is Plato’s “pre-dialectical cure” for this vulnerability (41). Since the pre-philosophical education of Callipolis discourages engagement with the critical reasoning one exercises in dialectic, mathematics education is necessary to help the guardians develop “…an unshakeable trust in rationality and in their own ability to solve problems by reasoning” (51).
Broadie offers this proposal as an alternative to two other influential answers to the question of the propaedeutic nature of mathematics. One view is that mathematical expertise, in the form of familiarity with mathematical practice and knowledge of theorems, is a pre-requisite for dialectic because dialectic grounds the hypotheses that make mathematical reasoning possible. This is dialectic as “meta-mathematics,” a thesis that commonly draws upon Republic 510b-511d (27).[2] Broadie’s reading accepts that mathematics is a pre-requisite for dialectic but denies that “dialectical thinking itself directly calls upon and employs professional mathematical expertise” (27). Her reasons for rejecting this claim are twofold. One appears to be philological, as she concurs with Slings in the opinion that Rep. 511d1-2—an allegedly crucial piece of text for the view of dialectic as meta-mathematics—is likely an interpolation (29-31). A second consideration is that the meta-mathematical project of dialectic has no clear connection with helping guardians become good rulers.[3] The second influential reading argues that “ethical reality is, somehow, mathematical—mathematical in all its complex detail[…]” (33).[4] Broadie rejects this view on both textual and exegetical grounds. On the textual question, she contends that there is no direct evidence in the Republic that the form of the Good is mathematicised, or that guardians trained in dialectic acquire a more precise understanding of justice because it has been mathematicised in some way (37). On the exegetical question, she disputes a strategy that some proponents of this view have used to make their argument, viz., “filling in” certain of Socrates’ provocative silences with speculation about ideas that might have been floating around the early academy (34, n.20).
When read in dialogue with these other interpretations, Broadie’s proposal initially shows promise, as it avoids too much speculation about the precise connections between dialectic and the contents of the mathematical studies, while explaining the necessity of such a massive education in mathematics. One weak point of this reading, however, is that it pins both the selection and amount of mathematics required for turning the soul on the contingencies of (a) Plato’s theoretical city and (b) the undeveloped state of other intellectual disciplines in Plato’s time. In a summary remark, Broadie writes:
I am suggesting, then, that it is cultural differences between Plato and us that explain why his ideal rulers, unlike even the best of ours, need deep immersion in mathematics to mould them into a mind-set that unswervingly applies rational argument and analysis to questions of ethics and politics (41).
Broadie’s reading denies that there is something unique in the content of mathematics that encourages confidence in reading. In her view, other intellectual disciplines in Plato’s time simply had not matured to a state where they could afford experiences that build confidence in reason (53). However, I find that this description contradicts a common experience that people have even after studying allegedly mature disciplines. While fields such as legal theory, history, and even philosophy have achieved some consensus about methods of inquiry and argumentative standards, the presence of enduring controversies in these fields readily gives the impression that there are no answers to the questions these disciplines ask—only “interpretations.” It seems to me, rather, that any other intellectual discipline than mathematics would exploit the vulnerability of reason that, on Broadie’s reading, Plato hopes to cure. One might also doubt the accuracy of the comparison of the reactions that individuals of “closed” versus “open” societies have to the exercise of critical reason and to encounters with heterodox views. When I consider who is more vulnerable to attitudes such as nihilism and skepticism, I simply find it difficult to judge. Yet Broadie’s argument concerning mathematics in the Republic relies on granting this view about the effect of Callipolis’ culture on the guardians. The assumption seems to be that the guardians who encountered dialectical arguments that challenge Callipolean orthodoxy would find nihilism and skepticism irresistible on account of their minimal exposure to heterodox views and lack of acquaintance with critical reason. But I also find the opposite plausible: if the guardians are the most zealous adherents to the values of Callipolis, they may resist heterodox views more effectively out of a sense of pride, defiance, or fanaticism. Likewise, it may be that citizens of open societies are more vulnerable to skepticism and nihilism because, according to their societies’ own self-conception, basic principles of justice are the fruits of a social contract, and thus not entirely “real” or “natural.”[5] Seeing as Plato wrote for an audience in a democratic society, and as the main interlocutors whom he portrays in the Republic are products of that society, it may be that he envisioned mathematical education as a cure for his own society’s ills.
These considerations lead me to think that Broadie’s view would be more persuasive if it acknowledged the unique ontological status of mathematics and relied less on the eccentricities of the imaginary city of Callipolis. Indeed, she would have good exegetical reasons to do so. When Plato turns to a description of philosophers’ eros, he characterizes philosophers as lovers of “[…] that learning which discloses to them something of the being that is always and does not wander about, driven by generation and decay.”[6] In light of this passage, a plausible explanation of the program of mathematical studies is that Plato thought the uniqueness of mathematical objects suits this eros well. In other words, Plato chooses mathematics as the subject that can give students an experience of reasoning that builds their confidence precisely because there is something special about mathematical objects—not simply because mathematics was the only sufficiently developed art at Plato’s time. And in Book VII, Plato prefaces the allegory of the cave by saying that we are all like the prisoners described—a remark that readers may take as an indication that Plato offers the study of mathematics as a liberatory path for anyone, not just the guardians of Callipolis.[7] Although the discussion of the effect of education on the soul contributes to the argument that philosophers ought to rule Callipolis, there is good reason to think that Plato is not merely describing what is required to make guardians of Callipolis good rulers, but what it means for anyone to be a philosopher—or at least to participate in Plato’s particular vision of philosophy. In all fairness, Broadie acknowledges—but subsequently declines—this interpretation of the Allegory of the Cave as one of two “incompatible perspectives” that the Allegory adopts (25, n.11); however, reducing the Allegory of the Cave to an attempt to shore up vulnerabilities in the culture of Callipolis may strike some interpreters as unduly restrictive.
Broadie’s monograph was delivered as a lecture, so it is understandable that questions regarding the ontological status of mathematical objects in the Republic go unaddressed. However, I worry that the strategy of agnosticism about the status of mathematical objects in the Republic—although initially attractive—renders the explanans insufficient for the explanandum, i.e., Plato’s requiring such an intensive mathematical education as a preparation for dialectic. Moreover, Broadie overlooks Socrates’ concluding remark (531c8-d5) about the mathematical studies, from which we gather that propaedeutic function of his educational program is to gain insight into the “kinship” among the different branches of mathematics. That this synoptic view of the mathematical studies is paramount in the preparation for dialectic has been the guiding hermeneutical principle for other scholars who see tighter connections between the content of the mathematical studies, dialectic, and the Good.[8] Although they may be guilty of “mathematicising” the Good to a certain degree, their readings do not always align with Broadie’s description of the interpretive possibilities concerning the propaedeutic role of the mathematical studies.[9] Broadie crucially denies that the guardians “… need mathematical education because dialectical thinking itself directly calls upon and employs professional mathematical expertise” on the grounds that “the text itself furnishes virtually no evidence” of this view (27). On this particular point, I believe that Broadie’s omission of 531c8-d5 misconstrues the nature of the evidence for the view that she claims to refute. The scholars who hold the view that Broadie denies about the propaedeutic function of mathematics tend to regard 531c8-d5 as the primary evidence that dialectical thinking draws upon insights gleaned exclusively from a mathematical education—not, as Broadie suggests, 511d1-2. It may be that to understand Plato’s reasons for requiring such an intensive study of mathematics as a propaedeutic to dialectic we must achieve this synoptic view ourselves.
Notes
[1] Rep. 539b-c
[2] As proponents of this view, Broadie cites Myles Burnyeat, “Plato on Why Mathematics is Good for the Soul,” in T.J. Smiley ed., Mathematics and Necessity: Essays in the History of Philosophy (Oxford: Oxford University Press, 2000), 1-82 and David Sedley “Philosophy, the Forms, and the Art of Ruling,” in The Cambridge Companion to Plato’s Republic (New York: Cambridge University Press, 2007), 256-283.
[3] See also Republic 505e-506a for Socrates’ rationale for steering the conversation toward an account of the guardians’ study of the Good.
[4] As proponents of this view Broadie cites J.C.B. Gosling, Plato (London: Routledge, 1973); John Cooper, “The Psychology of Justice in Plato,” American Philosophical Quarterly no.14 (1977): 151-157; Myles Burnyeat, “Platonism and Mathematics: A Prelude to Discussion,” Explorations in Ancient and Modern Philosophy, 2:145–72 (Cambridge: Cambridge University Press, 2012); Burnyeat, “Plato on Why Mathematics is Good for the Soul”; and Sedley, “Philosophy, Forms, and the Art of Ruling.”
[5] As in Glaucon’s argument at 358e-359b.
[6] Republic, 485a-b (Bloom).
[7] Rep. 515a5.
[8] See Theokritos Kouremenos, The Unity of Mathematics in Plato’s Republic (Franz Steiner Verlag: 2015); Mitchell Miller, “Figure, Ratio, Form,” Apeiron 32, no.4 (199): 76; Mitchell Miller, “A More ‘Exact Grasp’ of the Soul? Tripartition in the Republic and Dialectic” in Truth: Studies of a Robust Presence, ed. Kurtz Pritzl (Washington, D.C.: CUA Press, 2010), 40-101; Ian Robins, “Mathematics and the Conversion of the Mind: Republic vii 522c1-531e3,” in Ancient Philosophy (Fall 1995), 359-391; Myles Burnyeat, “Plato on Why Mathematics is Good for the Soul,” Proceedings of the British Academy, 103, 1-81.
[9] E.g., Broadie, 27-31. Broadie relies on Burnyeat as the sole representative of readings that “mathematicise the good.” The other readings mentioned (n. 19) differ from Burnyeat’s view.