Is Mathematical Knowledge Ethical Knowledge?

Why, in the planning of the philosopher-kings’ education in the Republic, would Plato reserve an entire decade for the study of mathematics, beginning with arithmetic and proceeding (respectively) to plane geometry, solid geometry, astronomy and harmonics? In his article “Plato on Why Mathematics is Good for the Soul,” M. F. Burnyeat addresses this question, and arrives at the striking conclusion that learning mathematics is not merely instrumental in coming to know the form of Justice, but that knowledge of mathematics is part of knowledge of Justice, or as he puts it, “the content of mathematics is a constitutive part of ethical understanding” (p. 6). While this suggestion would certainly explain why the study of mathematics holds such a prominent place in the education of the philosopher-kings, it is in no way clear how mathematical knowledge could possibly be ethical knowledge; how can learning about triangles or prime numbers be learning about Justice or the Good?

Socrates tells us at 525c that the study of mathematics will be useful only if “it leads the soul forcibly upward and compels it to discuss the numbers themselves, never permitting anyone to propose for discussion numbers attached to visible or tangible bodies.” A fundamental reason, then, that the study of mathematics is critical in the development of philosopher-kings, is that it is instrumental in learning to looking beyond the realm of sensible particulars and toward the world of the forms. Also, studying mathematics is especially useful for rulers, for example in planning a battle (522d), or in understanding the shifting seasons (527d). But Plato is clear on this: Socrates explains again and again (at Glaucon’s expense, for example at 525c, 527d-e and 528e-529) that such practical ends are not why the future philosophers ought to study mathematics.

What Socrates has to say about astronomy and harmonics helps us see how mathematical knowledge might be constitutive of ethical knowledge. Socrates holds that, as currently practiced, astronomy will be of no use in the education of the philosopher-kings. Instead, the future philosophers will study a new science of astronomy, which will concentrate on the nature of motion and movement of solid bodies in general, and thus Socrates suggests: “let’s study astronomy be means of problems, as we do geometry, and leave the things in the sky alone.”

Similarly, of mathematical harmonics, Socrates says that since the current practitioners are “measuring audible consonances and sounds against one another, they labor in vain, just like present-day astronomers” (531a). The philosophers-to-be shall abandon the empirical methodology of “[seeking] out the numbers that are to be found in these audible consonances,” and instead, once again, “make the ascent to the problems,” and investigate “which numbers are consonant and which aren’t or what the explanation is of each” (531c).

The future philosophers will study the harmony of the mathematical ratios (which are used in Greek music theory to represent harmonic relationships), rather than studying the harmonics of the audible sounds themselves. Why study the mathematics of concord and consonance in themselves, independent of sound?

Anyone who has read the Republic will have noticed that the idea of concord plays a large roll throughout the arguments. There are multiple passages where this of concord plays a crucial function. Drawing from both (410a-412b) and (441e-442a), Burnyeat maintains that purpose of their musical and gymnastic training is that it “will harmonise the two elements in their soul, the spirited and the philosophical (as if they were strings on a lyre), relaxing and tightening them as necessary to ‘tune’ the soul to be both brave and temperate” (p. 53-4). More crucially, Burnyeat points out (p. 54) that in Book IV, temperance gets defined as “a kind of harmony” (431e), and that justice gets defined as a well-ordered attunement of the soul (443c-e):

“[Justice] isn’t concerned with someone’s doing his own externally, but with what is inside him, with what is truly himself and his own... He puts himself in order, is his own friend, and harmonizes the three parts of himself like three limiting notes in a musical scale — high, low, and middle. He binds together those parts...and from having been many things he becomes entirely one, moderate and harmonious. Only then does he act. And when he is doing
anything... the action is just and fine that preserves this inner harmony...”

Passages like these explain why, when Socrates discusses the study of harmonics, he denies that harmony must be a relation between sounds. Harmony should be given a systematic, general mathematical treatment because, as Burnyeat argues (p. 56), “the important task of ruling is not day-to-day decision-making, but establishing and maintaining good structures, both institutional and psychological.” If the philosophers are to create and maintain harmonious and concordant institutional structures, they will require a thorough understanding of what harmony is, not as it is sometimes instantiated in audible sound, but its general mathematical nature.

And yet, Burnyeat’s suggestion here is not so different in kind than Glaucon’s many praises of mathematics for its usefulness to the rulers. I think the more important point is the following epistemological one. If the philosophers are to know what Justice is, then they must be able to give a full account of it, for being able to give an account is a necessary condition for knowledge for Plato (as in the dialectic). But, according to the treatment of Justice in the Republic, given in terms of harmony and attunement, one would need a thorough understanding of these notions central to the very nature of Justice, viz., harmony and attunement. Thus, to be able to give a full account of moral forms such as Justice, the philosophers-in-training must learn quite a bit about the main components of the idea of Justice. This explains the intensity of the philosopher-king’s mathematical regimine, as prescribed by Socrates, because it explains why the content of mathematics is relevant, and not merely that it requires abstracting away from sensible objects, or that it will be indispensable to the practical tasks of ruling.

In sum, Plato’s reserving of an entire decade for the study of mathematics is explained by knowledge of mathematics being constitutive of knowledge of the moral forms. Since the moral forms are defined by properties like harmony, one must understand harmony in order to give a full account (a necessary condition for knowledge) of Justice. This explains why the future philosopher-kings end their study of mathematics with harmonics, the general mathematical study of concord. Burnyeat point is very interesting because we no longer have to look at the Republic's definition of Justice at the soul as mere metaphor where there should be argument: the talk about harmonics here can be taken literally.