The Parthenon and Liberal Education

Geoff Lehman, Michael Weinman

• Éditeur: SUNY Press
• Année: 2018
• ISBN:
  9781438468433
• Sujet: Philosophy
• Sous-sujet: Philosophy History & Theory

The Parthenon and Liberal Education seeks to restore the study of mathematics to its original place of prominence in the liberal arts. To build this case, Geoff Lehman and Michael Weinman turn to Philolaus, a near contemporary of Socrates. The authors demonstrate the influence of his work involving number theory, astronomy, and harmonics on Plato's Republic and Timaeus, and outline its resonance with the program of study in the early Academy and with the architecture of the Parthenon. Lehman and Weinman argue that the Parthenon can be seen as the foremost embodiment of the practical working through of mathematical knowledge in its time, serving as a mediator between the early reception of Ancient Near-Eastern mathematical ideas and their integration into Greek thought as a form of liberal education, as the latter came to be defined by Plato and his followers. With its Doric architecture characterized by symmetria (commensurability) and harmonia (harmony; joining together), concepts explored contemporaneously by Philolaus, the Parthenon engages dialectical thought in ways that are of enduring relevance for the project of liberal education.

Part I

Plato on Dialectic and the Problem-Based Study of Mathematics

MICHAEL WEINMAN

Three things must be true for the central argument in this book—that the Parthenon is the most important “vanishing mediator1” between the archaic and largely “illegible” reception of Near-Eastern knowledge practices in Greece during the seventh and sixth centuries BCE and the creation of Plato’s Academy—to hold. First, it must be true both that there was such a reception of earlier technical, especially practical mathematical, Near-Eastern knowledge and that it was of real importance to the development of Greek mathematical thought in the fifth century and after. Second, it must be true that the design of the Parthenon is a principal site in which we can read this legacy; that is, it must be possible to find in the principles of the organization of the temple clear links to the objects and practices in earlier Greek mathematics, which themselves must be clearly connected to the Near-Eastern antecedents. Finally, we must be able to show that those features of the Parthenon that prove its connection to the Near-Eastern legacy are themselves integral to the kind of education Plato set out to offer in the Academy.

It was the work of the introduction to demonstrate the first of these three claims, and it will be the work of part II to demonstrate the second. It is our present task here in part I to substantiate the third. To bring this about, we here provide an analysis of how the “constructivism vs. realism” debate about the foundations of mathematics and the nature of mathematical objects of knowledge that we can be confident (thanks to Proclus) was already raging in the Academy by the time of Plato’s death can be traced back to the “problems” that partially motivated the design of the Parthenon, about seventy years earlier.

The textual basis for our attempted reconstruction of this debate begins with the place2 where Plato has Socrates argue that the life of the just man is exactly 729 times more pleasant than the life of the unjust man. In what follows, we will show why we find this seemingly comic and surely not literally intended claim at the moment of greatest drama in the dialogue, the place where Socrates will, once and for all, most decisively answer the central question of the Republic, which is not “What is justice?” but rather “Why be just?” In presenting our account, we offer a perspective on the broader problematic of the relationship between dialectic and mathematics in the dialogues, which we hold to be fundamentally linked with the debate in which Plato was participating about the reality and ideality of mathematical objects. To pursue the reading we hope to provide for this passage, we first bring it into conversation with (1) the interpretation of the Republic more generally (chapter 1) and (2) the account of the mathematical basis of physical reality in Timaeus (chapter 2), before reaching our conclusion concerning the “nesting” (or placement) of mathematics with respect to dialectic (chapter 3).

Chapter 1

Dialectic and the Mathematical Arts in Republic (9.587b–588a)

Philolaus’s Scale and the Final Bout between the Just and Unjust Souls

Of all the major interpretive questions that face readers of Plato’s dialogues, perhaps none are more pressing or less resolved3 than the questions we aim to address here. First, just how mathematically inclined, and how mathematically able, were Plato and his followers in the Academy? Second, just how influential were the beliefs, practices, and methods of “the Pythagoreans” to Plato’s views, as reported in the dialogues, and as (perhaps) otherwise or more expansively taught in the Academy? Indeed, we can already see the intertwinement of these questions, and their importance for the project of understanding Plato’s influence, in Aristotle’s presentation of Plato’s views and those of the early Academics in Metaphysics.4 Despite this long and persistent concern, there remain pockets of the corpus of dialogues that are both understudied and possibly telling—though surely not determinately so—for these questions.

One such pocket is our focus here: the passage of Resp. 9 (i.e., 587b–588a) that concludes the central argument of the dialogue, which stretches all the way back to the challenge to defend justice by means alone of what it brings about in the soul of the one who possesses it, put to Socrates by Adeimantus and Glaucon at the beginning of Book 2. Here Socrates reports that not only does the just man lead a more pleasant life than the unjust man, but precisely a 729 times more pleasant life. Given this placement at the conclusion of such a momentous stream of discourse, and that it immediately issues in the famous “image of the soul in speech,” we should expect this passage to be relevant for “big picture” interpretive questions like ours, all the more so since Plato here goes out of his way to “make math an issue” by attaching this coda in which we can precisely calculate the extent of the difference by which the just life is preferable to the unjust life.5 All the same, this moment has received a good deal less notice than Resp. 7, 522–532, where the “mathematical disciplines” are discussed in detail,6 and the Timaeus,7 which seems to present—although very ambivalently—Plato’s most sustained treatment of “the role of the mathematical arts” in the dialogues.

Given all the attention these interlocked interpretive problems have justly received in the service of trying to answer these profound questions about (1) the relevance of the dialogue form for the practice of dialectic and (2) the relationship of dialectic and mathematics for Plato,8 and that such scholarly results have not always converged toward a consistent “state of the art,” we cannot be utterly silent on Resp. 7 and Timaeus. We propose, however, to make our way (lightly) into these stormy seas by way of the manner in which (as it seems to us) the passage that is our focus here calls on and comments on this earlier discussion of the mathematical arts as “the prelude to the song of dialectic.” Our focus will be on the manner in which the salience of the number 729 is elucidated for each of the five mathematical arts named in Book 7, with the crucial exception of what is presented in Book 7 as the highest art, harmony. The question, then, will not so much be “Why 729?,” which we discuss to ask a more telling question; that question is, “Why does Plato have Socrates leave out the significance of 729 for harmony, after going pointedly through its relevance for arithmetic, geometry, solid geometry, and astronomy?”

We believe that the ultimate relevance of this number is that the ratio 729:5129 is an excellent approximation of the ratio of

: 1. That is, viewed with respect to harmonics, 729 is the best number to work with to show that “closing the circle” of the musical scale is impossible without introducing irrational numbers, which is precisely what was ruled out by the commitments, ontological rather than formal, of the Pythagorean harmonists who built the scale with which Plato is working here.10 If so, we are faced with a question quite relevant for our overarching interpretive concerns: why did Plato leave this out? Our suggestion is that by doing so he inscribes into the dialogue the incompleteness that is integral to the progress of dialectic.11 This brings us to our suggestion for how we ought to understand the relationship between philosophy and mathematics. Namely, mathematics provides the tools by which we can make ever more determinate the problems on which we wish to work—here, for instance, the problem of halving the whole tone as a special case of the problem of bringing into ratio (into logos) the irrational (alogon)—in the service of dialectic. This is the relationship of mathematics to philosophy and the meaning of the suggestion that the mathematical arts are a “prelude” to the song of dialectic (Resp., 7.531d).

There are, however, two relevant senses—call them the therapeutic and the transcendental—in which the impossibility of rationalizing the irrational as understood by mathematics might be the “prelude” to the “song” that would be the properly dialectical understanding of this impossibility. In both these senses the basic structure of the claim is the same: mathematics is something you must not fail to do before dialectic, but also something that you must not fail to move beyond. The difference between the two, though, is as follows. If the therapeutic sense is correct, then the point of the transition from mathematics to dialectic is one in which you “get beyond” the mathematical moment by getting over it, specifically by consciously rejecting the project of mathematics as the means by which the world of sense can be brought within the dominion of reason. If, on the other hand, the transcendental sense is correct, then the way the dialectical moment gets beyond the mathematical is one in which the project of mathematics as the means by which the world of sense can be brought within the dominion of reason is somehow preserved even as it is transcended.12 We believe that our basic interpretive picture of Plato’s reception of Philolaus’s musical scale in both Republic and Timaeus holds regardless of which of these two senses that we consider of how, exactly, mathematics is the prelude to dialectic’s song, and so we remain neutral concerning which of these two senses ultimately holds while offering our readings of Resp. 9.587b–588a and Tim., 35b–36c. When offering our reading of Platonic dialectic as the cornerstone of his vision of liberal arts education in chapter 3, however, we offer grounds to believe that the “transcendental” interpretation of the prelude-song relationship between mathematics and dialectic is in fact closer to the spirit of Platonic dialectic.

If this is right, then this would mean that the answers to our biggest questions would be something like (1) Plato thinks that doing mathematics is important, even integral, to cultivating a philosophical disposition, but does not in itself constitute a philosophical disposition; (2) thus, we would best understand the legacy of the Pythagoreans as predecessors who helped articulate the problems that would-be philosophers should try to cope with, but not as models for philosophical practice itself. Put another way, we should acquaint ourselves as well as possible with figures such as Plato’s philosopher-king and Timaeus, but if we want to be philosophers, we should understand our work as responding to them in a properly dialectical way, rather than as trying to emulate them.

1. The Two Interpretive Principles We Bring to Plato’s Dialogues

Before descending into the details of this passage, we ought first to justify as far as possible our two interpretive principles in approaching it. First our reading follows in the wake of others that insist that we must pay careful attention to what Plato has Socrates (and other characters) not say in the dialogues in order to understand what he does say—especially when he has that character call attention to the fact that there is something that he has to say that he is in fact not saying. Call this the “leaving things out” principle. Second, our reading joins those that hold that it is impossible to understand what Plato is trying to establish argumentatively—that is, what he means by dialectic—without paying careful attention to the dramatic context in which those arguments are offered. Call this the “dialectic in and through drama” principle; this must be understood against another interpretative principle (call it that of the “dialectic in and through dogma” reading tradition) that holds that, however great a stylist Plato was, his philosophical positions (his dogma) can and must be isolated from the dramatic form in which he offered them, and the work of philosophical interpreters is to “get at” the philosophical content (the dogma) by stripping away, as it were, the dramatic context in which it is presented.

Leaving Things Out

Integral to our understanding of this passage is that Plato consciously chooses both to cast the passage in terms of the five mathematical arts articulated in Book 7 and to leave out the relevance of the fifth, and highest, of these arts: harmonics.13 Making sense of this intentional leaving-out requires reflecting on Socrates’s famously enigmatic statement, in a crucial moment of the dialogue immediately preceding the presentation of the divided line and the allegory of the cave, that contrary to Glaucon’s wish that he give his account of the “likeness of the sun” (that is, the good) in full, leaving nothing out, “I, of course, am leaving out a throng of things [συχνά γε ἀπολείπω]” (Resp. 7.509c). One thing we cannot fail to notice is that this statement comes immediately before the presentation of the divided line, with which Book 6 concludes, and the image of the cave, with which Book 7 opens. Precisely what it means that these two justly famous moments in Plato’s masterpiece immediately follow the claim that the account of which they are part is going to be knowingly incomplete remains a source of intense debate. Some commentators are quite happy to take the account of the good in Resp. 6 and 7 as “more or less” Plato’s view of the form of the good; others insist that what we have in the dialogue is a mere shadow of his true view, which he couldn’t possibly provide in the dialogue for reasons that emerge in the conversation with Glaucon and Adeimantus, and also recur in other “late” dialogues and in the Seventh Letter.

What we will say about what is left out in “the 729 times more pleasantly” passage owes a lot to how Miller (2007), McNeill (2010), and Roochnik (2003) understand the role of “leaving things out” in (Plato’s) Socrates’s dialectical method more generally. Against this approach, there is a “stronger” reading of Resp., 6.509c, which sees in it an express reference to the precise content of Plato’s “unwritten teachings,” which (according to this “Tuebingen School” reading) were the true views of Plato on metaphysics and epistemology, consciously obscured in the dialogues for pedagogical and political reasons.14 And there is a more “deflationary” reading that holds that this concern with this passage is protesting too much and that Plato’s understanding of the form of the good is more or less legible from Resp.15 We hope that our particular interpretive path, a sort of “third way” between the deflationary and stronger readings of the “leaving things out” claim, will be vindicated by the results of our reading,16 but let us add a last prefatory justification of the “leaving things out” principle.

We hold the following claims to be true:

1.A satisfactory answer to the puzzle “Why 729?” should be held to be a significant desideratum for readers of Resp., given the passage’s importance for the central argument of the dialogue.

2.There is, as yet, no such satisfactory answer, neither from those who advance a more “dialectical” (sometimes, “esoteric”) reading, nor from t...

Table des matières

• Cover
• Title
• Copyright
• Contents
• List of Illustrations
• Acknowledgements
• Introduction Thinking the Parthenon and Liberal Arts Education Together
• Part I Plato on Dialectic and the Problem-Based Study of Mathematics
• Part II Harmonia and Symmetria of the Parthenon
• Afterword
• Appendix A Pythagorean Musical Ratios
• Appendix B Principal Measurements of the Parthenon
• Appendix C Elements of the Doric Order
• Appendix D Ground Plan of the Parthenon
• Appendix E Glossary of Technical Terms
• Notes
• Works Cited
• Index
• Back Cover