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The Parthenon as a Mediator between Greek Mathematics and Liberal Education

An excerpt from Michael Weinman and Geoff Lehman’s latest book

On April 3rd, 2018, Public Seminar will discuss The Parthenon and Liberal Education with Michael Weinman. The discussion will begin at 6:00 p.m. in the Wolff Conference room (D1103) of Albert and Vera List Academic Center (6 East 16th Street, the 11th Floor.)

We propose here to pursue a method of speculative reconstruction to detail what can be learned about the “state of the art” in the early development of “liberal education” in fifth-century Greece. One needs to be cautious in speaking about such a development at such a time, which predates the establishment of any independently operating institution that might naturally be thought to pursue such an educational project in today’s terms. The Parthenon, the foremost example of the practical application of mathematical knowledge in the mid-fifth century, insofar as it displays the cultural milieu in which mathematical knowledge was growing in both sophistication and in audience at the relevant time, can be understood as such an institution for liberal arts education. Specifically, coming to see the Parthenon as a manifestation in material form of the quest to achieve a formal integration of the mathematical arts points to a way in which liberal education has been, and could now be, a vital part of the civic life of a democratic society. The Parthenon is both the work of a well- educated group of theoretician-practitioners of mathematical knowledge and a work for the cultivation of a certain kind of generally educated citizen. Understood in this light, it helps us see the roots of the trivium and quadrivium as they later came to be classically conceived.

Before we attempt the comparative analysis of the Parthenon’s design features as a mediator between the earliest, scarcely documented sources of Greek mathematics and the liberal arts curriculum in the Academy, we ought perhaps to say a word about why its status as such a mediator did in fact vanish. That is, if the elements that emerge from our reading of the Parthenon are really there, why don’t the innovations in the Parthenon produce an explicit and immediate textual response? We feel this dilemma relates directly to the nature of the building as art object and as sacred space. As we will discuss with specific reference to elements we focus on later, the building does “theoretical work” in ways specific to the experience of a work of art, ways that need not, and ultimately cannot, be fully articulated in words. In that sense, we would not expect to find contemporaneous textual discussion of the theoretical work that the building is doing. Rather, as we see in Plato’s dialogues, analogous theoretical problems emerge in later texts, not so much through direct “influence” but through broader and more indirect connections that arise from their shared intellectual culture. This is similar to the situation in Renaissance Europe, when perspective pictures, made for a sacred context, involve specifically pictorial theological interpretations (i.e., not merely duplicating what texts can do), and also implicate epistemological paradigms that would emerge in more fully theorized and elaborated forms in later centuries (e.g., in Cartesian epistemology).

With so much said for the basic orientation we bring to the Parthenon, let us now reflect briefly on the history of the liberal arts as Plato came to give them determinate form. “Pythagoras introduced the quadrivium to Greece.” This traditional understanding – this “creation myth” – of Pythagoras as the first philosopher is attested very early in the classical canon. Indeed, though the text is very understated, Plato’s observation in Book 10 of the Republic that Pythagoras, like Homer, was hailed as a “master of education” seems to point to an already-established view that holds Pythagoras as a model of what Aristotle already refers to as “liberal education.” Our suggestion is that the procedure we will follow in our analysis – testing a formal understanding (here: the mathematical theme of reconciling arithmetic and geometry through harmonics) against a material object (here: the Parthenon itself in its architectural and sculptural program) – is precisely the model that Pythagoras introduced as a “model educator,” and the one that inspired the design of the Parthenon.

To cite just one (especially illuminating) example: if we look at the dimensions of the Parthenon’s stylobate, we see that they were quite likely determined by a method, standard for Doric architecture in the first half of the fifth century, based on intercolumniations five times the width of the triglyph, that is, on a 5:1 (80:16) ratio of intercolumniation to triglyph. There is an important difference, however, in the case of the Parthenon: the continuous proportion from which the façades and flanks of the building were constructed gives a 81:16 ratio between these elements, as two elements in a continuous proportion of intercolumniation to lower column diameter to triglyph width, where the full expression is 81:36:16 (this is a continuous proportion since 81 and 36 are in the same ratio as 36 and 16). The refinements involved with fitting these two slightly different constructive principles together – that is, these two different forms of symmetria (commensurability) – is a first instance, among many others we will investigate in detail, where we encounter the problem of harmonia (harmony, i.e., “joining together”) in the Parthenon.

One important consequence of the repeated use of this particular continuous proportion, at various scales, is that it allowed for the building’s overall cubic proportions, in the continuous proportion of 81 (length): 36 (width): 16 (height) to have as its unit the real, visible triglyph module. Focusing for the moment only on the two most remarkable features of this innovation – adapting the 5:1 (80:16) ratio to the mathematically much more interesting ratio of 81:16 and the even more remarkable offering of the visible triglyph module as the unit for the building as a whole – two points emerge. First, the remarkable sophistication of the theoretical reflection at work in the monument’s design becomes in a literal sense visible. Still more strikingly, we believe, the designers made this sophistication accessible to the temple’s audience, which is ultimately the whole city, by weaving these formal features into our sensory and embodied experience of the building.

Through this careful consideration of the educational program behind the design of the Parthenon and in its role as a form of civic education, we hope to show that the practical arts played a key part in the birth of liberal arts education. The well-rounded education that came to be programmatic in the Academy has as its proximal antecedent the practical, but not merely practical, education in the arts that the planners of the Parthenon brought to bear in and through its construction. More than anything else, this antecedence manifests itself in the elegant interrelation of the soon-to-be-canonized mathematical arts of arithmetic, geometry, astronomy, and harmonics in the building’s constructive program.

Both to shed light on the notion that the Parthenon is a vanishing mediator in this sense, and by way of concluding this statement concerning the significance of our project at large, we would like to address three fundamental criticisms to which our entire method of speculative reconstruction can reasonably be subjected.

First, a more hard-headed historian might object that even if we can “read” the Parthenon as it stands as being the site of the “integrated mathematical arts” as they are canonized in the fourth century, this does not, in light of the total absence of other primary source documentation, give us reason to be certain that any significant portion of the people who designed and built the temple had any awareness of the presence of these features or the capacity to appreciate them. Even less, the criticism could continue, do we have grounds to believe such features to be among the principles of its organization. In response, we would point to the intensity of the reflective awareness the design program displays, also in comparison with Doric temple design before and after the Parthenon. This suggests that the problems we will discuss in detail below were on the minds of those responsible for the building, and are not a projection back onto it. If that is possible, then we hope to show it is also plausible that some significant portion of the “knowledge workers” assigned to this commission had a reasonably advanced understanding of principles not yet recorded in the works of theoretical mathematics from the mid-fifth century. We also aim to show that the Parthenon as designed intends for its audience – or at least a considerable part (the “educated” or cultivated part, those versed in mousike, the works of the muses) of that audience – first, to recognize the presence of these problems, and then, having recognized them, to “educate themselves” in a manner not dissimilar to how Socrates defines dialectic as the “art of turning around the whole soul” ( Resp., 7.518d).

But, our interlocutor might insist, would it really have been the case that any number of people involved either in the design and the construction of the temple, or in visiting and making use of the space once built, would have had any access to, or interest in, the features on which we focus here? To this we would reply: the character of the Parthenon as a work of art, and not a theoretical written text, is crucial (and we mean “work of art” here in the broadest sense, not “art for art’s sake” but an idea of art inclusive of the building’s religious meaning and function). The building creates an encounter with every receptive viewer, whatever his or her educational background or degree of specialized knowledge, an encounter that is ultimately irreducible to an entirely verbal, or entirely mathematical, articulation. That encounter is first and foremost an embodied and sensory one, within which the mathematical and ontological questions the building raises are embedded, but it is never fully reducible to those questions. The analogy with music may be helpful here: in music, one can experience harmony, and have an emotional response to it, without understanding the mathematical principles involved; likewise, those mathematical principles themselves are not adequate to explain the ineffable character of music, even if they are its foundation. Thus, we can imagine that one viewer may experience symmetria and harmonia in a strictly intuitive way when encountering the Parthenon (symmetria in the well-ordered and pleasing proportions of the design, harmonia in the sense of a complex of parts holding together as one thing and in the beauty of the whole); another may connect those experiences to the religious and/or civic significance of the monument; another may speculate on the building’s mathematical character and even be inspired to count and measure; and another may consider the relationship between arithmetic and geometry, reflect on the philosophical question of harmonia, and be led toward dialectical thought.

This last group would probably be a small number of people, and the first group would probably be the largest. Also, many of those involved in making the Parthenon may have had specific technical knowledge that need not have involved awareness of all the larger philosophical questions we raise in the book. Still, such knowledge – for instance, a stonecutter’s knowledge of the required proportions and refinements of individual stones, and how to produce them – is a first step in that direction. Certainly, any account of the specific number of people who had access to these different kinds of knowledge would be purely speculative, at least within the scope of this project, but one’s intuition that the fullest intellectual and philosophical engagement with the building would have inevitably involved a relatively small number of people (Pythagoreans or otherwise) seems right. All the same, the range of experiences that the building produces, from the most direct and unconscious to the most reflective and theoretical, seem crucially related.

Thus, understanding the Parthenon rightly is possible only when we appreciate the role of practical exposure to “problems in the arts” in a liberal education generally. This, we want to suggest, holds not only for this “institution of liberal education”; actually all institutions pursuing such a program of education have an interest in exposing the students in their care to such problems. Education in the liberal arts originated from a dialectical reflection on problems in the practical arts and more abstract thinking about them, as found in what we would now call “the exact sciences.” Such education, in principle, ought always to be versed in such reflection. Or, more baldly still: humanities students ought to have enough quantitative competence to understand what questions in the exact sciences remain open and why they remain open. That, we believe, is the role of the problem-based study of mathematics in the Parthenon, and it is relevant as much for us today as for those who designed, built, and worshipped in the Parthenon.

Even if one is willing to accept our basic two-point hypothesis about the Parthenon as an institution of liberal education, and its corollary for liberal education more generally, though, there remains the following worry: what does the scholarly consensus tell us about the state of the art in Greek mathematical knowledge in the mid-fifth century, and does that consensus tell against our hypothesis? Is it really the case that much of what was known by the time Plato wrote the Republic (say 380 BCE) was in fact already known by a fair number of skilled artisans by the time the Parthenon was designed and built some sixty-five years earlier? Our reply begins by noticing that perhaps it was known but not demonstratively known, or put another way, known but not yet subject to deductive proof. This last concession is potentially decisive, as we hope to show. Scholarly consensus holds that formalized proof like what we read in Euclid was entirely absent from Greek mathematics until at least the time of Plato’s death. This does not, however, tell against our hypothesis that many of the most important findings first demonstrated in Elements – and crucial among these for us would be the propositions concerning mean proportionality, continuous proportion, and how these relate to square and cubic numbers – were in fact widely but perhaps not “demonstratively” or “formally” known by the middle of the fifth-century.

Finally, even if our less-given-to-speculation colleague is convinced that our approach survives these two plausibility tests, there remains the following concern: such a well-developed community of practitioners of such knowledge would surely have produced some kind of traceable work that should inform us of who they were and what their research problems and possible solutions were. Why then, the objection would go, are we entirely without any documentation of these groups, of their participants’ names and their findings? To this objection, we have two replies. First, the Parthenon itself is the primary source documentation of the community of researchers, whose research method was to work on the problem by designing the structure, and thus “publishing” their results not in a journal for specialists, but for everyone to experience in their civic, religious, and individual encounter with the building (which was a temple, built for the city as a whole with funding that the city secured from a mix of public and private sources). Second, while we believe that existence of one or more treatises having been written by the principal designers is entirely unnecessary to the argument, since the Parthenon speaks for itself, it does merit notice that Vitruvius refers to a book on the Parthenon by Iktinos and “Karpion” among a list of ancient architectural treatises, all now lost, at the beginning of Book VII of De architectura. We will probably never know, but it is not impossible that this treatise was (or was in part) something like a guidebook to understanding the Parthenon as a site for solving problems in the interdisciplinary practice of mathematical arts. If this is so, then the formal analysis we will provide in part II might best be understood as akin to what this treatise would have presented. In short, we hope to show that the designers of the Parthenon saw their creation this way, and that they did so because of their vision of what we might call a liberal education.

Michael Weinman is Professor of Philosophy at Bard College Berlin and the author of two earlier books: Language, Time, and Identity in Woolf’s The Waves (Lexington 2012) and Pleasure in Aristotle’s Ethics (Continuum 2007).

Geoff Lehman is on the faculty of Art History at Bard College Berlin.

The Parthenon and Liberal Education was published by SUNY Press in March 2018. A PDF document of the first chapter can be found here.

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