Tutors Talk Books: Dean Emily Langston on Euclid’s “Elements”
The following is excerpted from Dean Emily Langston’s welcoming speech to the new class of Graduate Institute students starting in spring 2019.
Mathematics. What can we learn from considering the word itself? Our English word “mathematics” comes from the Greek noun, “ta mathemata,” which in turn is related to the verb “manthano,” meaning “I learn, I perceive, I understand, I know.”
“Ta matamata,” then—or in English, “mathematics”—are the most characteristically knowable things. But this is certainly not everyone’s experience of math! I often have conversations with students who tell me that they are “not math people,” that they “just don’t get math.” They are frustrated because they have encountered a subject that, though still today is seen as a model of the knowable, seems unknowable and even alien. Still, most of those who are initially hesitant give it a try. And, as we begin working our way through book 1 of Euclid’s Elements, students almost invariably begin to find that somehow these things are “knowable” after all. Seeing the joyful response of students to this realization is one of the reasons I love leading this tutorial.
So can we say more about how this great thing comes to pass? The fact that mathematics claims to be about the “knowable,” and the fact that we (and I do mean all of us!) can experience it as knowable, begs one of the most profound questions that we confront in every segment of the GI curriculum: What is it to know? What do we mean when we say something is “knowable?”
The question is placed squarely in front of us by Plato’s dialogue Meno, the one text I know we all have in common; at the very heart of this dialogue we find a mathematical example functioning as a case study in what it is to learn and to know. The myth he recounts claims, and the demonstration involving the slave boy is meant to illustrate, that learning is really recollection of something that was already within us. We see it in the dialogue as, at various points, Socrates asks the boy questions about the diagram he is drawing. And the boy, though he has never been taught geometry, is somehow able to look at the diagram Socrates sketches and then turn to something within himself to make a judgment about what has been proposed. The example makes the case that learning involves something like recollection and recognition.
Let’s take this suggestion back to the material of the mathematics tutorial, to our reading of Euclid’s Elements. Perhaps the place where Euclid most explicitly demands that we check something within ourselves and give our assent is in the postulates. The very word postulate, which comes from the Latin “postulare,” meaning to ask or pray, makes the case clear. The claims made in the postulates are not proven to us; we are simply asked to accept them.
Like the slave boy, who consults something within himself when questioned about various candidates for the side of the double square, we consult something within ourselves when confronted with Euclid’s fifth postulate. The postulate tells us that two straight lines angled so that they are sloping toward one another will eventually meet. It’s hard to say exactly what within ourselves we consult when we are asked to assent to the truth of this statement, or where it comes from. Whether it is there due to some previous experience or there inherently, I will not here speculate. But surely it is not something that Euclid taught us; it was there already. It seems to be something embedded in the structure of our visual imaginations; and that the act we perform when we ask ourselves whether this postulate is true is more an act of visualization than of reasoning. If I sweep my inner eye far enough along these straight lines that are angled toward one another, I seem to “see” that they must meet. Our recognition of the congruence between what Euclid proposes and what we see with the inner eye is very satisfying and feels like a type of knowledge.
Those of you who have taken the tutorial may object. This emphasis on recollection and recognition work may work well enough with the postulates, which are a special case. But what about the material with which we actually begin the mathematics tutorial—not Euclid’s postulates, but his definitions? To take the first definition, it doesn’t seem true to say that we “already know” a point is that which has no part. Going on to definition two, I certainly can’t visualize a line that is “breadthless length.” Does it make sense to say that in seeking to understand these we refer to our spatial imaginations?
Based on my own experience, and on conversations with students over many years, part of what happens in the encounter with Euclid in the math tutorial is this: Somehow, as we examine the possibilities determined for us by the definitions, common notions and postulates, and move proposition by proposition through book one of the Elements, a spatial world is described which seems to coincide with our own lived and intuited experience of space. The clumsy attempts at straight lines we draw on the chalkboard, and the only somewhat more precise ones that mark the edge of the board, behave (always within the limits of their gross imperfections) like Euclidian “breadthless lengths” would behave. The love affair that almost every student has with Euclid springs partly from the fact that Euclid takes our own sense of interior and exterior space and re-presents it to us for our delighted recognition.
I’m making everything overly simple, of course. There are elements to understanding a Euclid proposition that even analogously are not “seeing.” And even in book one, Euclid presents me with truths that are confounding to my visual imagination; for example, in proposition 36, we learn that the two longer sides of a parallelogram can approach each other indefinitely, yet never meet, as the shape is stretched between two parallels. In the end however, contemplating such unexpected truths does not violate our sense of space; rather, it adds an additional element of satisfaction to our reading of Euclid, as we realize that we can learn more than we would ever have seen ourselves about the structure of space by moving step by step through the propositions. Our intuitions about space and our reason can inform each other.
These moments of recognition and increased understanding—when we “see” what Euclid means and agree that space we know really is like that—are moments of exhilaration. These things were in us all along, even in confirmed non-math people! It’s hard to know why we never realized this before. Perhaps it’s simply that no one asked us the right questions. But one of the defining characteristics of Great Books, in the Mathematics and Natural Science segment and in every part of the program, is that they do ask the right questions, and by doing so engage us on many levels.
The coherence we sense between the space Euclid describes and the space we perceive within us, and within which we perceive, is not simply a source of joy—it is a source of wonder, wonder that this should be so. Many students confront this wonder most directly when, somewhat more than halfway through the tutorial, we leave behind the world of Euclid and enter the world of the Russian geometer Nicolai Lobachevski. He disrupts entirely the easy alliance between our study of geometry and our intuitions about space. And yet this doesn’t keep us from moving forward with Lobachevski, reasoning carefully from one step to another in a sort of space which feels like it is quite definitely not our own.
The fact that we can do this opens the door to a host of new questions: about space; about the relationship between knowing and seeing, and about the nature of knowledge itself.