Toast to the Tutors

Toast to the Tutors, Summer 2016
Graduate Institute (Annapolis)
Delivered 3 August 2016
       
It is perhaps common, and indeed even laudatory, for these toasts to ask some variation of an entirely worthwhile question, what do the tutors do for us? or what purpose they serve with their questions. But I want to approach this toast from a slightly different angle, not so much what the tutors give us or do for us; but what they show us about the very act of tutoring; that is, the act implicit in the title of tutor. This is, I freely confess, a selfish question from someone who is by profession a teacher, even one called upon to run “seminars” and have my students dialogue with the great thinkers and writers our civilization (the catch--when they can barely dialogue with themselves!). In these circumstances I am quite fortunate as a teacher, even if I look with envy at the rarefied air of St. John’s method and practice, so wholly focused and intent on learning from the great books that even the happiest teacher should be jealous.

In as much as I have come to St. John’s the last four summers as a student, I have also come as a teacher to this “teacher summer camp.” And here I would suggest that for those interested in developing and practicing their craft of teaching, there are no better models than our dear tutors we are fortunate to have here at St. John’s. The poor student cannot be a good teacher--and if the poor student has the appearance of being a good teacher, we might correctly judge them to be false teachers, who at best might find their way at times but cannot truly instruct. One of the etymologies given for tutor--and here the secondary teacher shows his true colors, for words mean things and who can argue with the Oxford English Dictionary when cited in the classroom--we see the Latin root suggests the idea of guardianship, and so the tutors are the guardians and defenders of the great books and the ideas contained within! But that is not quite right in our usage, for we operate here with the assumption that all can consider and learn from the text, and surely such great books do not need custodians to illuminate their meaning.

Looking further we see the early English usage, a fellow of Oxford or Cambridge assigned to the supervision of undergraduates, or even “a senior boy appointed to help a junior boy in his studies.” This strikes me as far closer to what we are aiming for at St. John’s when we describe the faculty of the college as tutors, students helping students wrestle with the great texts that they themselves wrestle with. To be sure, they are highly qualified and credentialed students, experienced and well-read far beyond ordinary students. But they are students none the less, wrestling with the same texts and the same questions that we as students wrestle with in our first reading. And in that struggle, we may learn from them just as much as we learn from the text. This is not to discount the program that gives life to the college, but to suggest that the diverse approaches to dialogue practiced by the tutors is their most explicit act of teaching. We learn from them how to approach texts, how to dialogue about texts, and how to reflect on texts. In this tutors serve as the models of the community of learning that St. John’s desires and practices; tutors do not teach what is to be learned but teach how we ourselves ought to learn.

Opening questions perhaps are the most obvious example of how tutors teach us how to how to approach a text, and who here does not remember fondly many opening questions that have haunted or amused them. But the opening question only to set the stage for dialogue, not determine it; the follow up questions and references to the text are what make the drama and learning of dialogue so valuable. Here the perspectives of the tutors is most useful, and their knowledge of how to read most beneficial. They call us back to the text when we have gone awry, and introduce connections that might be missed in first or second readings. Everyone in seminar, of course, has read the text and considered it in some capacity; tutors help us see how we might do so in deeper and more fruitful ways.

Tutors also show us how to dialogue about text. In his “Notes on Dialogue” Stringfellow Barr remarks that “‘participational democracy’ consists in everybody's listening intently; it does not consist in what commercial television calls equal time.” This is quite contrary to the common norms of so-called democratic education, where everyone’s voice is equal. If understanding a text is the purpose of dialogue, however, listening becomes far more important. And in this our tutors are masters, listening closely both to the text and the discussion, and responding and questioning not for their own sake but for the sake of understanding. Barr compares good dialogue to a basketball team, where the ball is advanced and passed not for personal glory but for the goal of a basket. In our seminars there is no opposing team, only the goal of understanding, and here tutors lead by their consistent example.

Both reading and listening are forms of reflection on the text, in that they require the humility to hear what is being said and carefully consider it against the argument itself and not oneself. To reflect is to wonder, to ponder what makes something so, to always consider it anew. I fondly remember a certain tutor--who is perhaps speaking at commencement on Friday--expressing delight at a fairly unoriginal and mundane observation in Virgil. “I’d never thought of it that way!” he said with manifest joy. “Liar,” I thought to myself, “you’ve read it a hundred times.” How foolish I was to think this, to assume that a text could ever be so familiar as to be beyond reflection. To be a tutor is to approach the text with new eyes on a daily basis, to ask what else it might have to teach us today. We all have sincerely wondered this when faced with Kant or Vico or Plato for the first time; but to maintain this spirit of inquiry as a mode of living is the mark of a true student. As students we know this; and those of us who are teachers must take it to heart lest become stale and false in our instruction.
Please raise your glasses and join me, classmates, in a toast to these students and models of inquiry, our dear tutors and true students of learning:

To the tutors, who teach us to read the text and just the text;

To the tutors, who ask us why? and thereby bring us into the community of dialogue that we might
learn alongside them;

To the tutors, who might read a book twenty times, have had many seminars and papers on said book; or even have written their own book about it; and in rereading  a particular passage, may come know it for the first time;

To the tutors, the best students of the best books, who choose to spend their summers dialoguing with us for the sake of learning, the best teachers;

Cheers!

Euclid's Fifth Postulate

Euclid's Fifth Postulate is puzzling for several reasons, among them the uncertainty of its precise subject. Following the Fourth Postulate (“all right angles are equal to one another”), Postulate 5 takes a different tone: “That, if [emphasis added] a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” Several questions arise: why does the postulate begin (and end) with a conditional statement? Is Euclid working towards what he later proves in Book I, Proposition 32, that the three interior angles of a triangle are equal to two right angles? Or is this a statement about the nature of parallel lines (or apparent parallel lines), as suggested by the use of “produced indefinitely?” Above all, why is this “begged for” in the opening postulates?

A postulate is an opening presupposition in an argument, a premise to be accepted by all parties in the course of a debate or logical proof. Fundamentally, it is “something asked for or demanded,” sharing a common Latin root with the English word petition (Oxford English Dictionary, 3^rd Edition, 2006). Euclid does not—or cannot—prove the postulates, but asks us to accept them at the beginning of his geometry as matters of common understanding. Postulate 5 is first cited in support of Proposition 29, concerning the equal interior angles created when a straight line falls across two parallel lines. This is followed by Proposition 30, which depends on Proposition 29 to show that lines parallel to the same line are parallel to one another. Although this would seem obvious from both Definition 23 (“Parallel lines”) and Common Notion 1 (“Things which are equal to the same thing are also equal to one another”), Euclid proves this by comparing the angles created by a straight line falling on three parallel lines. To do so, he relies on Proposition 29, which relies on Postulate 5. Every remaining proposition in Book I, except the construction in Proposition 31, will depend on Proposition 29 and Postulate 5 in some manner.

Clearly Postulate 5 plays an important role in Euclid's system, but is it truly necessary as a postulate? Its complexity compared to the other postulates—and the conditional nature of its claims—make it difficult to accept at face value. Consider Euclid's overall treatment of parallel lines: Definition 23 defines parallel straight lines as “straight lines, which being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.” This definition does not, on the surface, tell us anything about the relationship between two parallel lines beyond the fact that they do not meet. They could be long or short lines, close or very distant from one another, but by definition they will not meet, no matter how far they are extended in either direction.

The use of “indeterminate” confuses the concept of parallel lines further. By Definition 23, parallel lines are straight but indefinite in terms of length, the opposite of having the definite and plainly understood limited lengths of line segments. We might recognize particular line segments AB and CD as being parallel within the terms of a proposition; but cannot project and further extend the segments to determine whether they may or may not intersect at some distant point. Postulate 5 addresses this challenge through a consideration of the angles immediately within our perspective: if two lines (which may or may not be parallel) crossed by a third line form interior angles adding up to less than two right angles, the lines will will intersect at some unknown point on the same side as the angles. Euclid is strangely silent about drawing conclusions from this postulate. He does not conclude that the lines are not parallel, nor does he reach the conclusion of Proposition 29 that the lines are parallel (if they do form two interior angles equally two right angles). This strange conditional formulation of the postulate, and its hesitancy to affirm anything but the intersection of the two lines gives the Postulate a sense of specific purpose that might not conform with broader uses throughout the Elements.

If the conditions of Postulate 5 are met, a triangle is created by the three crossing lines. Euclid leaves this conclusion unstated, but it easily follows that if two lines, both intersected by a third line, having interior angles less than two right angles, form an angle at the spot where they meet one another and so enclose the three sides of a triangle. But Euclid is evasive about this conclusion, keeping to the language he employed to define parallels. “[Lines] produced indefinitely...do not meet” (Definition 23), whereas in Postulate 5 the lines will “meet on that side.” In both cases, the lines are indefinitely produced, even beyond the point of comprehension: lines that never touch no matter how far into a plane they are extended or lines conceivably forming a triangle of great dimensions but minuscule magnitude. Euclid has already defined the basic terms of a triangle, but does not fully deal with the relationship between two right angles and the sum of the three angles of triangle until Book I, Proposition 32, just two propositions after he invoked Postulate 5. The proof of Proposition 32 is simple and direct, relying on the identical angles created by parallel lines set up on a straight line—which in turn is largely supported by Postulate 5.

Euclid does hint at this possibility in Proposition 13, where he establishes that the angles created when one straight line is set up on another straight line equal two right angles. It is possible to deduce from this that if the interior angles established in Postulate 5 are less than two right angles than the line segments may not truly stand in straight lines, but it is not possible to prove that they necessarily meet to form a third angle on the side of the interior angles. Proposition 14 expands on this by establishing that if the angles formed when two line segments meet on a third straight line on one side are equal to two right angles, the line segments are themselves in a straight line with each other. However, this cannot be used to deduce their relationship if they do not equal two right angles. Postulate 5 alone speaks to this situation in the absence of the conclusions of Proposition 32.

Proposition 17 comes closer to reaching the same conclusion of the Fifth Postulate, showing that “in any triangle two angles taken together in any manner are less than two right angles.” This touches on the essence of the Postulate in that it accurately describes the relationship of the angles created by three intersecting lines with the given of an existing triange, but lacks the specificity of which angles will be less than two right angles. Obviously they will be the interior angles of the triangle, but not necessarily the two interior angles of the straight line falling on the other two lines as described in Postulate 5. Moreover, Proposition 17 assumes the existence of a triangle, whereas Postulate 5 suggests that if the sum of the two interior angles are less than the sum of two right angles, a triangle will be formed. This distinction is crucial in the formulation of Proposition 29, a reductio ad absurdum proof that posits the interior angles created when parallel straight lines are crossed by a third line are equal to two right angles, a conclusion rejected on the basis of Postulate 5 and Proposition 27.

A final possibility for avoiding Postulate 5 appears in Propositions 27 & 28, where Euclid finally brings together several points to address the angles created when a third line falls on parallel lines. Proposition 27 argues if the alternate angles created when straight line A falls on straight lines B and C are equal, then B and C parallel to one another. Proposition 28 establishes similar relationships for the exterior and interior angles (via Proposition 15). While neither of these propositions specifically address right angles, Proposition 27 relies on a reductio proof similar to Postulate 5, drawing the parallel lines together to create a hypothetical triangle an opposite interior angle equal to the exterior angle. This conclusion is against both the definition of parallel lines and Proposition 16, and thus rejected. Euclid takes the conclusion of Proposition 27 further in Proposition 29, where he shows via Postulate 5 (and without the use of Proposition 27) the several equal angles formed when parallel lines are crossed by a straight line (and the interior angles on the same side are equal to two right angles). While Euclid can logically make numerous claims about equal angles and parallel lines without the use of Postulate 5, he cannot make specific claims about the magnitude of those angles—especially what two alternate angles less than two right angles will produce when their lines are indefinitely extended.

Much of the difficulty returns to what can be known with certainty about parallel lines. Postulate 5 attempts to establish greater certainty by setting forth measurable circumstances whereby lines may be determined to not be parallel, providing a contrast to the declarative “do not meet one another” of Definition 23. But if the Postulate is really about the nature of parallel lines, why doesn't Euclid say so? He simply says that the lines will meet one the side with the interior angles less than two right angles, without identifying the shape produced as a triangle. The absence of both these conclusions suggests that the Postulate is crafted in such a confined manner so as to limit its direct usage to only that of absolute necessity. And despite his heavy reliance on Proposition 29, Euclid only cites the Postulate directly once more in Book I: Proposition 44, concerning the construction of parallelograms.

This little citation lends credence the suggestion that Postulate 5 must be begged for due to the nature of parallel lines. The majority of Book I following Proposition 29 is concerned with parallel lines, and the proof that the sum of the angles of a triangle equal two right triangles is itself completed using the properties of parallel lines taught in Postulate 5. Despite many difficulties, it may be easier for us to grant Euclid's Postulate 5 than other more absolute--but equally questionable--pronouncements regarding the strange and infinite nature of parallel lines.

Submitted 10 July 2014; corrected 17 July 2014.

Only at St. John's

Part of a containing series. At the bottom of an email from the Bookstore Director advising us of a change in bookstore hours:

Before printing, please think about the environment
Respectez l’environnement, réfléchissez avant d’imprimer

The first line is fashionable in socially aware circles; the second is just in French. What, you don't speak French? The bookstore director at SJC does. I should mention that French is part of the undergrad Program in the third and forth year.