Polytropy

Index to this series

In the Platonic dialogues, Socrates frequently mentions τέχνη (technê), which is art in the archaic sense: skill or craft. The concern of this post is how one develops a skill, and what it means to have one in the first place.

Dictionaries may not bring it out, but an essential feature of craft would seem to be the following of a plan or model. Hannah Arendt makes the point in The Human Condition (1998/1958), where, on the subject of “Plato’s doctrine of ideas,” concerning “its content as well as … its terminology and exemplifications,” she says,

These reside in the experiences of the craftsman, who sees before his inner eye the shape of the model according to which he fabricates his object. To Plato, this model, which craftsmanship can only imitate but not create, is no product of the human mind but given to it.

That is on pages 302–3, in the twelfth and last of the readings that have been taken up on this blog. For the fourth reading, I made some notes on the etymology of τέχνη and related words in English. Arendt remarks there, on page 90, that “unskilled work is a con­tradiction in terms.”

Skill is exercised in both mathematics and English. In my own classes in the mathematics department of a university, if the students do not actually create the mental models that guide them as they solve problems, I hope at least they will recognize that they find the models within themselves. In his classes on English composition, Robert Pirsig wanted his own students to do the same thing. Plato teaches the possibility in the Meno, when he leads a person, raised as a slave, to recognize that he already knows how to double a square: he need only take a side that is not double the original, nor half again as long, but equal to the diagonal of the original square.

Out of our own resources, we can learn to solve problems that we have never faced before, and we can know for ourselves that we have solved the problems.

In calculus class, nearly every exercise in integration is new to the student, who can still solve the exercise and check the result by differentiation.

Every new sentence that we utter is a solution to the problem of saying whatever it was that we wanted to say. If we care, we have to decide whether it was a good solution. This may be where, in Plato’s metaphor, we look to the model given to us from some divine source. Unfortunately the students of William Deresiewicz were not interested in doing this:

I had asked them to introduce themselves and talk about their strengths and weaknesses as writers. Many had said some version of “I’m good at writing naturally” or “I’m good at writing conversationally,” “but I’m not good at revising” or “I’m not good at editing” … they thought of writing as something that just happens … they had never been asked to pay attention to their sentences as conscious constructions.

That’s from “American education’s new dark age” (Unherd, March 21, 2022), and I talked more about it in my notes on the tenth reading in The Human Condition.

I am not sure I had better luck than Deresiewicz, when I gave my own students the project that I described in the blog post called “An Exercise in Analytic Geometry.” I did have luck in publishing a paper about the project. Two referees disagreed on the merit of the paper, but it was accepted by a third and published as “Conics in Place” (Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia, May 2, 2022).

In the paper, I managed to include a couple of paragraphs about Pirsig, whose students would hand in a writing assignment and ask, “Is this what you want?” Pirsig commented,

An instructor often gets the feeling that he could spend the rest of his life telling the student what he wanted and never get anywhere precisely because the student is trying to produce what the instructor wants rather than what is good.

I quoted this also in “Poetry and Mathematics.” The passage is from a letter shared in 2018 by Pirsig’s widow, Wendy Pirsig, with Henry Gurr, who posted it on line as Quality In Freshman Writing 1961, A Prescient Formative Precursor Of ZMM.

The link to Pirsig’s letter was not working when I first included it here. Neither had the Internet Archive saved the letter, though it had saved the parent page. Now that the link is working again, I have had the Archive save the linked page. I note this, as a small example of accomplishing something that we might not have imagined we could do.

We might not imagine that we are creators, just by composing sentences that have never been heard before. It is true though, and those sentences are works of art. If only as the slave was led by Socrates, I have been led, by Collingwood in The Principles of Art (1938), to recognize that language and art are the same thing.

This is now art in the modern sense. Mary Midgley was not led to agree with Collingwood about it. In “Creation and Individuality,” included as the third chapter of Heart and Mind (2003/1981), she writes of him,

He sees the absence of a ‘preconceived end’ as a mark of real art, a mark which distinguishes it from mere craft. But if you really do not know what you are trying to bring about, it is hard to see how you can do it, and harder still to see how you can be called responsible.

Yes, it’s hard to see how you can do it. Is it so hard to see that you can do it? Socrates argues that the slave must have known all along how to double a square; he just needed his memory stirred. But did Midgley’s own essay exist all along? Surely it began as a vague idea that she brought into shape.

I say that as somebody who does not speak fluently, or—to put it positively—as somebody who speaks deliberately. There are times when we know there is something to be said, and yet we do not know what it is until we actually say it—and even then, we may not have got it right.

This must be true, whenever we speak, unless we are on autopilot. But then Augustine learned to speak on autopilot, by the account in the Confessions that Wittgenstein quotes at the beginning of the Philosophical Investigations (I have the third edition, 2001, with the translation by G.E.M. Anscombe):

When they (my elders) named some object, and accordingly moved towards something, I saw this and I grasped that the thing was called by the sound they uttered when they meant to point it out. Their intention was shewn by their bodily movements, as it were the natural language of all peoples: the expression of the face, the play of the eyes, the movement of other parts of the body, and the tone of voice which expresses our state of mind in seeking, having, rejecting, or avoiding something. Thus, as I heard words repeatedly used in their proper places in various sentences, I gradually learnt to understand what objects they signified; and after I had trained my mouth to form these signs, I used them to express my own desires.

It seems to me that so-called “machine learning” is Augustinian learning. I must once have accepted Augustine’s account as accurate, as I accepted Wittgenstein’s own ensuing account of language:

I send someone shopping. I give him a slip marked “five red apples”. He takes the slip to the shopkeeper, who opens the drawer marked “apples”; then he looks up the word “red” in a table and finds a colour sample opposite it; then he says the series of cardinal numbers—I assume that he knows them by heart—up to the word “five” and for each number he takes an apple of the same colour as the sample out of the drawer.

At first I took this naïvely as a fair analysis of how language works. I needed a teacher—Georgia Knight—to point out that language is more mysterious than that, and Wittgenstein knows this.

We were reading Wittgenstein in the language tutorial of the senior year at St John’s College in Santa Fe. In the language tutorial of my freshman year, in Annapolis, Chaninah Maschler had recommended my reading Wittgenstein, in part because my writing, like his, gave the reader little indication of where it was going.

Where I am going now is back to Hannah Arendt; but I shall use the help of

• Caroline Alexander, on the Iliad;
• Henry David Thoreau, on the poet in general, and Ossian in particular;
• Amy Mandelker, quoting Tolstoy on War and Peace;
• Simone Weil on the Iliad.

My concern is the mystery of our creativity. I have tried to say why recognizing it is essential to education. In mathematics, when given the exercise of proving a particular statement, some students go to the web to find a previously written proof. In an earlier day, they went to the library. This was not what had been intended by the exercise. Nonetheless, it is what some of my classmates did in graduate school. I confess to having made use of their labor. For an exercise that I had not been able to solve alone, I read a published proof that my classmates had found. I then wrote out what I would call my own proof, based on the understanding that I had derived from reading. Some classmates must have just copied from the book, without understanding; for the teacher (Jeffrey Adams) complained about this in class (and said he was not giving them credit).

Copying with understanding is a part of education, by the account of Collingwood, as I would transpose it to mathematics from its original setting in Outlines of a Philosophy of Art (1925):

But the imitativeness of students’ work is of value in developing and strengthening the creative faculty precisely because it is not pure imitation but disguised creation. You can make a novice climb by roping him and leading him; he supports himself because he thinks some one else is supporting him. This applies to all education. The student is really painting pictures of his own; his own hand and eye are doing the work, and the copy is only setting the pace. When the student awakes to the consciousness of this truth, he is no longer a student, but has graduated as a master.

Copying somebody else’s words may be part of learning to write. In sixth grade we were given poems to copy into our notebooks, and we illuminated them as we wished with colored pencils; I suppose this was a lesson in both penmanship and literature. I don’t think it would be of much mathematical use to copy, say, Euclid’s proofs. However, it might be useful to translate them, whether from Greek to English, or from the English of Heath’s translation into one’s own English, or from any verbal language into the colors of Oliver Byrne’s edition, or from the printed pages of Byrne into the amazing website of Nicholas Rougeux.

The words of Collingwood above are his translation from a draft that read as follows:

This phase of doing something at unawares is the essence of education. The function of the teacher is to help his pupil to do what he couldn’t do alone—which is a paradox, because if one can do it one can do it, and if one can’t one can’t: but it is the paradox of all education. The rope doesn’t take the weight of a climber, he has to climb for himself; but a better climber can coax him up places that he couldn’t get up without the “moral support” of the rope because he doesn’t know he could get up them. Similar is the function of example in morality, etc.

That is from Collingwood’s handwritten lecture notes of 1924, posthumously published in The Philosophy of Enchantment (2005; page 66, note 30). Not having known of its existence, I discovered the book in the Beyoğlu outlet of Pandora Kitabevi; this later moved across the street, and then a Nişantaşı branch was opened, and now the Beyoğlu branch is closing.

I suppose Collingwood’s “paradox of all education” is akin to the one expressed by the title character of the Meno (80d):

Why, on what lines will you look, Socrates, for a thing [namely virtue] of whose nature you know nothing at all? Pray, what sort of thing, amongst those that you know not, will you treat us to as the object of your search? Or even supposing, at the best, that you hit upon it, how will you know it is the thing you did not know?

Plato himself had a paradoxical or ambivalent attitude towards craft, by the account of Hannah Arendt. Here she is in the tenth reading in The Human Condition, on page 229:

Plato and, to a lesser degree, Aristotle, who thought craftsmen not even worthy of full-fledged citizenship, were the first to propose handling political matters and ruling political bodies in the mode of fabrication. This seeming contradiction clearly indicates the depth of the authentic perplexities inherent in the human capacity for action …

Again, the key point seems to be that the craftsperson works from a plan or model. Arendt made that clear on page 226, talking about what we call the philosopher-king (I noted elsewhere that Socrates does not actually use the term):

It is only when he returns to the dark cave of human affairs to live once more with his fellow men that he needs the ideas for guidance as standards and rules by which to measure and under which to subsume the varied multitude of human deeds and words with the same absolute, “objective” certainty with which the craftsman can be guided in making and the layman in judging individual beds by using the unwavering ever-present model, the “idea” of bed in general.

Have there always been carpenters to make beds? Has Plato no sense that the full complement of crafts performed in a community has ever changed? He himself enlarged the library of writing available to us. Where does a Platonic dialogue come from—was Plato taking dictation from an angel, as the Prophet Muhammad is supposed to have done?

Marx had an explanation: we extrude literature as the silkworm does silk. Arendt mentions this twice, first in note 36 on pages 99–100, in the fourth reading:

“A spider conducts operations that resemble those of a weaver, and a bee puts to shame many an architect in the construction of her cells. But what distinguishes the worst architect from the best of bees is this, that the architect raises his structure in imagination before he erects it in reality …” … Obviously, Marx no longer speaks of labor, but of work—with which he is not concerned; and the best proof of this is that the apparently all-important element of “imagination” plays no role whatsoever in his labor theory … [99] … Marx remained convinced that “Milton produced Paradise Lost for the same reason a silk worm produces silk.”

This is one of only two referents for Imagination in the Index, which (according to the Publisher’s Note on page 328) is more detailed than the one that Arendt herself saw made for her book. The other referent for Imagination is on page 310, again in the twelfth reading:

While the ancients had relied upon imagination and memory, the imagination of pains from which they were free or the memory of past pleasures in situations of acute painfulness, to convince themselves of their happiness, the moderns needed the calculus of pleasure or the puritan moral bookkeeping of merits and transgres­sions to arrive at some illusory mathematical certainty of happiness or salvation.

That is interesting, but perhaps not illuminating as to the source of literature and of art in general. The silkworm returns later, on page 321:

Socialized mankind is that state of society where only one interest rules, and the subject of this interest is either classes or man-kind, but neither man nor men … now even the last trace of action in what men were doing, the motive implied in self-interest, disappeared. What was left was a “natural force” … What was not needed … could be justified only in terms of a peculiarity of human as distinguished from other animal life—so that Milton was considered to have written his Paradise Lost for the same reasons and out of similar urges that compel the silkworm to produce silk.

In the simile of silkworm and artist, Arendt thinks there’s something funny, but I’m not sure what it is. To my mind, the difference is that the artist creates something, or makes something new. The silkworm is doing only what her ancestors have been doing for countless generations. Better to liken the artist to Nature herself, who, through the mysterious process of evolution, has given us a fiber that we can weave into warm and elegant clothing.

Scientifically though, nature has no purpose; it just happens. To make sense of it, we think in terms of purposes, so that silk evolved to serve the purpose of protecting the larva as it metamorphosed into pupa and imago.

I remain perplexed by Arendt’s treatment of the artist, even though she was apparently one herself, in the sense of writing poems. On page 170, in the seventh reading, she says,

Of all things of thought, poetry is closest to thought, and a poem is less a thing than any other work of art; yet even a poem, no matter how long it existed as a living spoken word in the recollection of the bard and those who listened to him, will eventually be “made,” that is, written down and transformed into a tangible thing among things, because remembrance and the gift of recollection, from which all desire for imperishability springs, need tangible things to remind them, lest they perish themselves.

I suppose Arendt is alluding here to the composition of the Homeric epics, which may have happened without writing. In this regard, let me note the plausible assertion that the characters of the epics are uniquely developed individuals, such as would not be found in a purely oral tradition. This is what Caroline Alexander says in the Introduction of her 2015 translation of the Iliad:

My own views are shaped by my experience in the 1980s establishing a small department of classics at the University of Malawi, in southeast Africa. In discussing Homer, my Malawian students and colleagues, who had grown up with genuine, living oral traditions and knew the genre intimately, were emphatic that the Iliad did not “feel” like an oral poem. To their sensibilities, despite the obvious evidence of an oral legacy, Homer was a literary poet. He did not honor oral conventions. In particular, his characters are “round,” which is to say fully formed. The Iliad’s dramatic speeches serve as much to reveal a speaker’s character as to further epic action, for example, while traditional oral poetry, being intensely communal, is not similarly invested in individual characterization. Homer is celebrated by literary people in literary cultures, my associates maintained, because his compositions meet literary expectations.

What Alexander goes on to say below is like what I would say to Arendt. Whether a poem is only spoken or also written down, it is always a tangible thing, because it makes an impression on organs of sense, be they fingers, eyes, or ears. Even if the poet never utters a poem out loud, but keeps it in the isolation of his or her own mind, neuroscientists should be able to extract it, the way they are learning to communicate with patients who have locked-in syndrome.

The key point is that the neuroscientists would not be able to compose the poet’s own poem, just by studying the poet’s brain. Somebody who knew the poet’s existing poems might produce a forgery of the poet’s work, but specific knowledge of the physical features of the brain would be of no help in this. Words of Thoreau are apropos:

The true poem is not that which the public read. There is always a poem not printed on paper, coincident with the production of this, stereotyped in the poet’s life. It is what he has become through his work. Not how is the idea expressed in stone, or on canvas or paper, is the question, but how far it has obtained form and expression in the life of the artist. His true work will not stand in any prince’s gallery.

That is from “Friday” in A Week on the Concord and Merrimack Rivers. I note the passage just now, as I remind myself how Thoreau was been taken in by the forgeries attributed to an ancient Gaelic poet called Ossian (whom I hadn’t heard of, before reading Thoreau):

The genuine remains of Ossian, or those ancient poems which bear his name, though of less fame and extent, are, in many respects, of the same stamp with the Iliad itself. He asserts the dignity of the bard no less than Homer, and in his era we hear of no other priest than he …

Ossian reminds us of the most refined and rudest eras, of Homer, Pindar, Isaiah, and the American Indian. In his poetry, as in Homer’s, only the simplest and most enduring features of humanity are seen, such essential parts of a man as Stonehenge exhibits of a temple; we see the circles of stone, and the upright shaft alone.

I think I would be in concord with Caroline Alexander to say that the Iliad is more Gothic cathedral than Stonehenge: not stripped down, but as intricately developed as War and Peace.

Tolstoy was creating something new in that epic; I learned this from Amy Mandelker in the Introduction to the Oxford World’s Classics edition:

Tolstoy was also writing within a European tradition in which the Napoleonic war had already acquired mythopoetic grandeur … By Tolstoy’s own account, the anxiety of influence and the pressure of literary precedent and convention was unendurable: ‘time and my strength were flowing away with every hour, and I knew that nobody would ever tell what I had to tell … Above all, traditions both of form and content oppressed me. I was afraid to write in a language different from that in which everybody writes. I was afraid that my writing would fall into no existing genre, neither novel, nor tale, nor epic, nor history …’ The key to artistic freedom was to reject any formal or stylistic requirements of literary genres, which Tolstoy happily found could be accomplished through an appeal to his own native Russian literary tradition, noted for its experimental character and flouting of literary convention. ‘We Russians don’t know how to write novels in the European sense of the word,’ he announced, proudly and provocatively …

Let me return to Caroline Alexander and what she says about writing in a tradition:

The attention lavished on the question of whether the Homeric poems owe their final form to oral or to literary composition has focused excessively on mechanics, on the physical act of recitation versus that of writing. The more interesting question is not whether a traditional poem was ultimately recorded by the spoken or the written word, but rather in what relationship the final poet stood to his tradition. Did Homer see himself as simply one poet in a long line of traditional poets, improvising and transmitting the tradition he had inherited, more or less as it had always been done? Or did he see himself as standing in a different relationship to the traditional material than the poets before him? Regardless of whether he sang, dictated, or wrote—did he see himself as doing something with the traditional material that had never been done before? This, it seems to me, is the fundamental Homeric Question.

I recall that for Arendt, the poet’s job was to give eternal life to the Greek warriors, and the πόλις (polis, city) was there to make sure he did that job. This is from the eighth reading, pages 197–8:

Homer was not only a shining example of the poet’s political function, and therefore the “educator of all Hellas”; the very fact that so great an enterprise as the Trojan War could have been forgotten without a poet to immortalize it several hundred years later offered only too good an example of what could happen to human greatness if it had nothing but poets to rely on for its permanence.

… it is as though the men who returned from the Trojan War had wished to make permanent the space of action which had arisen from their deeds and sufferings, to prevent its perishing with their dispersal and return to their isolated home­steads.

I cannot fault Arendt for not sharing my own interests; but it seems important that some of the best humans in the Iliad—Hector, Priam, Sarpedon—are on the enemy side. For helping me to see this, I am going to credit Simone Weil, whose essay on the Iliad—“The Poem of Force”—was translated by Arendt’s friend Mary McCarthy:

The Greeks, generally speaking, were endowed with spiritual force that allowed them to avoid self-deception. The rewards of this were great; they discovered how to achieve in all their acts the greatest lucidity, purity, and simplicity. But the spirit that was transmitted from the Iliad to the Gospels by way of the tragic poets never jumped the borders of Greek civilization …

Throughout twenty centuries of Christianity, the Romans and the Hebrews have been admired, read, imitated, both in deed and word; their masterpieces have yielded an appropriate quotation every time anybody had a crime he wanted to justify.

Homer is as great an actor—a person who acts—as anybody he sings about.