Polytropy

A Twitter friend living here in Istanbul announced (on June 16) her pleasure in having a memoir published in Meanjin.

Since this Australian literary publication is not giving away its articles for free, I bought the relevant issue (Winter 2020). In “Unpacking Home: Thoughts of a Displaced Traveller,” Lisa Morrow writes of teaching students of English to distinguish house and home, even as Australia is no longer her home, particularly now that her parents are dead and she has no relations with her two siblings.

The furniture and belongings I carry with me from country to country are the only relics of my past that remain. The heaviest component, my library, fits into 29 boxes, each large enough to hold a dozen bottles of wine. I only drank some of them myself, but I have read every title in this collection, which spans more than 30 years of my life. Who I am is written in the pages. I just have to find the right one.

That sounds a bit like me, although I haven’t changed countries in twenty years. Since then, I have brought more and more of my books to Turkey from the US, not in wine crates, but in Rubbermaid storage bins. One of these, sitting on a table, serves me as the standing desk at which I am writing now.

I remembered somehow discovering that an Australian literary magazine had had an issue on the theme of mathematics. That magazine turned out to be the one called Cordite, which I had made notes about on April 5, after an email friend shared a poem by Wendell Berry called “The Peace of Wild Things”:

When despair for the world grows in me
and I wake in the night at the least sound
in fear of what my life and my children’s lives may be,
I go and lie down where the wood drake
rests in his beauty on the water, and the great heron feeds.
I come into the peace of wild things
who do not tax their lives with forethought
of grief. I come into the presence of still water.
And I feel above me the day-blind stars
waiting with their light. For a time
I rest in the grace of the world, and am free.

I could best do as Berry does in the valley of the North River, a tributary of the Cacapon River, a tributary of the Potomac. We buried my mother’s ashes there, six years ago.

I spent time there alone after graduating from college. From the books in the house, I picked up Randall Jarrell, The Animal Family, and I recall from this the Mermaid’s account of how little fish are willing to get close to the predators when the former know that the latter are not hungry.

(On November 11, 2021, I shall think to check for the book on the Internet Archive. The passage I am recalling is probably this:

The only things the mermaid was afraid of were big sharks and killer whales. “They kill us if they catch us,” she told him. But she was afraid of them in a matter-of-fact, indifferent way, and said none of her people really thought about them or hated them. “Why should we?” she said. “They eat us the way we eat fish. The fish don’t hate us. They swim next to us when they know we’re not hungry, and when we’re hungry they get away from us if they can. Everything lives on everything.”

I have not remembered specifically that the mer-people are the predators.)

That nature is in an eternal struggle for survival: this is one way of looking at nature, but it is our way, and it is not our only one. Mary Midgley points this out in Evolution as a Religion (1985/2002).

Looking at seagulls eying each other on the next roof over, here in Istanbul, I can be reminded of the two male classmates who lived on my corridor in our senior year of college. When near one another, they would put their hands over their crotches for protection.

“Wild things,” say Berry, “do not tax their lives with forethought of grief.” I imagine that’s true. By living in a country without a Protestant work ethic, I also imagine I’ve learned how we may not tax ourselves with forethought of grief.

A problem with seeking refuge in nature is that there will always be somebody wanting to pave your paradise and put up a parking lot.

A difficulty I have with Berry’s poem is the unexpected passage from night to day. Berry wakes in the night, but then, when he goes and lies down “where the wood drake rests in his beauty on the water,” day seems to have come; for the stars are “day-blind,” and I think this means the sun’s light has made them invisible. Berry could mean the stars are hemeralopic (I just learned that word), in the sense that always, even at night, they carry the property of being invisible during the day; but the stars are also “waiting with their light,” and I suppose this means they are waiting for night.

Berry feels the stars above him, but I suppose you can do this at night, as I recall from a poem whose text I have not got at hand, but it was by James Beall, a tutor at St John’s College. The poet puts his hand to a windowpane and feels starlight hitting it. If I recall correctly, he is in bed with a mate, and he likens the stars’ photons to his own seed.

I happened to encounter a poem called “God” that begins in bed and passes to the heavens:

God is a daybed, on which we lie outstretched in the universe

pure as angels, with saint-blue eyes answering the salutation of the stars;

god is a pillow on which we rest our head,

god is a support for our feet;

god is a store of strength and a virginal darkness;

god is the immaculate soul of the unseen and the already decayed body of the unthought;

god is the stagnant water of eternity;

god is the fertile seed of nothingness and the handful of ash from burnt-down worlds;

god is the myriad insects and the ecstasy of the rose;

god is an empty swing between the nothing and the universe;

god is a prison for all free souls;

god is a harp for the mightiest hand of wrath;

god is what longing can persuade to descend upon the earth!

That’s Edith Södergran, as translated by Nicholas Lawrence (2016).

A girl called Agnes reads Södergran in Lukas Moodysson’s wonderful Fucking Åmål (or Show Me Love for Anglophone audiences). We happened to watch it the other night, or rather two nights; we have been turning on the home entertainment system about 45 minutes a night, which means we spent four nights watching Malcolm X, and we went on to spend the same watching Blue Is the Warmest Color. Like Åmål, Blue features high-schoolers, at least to start with, but they are not so convincing as in Moodysson’s film. Maybe French children mature earlier than Swedish children, or at least are better trained to pretend.

I don’t know what to think of Södergran, as I don’t know what to think of a book I have picked up, Thomas J.J. Altizer, The Gospel of Christian Atheism (1966). I have my father’s copy, but I don’t recall his talking about it. I kept the book, maybe in part because the bio on the back says the author “attended St. John’s College, Annapolis, Maryland” (though he “received his degrees of A.B., A.M., and Ph.D. at the University of Chicago”).

If I understand Altizer’s basic point, it might be more simply expressed. Probably then I don’t understand it, but it seems to be that the Gospels might as well end with Good Friday.

The Australian poetry review called Cordite where I found Södergran turns out to have had an issue on the theme of mathematics. I am not sure what to make of this either. Mathematics is already poetic. As the graphic version of my friend Özlem says in two speech balloons, as she is trying to help Özge graduate in mathematics from Boğaziçi University, in Özge Samancı’s delightful graphic memoir, Dare to Disappoint (i.e. dare not to be what your parents want you to be):

A subset S of a topological space X is compact,
if every open cover of S has a finite subcover.

Try to imagine it.
It is so poetic.

I tried to convey the poetry of a particular compact set in the lavishly illustrated “Tree of Life”; I effectively proved its compactness in “Boolean Arithmetic.” However, when I proposed this compactness as a topic for a student who wanted to do her diploma project with me, and I recommended a book on the subject (in Turkish, by Özlem’s husband), the student worked for some months before leaving me for an easier topic with an easier teacher.

She wasn’t really a mathematician, maybe. When my wife Ayşe eulogized an old teacher of hers who had died, she recalled his saying that the mathematician was somebody who, when reaching a fork in the road, continued on the more difficult path.

Says Fiona Hile, editor of the Cordite issue on mathematics (November, 2017):

I still don’t know how to “do” mathematics but in reading through the twelve hundred or so poems submitted to this special issue of Cordite I was looking for traces of the various ways in which it can make its presence felt.

It intrigues me that Hile goes on to mention another mathematician friend of mine, Maryanthe (she once came to our flat for a beer during a conference here in Istanbul):

… of particular interest to me here is the way in which Malliaris and Shelah stumbled onto their discovery. In his account of Badiou’s philosophical edifice, Norris explains how a subject’s fidelity to a generic truth procedure “can make room, via these concepts of the generic and indiscernible, for the advent of truths that as yet lie beyond the compass of achieved (or achievable) knowledge.” What at first seems insoluble or paradoxical can be turned via Cohen’s technique of forcing “into a fully operative concept”. In proving that the two properties they were working on were both maximally complex, Malliaris and Shelah were also able to show that two infinities (p and t) that were thought to be of different sizes were in fact equal. They did this by “cutting a path between set theory and model theory” in a move that deployed Paul Cohen’s method of “forcing” to solve one of the remaining problems of the continuum hypothesis. The move is reminiscent (in terms of audacity if not scope) of Cantor’s realisation that “the scandal of the infinite – of a part that must somehow be conceived as equal to the whole – could in fact serve as its very definition or distinguishing mark.”

I’m afraid I have not been able to retain an understanding of forcing, though I worked through Cohen’s book on the subject with several students in Ankara.

Before I could recover what I wrote above about Cordite, I found a post on mathematics in the Meanjin blog.

Anupama Pilbrow knows our subject as a suffering that is also a pleasure. She seems to have majored in it. Four months after graduation, she returns to it with a friend, to study a book that one of my non-math friends once paged through with bewilderment, when I was in grad school in the 1990s:

After our two-person seminar, my brain physically hurts and I want to have a nap. I try to do other work but it feels like there is glue in my head. I can barely read. Later in the afternoon, I write to Nick, “Can we please do that again next week? That was so fun, I miss maths so much.”

She misses it, because what she concentrates on now is poetry:

As a poet, I’ve been asked a few times to comment on the intersections between mathematics and poetry. I tend to hesitate in my reply. I flip-flop. I say either “they are very alike!” or “they are hardly alike!” Mathematics and poetry push the limits of language a lot – in terms of what words mean, how new words and concepts rise up out of imagery, how meaning shifts to accommodate advances in understanding. And although the points of likeness and difference between the two disciplines are ripe, I don’t consider these things in a conscious way when I’m doing maths, or poetry. Because, for me, to do one means to decide, if temporarily, against the other.

I addressed the author in a tweet:

One reason to like your article is your acknowledgment that there are different ways of thinking (and thus doing).

I think the real “mind-body problem” is this: how can one person do such different things as, say, poetry and mathematics?

I continue with this. There are two fledgling yellow-legged gulls being raised on the roof that our balcony overlooks; so far they flap their wings only in frustration, unable to get much lift. They are two gulls; but the mind and body are not two in this way. They are two ways (or classes of ways) of thinking about ourselves.

Apparently this is not a popular idea. I wonder if this has to do with the difficulty of recognizing oneself as even one kind of thinker, let alone two.

I see this difficulty in my students, whom I could hardly get to know before the pandemic closed the universities this spring. I was to teach them analytic geometry, but didn’t know how to do this online, and didn’t think I could demand much from the students anyway. I seem still to have ended up demanding a lot: that the students choose non-orthogonal conjugate diameters for an hyperbola or ellipse, then find the orthogonal conjugate diameters – the axes – of the same conic. (I went on to write about this in “An Exercise in Analytic Geometry.”)

I gave the students my own examples, in print and in my own hand. Some students were able to imitate my work, using their own chosen parameters, and achieving correct results.

Ideally students would understand the work, then do it in their own way; but it is still an achievement to be able to copy me line by line, except for the numbers, which the students supplied and worked with.

I told the students what was true, that I myself easily made mistakes. I pointed out that they could check their work by plotting points on a graph. This was how I had checked my own work. However, this also turned out to be an aspect of my examples that practically no student could imitate.

Neither could the students cheat, as far as I could tell; so that was good. WolframAlpha will apparently not analyze your equation as required for my exercise. I ended up writing my own program, in TeX, in order to compute answers, and draw graphs, for each student’s parameters. This was a lot of work, and I might have saved time, had I had knowledge of existing mathematical software; but I have never cared to acquire this knowledge. Neither did any student seem to have done this, or to have a confederate who had.

One student did use a program to draw her ellipse and to plot the endpoints of the axes that she had computed by hand. Those endpoints did not actually lie on the ellipse. The student should then have gone back to her computations, in order to find her error. At least she could have said, “I thus know my computations are in error, but I have no time to check them.”

She did not do this. I understand not wanting to go back over your computations. But to my mind, an attractive point of the exercise was that, done correctly, the computations would give you something nice to look at. This “something nice,” the graph, would also allow the student to know that she was correct, without having to ask the teacher.

To learn that one can know one is correct, without having to consult anybody else: I think this is the point of studying mathematics.

The like may be true in art. You can be satisfied with your poem or your painting, regardless of what anybody else thinks. But many students seem to want to avoid the responsibility of being satisfied, as much in English as in mathematics. Robert Pirsig wrote about them in a letter to another English professor:

The problem being fought is the old problem that is renewed each time a student brings in a rewritten paper saying, “Is this what you want?” The question seems ordinary enough to the student but every time one tries to answer it honestly it becomes a frustrating and subtly maddening question. 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.

There seems to be a difference though. In art, you may care what others think; but you don’t have to, the way you do in mathematics. Here, if somebody disputes your computation or your proof, you must resolve the disagreement. The student should learn this too. You cannot debate, much less come to blows, but must work together for mutual understanding and agreement.

That’s the ideal. Maybe as the shaky foundations of mathematics were being exposed in the nineteenth century, or even more recently, conferences experienced such brawls as are described in this memoir of an Edgar Allen Poe conference:

so the place goes dead fucking silent as every giant ass poe stan in the room is immediately thrust into a series of war flashbacks: the orangutan argument, violently carried out over seminar tables, in literary journals, at graduate student house parties, the spittle flying, the wine and coffee spilled, the friendships torn – the red faces and bulging veins – curses thrown and teaching posts abandoned – panels just like this one fallen into chaos – distant sirens, skies falling, the dog-eared norton critical editions slicing through the air like sabres – the textual support! o, the quotes!

I may have been too naïve to recognize it then, but I don’t think there was such a contentious spirit at St John’s, where faculty were not pressed to publish original research. I say that attending the College was the best thing I ever did, besides getting together with my wife (at the Fields Institute for Research in Mathematical Sciences in Toronto). To these two important actions, I should probably add getting to know Collingwood, who was a polymath, but didn’t know much of mathematics.

I had read on Wikipedia that Collingwood had been an influence on Michael Oakeshott. I have been moved to investigate this influence. I read the article on Oakeshott in the Stanford Encyclopedia of Philosophy.

At least I read the sections that mentioned Collingwood. I am not a philosopher, because I am not interested in struggling with philosophical writing that is not somehow personally appealing.

I can do this with mathematical writing, if the mathematics behind the writing is appealing. In writing a review of an article recently, I’ve enjoyed working out examples to illustrate the main theorem in a way that the authors didn’t do.

In the SEP article on Oakeshott, I read a lot that sounded like what I had learned from Collingwood:

Philosophers have used the word “mode” to refer to an attribute that a thing can possess or the form a substance can take. For Oakeshott, this thing or substance is experience … experience involves thinking and therefore ideas … It is an “autonomous” kind of thinking … A puzzle, then, is how the modes can talk to one another, and the solution is that as modes they don’t …

… Because propositions in one mode of discourse have no standing in another, truth is coherence, however defined, within a given mode. To argue across a modal boundary is to commit the fallacy of ignoratio elenchi (irrelevance). If there is any relationship between the modes it is conversational, not argumentative … there is no extra-modal definition of reason or rationality. The illusion that there is arises from privileging what counts as reasonable within a given mode and denigrating what is considered reasonable in other modes. This illusion of superiority generates the narrowness, and at times hubris, characteristic of each mode connoted by the labels “historicism”, “scientism”, “pragmatism”, and “aestheticism”. A conversational as opposed to argumentative juxtaposition of modal voices is respectful of differences and for that reason inherently civilized, which means that to insist on the primacy of any single mode is not only boorish but barbaric.

Right on. We must know and respect that there are different ways of thinking.

And yet sometimes it is said that we must not respect this. I have recalled the words of the man who spoke (though I do not recall how he was chosen) at my college graduation ceremony:

When President Ronald Reagan called the Soviet Union “the evil empire,” right-thinking persons joined in an angry chorus of protest against such provocative rhetoric. At other times Mr. Reagan has said that the United States and the Soviet Union “have different values” (italics added), an assertion that those same persons greet at worst with silence and frequently with approval. I believe he thought he was saying the same thing in both instances …

Thus Allan Bloom in The Closing of the American Mind (1987). Reagan may have thought that way; and yet it’s not evil to be a poet, or a mathematician, or a programmer, or a physicist, even though they all value different things.

The Stanford Encyclopedia article on Oakeshott (written by Terry Nardin) mentions Collingwood’s Speculum Mentis (1924) as being concerned with five modes:

R.G. Collingwood … begins with Hegel’s triad of art, religion, and philosophy, identifying philosophy broadly defined with “knowledge” and distinguishing three kinds of knowledge – science, history, and philosophy narrowly defined – to generate a fivefold hierarchy of modes … Oakeshott, partly in response to Collingwood, folds art and religion into practice, denies that modes can be ordered hierarchically, and defines philosophy as the activity of interrogating presuppositions, including its own, and therefore not itself a mode.

Somehow, I don’t think there’s a dispute here that needs resolving. We are not doing mathematics. For Collingwood, the “hierarchy” of modes, if there really is one, is in the order of development. Paleolithic humans had art; neolithic, religion; the Greeks, science. The child develops first an artistic sense, then a religious, then a scientific; then in adolescence starts over again, except it’s not really “again,” since an adolescent’s art is not a baby’s.

Speculum Mentis is a kind of abstract autobiography. The author writes:

If the person to whom is committed the task of judging between art, religion, science, history, and philosophy can prove to us that he has lived these lives for himself, has graduated successively in each of these schools, then we shall listen with respect to his opinion.

Collingwood is implicitly such a person. His parents were artists; he considered being a professional violinist (as I reported in “Anthropology of Mathematics,” citing Ray Monk); later in life, he regretted not having been a writer. In Speculum Mentis he alludes to his scholarship on Roman Britain:

… periods of history thus individualized are necessarily beset by “loose ends” and fallacies arising from ignorance or error of their context. Now there is and can be no limit to the extent to which a “special history” may be falsified by these elements. The writer insists upon this difficulty not as a hostile and unsympathetic critic of historians, but as an historian himself, one who takes a special delight in historical research and inquiry; not only in the reading of history-books but in the attempt to solve problems which the writers of history-books do not attack. But as a specialist in one particular period he is acutely conscious that his ignorance of the antecedents of that period introduces a coefficient of error into his work of whose magnitude he can never be aware …

I haven’t understood why Oakeshott wrote Experience and its Modes (1933), whose Introduction begins:

An interest in philosophy is often first aroused by an irrelevant impulse to see the world and ourselves better than we find them. We seek in philosophy what wiser men would look for in a gospel, some guidance as to le prix des choses, some convincing proof that there is nothing degrading in one’s being alive …

Somehow it makes sense that Oakeshott would be considered a forefather of postmodern conservatism.

Thinking about what we do is apparently hard. According to Galen Strawson,

we know exactly what consciousness is – where by “consciousness” I mean what most people mean in this debate: experience of any kind whatever. It’s the most familiar thing there is, whether it’s experience of emotion, pain, understanding what someone is saying, seeing, hearing, touching, tasting or feeling. It is in fact the only thing in the universe whose ultimate intrinsic nature we can claim to know. It is utterly unmysterious.

It seems to me the familiar is not always unmysterious. I have an urge now to write some words in response to words that I have read lately. I satisfy the urge by writing; but I cannot know the writing is satisfactory until it is done; and perhaps it is never finally done. How is this possible? It is a mystery, but it is my experience.

Strawson’s “experience” seems passive, except maybe for the part about “understanding what someone is saying”; but if Strawson recognized this part of experience as being especially active, he might have said so.

Strawson continues:

The nature of physical stuff, by contrast, is deeply mysterious, and physics grows stranger by the hour. (Richard Feynman’s remark about quantum theory – “I think I can safely say that nobody understands quantum mechanics” – seems as true as ever.) Or rather, more carefully: The nature of physical stuff is mysterious except insofar as consciousness is itself a form of physical stuff. This point, which is at first extremely startling, was well put by Bertrand Russell in the 1950s in his essay “Mind and Matter”.

This would make sense, if the second and third instances of the adjective “physical” were left out: “The nature of stuff is mysterious except insofar as consciousness is itself a form of stuff.” But I don’t see the point of applying the adjective “physical” to stuff that is not an object of the sciences called physical; and consciousness is not such an object.

According to Strawson,

The German philosopher Gottfried Wilhelm Leibniz made the point vividly in 1714. Perception or consciousness, he wrote, is “inexplicable on mechanical principles …”

… Leibniz’s basic point remains untouched. His mistake is to go further, and conclude that physical goings-on can’t possibly be conscious goings-on. Many make the same mistake today – the Very Large Mistake …

A physical going-on is studied by physics, which is itself a conscious going-on. To say these goings-on are the same kind of thing is like saying poetry and mathematics are the same. They are not, even though the same person can do them. To see this, just do them.