Polytropy

My two recent courses at the Nesin Mathematics Village had a common theme. I want to describe the theme here, as simply as I can—I mean, by using as little technical knowledge of mathematics as I can. But I shall talk also about related poetry and philosophy, of T. S. Eliot and R. G. Collingwood respectively.

[The top image shows a page from the Echternach Gospels and illustrates the first episode, “The Skin of Our Teeth,” of Kenneth Clark’s Civilisation series on BBC (1969); I take the jpg file from Wikimedia, as supplied by James Steakley from Jean Hubert et al., Europe in the Dark Ages (London: Thames & Hudson, 1969), p. 163. I created the middle image, of an elaborate binary tree, with LaTeX and pstricks and related packages. I took the photograph at the bottom at the new year’s party in my department, Friday, December 28, 2012]

If you don’t know the mathematical lingo, you can skip the present paragraph; if you do know the lingo, my mathematical subject will be the space of branches of the full binary tree of height ω. I shall call this tree simply the Binary Tree. The space of its branches is homeomorphic to the Cantor ternary set, here just the Cantor Set: this was the subject of my first course in Şirince. The space is also homeomorphic to the space of dyadic integers, which arise as a Cartesian factor of the absolute Galois group of a finite field. These fields were the subject of my second course, though I had not expected to get as far as their Galois groups.

[On the left, the Battersea Shield, dredged from the Thames River in 1857, now in the British Museum. On the right, a hu or ritual vessel, said to be from the Zhou or Han Dynasty in China, now in the Victoria & Albert Museum, although I could not confirm this online; I photographed the page of 30,000 Years of Art (London: Phaidon, 2007) where the vessel appears, opposite the Battersea Shield. Both objects are dated to around 200 BCE]

Trying to forget all of the technicalities, I propose that the common theme of my courses was, in a word, time. In a few more words, it was the choices we make in time.

[A bare tree in Şirince, January, 2018; from my previous article]

Choices in time are a theme of Eliot’s Four Quartets. Begins the first Quartet, “Burnt Norton”:

Time present and time past
Are both perhaps present in time future,
And time future contained in time past.
If all time is eternally present
All time is irredeemable.
What might have been is an abstraction
Remaining a perpetual possibility
Only in a world of speculation.
What might have been and what has been
Point to one end, which is always present.

[I printed and bound “Burnt Norton” using ideas from Ellen Lupton, Indie Publishing (Princeton Architectural Press, 2009), June 29, 2011]

We shall be considering all of the possibilities of what might have been. We shall consider each of them abstractly, as the bare possibility of having turned the other way at a fork in the road. Considered thus, life becomes an infinite sequence of choices, between left and right, or perhaps up and down. For us, these possibilities will constitute the Binary Tree. This is the mathematical aspect of our theme.

[A representation of the Binary Tree in proportion 3 : 2]

Strictly speaking, the Four Quartets begin not with the words above, but with an epigraph comprising two of the fragments of Heraclitus of Ephesus, as collected in Diels’s Fragmente der Vorsokratiker. Eliot selects numbers 2 and 60. The latter reads:

ὁδὸς ἄνω κάτω μία καὶ ὡυτή.

Eliot does not translate. Either you are a scholar who recognizes the meaning of the Greek, or you live with obscurity. Or perhaps you choose to do some research. Even in ancient times, to people who spoke Greek, Heraclitus was known as “The Obscure.” We may confuse the obscure with the poetic.

[On the right, the Desborough Mirror from the British Museum; on the left, portrait bust of a Flavian woman, from the Capitoline Museums; image from Wikimedia Commons. Again the objects are shown on facing pages of 30,000 Years of Art; they are dated to the first century CE]

In English, what Heraclitus says is, “The way up and the way down are one and the same.” Eliot will say something like this, in the third movement of the third Quartet, “The Dry Salvages”:

I sometimes wonder if that is what Krishna meant—
Among other things—or one way of putting the same thing:
That the future is a faded song, a Royal Rose or a lavender spray
Of wistful regret for those who are not yet here to regret,
Pressed between yellow leaves of a book that has never been opened.
And the way up is the way down, the way forward is the way back.
You cannot face it steadily, but this thing is sure,
That time is no healer: the patient is no longer there.

The original meaning of Heraclitus need not be so recondite. If two friends live at opposite ends of a road, one of them may go up to visit the other, while next time, the other goes down. They both use the same road.

[Koran stand from Konya, around 1270–80, now in the Pergamon Museum, Berlin]

I return to “Burnt Norton,” in its second movement:

At the still point of the turning world. Neither flesh nor fleshless;
Neither from nor towards; at the still point, there the dance is,
But neither arrest nor movement. And do not call it fixity,
Where past and future are gathered. Neither movement from nor towards,
Neither ascent nor decline. Except for the point, the still point,
There would be no dance, and there is only the dance.

Our dance will be along or within the the Binary Tree.

[A variant in the same ratio, namely the Golden Ratio, of the first binary tree above]

As I noted elsewhere (and it is apparently confirmed in various places, such as the new edition of Collingwood’s autobiography), Eliot attended Collingwood’s Oxford lectures on the Aristotelian treatise, called in Latin De Anima, about what animates us: ψυχή in Greek, and in English, soul. Collingwood wrote about dance in The Principles of Art (page 55):

[S]uppose an artist wanted to reproduce the emotional effect of a ritual dance in which the dancers trace a pattern on the ground. The modern traveller would photograph the dancers as they stand at a given moment. A conventional modern artist, with a mind debauched by naturalism, would draw them in the same kind of way. This would be a silly thing to do, because the emotional effect of the dance depends not on any instantaneous posture but on the traced pattern. The sensible thing would be to leave out the dancers altogether, and draw the pattern by itself.

A dance through the Binary Tree is a pattern of turns. It is a sequence of turns, each turn being one of two ways. The ways may be left and right, or perhaps up or down. We might take warning from the second movement of “The Dry Salvages”:

It seems, as one becomes older,
That the past has another pattern, and ceases to be a mere sequence—
Or even development: the latter a partial fallacy
Encouraged by superficial notions of evolution,
Which becomes, in the popular mind, a means of disowning the past.

I can only say that every dance through the Binary Tree will preserve a record of where it has been.

[Also from the new year’s party in Bomonti, December 28, 2012]

The dance is a pattern or sequence of choices between two options. We shall label these options, for convenience, as 0 or 1. The labels could just as well be something else, such as the noughts and crosses in a game of tick-tack-toe. All that is really needed is that they be distinct from one another, like the nine digits used in a Sudoku puzzle.

[The Binary Tree in standard fashion, made with \pstree in pstricks]

Are the choices of life so meaningless? Some psychologists study of free will in laboratories by asking subjects to push buttons that do nothing. I have questioned this practice. However, if I am anxious to aver that life is full of more serious choices, as between pleasure and work, then maybe I sense, or I suffer from, the moral disease that Collingwood diagnoses in The Principles of Art (page 96):

Among its symptoms are the unprecedented growth of the amusement trade, to meet what has become an insatiable craving; an almost universal agreement that the kinds of work on which the existence of a civilization like ours most obviously depends…is an intolerable drudgery; the discovery that what makes this intolerable is not the pinch of poverty or bad housing or disease but the nature of the work itself in the conditions our civilization has created; the demand arising out of this discovery, and universally accepted as reasonable, for an increased provision of leisure, which means opportunity for amusement, and of amusements to fill it; the use of alcohol, tobacco, and many other drugs, not for ritual purposes, but to deaden the nerves and distract the mind from the tedious and irritating concerns of ordinary life…

Collingwood is writing in 1937, but I am not sure how much has changed since then. Artists are always faced with a choice. I do not know the current status of traditional standards of technical competence; but for Collingwood (page 333),

the question is whether this ideal of artistic competence is directed backwards into the blind alley of nineteenth century individualism, where the artist’s only purpose was to express ‘himself’, or forwards into a new path where the artist, laying aside his individualistic pretensions, walks as the spokesman of his audience.

In its “Aesthetics” article, the Internet Encyclopedia of Philosophy is mistaken to say that Collingwood “took art to be a matter of self-expression.” In literature, he thought, artists had turned away from self-expression as such: this was good, and

credit for this belongs in the main to one great poet, who has set the example by taking as his theme in a long series of poems a subject that interests every one, the decay of our civilization.

The poet was Eliot, and his series of poems had culminated in The Waste Land of 1922. “Burnt Norton” seems to have been published first in 1936.

[Mshatta Palace façade, Jordan, commissioned by Umayyad caliph al-Walid II; never completed; rediscovered in the 19th century and given to Kaiser Wilhelm II by Sultan Abdul Hamid II (who suppressed the first Ottoman parliament, featured in “Bosphorus Sky”); now in the Pergamon Museum]

I want to return to, and develop, the abstraction of the Binary Tree. I am faced with some choices. I can, but need not, reduce the tree to the numbers that make up the so-called Cantor Set. I allude here not to the zeros and ones already mentioned, but to how each infinite sequence of zeros and ones can be used to determine a different real number on the number-line. Real numbers are the subject of most of high-school mathematics. In this way, at least, they should be familiar to just about every reader.

[Every path upwards through the tree leads to a different point of the Cantor Ternary Set. Some points along the way are given their values in ternary notation; along the top, values are given as proper fractions in base ten. You probably have to click on the image to see these; the source file has size A0 and was made using the a0poster package for LaTeX]

There are two reasons not to reduce the Binary Tree to the Cantor Set. One reason is not to encourage the common belief that mathematics is numbers, and equations of numbers, or of letters standing mysteriously for numbers. Mathematics is not always so. In writing about Thales of Miletus, I suggested that an equation like

A = πr²

ought indeed to be frightening, because it is a mask for the most difficult mathematics in all of the thirteen books of Euclid’s Elements.

An equation may serve as a mask that obscures one’s own ignorance. In high-school chemistry, I became adept at manipulating the equation

PV = nRT,

which is the form in which the Ideal Gas Law is applied to numerical problems. The letters in the equation stand respectively for (1) pressure; (2) volume; (3) number of molecules, usually expressed as a multiple of Avogadro’s number; (4) a constant of proportionality; and (5) temperature. It may be clear enough how to add together two different pressures, or volumes, or (especially) numbers of molecules, theoretically.

[Called “Rug with Interlaced Rosettes” in Pergamon Museum, Berlin: 66 Masterpieces (London: Scala, 2005), this carpet, dated to 1500 in Turkey, is said to have been purchased by Wilhelm von Bode in 1888, “presumably in Italy”; it was hard to find on the museum website—for example, I had to search for teppich though search instructions were given in English]

How can you add temperatures? On a hot summer day, in a country using the metric system—almost any country, other than the United States—, you can watch a thermometer rise from 20 degrees in the early morning, to 30 degrees in the late morning. Then you can see the passage from 30 up to 40 degrees in the afternoon. In either case, you have seen a ten-degree change. What was really the same about the two changes? Where are those ten degrees?

[Berliner Dom, September 14, 2007, during the visit when I saw other objects shown here, though I did not photograph them]

We can add two centimeters to five centimeters by placing them at the end. If we want to add two degrees to our indoor temperature in winter, how do we obtain those two degrees, and where do we put them?

There may be answers to these questions. Without such answers, temperature will still be useful for us, as an example of a linear ordering that is not obviously numerical in the fullest sense. We can say of two different temperatures that one is greater than the other, even if we have no way to subtract the less from the greater.

Apparently there is technical terminology for this situation: temperature is an ordinal scale, but not obviously a proper interval scale, much less a ratio scale.

I said there were two reasons not to reduce the Binary Tree to the Cantor Set. A second reason is the symmetry of the Tree, due to the interchangeability of 0 and 1. Symmetry is lost when the Tree is embedded (or rather, its branches are embedded) in the real number line.

Let me give a precise mathematical description of the Tree. It should help to make possible the aesthetic pleasure of stepping back for a bigger picture, and then stepping back again. This is like the pleasure that I described in “Some Say Poetry.” There the pleasure is of stepping in: into a poem of Sylvia Plath, or into the Ayasofya. The stepping in shows you an inside, larger than you imagined from outside.

The Binary Tree has branching points, which we may call nodes. Each node is labelled with a different finite sequence of zeros and ones. There is a root node, labelled with the empty sequence, ( ); the height of this node is 0. The root is followed by the nodes (0) and (1), having height 1. These nodes are followed respectively by nodes of height 2: (0), by (0,0) and (0,1); (1), by (1,0) and (1,1). Then come nodes of height 3, and so on. We can compare the heights of two different nodes, since heights are natural numbers, and these are linearly ordered. However, different nodes can have the same height. A node itself is above another if its label has the label of the other as an initial segment. Thus (0,1,1) is above ( ), (0), and (0,1), but not above (1,1).

The binary tree is an example of the kind of structure called simply a tree; and a tree is an example of an ordered set or simply an order. Some persons may restrict this term to what I am calling a linear order; for such persons, what I call an order is a partial order, even though such a thing is allowed to be total or, as I am saying, linear.

An order then consists of some things called elements or nodes or points or something, and when two of these are selected, sometimes it is possible to say that one of them is less than the other—or below the other, or to the left of the other, or something like that. If A is less than B (or to the left of B, or whatever), and B is less than C, then A must be considered as being less than C. In a word, the ordering of the points is transitive. However, no point is less than itself: the ordering is irreflexive. Therefore the same point cannot be both less than and greater than another, since otherwise, by transitivity, it would be less than itself.

It is not necessary to think of examples of orders as such, besides the Binary Tree. However, we shall want to think of linear orders, which are orders in which every two elements are comparable. For a simpler term, we may refer to a linear order as a chain.

Every node of the Binary Tree has a height, which is some natural number n. Then the node itself is labelled with, or simply is, a sequence

(e₀, e₁, e₂, e₃, …, e_n−1).

Each entry in this sequence is e_k, where k is a natural number less than n; and e_k is either 0 or 1. The node and the nodes below it constitute a chain. This chain then consists of the nodes

( ), (e₀), (e₀, e₁), (e₀, e₁, e₂), …, (e₀, e₁, e₂, e₃, …, e_n−1).

There are longer chains. For example, to the chain we have, we can add one of the nodes (e₀, e₁, e₂, e₃, …, e_n−1,0) and (e₀, e₁, e₂, e₃, …, e_n−1,1). We cannot add both; we have to make a choice.

The Binary Tree has maximal chains: chains to which no node can be added. Such a chain is called a branch of the tree. An example is the chain comprising all of the nodes in which zero is the only entry. Another branch follows the pattern

( ), (0), (0,1), (0,1,0), (0,1,0,1), (0,1,0,1,0), …

The essential observation is that every branch of the binary tree consists of the finite initial segments of some infinite sequence of zeros and ones. (Moreover, the finite initial segments of any such sequence always constitute a branch.) The branch that we just described consists of the finite initial segments of the sequence

(0,1,0,1,0,…).

Here is where we are stepping back for a broader view. The binary tree constitutes one structure, namely the order that we have described. The branches of this tree constitute a new structure.

There is no real difference between two branches of the tree. A difference arises when we draw the tree; but then we may remember what Heraclitus said. The ways up and down, or left and right, are the same. It doesn’t matter which way we go.

This makes the Binary Tree a perfect metaphor for life, even though it may matter a great deal what we do in life. There may be any number of reasons why we ought to do this or that; there is no reason why we actually do it. This is a theme of Chapter XIII, “Choice,” of Collingwood’s New Leviathan (1942):

13. 1. A man about to choose finds himself aware of a situation in which alternative courses of action are open to him. It is between these that he chooses.

13. 11. I distinguish choice from decision only as two words which mean nearly enough the same thing to be left here undistinguished.

13. 12. The kind of choice with which I am concerned in this chapter is only one kind: the simplest; mere choice or mere decision, uncomplicated by any reason why it should be made in this way and not that; in fact, caprice.

Caprice is possible. It is how we become free to make choices in the first place:

13. 2. The problem of free will is not whether men are free (for every one is free who has reached the level of development that enables him to choose) but, how does a man become free? For he must be free before he can make a choice; consequently no man can become free by choosing.

13. 21. The act of becoming free cannot be done to a man by anything other than himself. Let us call it, then, an act of self-liberation. This act cannot be voluntary.

The act of self-liberation is an act of denying oneself and thereby gaining self-respect; but this does not explain the act:

13. 32. The man who denies himself and gains self-respect is richly rewarded; but that is not why he does it. His act of self-denial, not being a voluntary act (13. 21), cannot be a utilitarian act, the exchange of one thing for something more valuable.

13. 33. And if he knew what he stands to gain, he would not value it. What charm has self-respect for a man whose desires are concentrated on happiness?

I have already devoted an article to Collingwood’s chapter on choice, but there I quoted different paragraphs, except in one case. The idea that is now relevant goes back to Collingwood’s first published work (as far as I know): “The Devil,” an essay forming part of a collection with eleven authors called Concerning Prayer (1916). The Devil cannot exist as an explanation of evil (page 459):

The truth is that evil neither requires nor admits any explanation whatever. To the question, “Why do people do wrong?” the only answer is, “Because they choose to.”

Let the Binary Tree be a map of all of the courses that one’s life can take or could have taken. If all of those courses look the same, it is because there is or was no reason—or no external reason—to take one course, rather than another. Collingwood elaborates.

To a mind obsessed by the idea of causation, the idea that everything must be explained by something else, this answer seems inadequate. But action is precisely that which is not caused; the will of a person acting determines itself and is not determined by anything outside itself. Causation has doubtless its proper sphere. In certain studies it may be true, or true enough for scientific purposes, to describe one event as entirely due to another. But if the Law of Causation is a good servant, it is a bad master. It cannot be applied to the activity of the will without explicitly falsifying the whole nature of that activity. An act of the will is its own cause and its own explanation; to seek its explanation in something else is to treat it not as an act but as a mechanical event.

An act may be good or bad. Good and bad are not opposites, not in the sense of arising from opposite parts of ourselves. As Collingwood puts it (pages 464–5),

That which acts is never one part of the self; it is the whole self. It is impossible to split up a man into two parts and ascribe his good actions to one part—his soul, his reason, his spirit, his altruistic impulses—and his bad actions to another. Each action is done by him, by his one indivisible will. Call that will anything you like; say that his self is desire, and you must distinguish between right desires and wrong desires; say that it is spirit, and you must add that spirit may be good or bad. The essence of his good acts is that he might have done a bad one: the essence of his bad, that he—the same he—might have done a good. It is impossible to distinguish between any two categories one of which is necessarily bad and the other necessarily good.

Escher can make a picture showing angels and devils making up separate parts of a whole world. The Binary Tree can be drawn on such a pattern as Escher might have developed. There are left turns and right turns; however, there is no labelling of one kind of turn as sinister, the other as righteous.

Escher, Circle Limit IV

Time assigns a linear ordering to one person’s path of life. However, time gives no way to compare two different paths. If two souls are shinning up a tree, then one of them may be higher than the other, in the sense that the latter can reach the former by continuing to climb. However, if the two climbers have passed the same node, but chosen different branches, then their positions are now incomparable.

I say there is no real difference between branches of the binary tree. The most remarkable feature of the branches may be not what they are, but how many there are.

We can assign a linear ordering to the nodes of the binary tree by treating them as “words,” written in the alphabet whose “letters” are 0 and 1, and where 0 comes before 1 in the alphabetical order. In an English dictionary, nothing comes before A. This means the indefinite article a must be listed before the word aa for rough lava, because the word a consists of the letter A, followed by—nothing. However, all other single-letter words, such as I and O, come after all words that begin with A. If we tried to order the nodes of the binary tree this way, we would start out

( ), (0), (0,0), (0,0,0), (0,0,0,0),

and we would never reach a “word” in which the “letter” 1 occurred. So what we do instead is list all of the words of length n before the words of length n + 1. Thus we get

( ), (0), (1), (0,0), (0,1), (1,0), (1,1), (0,0,0), (0,0,1), (0,1,0),

and so on. Every node of the Binary Tree is somewhere on this list.

[Another variant of the tree in the Golden Rectangle; full size is A2]

The branches of the binary tree cannot be listed in this way. I give the proof, while confessing that I did not accept it when I first saw it, early in high school. I might have been reading Lillian R. Lieber’s Infinity: Beyond the Beyond the Beyond, originally published in 1953, edited by Barry Mazur and published by Paul Dry Books in 2007. In any case, the proof that I saw when I was young almost certainly involved decimal expansions of real numbers, not zero-one sequences: the latter case may be conceptually easier.

[Books used for this article]

Every branch of the binary tree corresponds to an infinite sequence

(e₀, e₁, e₂, e₃, …)

of zeros and ones. Then we can write out any list of branches in the form

(e_0,0, e_0,1, e_0,2, e_0,3, …),
(e_1,0, e_1,1, e_1,2, e_1,3, …),
(e_2,0, e_2,1, e_2,2, e_2,3, …),
(e_3,0, e_3,1, e_3,2, e_3,3, …),
…

The entries along the diagonal of this array compose the branch

(e_0,0, e_1,1, e_2,2, e_3,3, …).

This branch has an “opposite”: a branch (f₀, f₁, f₂, f₃, …) in which f₀ is different from e_0,0, and f₁ is different from e_1,1, and so on. The new branch cannot be anywhere on the old list.

In a word, while the nodes of the Binary Tree are countable, in the sense that you can put them in a list so that you can “count up” to any one of them, the branches are not countable.

As a consequence, if the branches are divided into two collections, one of these must be uncountable. In particular, if there is a list of recommended courses of life, then there will be too many discouraged courses of life to form a list. This means you may never be able to tell whether a proposed course is recommended or discouraged. There might be lists of nodes to aim for and to avoid; but this is not sufficient guidance, at least if one believes Jimmy Page and Robert Plant:

Yes there are two paths you can go by, but in the long run,
there’s still time to change the road you’re on.

If you are somewhere on the zero path, (0, 0, 0, …), you can always still pass through one of the nodes in which 1 occurs, if these are on the recommended list; but with that thought, you may never actually do it.

[Shiva Nataraja, Seigneur de la danse, Musée Guimet, Paris, June 4, 2011]

There is more to day about the Binary Tree. As I said, we can step back again. This will be to see the structure of sets of branches. This will have to go in another article.