If you roll out a lump of clay into a snake, then tie a string loosely around it, can you contort the ends of the snake, without actually pressing them together, so that you cannot get the string off?

You can stretch the clay into a Medusa’s head of snakes, and tangle them as you like, again without letting them touch. If you are allowed to rest the string on the surface of the clay, then you can get it off: you just slide it around and over what was an end of the original snake.

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More of What It Is

I say that mathematics is the deductive science; and yet there would seem to be mathematicians who disagree. I take up two cases here.

From Archimedes, De Planorum Aequilibriis,
in Heiberg’s edition (Leipzig: Teubner, 1881)

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What Mathematics Is

Mathematics “has no generally accepted definition,” according to Wikipedia today. Two references are given for the assertion. I suggest that what has no generally accepted definition is the subject of mathematics: the object of study, what mathematics is about. Mathematics itself can be defined by its method. As Wikipedia currently says also,

it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions.

I would put it more simply. Mathematics is the science whose findings are proved by deduction.

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This is a little about mathematics, and a little about writing for the web, but mostly about the nuts and bolts of putting mathematics on the web. I want to record how, mainly with the pandoc program, I have converted some mathematics from a LaTeX file into html. Like “Computer Recovery” then, this post is a laboratory notebook.

The mathematics is a proof of Dirichlet’s 1837 theorem on primes in arithmetic progressions. This is the theorem that, if to some number you keep adding a number that is prime to it, there will be no end to the primes that you encounter in this way.

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Map of Art

The bulk of this post is a summary of the chapter on art in Collingwood’s Speculum Mentis: or The Map of Knowledge (1924). The motto of the book is the first clause of I Corinthians 13:12:

Βλέπομεν γὰρ ἄρτι δι’ ἐσόπτρου ἐν αἰνίγματι

For now we see through a glass, darkly

The chapter “Art” has eight sections:

  1. Art as Pure Imagination

  2. The Work of Art

  3. The Monadism of Art

  4. Meaning in Art

  5. Knowledge as Question and Answer

  6. Art as a form of Error

  7. The Dialectic of Art

  8. Play

The summary that I give below is arranged according to these sections.

Art is one instance of an activity that may succeed or fail. In the strictest sense, every activity has this property; for there is a distinction between what we are doing and what we are trying to do.

I begin by observing that, with reference to quoted words of Agnes Callard about Aristotle. Callard happens to be named in the acknowledgments of Lost in Thought by Zena Hitz, whose words about analytic philosophy I shall ultimately quote as well.

Editing this post from a certain email that I drafted on July 26, I have removed (as not having been intended for publication) some words of other persons that I had quoted from a 2012 email discussion of art and Speculum Mentis.

On July 21, Agnes Callard had a good essay in the New York Times, “Should We Cancel Aristotle?” Sure, Aristotle liked slavery and stuff; but reading him – reading him literally – doesn’t mean we agree with him. Neither does it make us agree with him, unless perhaps he is right after all. In Callard’s words:

Yet I would defend Aristotle, and his place on philosophy syllabuses, by pointing to the benefits of engaging with him. He can help us identify the grounds of our own egalitarian commitments; and his ethical system may capture truths – for instance, about the importance of aiming for extraordinary excellence – that we have yet to incorporate into our own.

And I want to go a step further, and make an even stronger claim on behalf of Aristotle. It is not only that the benefits of reading Aristotle counteract the costs, but that there are no costs. In fact we have no reason at all to cancel Aristotle. Aristotle is simply not our enemy.

I, like Aristotle, am a philosopher, and we philosophers must countenance the possibility of radical disagreement on the most fundamental questions. Philosophers hold up as an ideal the aim of never treating our interlocutor as a hostile combatant. But if someone puts forward views that directly contradict your moral sensibilities, how can you avoid hostility? The answer is to take him literally – which is to say, read his words purely as vehicles for the contents of his beliefs.

Aristotle urges “the importance of aiming for extraordinary excellence.” As philosophers, he and Callard “hold up as an ideal the aim of never treating our interlocutor as a hostile combatant.” They have aims; we have aims; and yet we may not achieve them.

In particular, the words of Aristotle are not simply “vehicles for the contents of his beliefs”; not boxcars in which the Philosopher loaded his beliefs in order to deliver them to us. Before the loading, what could the beliefs have been? The words represent an attempt to work out the beliefs. The attempt may have been only partially successful.

The idea is in Collingwood, who says for example in The Principles of Art (1938):

The proper meaning of a word … is never something upon which the word sits perched like a gull on a stone; it is something over which the word hovers like a gull over a ship’s stern. Trying to fix the proper meaning in our minds is like coaxing the gull to settle in the rigging, with the rule that the gull must be alive when it settles: one must not shoot it and tie it there. The way to discover the proper meaning is to ask not, ‘What do we mean?’ but, ‘What are we trying to mean?’

The word under consideration, by Collingwood then and by me now, is art.

A beginning of such considerations is recorded in “Poetry and Mathematics,” a post beginning with talk about writings in a couple of Australian literary publications. One of the essays was about poetry and mathematics, by a poet who had studied mathematics.

I found some reason to observe how Michael Oakeshott had worked on the theme of Collingwood’s Speculum Mentis: or The Map of Knowledge of 1924.

The book can be classified, with Religion and Philosophy of 1916, as belonging to Collingwood’s juvenilia. This could have made it more accessible than a mature work; but it didn’t.

Hegel’s Phenomenology of Spirit was inaccessible. In my day at St John’s College, we read short excerpts of this book, and if memory serves, one of our two seminar leaders admitted from the start that he didn’t know what the book was about.

The other seminar leader had read the tome in graduate school, but perhaps didn’t feel much more confident about it. He did suggest that just reading the whole thing might be the way to go, if reading little bits at a time did not clear it up.

Nonetheless, reading Speculum Mentis section by section seems to clear some things up for me.

Here is my attempt to summarize the chapter called “Art.”

“Art” is Chapter III, the first two chapters being

  • “Prologue,” which begins, “All thought exists for the sake of action”;

  • “Speculum Mentis,” which begins, “Our task, then, is the construction of a map of knowledge” – spoiler alert, the map never gets constructed.

Chapter III has eight sections. Collingwood will not explain till § 4 that the first three sections give a one-sided account of art, based on Vico and Croce: an account that, since the Renaissance, has tended to replace the ancient account.

The mode of thinking in the chapter then would seem to be hypothetical, so to speak, as art itself is.

§ 1. “Art as pure imagination”

“Art is the simplest and most primitive, the least sophisticated, of all possible frames of mind,” not because only children and “savages” make art or are even best at it, but because art imposes no requirement of literal truth.

For example, it is irrelevant to Cymbeline the play, as a work of art, whether there really was a king called Cymbeline who acted as Shakespeare’s character does.

“A philosophical theory must be capable of being conceived as a whole, a historical narrative, of being narrated as a whole – narrated, that is, as true – a work of art, of being imagined as a whole.”

“Art, then, is pure imagination.”

§ 2. “The Work of Art”

Nonetheless, art is trying to be something: not literally true, but beautiful. Art can fail to be this. The possibility of failure “is the standing refutation of all emotional and sensationalistic theories of art.”

Nonetheless, beauty is not a concept. There are no laws or principles for achieving it. If you try to explain the beautiful in terms of its form, you fail.

This is a paradox, whose resolution is that beauty is not a concept, but “the guise under which concepts in general appear to the aesthetic consciousness.”

The paradox seems real to me. A way I try to understand the resolution is to think of Collingwood’s 1916 essay “The Devil,” in which evil is found to be neither the negation of good, nor the opposite of good, but the counterfeit of good. As with being beautiful, so with being good, there is no law for it. “It is a duty, indeed it is the spring of all moral advance, to criticise current standards of morality,” and yet the essence of evil is also to engage in this criticism.

Meanwhile, being an act of imagination, the work of art is not a physical object. The physical object may help the rest of us achieve the aesthetic activity that the artist has accomplished; but this is not really true, since “one never sees anything in anybody’s work but what one brings to it.”

I would object here that Collingwood (or rather Croce, as Collingwood seems to write in his guise) discounts the possibility that the physical work of art teaches us how to see. We shall come back to this in § 4.

§ 3. “The Monadism of Art”

“Every fresh aesthetic act creates a new work of art, though one such act may last for five years at a time.”

“Works of art always ignore one another and begin each from the beginning: they are windowless monads.”

“Art in its pure form is therefore unaware even that it is imagination; the monad does not know that it is windowless; the artist does not say ‘I am only imagining’, for that would be to distinguish imagination from knowledge, and this he does not do. Hence the aesthetic life of children and uneducated people results in what an unintelligent critic calls lying and hallucination.”

I would suggest, as an example, that the current POTUS does not lie, because he has no conception of the truth. He has been allowed to continue living in the dream-world of a child (and this is a terrible indictment of the United States).

§ 4. “Meaning in Art”

Now Collingwood points out what I said at the beginning, that the foregoing account of art is one-sided.

The “special problem of the philosophy of art to-day” is to reconcile that account with the older one, whereby art could teach moral, religious, or philosophical truths.

Imagination must go to work on something:

“Our dreams have a certain continuity with our waking life, and imagination never cuts itself wholly adrift from fact.”

The idea of art as a kind of dreaming comes back in § 7.

§ 5. “Knowledge as Question and Answer”

To imagine is to suppose, which is to question. To answer is to assert.

“Questioning is the cutting edge of knowledge; assertion is the dead weight behind the edge that gives it driving force.”

I note that Pirsig uses a similar metaphor in Zen and the Art of Motorcycle Maintenance. What he calls Romantic Knowledge is the leading edge of the train of knowledge; Classical Knowledge is the train itself, the engine and cars.

§ 6. “Art as a form of Error”

Art wants only to ask questions. In life you can’t do that. “The artist is an artist only for short times; he turns artist for a while, like a werewolf.”

Thus the life of art is unstable. For example, a school of art, once founded, declines. (Herbert Read opens his Concise History of Modern Painting—which I first read while working on a farm in 1988—by quoting Collingwood on this decline.)

Collingwood does not spell out the reason that I would give for the decline. Founding a school of art (such as Impressionism, or Abstract Expressionism) means figuring out how to do something (e.g. paint shadows with colors, or just splash paint on canvas). This means finding a technique. Once you have the technique, you can go on applying it without any fresh act of imagination.

§ 7. “The Dialectic of Art”

I think Collingwood will suggest such an idea about technique in The Principles of Art. There he finds it important to distinguish art from craft. They are not really separable in fact, and I think Collingwood says this now (in § 7 of Speculum Mentis), though without referring to craft as such:

“The cleavage between means and end, technique and inspiration, talent and genius, materials and result, is inevitable in the life of art, and only those who idealize that life by looking at it from the outside deny the dualism. The artist knows that he can only get his work of art by passing through a non-aesthetic world which is that of facts, training, daily life, and so forth.”

In The Principles of Art, the presence of distinctions between (i) means and end and (ii) raw material and finished product will be two of the differentiae of craft as distinct from art.

Collingwood considers dreams as works of art. A dream has a structure: this is why we can call it a dream, one dream, as opposed to some kind of un-unified mélange. The structure is not “unconscious,” since everything we can know about the dream comes from our being conscious of it. But the structure is implicit until made explicit by psycho-analysis.

Knowing the possibility of such analysis, if you proceed to construct your “dreams” consciously, you have become an artist.

I put “dreams” in quotes because I take the word here to refer to any work of the imagination. The artist “conceives [the work of art] in advance of imagining it, in the sense that at any given moment in the process of creation he has in his mind a criterion which enables him to distinguish between the right and the wrong way of continuing the process of imagination itself.”

The artist’s awareness of such a criterion is why Collingwood will later (in The Principles of Art) coin the term “criteriological” for such sciences as aesthetics (also logic, ethics, and economics; I add the example of grammar, which I used in my best attempt to explicate the notion of criteriological science).

In art there is a distinction between form and content, but if it is “developed,” the work of art ceases to be that: for example, poetry becomes prose.

I mentioned the Australian poet (her name is Anupama Pilbrow) who had studied and written about mathematics. There may be poetry in mathematics, but the way mathematics is communicated is entirely prose.

§ 8. “Play”

As art is the most rudimentary thought, play is the most rudimentary action.

We can find reasons for play, whether childish or adult; but “all such explanations of play are in part mythological and forced.”

“God may be pictured as an artist, or as playing, with far more verisimilitude than as a scientist or a business man.”

Still people will ask why we play. “What is the use?” they ask.

It is no excuse “that the overstrained spirit must be allowed some relief from the burden of its responsibilities; for these responsibilities, properly understood, are nothing but its own highest and freest life, and to face them is to find, not to sacrifice, our happiness.”

Life is always an adventure, and “play, which is identical with art, is the attitude which looks at the world as an infinite and indeterminate field for activity, a perpetual adventure.”

In her recent book Lost in Thought: The Hidden Pleasures of an Intellectual Life (Princeton University, 2020), St John’s College alumna and tutor Zena Hitz says that in grad school, one thing she learned was

the delightful mental gymnastics of analytic philosophy, in which any manner of thesis whatsoever is defended, explored, and almost always refuted.

That is the comic version of a tragedy that Collingwood tells of a senior colleague at Oxford, now counted among the analytic philosophers:

[H. A.] Prichard, developing his extraordinary gift for destructive criticism, by degrees destroyed not only the ‘idealism’ he at first set out to destroy but the ‘realism’ in whose interest he set out to destroy it, and described a path converging visibly, as years went by, with the zero-line of complete scepticism.

That’s from An Autobiography (1939). Speculum Mentis is itself autobiographical, in the sense of reviewing careers that Collingwood might have selected. The book covers in turn art, religion, science, history, and philosophy. In 1939 (the autobiography being with the publisher), Collingwood wrote his wife from Java that he wished he had been a novelist.

Zena Hitz starts her book with a Prologue (“How Washing Dishes Restored My Intellectual Life”), which reviews her life story: growing up in San Francisco in a house full of books; heading east to Annapolis for college; grad school in philosophy; a reconsideration after 2001/09/11; staying with philosophy and getting a job in it; joining the Roman Catholic Church; investigating some convents and moving to a sort of Catholic commune (Madonna House) in Ontario; finding her calling to be back at St John’s.

Discrete Logarithms

In the fall of 2017, I created what I propose to consider as being both art and mathematics. Call the art conceptual; the mathematics, expository; here it is, as a booklet of 88 pages, size A5, in pdf format.

More precisely, the work to be considered as both art and mathematics is the middle of the three chapters that make up the booklet. The first chapter is an essay on art, ultimately considering some examples that inspire my own. The last chapter establishes the principle whereby the lists of numbers in Chapter 2 are created.

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An Exercise in Analytic Geometry

This past spring, when my university in Istanbul was closed (like all others in Turkey) against the spread of the novel coronavirus, I created for my students an exercise, to serve at least as a distraction for those who could find distraction in learning.

From Weeks & Adkins, Second Course in Algebra, p. 395

The exercise uses no more mathematical tools than may be found in an algebra course in high school; yet it serves the purposes of university mathematics, as I understand them.

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Be Sex Binary, We Are Not

Content warning: suicide.

The following sentence is bold in the last paragraph of an essay: “the science is clear and conclusive: sex is not binary, transgender people are real.” I don’t know what the writer means by this. As far as I can tell, as a biological concept used for explaining reproduction, sex has two kinds or parts or sides or aspects, and the essay tacitly affirms this; at the same time, obviously persons called transgender exist.

☾ ♂ ☿ ♃ ♀ ♄ ☉

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Poetry and Mathematics

This reviews some reading and thinking of recent weeks, pertaining more or less to the title subjects, of which it may be worth noting that

  • poetry is from ποιέω “make”;

  • mathematics is from μανθάνω “learn.”

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Return to Narnia


My subject is the Chronicles of Narnia of C. S. Lewis (1898–1963). I consider this heptad of books (published 1950–6) as constituting (1) literature (2) for children (3) that I enjoyed in my first decade and continue to enjoy in my sixth.

  1. By literature, I mean a work of art whose medium is prose. Prose may also be a work of craft, intended to fulfil some purpose. This purpose could be to serve a market for fantasy or children’s books. Art as such has no purpose that can be specified in advance.

  2. Writing for children may take certain liberties that annoy adults.

  3. As with any post in this blog, I write out of my own personal interest. As a child, I read other fantasies, such as those of Lloyd Alexander, John Christopher, Ursula LeGuin, and J.R.R. Tolkien. Only the works of C. S. Lewis have stayed with me. This essay may be considered as an exploration of why, or least an example of how.

The seven books of the Chronicles of Narnia, Collier edition
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