Category Archives: Collingwood

Concerning the philosopher R. G. Collingwood (1889–1943). Many if not most of my posts concern Collingwood somehow, so this category may not be of much use. See Articles on Collingwood for some articles by other persons

What Mathematics Is

Mathematics “has no generally accepted definition,” according to Wikipedia on September 15, 2020, with two references. On September 14, 2023, the assertion is, “There is no general consensus among mathematicians about a common definition for their academic discipline”; this time, there are no references.

I suggest that what really 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 says also (as of either date given above),

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.

A 7×7 grid of squares, divided into four 3×4 rectangles arranged symmetrically about one square; the rectangles are divided in two by diagonals, which themselves describe a square
The right triangle whose legs are 3 and 4 has hypotenuse 5, because the square on it is
(4 − 3)2 + 2 ⋅ (4 ⋅ 3),
which is indeed 25 or 52. This is also
42 + 32.

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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:

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Return to Narnia

1

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 continue to enjoy in my sixth decade, having started in my first.

  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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Thoreau and Anacreon

Note added, October 5, 2023. At the end of this post, from sunny days in the first spring of the Covid pandemic, I take up Anacreon’s poem “The Thracian Filly,” translated as “To a Colt” in A Week on the Concord and Merrimack Rivers. Anacreon was from Teos, whose ruins I have visited. Thoreau may not be sensitive to the sexual connotations of the poem. First I review Thoreau’s book, noting in particular:

  • The book is written, like my blog posts, to please the author, who would rather do without money than sell stuff to get it.
  • The author’s relative indifference to human affairs in the face of nature is becoming less tenable, when a Pacific island inhabited by 40 persons and visited once a month by a boat (and once for all, probably, by a travel writer) is losing its palms to an invasive beetle.

Other books discussed or mentioned (and in my physical library) are

  • Bean, Aegean Turkey;
  • Collingwood, The First Mate’s Log;
  • Lawson, The Drinkers’ Guide to the Middle East;
  • Schalansky, Pocket Atlas of Remote Islands;
  • Thoreau, Walden;
  • Trypanis (ed.), The Penguin Book of Greek Verse;
  • Walls, Henry David Thoreau: A Life.

Gray clouds over blue sky over white clouds over buildings

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Doing and Suffering

To do injustice is worse than to suffer it. Socrates proves this to Polus and Callicles in the dialogue of Plato called the Gorgias.

I wish to review the proofs, because I think they are correct, and their result is worth knowing.

Loeb Plato III cover

Or is the result already clear to everybody?

Whom would you rather be: a Muslim in India, under attack by a Hindu mob, or a member of that mob?

You would rather not be involved; but if you had to choose, which option would be less bad: to be driven to an insane murderous fury, or to be the object of that fury?

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Evolution of Reality

I enjoy and recommend Robert Wright’s Nonzero Newsletter, which presents thought on both American politics and thought itself.

Tiny green plants on red tile roof, cloudy day

In a 2017 post of this blog, I quoted Wright’s 1988 article in The Atlantic Monthly about Edward Fredkin. Somewhat differently from Fredkin, I spelled out my title, “What Philosophy Is,” without actually being a professional philosopher. I touched on a theme that I shall take up now: that thinkers today could benefit from knowing the thought of R. G. Collingwood.

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On the Odyssey, Book I

  • In reading his rendition of the Iliad, having enjoyed hearing Chapman speak out loud and bold;

  • having enjoyed writing here about each book, particularly the last ten books in ten days on an Aegean beach in September of this year (2019);

  • having taken the name of this blog from the first line of the Odyssey;

  • having obtained, from Homer Books here in Istanbul, Emily Wilson’s recent translation (New York: Norton, 2018);

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On the Idea of History

Note added, March 10–11, 2021. The bulk of this post concerns race in the theory of history, particularly the theory attributed to Johann Gottfried Herder (1744–1803). Not having read Herder for myself, I rely on the accounts of

  • R. G. Collingwood in § 2, “Herder,” of Part III of The Idea of History (1946),
  • Michael Forster in “Johann Gottfried von Herder,” Stanford Encyclopedia of Philosophy (summer 2019).

Somebody like Herder may introduce race as an hypothesis to explain history, but ultimately the hypothesis fails, by denying us the freedom that is essential to history as such. Nonetheless, Forster defends Herder as having

an impartial concern for all human beings … Herder does also insist on respecting, preserving, and advancing national groupings. However, this is entirely unalarming,

because, for one thing, “The ‘nation’ in question is not racial but linguistic and cultural.”

Change Collingwood’s word “race” to “linguistic and cultural grouping” then. I think his conclusion remains sound: “Once Herder’s theory of race is accepted, there is no escaping the Nazi marriage laws.”

More detail is in the post below. I go on to review the philosophy of history that Collingwood presents in the Introduction of The Idea of History. This book provided me with a title for the post.

I wrote a lot in this post, as I often do. Growing self-conscious for being opinionated about the theory of history, I listed the published evidence of my actually being an historian (an historian of ancient Greek mathematics in particular).

I originally wrote that my research had been inspired by a tweet. The author of that tweet also wrote the nice long comment on this post. However, although the tweet can be found on the Internet Archive, the author later deleted his Twitter account, and so the tweet appears on Twitter today as a gap in the thread above my own tweet in response to the other tweet.

That missing tweet referred to another tweet of the author, but the Internet Archive seems not to have saved it. It did save the present post; so if one were curious, one could see the changes that I have made since initial publication, or rather since September 29, 2020, when the Archive took the first snapshot.

The changes are for style and local clarity. Any large-scale changes would need me to recover the spirit that possessed me when I originally wrote.

I return to this post now, simply because a friend mentioned reading Middlemarch, and I remembered quoting George Eliot’s novel in a blog post, and that post turned out to be this one.

Had somebody mentioned reading Herder, I might have recalled writing about him in a blog post; that would be this post too.


Our environment may influence our feelings, but what we make of those feelings is up to us. Thus we are free; we are not constrained by some fixed “human nature”—or if we are, who is to say what its limits are?


Rembrandt van Rijn (and Workshop?), Dutch, 1606–1669,
The Apostle Paul, c. 1657, oil on canvas,
Widener Collection, National Gallery of Art

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Anthropology of Mathematics

This essay was long when originally published; now, on November 30, 2019, I have made it longer, in an attempt to clarify some points.

The essay begins with two brief quotations, from Collingwood and Pirsig respectively, about what it takes to know people.

  • The Pirsig quote is from Lila, which is somewhat interesting as a novel, but naive about metaphysics; it might have benefited from an understanding of Collingwood’s Essay on Metaphysics.

  • A recent article by Ray Monk in Prospect seems to justify my interest in Collingwood; eventually I have a look at the article.

Ideas that come up along the way include the following.

  1. For C. S. Lewis, the reality of moral truth shows there is something beyond the scope of natural science.

  2. I say the same for mathematical truth.

  3. Truths we learn as children are open to question. In their educational childhoods, mathematicians have often learned wrongly the techniques of induction and recursion.

  4. The philosophical thesis of physicalism is of doubtful value.

  5. Mathematicians and philosophers who ape them (as in a particular definition of physicalism) use “iff” needlessly.

  6. A pair of mathematicians who use “iff” needlessly seem also to misunderstand induction and recursion.

  7. Their work is nonetheless admirable, like the famous expression of universal equality by the slave-driving Thomas Jefferson.

  8. Mathematical truth is discovered and confirmed by thought.

  9. Truth is a product of every kind of science; it is not an object of natural science.

  10. The distinction between thinking and feeling is a theme of Collingwood.

  11. In particular, thought is self-critical: it judges whether itself is going well.

  12. Students of mathematics must learn their right to judge what is correct, along with their responsibility to reach agreement with others about what is correct. I say this.

  13. Students of English must learn not only to judge their own work, but even that they can judge it. Pirsig says this.

  14. For Monk, Collingwood’s demise has meant Ryle’s rise: unfortunately so since, for one thing, Ryle has no interest in the past.

  15. In a metaphor developed by Matthew Arnold, Collingwood and Pirsig are two of my touchstones.

  16. Thoreau is another. He affects indifference to the past, but his real views are more subtle.

  17. According to Monk, Collingwood could have been a professional violinist; Ryle had “no ear for tunes.”

  18. For Collingwood, Victoria’s memorial to Albert was hideous; for Pirsig, Victorian America was the same.

  19. Again according to Monk, some persons might mistake Collingwood for Wittgenstein.

  20. My method of gathering together ideas, as outlined above, resembles Pirsig’s method, described in Lila, of collecting ideas on index cards.

  21. Our problems are not vague, but precise.


When Donald Trump won the 2016 U.S. Presidential election, which opinion polls had said he would lose, I wrote a post here called “How To Learn about People.” I thought for example that just calling people up and asking whom they would vote for was not a great way to learn about them, even if all you wanted to know was whom they would vote for. Why should people tell you the truth?

Saturn eclipse mosaic from Cassini

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On Translation

Achilles is found singing to a lyre, in a passage of Book IX of the Iliad. Homer sets the scene in five dactylic hexameters; George Chapman translates them into four couplets of fourteeners.

I wrote a post about each book of the Iliad, in Chapman’s version of 1611. As I said at the end, I look forward to reading Emily Wilson’s version. Meanwhile, here I examine the vignette of the lyre in several existing English translations, as well as in the original.

Three books mentioned in the text Continue reading