Ordinals

This is about the ordinal numbers, which (except for the finite ones) are less well known than the real numbers, although theoretically simpler.

The numbers of either kind compose a linear order: they can be arranged in a line, from less to greater. The orders have similarities and differences:

  • Of real numbers,

    • there is no greatest,

    • there is no least,

    • there is a countable dense set (namely the rational numbers),

    • every nonempty set with an upper bound has a least upper bound.

  • Of ordinal numbers,

    • there is no greatest,

    • every nonempty set has a least element,

    • those less than a given one compose a set,

    • every set has a least upper bound.

One can conclude in particular that the ordinals as a whole do not compose a set; they are a proper class. This is the Burali-Forti Paradox.

Diagram of reals as a solid line without endpoints; the ordinals as a sequence of dots, periodically coming to a limit Continue reading

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);

  • Book on table, Wilson's Odyssey Continue reading

Computer Recovery

I record here how I fixed my computer, because

  • I am pleased to have been able to do it, and

  • I may have to do it again.

Briefly, when Windows on my laptop failed, I installed Ubuntu, but this failed. Somebody else installed Ubuntu again, and this worked for a while before failing. I managed to fix that problem for myself; but later an upgrade failed. Now I have fixed that.

Computer on table by window at dawn

This post is some kind of laboratory notebook. Continue reading

On the Idea of History

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

Insofar as we humans have come to recognize our freedom, we have done so after thinking that what we did depended on our class—our kind, our sort, even our “race.” We might distinguish three stages of thought about ourselves.

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Anthropology of Mathematics

When Donald Trump won the 2016 U.S. Presidential election, which opinion polls had said he would lose, I wrote a post called “How To Learn about People.” I was thinking of their politics, not their occupations.

Saturn eclipse mosaic from Cassini

If however you wanted to understand people whose occupation happened to be mathematics, you would need to learn what it meant to prove a theorem. Mere observation would not be enough:

  • In the words of R. G. Collingwood in Religion and Philosophy (1916, page 42), quoted in An Autobiography (1939, page 93) as well as in the earlier post here, “The mind, regarded in this external way, really ceases to be a mind at all.”

  • In the words of English teacher and anthropologist Verne Dusenberry, quoted by Robert Pirsig in Lila (1991, page 35), “The trouble with the objective approach is that you don’t learn much that way.”

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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

On Chapman’s Homer’s Iliad, Book XXIV

One man kills another, legally, according to the laws of war, such as they are. The two sides fight over the body, which might be ransomed, if taken by the killer’s side; however, the body is not so taken. The friend of the slain man kills the killer and takes his body to mutilate, though this be sacrilege.

The father of the newly slain man crosses enemy lines to ransom his son’s body. He puts his lips to the hand of the killer, who agrees to give up the body, even coming to admire the father, who in turn admires him.

Rembrandt van Rijn (Dutch, 1606-69), Lucretia, 1664, oil on canvas, Andrew W. Mellon Collection. National Gallery of Art, Washington

Such are the emotions of the Iliad. Homer depicts them as terrifically as Rembrandt does those of a woman, Lucretia, about to kill herself in shame for having been raped. One might consider these works as “emotion porn,” where the second element of this phrase denotes

written or visual material that emphasizes the sensuous or sensational aspects of a non-sexual subject, appealing to its audience in a manner likened to the titillating effect of pornography

—in the words of the third edition of the Oxford English Dictionary, as quoted by Arnold Zwicky in a blog article, “X porn.” Continue reading

On Chapman’s Homer’s Iliad, Book XXIII

Book XXII of the Iliad is rich in human emotion; Book XXIII, in anthropological detail. The books form a natural sequence:

  1. Defiance, flight, fight, and death of a man.
  2. Funeral and memorial games for a man.

That the man is different in either case creates tension, to be resolved in the next and final book (whose emotions I once took up in “Homer for the Civilian”).

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On Chapman’s Homer’s Iliad, Book XXII

Andromache draws a hot bath, for Hector to slip into when he comes home from the war. Actually she has her maids heat the water, while she herself weaves flowers into a tapestry.

Mouth of stream forming border between Balıkesir and İzmir

All the Trojans managed to slip into the safety of Troy, while Achilles was distracted in Book XXI of the Iliad. Only Hector and Deiphobus have stayed outside. Hector is really glad to have his brother along to confront Achilles.

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On Chapman’s Homer’s Iliad, Book XXI

Jove allowed the gods to aid whom they would, in the previous book of the Iliad; now, in Book XXI, they fight with one another. The god of fire attacks a river god; the god of war, the goddess of wisdom. This calls into question the notion of gods as personifications of abstract concepts.

Road to beach, shaded by pines

One may say that of course fire is at war with water, and war with wisdom. I would say rather that we can be at war with one another, because none of us is simply one thing distinct from any other.

No man is an island, entire of itself; every man is a piece of the continent, a part of the main.

We have similarities and differences, and so do the gods. We can all do many things.

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