Category Archives: Art


The post below is a way to record a passage in the Euthyphro where Socrates say something true and important about mathematics. The passage is on a list of Platonic passages that I recently found, having written it in a notebook on May 23, 2018. The other passages are in the Republic; Continue reading

Logic of Elliptic Curves

In my 1997 doctoral dissertation, the main idea came as I was lying in bed one Sunday morning. Continue reading

On Gödel’s Incompleteness Theorem

This is an appreciation of Gödel’s Incompleteness Theorem of 1931. I am provoked by a depreciation of the theorem.

In the “Gödel for Dummies” version of the Theorem, there are mathematical sentences that are both true and unprovable. This requires two points of clarification. Continue reading

NL XLII: The First Barbarism: The Saracens

Index to this series

Executive summary: The barbarians who overran the Western Roman Empire were not barbarists in Collingwood’s technical sense. However, “in the seventh century a movement inspired by hostility towards everything Roman … and everything Christian, flared up on the south-eastern frontier of the Roman world” (42. 22). This movement was therefore barbarist. Failing to conquer Europe, either from the east at Constantinople, or from the west at Tours, the movement settled down and ceased being barbarist—by the account in Chapter XLII, “The First Barbarism: The Saracens,” and later, in Collingwood’s New Leviathan. I check this account against more recent sources; it is barbarist to think that the “movement” in question, or indeed any movement, must always be barbarist; I look at the “civilization” of the British Empire as portrayed in a story of Maugham, and I compare a character of the story to Collingwood.

Collingwood’s historical account of barbarisms is a minefield, if one wishes not to sound like a barbarist oneself. The four examples will be

  1. the Saracens,
  2. the “Albigensian Heresy” (or the Bogomils),
  3. the Turks, and
  4. the Germans.

The very formula “the X”—definite article followed by national or quasi-national adjective—this has a barbaric use in branding a people with indelible features. A retort then is “not all X,” as in “not all men.” Collingwood issues such a proviso himself:

45. 68. Please observe, Reader, that I am not talking about all Germans. I do not say that all Germans are liars. I know of some who are not; those heroes, for example, who continue in spite of everything the Nazis can do to run their secret wireless station and keep on printing Das Wahre Deutschland.

Das wahre Deutschland, from a Swiss antiquarian bookshop, Antiquariat Peter Petrej

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On Chapman’s Homer’s Iliad, Book XIV

Index to this series | Text of Chapman’s Homer’s Iliad

When Neptune was helping the Greeks stave off certain defeat, I tried to suggest that divine intervention in the course of events might be understood as human resolve to change that course. This was in Book XIII of the Iliad, where Neptune took the form of one of the Greeks—Calchas—in order to exhort the others. They would have listened to Calchas anyway; he was a prophet. Ajax Oileus said he could tell Calchas was “really” a god; we can read this to mean Calchas was inspiring. We can say this of somebody today, without meaning to suggest any supernatural influence.

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On Chapman’s Homer’s Iliad, Book XIII

Index to this series | Text of Chapman’s Homer’s Iliad

Usually when your defenses are breached, you are lost. Thus when the Ottoman Turks under Mehmet I found a way through the Theodosian Walls on Tuesday morning, May 29, 1453, the city of Constantinople was theirs.

Now the Trojans under Hector have breached the wall around the Greek ships, and—the ships are not theirs. How does Homer explain this?

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NL XXVIII: The Forms of Political Action

Index to this series

Executive summary (added September 12, 2018), To condemn political discussion is to wish for tyranny: this continues a thought from the previous chapter. The ruling class need not share their deliberations with the ruled class, but it is better if they do. As our understanding of the reason for an action evolves—at first the action is merely useful, but later it conforms to a rule—, so ruling, originally by decree, has evolved to include legislation. However, to be enforced, law does not require a formal structure; international law is an example. The best reason for an action is duty. Though the German Treitschke says our highest duty is to the state, he gives the state no duty, and so his politics are entirely utilitarian.

In Plato’s Republic, Socrates seeks understanding of the just human being through examination of the just state. In Collingwood’s New Leviathan, the order is reversed. What we first considered in somebody, we now look at in the “body politic.”

Narthex of the former Taksiyarhis Church (now museum), Ayvalık, Balıkesir, Turkey, August 30, 2018

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On Chapman’s Homer’s Iliad, Book XII

Index to this series | Text of Chapman’s Homer’s Iliad

Both first and last place may be prominent in a narrative. Occurring three-quarters of the way into Book XII of the Iliad, but presented last below, Sarpedon’s great speech on leadership ought to be known by everybody with authority and power.

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NL XXVII: Force in Politics

Index to this series

Executive summary (added September 12, 2018): When persons cannot rule themselves, they are ruled by force, as a duty, by other persons, for the benefit and pleasure of all. Force includes fraud and deceit; but their use must be limited, if those persons who are being ruled by force now will one day join the ruling class themselves. If a liberal and a conservative party take up respectively the ideals of democracy and aristocracy discussed in the last chapter, the parties must understand that each needs the other, in order to engage in the dialectic that aims for the best society. If somebody thinks the two parties waste energy, either in pretending to be in opposition to one another, or in actually being opposed, then that person is effectively wishing for tyranny.

In my last post on the New Leviathan (which was my first for this year), I said Collingwood would discuss the British parliament in Chapter XXVII. That chapter now my subject.

The ruling class must incorporate new members from time to time, whether anybody thinks about it or not (27. 75). Anybody who does think about it may take up one of two goals (27. 77).

27. 79. To hasten the percolation of liberty throughout every part of the body politic was the avowed aim of the Liberal party; to retard it was the avowed aim of the Conservative party.

27. 8. The relation between them was consciously dialectical. They were not fundamentally in disagreement. Both held it as an axiom that the process of percolation must go on. Both held that given certain circumstances, which might very well change from time to time, there was an optimum rate for it, discoverable within a reasonable margin of error by experiment.

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On Chapman’s Homer’s Iliad, Book XI

Index to this series | Text of Chapman’s Homer’s Iliad

After the active night of Book X comes the dawn of a thrilling day.

  1. AVrora, out of restfull bed, did from bright Tython rise,
  2. To bring each deathlesse essence light, and vse, to mortall eyes.

The deathless essence called Jove sends Discord to the Greeks. She lights on the ship of Ulysses, in the middle of the fleet, so all can hear as she belts out her “Orthian song.”

  1. And presently was bitter warre, more sweet a thousand times
  2. Then any choice in hollow keeles, to greet their natiu climes.

The sweetness of war is the theme of the book. Continue reading