Tag Archives: 2020

Law and History

I learned about Peter Turchin recently through his profile in the Atlantic by Graeme Wood. I had learned about the Atlantic article from historians on Twitter such as James Ryan, who does “Turkish history and other stuff,” according to his own Twitter profile, and who tweeted in response to Wood’s article,

This is really interesting research, but, uh, it is only history in the way that a particle physicist does history.

In response to that, a thread began:

Needless to say, no historian would find this “approach” acceptable. There’s a reason we spend so much time on historiography when new historians are trained; we have complex, rich debates that have continued for longer than any field except philosophy on how to approach history.

That was by Axel Çorlu, living in the US, but “Born in Izmir, Turkey, to a Levantine (Italian/Greek/French/Armenian) family” according to his Academia page.

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The Asıl of the Iliad

Etymologically speaking, the asıl of a thing is its root. The Arabic root of the Turkish word means bitki kökü, “vegetable root,” according to Sevan Nişanyan’s Turkish etymological dictionary.

In the Iliad, why is Achilles so affronted by Agamemnon as to refuse to help the Greeks, even as their attack on Troy is becoming a defensive war, at the wall that they have erected about their own ships? If the answer is to be found through study, then Book IX of the Iliad is what to study.

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One day during the Trojan War, Apollo and Athena decide to give the combatants a break. The general conflict is to be replaced with a one-on-one. The Olympians induce Helenus to tell his brother Hector to take on whichever of the Greeks is up for it.

Only Menelaus will accept the challenge at first. His brother Agamemnon makes him withdraw. When none of the other Greeks comes forward, Nestor chides them. After a story of his former prowess, he utters the words that Chapman renders as two couplets:

O that my youth were now as fresh, and all my powers as sound;
Soone should bold Hector be impugn’d: yet you that most are crownd
With fortitude, of all our hoast; euen you, me thinkes are slow,
Not free, and set on fire with lust, t’encounter such a foe.

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Pacifism is properly pacificism, the making of peace: not a belief or an attitude, but a practice. Mathematics then is pacifist, because learning it means learning that you cannot fight your way to the truth. Might does not make right. If others are going to agree with you, they will have to do it freely. Moreover, you cannot rest until they do agree with you, if you’ve got a piece of mathematics that you think is right; for you could be wrong, if others don’t agree.

The book *Dorothy Healey Remembers,* with photo of subject

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This directory has the following sections.

The links in the sections are sometimes to posts of this blog, but mostly to pages and media (especially pdf files).

Posts, pages, and media are what this blog has, because it uses the WordPress.org content management system, by virtue of being hosted by WordPress.com. You are now reading a post, as you can tell from its having an initial publication date both in its address and somewhere at the end of its body. The contents of any post can always be edited. I shall be editing this post in order to make available, or highlight, pages, verbal media (namely pdf files) and some series and categories of posts.

In principle, all of my posts can be seen at polytropy.com, in reverse chronological order. I have them listed in forward order, by year, on my About page. As I explain there, I try to keep track of posts with tags and categories. Moreover, if one post revisits a theme of another post, I try to link to that post. Every post shows (at the bottom) which other posts are linked to it.

A dream, never to be realized, would be to have all of my ideas as well-organized as in Wittgenstein’s Tractatus Logico-Philosophicus.

I don’t know how the random visitor can find my pages, although search engines find at least some of them. Much less do I know how one would find my media, although the media allowed by WordPress now include pdf files. I have uploaded a number of these, and created a number of pages.

It would be possible to have all top-level pages included automatically in the menu which now forms a horizontal list at the top of each post and page.

The directory proper now begins.



Poetry, in the broad etymological sense of something made; call it conceptual art, or whatever you like, but it’s all referred to in the post “Discrete Logarithms”:



Pages describing (as well as listing) categories of posts


There are series of posts about particular works of prose and poetry. I could make each one into a category, but so far I have not. One post in the series may serve as an index, or else a page may serve.


For my courses I normally prepare pages on my department’s server; but since I cannot access this from home, I may also use the blog.

  • Kümeler Kuramı (“Set theory,” including my attempts to summarize in html the contents of a course in axiomatic set theory for which I have also written a full text in LaTeX. For the summary of fall 2019, I had not yet discovered the usefulness of pandoc as described in “LaTeX to HTML”)

  • Analitik Geometri Özeti (“summary of analytic geometry,” for a course in spring 2020; as the Covid-19 lockdown took hold, the page just became the course page)

  • Ordinal Analiz (“ordinal analysis,” that is, set theory with emphasis on the ordinals as a structure analogous to the linearly ordered set of real numbers studied in so-called real analysis; the post “Ordinals” also takes up the analogy; I made the page for a course in Şirince, in case I wanted to change the page while I was there, though in the event I didn’t; notes from the second week, in English, are on a departmental page, along with the syllabus for a summer course in 2020 that was cancelled)

  • Öklid (Resources for the course Öklid geometrisine giriş, “introduction to Euclidean geometry,” fall 2020)

  • Topoloji (homepage for a graduate topology course, fall 2020, with links to weekly notes in English. The notes are mostly in pdf format, for A6 paper in landscape orientation; but one week I used html)

  • Ayşe Berkman’ın yedek sayfasıdır (for her spring 2020 course during the lockdown)

Writing of others

Sometimes annotated by me:

Updated January 3, 2023

Articles on Collingwood

This article gathers, and in some cases quotes and examines, popular articles about R. G. Collingwood (1889–1943).

  • By articles, I mean not blog posts like mine and others’, but essays by professionals in publications that have editors.

  • By popular, I mean written not for other professionals, but for the laity.

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Mathematics and Logic

Large parts of this post are taken up with two subjects:

  1. The notion (due to Collingwood) of criteriological sciences, logic being one of them.

  2. Gödel’s theorems of completeness and incompleteness, as examples of results in the science of logic.

Like the most recent in the current spate of mathematics posts, the present one has arisen from material originally drafted for the first post in this series.

In that post, I defined mathematics as the science whose findings are proved by deduction. This definition does not say what mathematics is about. We can say however what logic is about: it is about mathematics quâ deduction, and more generally about reasoning as such. This makes logic a criteriological science, because logic seeks, examines, clarifies and limits the criteria whereby we can make deductions. As examples of this activity, Gödel’s theorems are, in a crude sense to be refined below, that

  • everything true in all possible mathematical worlds can be deduced;

  • some things true in the world of numbers can never be deduced;

  • the latter theorem is one of those things.

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Multiplicity of Mathematics

I continue with the recent posts about mathematics, which so far have been as follows.

  1. What Mathematics Is”: As distinct from the natural sciences, mathematics is the science whose findings are proved by deduction. I say this myself, and I find it at least implicit in an address by Euphemia Lofton Haynes.

  2. More of What It Is”: Some mathematicians do not distinguish mathematics from physics.

  3. Knottedness”: Topologically speaking, there is a sphere whose outside is not that of a sphere. The example is Alexander’s Horned Sphere, but it cannot actually be physically constructed.

  4. Why It Works”: Why there can be such a thing as the horned sphere.

When I first drafted the first post above, I said a lot more than I eventually posted. I saved it for later, and later is starting to come now.

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Why It Works

The last post, “Knottedness,” constructed Alexander’s Horned Sphere and proved, or sketched the proof, that

  • the horned sphere itself is topologically a sphere, and in particular is simply connected, meaning

    • it’s path-connected: there’s a path from every point to every other point;

    • loops contract to points—are null-homotopic;

  • the space outside of the horned sphere is not simply connected.

This is paradoxical. You would think that if any loop sitting on the horned sphere can be drawn to a point, and any loop outside the horned sphere can be made to sit on the sphere and then drawn to a point, then we ought to be able to get the loop really close to the horned sphere, and let it contract it to a point, just the way it could, if it were actually on the horned sphere.

You would think that, but you would be wrong. Continue reading


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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