Category Archives: Education

Mathematics and Logic

I continue with the mathematics posts, taking up, as I did in the last, material originally drafted for the first.

Designated for its own post, material can grow, as has the material of this post in the drafting. Large parts of it are taken up with

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

  2. Gödel’s logical theorems of completeness and incompleteness.

I have 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. This makes logic a criteriological science, since it seeks, examines, clarifies and limits the criteria whereby we can make deductions. As examples of this activity, Gödel’s theorems are 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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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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On Being Given to Know

  1. What if we could upload books to our brains?
  2. What if a machine could tell us what was true?

We may speculate, and it is interesting that we do speculate, because I think the questions do not ultimately make sense—not the sense that seems to be intended anyway, whereby something can be got for nothing.

View from Şavşat

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NL XLV: The Germans

Index to this series

At the end of Collingwood’s New Leviathan (1942), we reach a chapter whose theme is that of my more recent articles on grammar.

By August Macke – The Yorck Project (2002) 10.000 Meisterwerke der Malerei (DVD-ROM), distributed by DIRECTMEDIA Publishing GmbH. ISBN: 3936122202., Public Domain, Link

As history, Collingwood’s last chapter is difficult, for the reasons that trouble Herbert Read at the beginning of his Concise History of Modern Painting (revised 1968, augmented 1974). Read opens his first chapter with a passage from Collingwood’s Speculum Mentis (1924):

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NL XXXVII: Civilization As Education

Index to this series

Knowing, from the previous chapter, what civilization is, we ask: How do we bring it about? Collingwood’s answer is to homeschool our children.

This is the New Leviathan’s first detailed piece of positive advice, and it may sound crazy. Rather than list reasons why, I want to see what sense can be made of the ideas.

Civility is respect (37. 15). To respect another person is to recognize their freedom (37. 14). To do this, one needs self-respect (37. 13).

Instead of respect, we may approach another person with servility, namely “the demeanour of a man lacking self-respect towards one whom he fears” (37. 17). The will to barbarism is just the will to servility (37. 19).


Mary Cassatt (American, 1844-1926)
The Boating Party, 1893/1894, oil on canvas
Chester Dale Collection, National Gallery of Art Continue reading

Writing Rules

Executive summary (added July 16, 2018): I have had enough of misrepresentation by experts of what other experts have to say about grammar.


An ongoing concern of this blog is the subject taught in school called grammar. See for example

  • the previous post, “Writing and Inversion,” about how a supposed rule against the passive voice might be better understood as a rule to avoid certain inversions of order (namely those inversions that add words and torpor);

  • the post before that, “A New Kind of Science,” presenting a theory that grammar is properly neither prescriptive nor descriptive, but “criteriological,” because it examines the criteria that we apply to our own speaking and writing;

  • an early expression (from six years ago) of some of those ideas: “Strunk and White.”

Grammar causes anxiety. Every aspect of school would seem to cause anxiety in somebody. Decades after they have left school, how many persons have nightmares of missing an examination? My mother and her brother were such a persons, as I learned when growing up. I seem not to be such a person, though I once dreamt of missing a plane.

How much support of current US President Donald Trump is due to memories of belittlement by teachers at school? Similar questions may be raised about

  • UK government minister Michael Gove’s saying, “people in this country have had enough of experts …”;

  • the rise in Turkey of Recep Tayyip Erdoğan, who has perceived a special threat from the Peace Academics.

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Writing and Inversion

Executive summary: The “voice” of a transitive verb may be active or passive. A piece of writing may be vigorous or torpid. There is not an exact correspondence between passive verbs and torpid writing. However, a passive verb is used to effect inversion of subject and object. One may also invert subject and auxiliary verb, subject and predicate, or two clauses, always adding new words. Each inversion may lead to torpid writing. This is what Strunk warned about in The Elements of Style, by issuing the command, “Use the active voice.” The command must be followed with discretion. Williams makes the same case, more elaborately, in Style: Towards Clarity and Grace. There is no foolproof executive summary of how to write well.


When E. B. White revised William Strunk’s original Elements of Style, he did not retain Strunk’s “Introductory,” whose first paragraph said of the book,

The experience of its writer has been that once past the essentials, students profit most by individual instruction based on the problems of their own work, and that each instructor has his own body of theory, which he may prefer to that offered by any textbook.

Perhaps many students today cannot receive individual instruction. They are just given textbooks that try to spell out everything. I have sensed this in mathematics, where new calculus books seem a lot bigger than those of 1950 and earlier. Continue reading

A New Kind of Science

Executive summary. Some sciences are called descriptive, empirical, or natural; others, prescriptive or normative. We should recognize a third kind of science, which studies the criteria as such that a thinking being imposes on itself as it tries to achieve success. I propose linguistics as an example. Collingwood introduced the term criteriological for the third kind of science. This was in The Principles of Art (1938), though I find the germ of the concept in earlier work, even in Collingwood’s first book, Religion and Philosophy (1916), in the passage on psychology that the author would recall in An Autobiography (1939).

Collingwood’s examples of criteriological sciences are logic, ethics, aesthetics, and economics. Pirsig effectively (and independently) works out rhetoric as an example in Zen and the Art of Motorcycle Maintenance (1974). We may benefit from clarity here, given how people can have a strong reaction to being lectured by experts. For Collingwood, such a reaction is found in Nazi Germany; see the last chapter of The New Leviathan (1942). Reactions to grammar are the subject of my own two ensuing articles, “Writing and Inversion” and “Writing Rules.”


Some sciences are not recognized for what they are. The sciences themselves are not new, but a proper understanding of them may be new to some of us, including myself.

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What Philosophy Is

With my presumptuous title, I imitate Arthur Danto’s What Art Is (2013), mentioned in my last post, “Some Say Poetry.” The book is fine, and I have learned from it; but Danto could have learned from Collingwood’s Principles of Art.

Picasso, The Tragedy (1903), National Gallery of Art, Washington Continue reading