Category Archives: Education

Creativity

In the Platonic dialogues, Socrates frequently mentions τέχνη (technê), which is art in the archaic sense: skill or craft. The concern of this post is how one develops a skill, and what it means to have one in the first place.

Books quoted or mentioned in the text, by Midgley, Simone Weil, Thoreau, Amy Mandelker (on Tolstoy), Oliver Byrne (on Euclid), Wittgenstein, Arendt, and Caroline Alexander (on Homer)

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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 be constructed physically.
  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.

Octahedron with edges divided in the Golden Ratio by the vertices of an icosahedron

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An Exercise in Analytic Geometry

This past spring (of 2020), 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.

Diagram from textbook page shows, centered at the origin of coordinates, a circle and an ellipse whose four points of intersection are traversed by two lines in red through the origin
From Weeks & Adkins, Second Course in Algebra, p. 395

Note added, April 17, 2023: An account of the mathematics involved in the exercise would ultimately be published as: Pierce, D. (2021). “Conics in Place.” Annales Universitatis Paedagogicae Cracoviensis | Studia Ad Didacticam Mathematicae Pertinentia, 13, 127–150.

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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.

Trees, clearings, and houses in the foreground; mountains receding into the background; clouds above
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? Quite a few, it would seem; see the evidence appended to this post. My mother and her brother were such 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.

On that last matter, see my blog essay of March, 2016, “Academic Freedom.”

In the blog generally, I may criticize some of my fellow academics; but I criticize them for their own criticism of fellow academics and thinkers. Thus in the article “Strunk and White” listed above, I say Geoffrey Pullum was stupid to decry, in 2009, the “50 Years of Stupid Grammar Advice” offered by Strunk and White’s Elements of Style.

Still I have respected Pullum’s recommendation of Style: Toward Clarity and Grace, by Joseph M. Williams. I bought Williams’s book, and in this post I focus on some of his advice.

I thought Williams’s book might be more “democratic” than Strunk and White’s, in the sense of being aimed at a broader audience. That broader audience might include students whose parents didn’t go to college or grow up speaking English.

Now I have doubts that Williams has such an audience in mind. In his final chapter, called “Usage,” Williams writes (on page 176) of

Three Kinds of Rules

1. Some rules characterize the basic structure of English … No native speaker of English has to think about these rules at all.

2. Some rules distinguish standard from nonstandard speech … The only writers and speakers who worry about these rules are those upwardly mobile types who are striving to join the educated class of writers and speakers …

3. Finally, some grammarians try to impose on those who already write educated standard English particular items of usage that they think those educated writers should observe – don’t split infinitives; use that, not which for restrictive clauses …

This may not be much evidence to go on; but judging from the style of #1, I’m not sure Williams has considered the possibility of having readers who are not native speakers of English. In #2, by referring pejoratively to “those upwardly mobile types,” Williams seems to think they are not readers either.

Appearances are corroborated on the next page, after Williams describes again his three kinds of rules:

1. Some rules account for the fundamental structure of English …

2. Some rules distinguish the dialects of the educated and the uneducated …

3. And some rules belong to that category of rules observed by some well-educated people, and ignored by others equally well-educated …

Ordinarily, the first set of rules concerns us not at all. And if you are interested in this book, you probably aren’t much concerned with the second set either. It is the third set of rules that concern – sometimes obsess – already competent but not entirely secure writers. They are the rules of usage out of which the Pop Grammarians have created their cottage industry.

In faithfully transcribing Williams’s words about rules of usage, I have noticed that they violate a certain rule: “Enclose parenthetic expressions between commas.” Indeed, in the first block quotation above, giving the first list of “Three Kinds of Rules,” look again at the last clause (itself a rule):

use that, not which for restrictive clauses.

Here the phrase “not which” is parenthetic, but is not printed that way. Since a comma precedes it, a comma ought also to follow it, at least if one agrees with the rule that I stated, “Enclose parenthetic expressions between commas.”

That rule happens to be Rule 3 of Strunk’s original eight “Elementary Rules of Usage.” It is still Rule 3 in the version of The Elements of Style edited by E. B. White, although some of the other rules have been changed.

I find the same rule also as part of Rule 12d in the Harbrace College Handbook (8th edition, 1977), used in the ninth-grade English class at my private, college-preparatory school for boys in Washington. According to the Handbook:

Commas set off nonrestrictive clauses and phrases and other parenthetical and miscellaneous elements, such as transitional expressions, items in dates, words used in direct address, and so on. Restrictive clauses and phrases are not set off by commas.

Surely I was taught this rule in earlier years too. The rule seems unobjectionable and even natural to me now, and I do not recall any difficulty with it.

Yellow cover of Harbrace College Handbook 8

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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. 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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