Category Archives: Prose

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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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 is 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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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

Effectiveness

Preface

First published May 17, 2018, this essay concerns Eugene Wigner’s 1960 article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” I wrote a lot, which I now propose to summarize by section. (The meditations also continue in the next article.)

  • Some things are miraculous. Among Wigner’s examples are
    • that mathematics is possible at all, and
    • that “regularities” in the physical world can be discovered, as by Galileo and Newton.

    For Wigner, we should be grateful for the undeserved gift of a mathematial formulation of the laws of physics. This makes no sense theologically—and here I agree with the character Larry Darrell in Somerset Maugham’s novel The Razor’s Edge. Wigner’s idea that our mathematical reasoning power has been brought to perfection makes no sense to me either.

  • Everything is miraculous. Here I agree with Collingwood in Religion and Philosophy. A miracle cannot be the breaking of a natural law, since such a thing cannot be broken. A great artist like Beethoven follows no rules in the first place, or makes them up as he goes along; and he is like God in this way.
  • Natural law. That it cannot be broken is part of the very concept of natural law. Quantum phenomena and the theory of relativity have not in fact been brought under a single law; for Wigner, it may not be possible.
  • Mystery. Not only can we not define miracles, but (as we should have observed in the first place) we cannot even say when they happen. If like Wigner we call something miraculous, this means it cleanses our own doors of perception, in the sense of William Blake.
  • Definitions. In his treatment of miracle in Religion and Philosophy, Collingwood shows the futility of trying to define a term when you are not sure how to use it. He makes this futility explicit in The Principles of Art. If we are going to think about the use of mathematics in natural science, this means we ought to be mathematician, natural scientist, and philosopher; and not just “natural scientist,” but physicist and biologist, since if mathematics is effective in physics, it would seem to be ineffective in biology.
  • Being a philosopher. We are all philosophers, in the sense that Maugham describes in the story “Appearance and Reality,” if only we think. All thought is for the sake of action. This does not mean that thought occurs separately from an action and is to be judged by the action. We may value “pure” thought, such as doing mathematics or making music or living the contemplative life of a monk. This however moves me to a give a thought to the disaster of contemporary politics.
  • Philosophizing about science. For present purposes, compart­ment­al­ization of knowledge is a problem. So is the dominance of analytic philosophy, for suggesting (as one cited person seems to think) that big problems can be broken into little ones and solved independently. In mathematics, students should learn their right to question somebody else’s solutions to problems. In philosophy, the problems themselves will be our own. Philosophy as such cannot decide what the problems of physics or biology are, though it may help to understand the “absolute presuppositions” that underlie the problems. Philosophers quâ metaphysicians cannot determine once for all what the general structure of the universe is. This does not mean they should do “experimental philosophy,” taking opinion polls about supposedly philosophical questions. What matters is not what people say, but what they mean and are trying to mean. As Collingwood observes, metaphysics is an historical science.

For more on the last points, see a more recent article, “Re-enactment.” (This Preface added June 3, 2018.)


I am writing from the Math Village, and here I happen to have read that Abraham Lincoln kept no known diary as such, but noted his thoughts on loose slips of paper. Admired because he “could simply sit down and write another of his eloquent public letters,”

Lincoln demurred. “I had it nearly all in there,” he said, pointing to an open desk drawer. “It was in disconnected thoughts, which I had jotted down from time to time on separate scraps of paper.” This was how he worked, the president explained. It was on such scraps of paper, accumulating over the years into a diaristic density, that Lincoln saved and assembled what he described to the visitor as his “best thoughts on the subject.”

Thus Ronald C. White, “Notes to Self,” Harper’s, February 2018. My own notes to self are normally in bound notebooks, and perhaps later in blog articles such as the present one, which is inspired by the 1960 article called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” by Eugene Wigner.
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Some Say Poetry

Potted palms with plaster farm animals on hillside behind

Kuzguncuk, 2017.11.05

I originally set out to preserve here, for future reference, a poetry review that I liked. A remark on being a student had drawn my attention:

In My Poets, a work of autobiographical criticism with occasional ventriloquial interludes, McLane recalls two “early impasses in reading,” freshman-year encounters with Charles Olson and Frank O’Hara. She writes about not “getting it” but wanting to get it, about a desire to get it that was left wanting by code-breaking and analysis and satisfied by hearing and feeling.

I shall try to say more about learning and creating, in poetry and also in mathematics, after quoting the review in its entirety. It constitutes the second half of a “New Books” column by Christine Smallwood, in the Reviews section of Harper’s, July 2017.

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Romance

At the end of Shakespeare’s romance called The Tempest, Prospero plans to retire to Milan, where “Every third thought shall be my grave.” I remember these words from reading the play in school and college. I also have thoughts of my grave, and their frequency may increase as the years pass. However, for each of those thoughts, I seem to have more thoughts that are based on memories of youth and childhood.


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NL XXV: The Three Laws of Politics

Index to this series

Executive summary (added September 11, 2018): The the three laws of politics are that (1) within the body politic, there is a ruling class, which is a society proper; (2) the ruling class can take in members from the ruled class; (3) the ruled and ruling classes will resemble one another, so that e.g. rulers of slaves will become slavish themselves. I compare such laws with physical laws, as discussed by Einstein; but on this subject, a look ahead to Chapter XXXI, “Classical Physics and Classical Politics,” would be in order. Meanwhile, by the Second Law, the body politic, or its ruling class, can be a permanent society; Nazi claims about the youth or senility of different states are bogus. There are further gradations within the ruling and ruled classes, according to strength of will; a weak will can be strengthened by another person’s stronger will through induction.


A pervading theme of the New Leviathan is freedom of will. Whether we actually have it is only a pseudo-problem (13. 17). Some persons have been fooled into thinking it a problem, perhaps by the misleading myth that free will is a divine gift, like life itself, breathed into our nostrils when, in Genesis 2:7, God forms us of the dust of the ground. As Collingwood observes at the end of Chapter XXIII, “The Family As a Society,” we are born neither free nor in chains. We have to grow up. Growing up is becoming free.

The Book of Genesis Illustrated by Robert Crumb (New York: Norton, 2009)

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NL XXI: Society as Joint Will

Index to this series

Executive summary (added September 10, 2018):

1.
I cannot say “I will” without recognizing the possibility of joining with others to say “we will.”
2.
A social consciousness consists of (1) a precise idea of one’s place in a society and (2) a vague sense of the society as a whole. The latter sense may be incorrect, having been foolishly accepted on the word of an authority.
3.
Properly understood, ruling, of itself and perhaps of a non-social community, may be all that a society does. It is the responsibility of the members alone.
4.
To form a society means (1) to form social relations and (2) to do this for some purpose. To focus on (1) yields the idea of a universal society—which cannot actually exist, despite foolish hopes for the League of Nations.
5.
The universal society cannot exist, because we produce a society by transforming an earlier community, and some trace of this must remain.
6.
Members of a society are equal, (1) in having the freedom to join and (2) in just being members. A society may create an inequality, as by delegating authority. There may be natural inequalities, not produced by the society itself; society may compensate for them, turn them into assets, or even depend on them, as in the case of initiative.
7.
Rule by force (in a non-social community) may be by rewards and punishments, that is, objects of desire and fear. These may be promised or threatened fraudulently. One who grows too accustomed to exerting force may lose freedom of will.
8.
Societies institute criminal law to mitigate members’ losses of freedom of will.
9.
Societies can be temporary or permanent.

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