Tag Archives: Bertrand Russell

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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Because Herman Wouk was going to put physicists in a novel, Richard Feynman advised him to learn calculus: “It’s the language God talks.” I think I know what Feynman meant. Calculus is the means by which we express the laws of the physical universe. This is the universe that, according to the mythology, God brought into existence with such commands as, “Let there be light.” Calculus has allowed us to refine those words of creation from the Biblical account. Credited as a discover of calculus, as well as of physical laws, Isaac Newton was given an epitaph (ultimately not used) by Alexander Pope:

Nature and Nature’s laws lay hid in night:
God said, Let Newton be! and all was light.

I don’t know, but maybe Steven Strogatz quotes Pope’s words in his 2019 book, Infinite Powers: How Calculus Reveals the Secrets of the Universe. This is where I found out about Wouk’s visit with Feynman. I saw the book recently (Saturday, February 22, 2020) in Pandora Kitabevi here in Istanbul. I looked in the book for a certain topic that was of interest to me, but did not find it; then I found a serious misunderstanding.

book cover: Steven Strogatz, Infinite Powers Continue reading

NL I: “Body and Mind” Again

Index to this series

“We are beginning an inquiry into civilization,” writes Collingwood, “and the revolt against it which is the most conspicuous thing going on at the present time.” The time is the early 1940s.

Human tourists photographing sculptured supine blue ape with chrome testicles outside the Intercontinental Hotel, Prague Continue reading

A Defense

Here is the defense (savunma) of Ayşe Berkman before the 36th Heavy Penalty Court (Ağır Ceza Mahkemesi) of Istanbul, January 10, 2019, against the charge of making propaganda for a terrorist organization (terör örgütü propagandası yapmak).

Crowd of mostly smiling people outside a courtroom

The crowd from the courtroom when the session was over.
From a tweet of the Peace Academics

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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.

I shall review the mathematics of the theorem, first in outline, later in more detail. The mathematics is difficult. I have trouble reproducing it at will and even just confirming what I have already written about it below (for I am adding these words a year after the original publication of this essay).

The difficulty of Gödel’s mathematics is part of the point of this essay. A person who thinks Gödel’s Theorem is unsurprising is probably a person who does not understand it.

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.

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NL XXX: War As the Breakdown of Policy

Index to this series

Humans have not always made war (30. 1); why do we make it now? War is said to be a continuation of policy (30. 14); but as Collingwood cleverly points out (30. 15), the saying due to Clausewitz (30. 69) is ambiguous: a continuation could be an extension or a breakdown (30. 16–17).

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NL XXIX: External Politics

Index to this series

Executive summary (added September 13, 2018, edited July 19, 2019): Dealing with other bodies politic is the third stage of political life, after societies’

  • formation (as if by marriage),

  • dominion over non-social communities (as if by having children).

Having been used in the first two stages, dialectic can be used in the third. Being the eristic of external politics, war has no psychological cause. War is a state of mind, which does not think non-agreement can become agreement. Pacifism has this state of mind.

External politics are international relations. These represent the third of the “stages” in political life (29. 1), which we enumerate:

  1. The joining of wills into a society, which rules itself (29. 11).

  2. Such a society’s ruling over a non-social community in a body politic (29. 12).

  3. Dealing with other bodies politic (29. 13).

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Fascism As Abetted by Realism

Fascism is class warfare waged on behalf of the capitalists. This should be realized by anybody who is attracted to accidental features of Fascism such as nationalism, racism, or militarism. The Fascists are not on your side, even if they share your nationality or “race” or fascination with weaponry and military discipline.

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NL XIV: “Reason”

Index to this series

Summary added January 29, 2019, revised May 8, 2019. Practical reason is the support of one intention by another; theoretical, one proposition by another. Reasoning is thus always “motivated reasoning”: we engage in it to relieve the distress of uncertainty. Reason is primarily practical, only secondarily theoretical; and the reason for saying this is the persistence of anthropomorphism in theoretical reasoning: by the Law of Primitive Survivals in Chapter IX, we tend to think even of inanimate objects as forming intentions the way we do.

Reasons for adding this summary of Chapter XIV of Collingwood’s New Leviathan include

  • the tortuousness of the following post on the chapter,
  • the provocation of a Guardian column by Oliver Burkeman on motivated reasoning.

Says Burkeman, whose “problem” is apparently motivated reasoning itself,

One of the sneakier forms of the problem, highlighted in a recent essay by the American ethicist Jennifer Zamzow, is “solution aversion”: people judge the seriousness of a social problem, it’s been found, partly based on how appetising or displeasing they find the proposed solution. Obviously, that’s illogical …

On the contrary, how we reason cannot be “illogical,” any more than how we speak can be “ungrammatical.” Logic is an account, or an analysis, of how we do actually reason; grammar, of how we speak. Of course we may make errors, by our own standards.

Rogier van der Weyden (Netherlandish, 1399/1400-1464), Portrait of a Lady, c. 1460, oil on panel, Andrew W. Mellon Collection
Rogier van der Weyden (Netherlandish, 1399/1400–1464),
Portrait of a Lady, c. 1460, oil on panel
National Gallery of Art, Washington; Andrew W. Mellon Collection


There was a rumor that Collingwood had become a communist. According to David Boucher, editor of the revised (1992) edition of The New Leviathan, the rumor was one of the “many reasons why [that book] failed to attract the acclaim which had been afforded Collingwood’s other major works.” Continue reading

Thinking & Feeling

This essay is written as a distraction from current events, though I make some reference to them. I am prompted by questions of analogy provoked by

  1. the similes of Homer, and

  2. a recent theater review in Harper’s that mentions the parables of Jesus.

DSC06654 (2)
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