Category Archives: Categorical Thinking

On Gödel’s Incompleteness Theorem

December 15, 2018 – 8:43 am

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.

Next to a slice from a tree trunk, a worn copy of the book Frege and Gödel: Two Fundamental Texts in Mathematical Logic, edited by Jean van Heijenoort

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By David Pierce | Also posted in Archimedes, Descartes, Euclid, Exposition, Hitchens, Mathematics, Philosophy, Philosophy of Mathematics, Science | Tagged , , , , , , , , , , , , , , , , , , , , , | Comments (16)

A New Kind of Science

July 10, 2018 – 5:51 pm

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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By David Pierce | Also posted in “ceases to be a mind”, Collingwood, Criteriological Science, Education, Grammar, Ontological Proof, Pirsig, Principles of Art, Science, Strunk and White | Tagged , , , , , , , , , , , , | Comments (27)

The Private, Unskilled One

May 19, 2017 – 5:55 am

I went into Istanbul’s Pandora Bookshop a month ago, looking for an English translation of War and Peace, since the Garnett translation I had read at college was falling apart. I was told the Oxford World’s Classics edition (with the Maude translation) was coming the next week, and it did come.

Elif Batuman, The Idiot, in Nesin Matematik Köyü, Kayser Dağı Mevkii, Şirince, Selçuk, İzmir, Turkey, 2017.05.18

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By David Pierce | Also posted in Education, Harper’s, Hitchens, Maugham, Nesin Mathematics Village, Thoreau | Tagged , , , , , , , , | Comments (5)

Facts (NL IX, ‘Retrospect,’ first 6 paragraphs)

August 1, 2014 – 9:43 am

Index to this series

A certain person says,

I am not better than you or more virtuous than you. If you see me on the right path, help me. If you see me on the wrong path, advise me and halt me. And obey me as far as I obey God.

How should one hear these words: as an eminently reasonable expression of benevolent humility such as any of us might honorably make? Well, no matter how qualified, the command obey me might be a warning sign. The words are in fact from a recently published video, as quoted in the Guardian Weekly (Vol 190 No 5, 11–17 July 2014, p. 4). The speaker is the man whose nom de guerre is Abu Bakr al-Baghdadi, on whose head the Department of State of the United States of America placed a ten-million-dollar bounty in 2011. He now styles himself Ibrahim, Caliph of the Islamic State, a new entity that is supposed to restore the lost Muslim glory of past centuries. This restoration is to be achieved through war. War requires military discipline, with punishments meted out for infractions like insubordination, not to mention the slaughter of those perceived as enemies. So al-Baghdadi’s request to be advised and halted if seen to be in the wrong must be interpreted rather carefully.

It is difficult to know how to interpret somebody’s words. With that I pass to the transitional chapter in the first part, “Man,” of Collingwood’s New Leviathan.

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By David Pierce | Also posted in Aristotle, Collingwood, Grammar, Mathematics, New Leviathan, Philosophy, Plato, Turkish, Visual Art | Tagged , , , , , | Comments (4)

NL II: “The Relation Between Body and Mind”

January 20, 2014 – 11:39 am

Index to this series

I continue making notes on The New Leviathan of R. G. Collingwood (1889–1943). Now my main concern is with the second chapter, “The Relation Between Body and Mind”; but I shall range widely, as I did for the first chapter.

Preliminaries

Some writers begin with an outline, which they proceed to fill out with words. At least, they do this if they do what they are taught in school, according to Robert Pirsig:

He showed how the aspect of Quality called unity, the hanging-togetherness of a story, could be improved with a technique called an outline. The authority of an argument could be jacked up with a technique called footnotes, which gives authoritative reference. Outlines and footnotes are standard things taught in all freshman composition classes, but now as devices for improving Quality they had a purpose.

That is from Zen and the Art of Motorcycle Maintenance, chapter 17.

Does anybody strictly follow the textbook method of writing? Continue reading →

By David Pierce | Also posted in absolute presuppositions, Collingwood, Descartes, Education, History, Logic, Mathematics, New Leviathan, Ontological Proof, overlap of classes, Philosophy, Philosophy of History, Pirsig, Plato, Science | Tagged , , , , , , , , , | Comments (13)

Psychology

August 28, 2013 – 9:40 am

Preface (January 17–18, 2019). This essay is built around two extended quotations from Collingwood:

  1. From the posthumous Idea of History (1946) with the core idea, “people do not know what they are doing until they have done it.”
  2. From An Essay on Philosophical Method (1933), about how logic is neither a purely descriptive nor a purely normative science.

The quotations pertain to the title subject of psychology for the following reasons.

  1. Psychological experiments show that we may not know what we are doing until we have done it.
  2. Psychology is a descriptive science.

Psychological experiments can tell us about what we do, only when we presuppose the general applicability of their findings. This is true for any descriptive science. Philosophy demands more. A philosophical science like logic is categorical, in the sense of the second listed quotation, because it is what Collingwood will later call criteriological. I go on to discuss criteriological sciences as such in “A New Kind of Science,” but not here.

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By David Pierce | Also posted in Collingwood, Criteriological Science, Euclid, Galileo, Logic, Ontological Proof, Philosophy, Principles of Art, Psychology, Science | Tagged , , , , , , | Comments (11)