December 3, 2002

  • From the Erlangen Program
    to Category Theory

    See the following, apparently all by Jean-Pierre Marquis, Département de Philosophie, Université de Montréal:

    See also the following by Marquis:

  • Symmetry, Invariance, and Objectivity


    The book Invariances: The Structure of the Objective World, by Harvard philosopher Robert Nozick, was reviewed in the New York Review of Books issue dated June 27, 2002.


    On page 76 of this book, published by Harvard University Press in 2001, Nozick writes:



    "An objective fact is invariant under various transformations. It is this invariance that constitutes something as an objective truth...."


    Compare this with Hermann Weyl's definition in his classic Symmetry (Princeton University Press, 1952, page 132):



    "Objectivity means invariance with respect to the group of automorphisms."


    It has finally been pointed out in the Review, by a professor at Göttingen, that Nozick's book should have included Weyl's definition.


    I pointed this out on June 10, 2002.


    For a survey of material on this topic, see this Google search on "nozick invariances weyl" (without the quotes).


    Nozick's omitting Weyl's definition amounts to blatant plagiarism of an idea.


    Of course, including Weyl's definition would have required Nozick to discuss seriously the concept of groups of automorphisms. Such a discussion would not have been compatible with the current level of philosophical discussion at Harvard, which apparently seldom rises above the level of cocktail-party chatter.


    A similarly low level of discourse is found in the essay "Geometrical Creatures," by Jim Holt, also in the issue of the New York Review of Books dated December 19, 2002. Holt at least writes well, and includes (if only in parentheses) a remark that is highly relevant to the Nozick-vs.-Weyl discussion of invariance elsewhere in the Review:



    "All the geometries ever imagined turn out to be variations on a single theme: how certain properties of a space remain unchanged when its points get rearranged."  (p. 69)


    This is perhaps suitable for intelligent but ignorant adolescents; even they, however, should be given some historical background. Holt is talking here about the Erlangen program of Felix Christian Klein, and should say so. For a more sophisticated and nuanced discussion, see this web page on Klein's Erlangen Program, apparently by Jean-Pierre Marquis, Département de Philosophie, Université de Montréal. For more by Marquis, see my later entry for today, "From the Erlangen Program to Category Theory."

December 2, 2002

December 1, 2002

  • Milestones in Catholic History


    From Dr. Mac's Cultural Calendar:



    • On this day in 1929, Bingo was invented
        by Edwin S. Lowe.
    The wording of this masterpiece of ecclesiastical history, apparently written by a Protestant (though not very Protestant), leaves something to be desired. See Bingo History for more details.

    Shamrock Bingo Angel

    "It never hurts to have an Irish angel on your team! This adorable red-headed fabric cherub, complete with sparkling golden wings and a shamrock necklace, just may be someone’s lucky charm."

    For a Jewish approach to this milestone of theology, see my note commemorating the death, on Christmas Day, 2000, of one of the twentieth century's great Scrooge figures, Willard van Orman Quine:



    On Linguistic Creation.

    As that note observes, we may imagine Quine to have escaped the torments of Hell.  For some further adventures, see my note Quine in Purgatory

November 30, 2002

  • X Day


    From the website Scotland: St. Andrew —









    Saint Andrew is the Patron Saint of Scotland, and St. Andrew's Day is celebrated by Scots around the world on the 30th November.

    The flag of Scotland is the Cross of St. Andrew, and this is widely displayed as a symbol of national identity.


    Xangans without Scots ancestry may still celebrate by displaying the following symbol:


  • Archetypal Criticism


    My previous note includes the following:


    "For a... literary antidote to postmodernist nihilism, see Archetypal Theory and Criticism, by Glen R. Gill."







    This week's
    Time Magazine cover
    suggests a followup to
    the Gill reference
    in defense of Jung and
    his theory of archetypes.



    Carl Gustav Jung, from a strongly Protestant background, has been vilified as an "Aryan Christ" by Catholics and Jews


    To counteract this vilification, here are two links:


November 29, 2002

  • A Logocentric Archetype


    Today we examine the relativist, nominalist, leftist, nihilist, despairing, depressing, absurd, and abominable work of Samuel Beckett, darling of the postmodernists.


    One lens through which to view Beckett is an essay by Jennifer Martin, "Beckettian Drama as Protest: A Postmodern Examination of the 'Delogocentering' of Language." Martin begins her essay with two quotations: one from the contemptible French twerp Jacques Derrida, and one from Beckett's masterpiece of stupidity, Molloy. For a logocentric deconstruction of Derrida, see my note, "The Shining of May 29," which demonstrates how Derrida attempts to convert a rather important mathematical result to his brand of nauseating and pretentious nonsense, and of course gets it wrong. For a logocentric deconstruction of Molloy, consider the following passage:

    "I took advantage of being at the seaside to lay in a store of sucking-stones. They were pebbles but I call them stones.... I distributed them equally among my four pockets, and sucked them turn and turn about. This raised a problem which I first solved in the following way. I had say sixteen stones, four in each of my four pockets these being the two pockets of my trousers and the two pockets of my greatcoat. Taking a stone from the right pocket of my greatcoat, and putting it in my mouth, I replaced it in the right pocket of my greatcoat by a stone from the right pocket of my trousers, which I replaced by a stone from the left pocket of my trousers, which I replaced by a stone from the left pocket of my greatcoat, which I replaced by the stone which was in my mouth, as soon as I had finished sucking it. Thus there were still four stones in each of my four pockets, but not quite the same stones....But this solution did not satisfy me fully. For it did not escape me that, by an extraordinary hazard, the four stones circulating thus might always be the same four."


    Beckett is describing, in great detail, how a damned moron might approach the extraordinarily beautiful mathematical discipline known as group theory, founded by the French anticleric and leftist Evariste Galois. Disciples of Derrida may play at mimicking the politics of Galois, but will never come close to imitating his genius. For a worthwhile discussion of permutation groups acting on a set of 16 elements, see R. D. Carmichael's masterly work, Introduction to the Theory of Groups of Finite Order, Ginn, Boston, 1937, reprinted by Dover, New York, 1956.


    There are at least two ways of approaching permutations on 16 elements in what Pascal calls "l'esprit géométrique." My website Diamond Theory discusses the action of the affine group in a four-dimensional finite geometry of 16 points. For a four-dimensional euclidean hypercube, or tesseract, with 16 vertices, see the highly logocentric movable illustration by Harry J. Smith. The concept of a tesseract was made famous, though seen through a glass darkly, by the Christian writer Madeleine L'Engle in her novel for children and young adults, A Wrinkle in Tme.


    This tesseract may serve as an archetype for what Pascal, Simone Weil (see my earlier notes), Harry J. Smith, and Madeleine L'Engle might, borrowing their enemies' language, call their "logocentric" philosophy.


    For a more literary antidote to postmodernist nihilism, see Archetypal Theory and Criticism, by Glen R. Gill.


    For a discussion of the full range of meaning of the word "logos," which has rational as well as religious connotations, click here.

  • On Madeleine L'Engle's birthday:


    There is such a thing as a tesseract.

November 27, 2002

  • Waiting for Logos


    Searching for background on the phrase "logos and logic" in yesterday's "Notes toward a Supreme Fact," I found this passage:



    "...a theory of psychology based on the idea of the soul as the dialectical, self-contradictory syzygy of a) soul as anima and b) soul as animus. Jungian and archetypal psychology appear to have taken heed more or less of only one half of the whole syzygy, predominantly serving an anima cut loose from her own Other, the animus as logos and logic (whose first and most extreme phenomenological image is the killer of the anima, Bluebeard). Thus psychology tends to defend the virginal innocence of the anima and her imagination..."


    -- Wolfgang Giegerich, "Once More the Reality/Irreality Issue: A Reply to Hillman's Reply," website 


    The anima and other Jungian concepts are used to analyze Wallace Stevens in an excellent essay by Michael Bryson, "The Quest for the Fiction of an Absolute." Part of Bryson's motivation in this essay is the conflict between the trendy leftist nominalism of postmodern critics and the conservative realism of more traditional critics:


    "David Jarraway, in his Stevens and the Question of Belief, writes about a Stevens figured as a proto-deconstructionist, insisting on 'Steven's insistence on dismantling the logocentric models of belief' (311) in 'An Ordinary Evening in New Haven.' In opposition to these readings comes a work like Janet McCann's Wallace Stevens Revisited: 'The Celestial Possible', in which the claim is made (speaking of the post-1940 period of Stevens' life) that 'God preoccupied him for the rest of his career.'"

    Here "logocentric" is a buzz word for "Christian." Stevens, unlike the postmodernists, was not anti-Christian. He did, however, see that the old structures of belief could not be maintained indefinitely, and pondered what could be found to replace them. "Notes toward a Supreme Fiction" deals with this problem. In his essay on Stevens' "Notes," Bryson emphasizes the "negative capability" of Keats as a contemplative technique:


    "The willingness to exist in a state of negative capability, to accept that sometimes what we are seeking is not that which reason can impose...."

    For some related material, see Simone Weil's remarks on Electra waiting for her brother Orestes. Simone Weil's brother was one of the greatest mathematicians of the past century, André Weil.



    "Electra did not seek Orestes, she waited for him..."


    -- Simone Weil


    "...at the end, she pulls it all together brilliantly in the story of Electra and Orestes, where the importance of waiting on God rather than seeking is brought home forcefully."


    -- Tom Hinkle, review of Waiting for God 


    Compare her remarks on waiting for Orestes with the following passage from Waiting for God:



    "We do not obtain the most precious gifts by going in search of them but by waiting for them. Man cannot discover them by his own powers, and if he sets out to seek for them he will find in their place counterfeits of which he will be unable to discern falsity.


    The solution of a geometry problem does not in itself constitute a precious gift, but the same law applies to it because it is the image of something precious. Being a little fragment of particular truth, it is a pure image of the unique, eternal, and living Truth, the very Truth that once in a human voice declared: "I am the Truth."


    Every school exercise, thought of in this way, is like a sacrament.


    In every school exercise there is a special way of waiting upon truth, setting our hearts upon it, yet not allowing ourselves to go out in search of it. There is a way of giving our attention to the data of a problem in geometry without trying to find the solution...."


    -- Simone Weil, "Reflections on the Right Use of School Studies with a View to the Love of  God"


    Weil concludes the preceding essay with the following passage:



    "Academic work is one of those fields containing a pearl so precious that it is worth while to sell all of our possessions, keeping nothing for ourselves, in order to be able to acquire it."


    This biblical metaphor is also echoed in the work of Pascal, who combined in one person the theological talent of Simone Weil and the mathematical talent of her brother. After discussing how proofs should be written, Pascal says


    "The method of not erring is sought by all the world. The logicians profess to guide to it, the geometricians alone attain it, and apart from their science, and the imitations of it, there are no true demonstrations. The whole art is included in the simple precepts that we have given; they alone are sufficient, they alone afford proofs; all other rules are useless or injurious. This I know by long experience of all kinds of books and persons.

    And on this point I pass the same judgment as those who say that geometricians give them nothing new by these rules, because they possessed them in reality, but confounded with a multitude of others, either useless or false, from which they could not discriminate them, as those who, seeking a diamond of great price amidst a number of false ones, but from which they know not how to distinguish it, should boast, in holding them all together, of possessing the true one equally with him who without pausing at this mass of rubbish lays his hand upon the costly stone which they are seeking and for which they do not throw away the rest."


    -- Blaise Pascal, The Art of Persuasion



    For more diamond metaphors and Jungian analysis, see

    The Diamond Archetype.

November 26, 2002

  • Andante Cantabile


    As we prepare to see publicity for Russell Crowe in a new role, that of Captain Jack Aubrey in "The Far Side of the World," based on Master and Commander by Patrick O'Brian, we bid farewell to Patti LaBelle and her Ya-Ya, and say hello to a piece more attuned to Aubrey's tastes.  This site's background music is now Mozart's Duo for Violin and Viola in Bb, K.424, 2, andante cantabile.