June 21, 2007

  • Structural Logic continued:

    Let No Man
    Write My Epigraph

    (See entries of June 19th.)


    "His graceful accounts of the Bach Suites for Unaccompanied Cello
    illuminated the works’ structural logic as well as their inner
    spirituality."

    --Allan Kozinn on Mstislav Rostropovich in The New York Times, quoted in Log24 on April 29, 2007


    "At that instant he saw, in one blaze of light, an image of
    unutterable conviction.... the core of life, the essential
    pattern whence all other things proceed, the kernel of eternity."

    -- Thomas Wolfe, Of Time and the River, quoted in Log24 on June 9, 2005

    "... the stabiliser of an octad preserves the affine space structure
    on its complement, and (from the construction) induces AGL(4,2) on it.
    (It induces A8 on the octad, the kernel of this action being the translation group of the affine space.)"

    -- Peter J. Cameron, "The Geometry of the Mathieu Groups" (pdf)

    "... donc Dieu existe, réponse!"

    -- Attributed, some say falsely,
    to Leonhard Euler


    "Only gradually did I discover
    what the mandala really is:
    'Formation, Transformation,
    Eternal Mind's eternal recreation'"

    (Faust, Part Two, as
    quoted by Jung in
    Memories, Dreams, Reflections)

    Wolfgang Pauli as Mephistopheles

    "Pauli as Mephistopheles

    in a 1932 parody of

    Goethe's Faust
    at Niels Bohr's

    institute in Copenhagen.

    The drawing is one of

    many by
    George Gamow

    illustrating the script."

    -- Physics Today

    "Borja dropped the mutilated book on the floor with the others. He was
    looking at the nine engravings and at the circle, checking strange
    correspondences between them.

    'To meet someone' was his enigmatic answer. 'To search for the stone
    that the Great Architect rejected, the philosopher's stone, the basis
    of the philosophical work. The stone of power. The devil likes
    metamorphoses, Corso.'"

    -- The Club Dumas, basis for the Roman Polanski film "The Ninth Gate" (See 12/24/05.)

    "Pauli linked this symbolism
    with the concept of automorphism."

    -- The Innermost Kernel
     (previous entry)

    And from
    "Symmetry in Mathematics
    and Mathematics of Symmetry    "
    (pdf), by Peter J. Cameron,
    a paper presented at the

    International Symmetry Conference    ,
    Edinburgh, Jan. 14-17, 2007,
    we have


    The Epigraph--

    Weyl on automorphisms
    (Here "whatever" should
    of course be "whenever.")

    Also from the
    Cameron paper:

    Local or global?

    Among other (mostly more vague) definitions of symmetry, the dictionary will typically list two, something like this:

    • exact correspondence of parts;
    • remaining unchanged by transformation.

    Mathematicians
    typically consider the second, global, notion, but what about the
    first, local, notion, and what is the relationship between them? 
    A structure M is homogeneous
    if every isomorphism between finite substructures of M can be extended
    to an automorphism of M; in other words, "any local symmetry is global."

    Some Log24 entries
    related to the above politically
    (women in mathematics)--

    Global and Local:
    One Small Step

    and mathematically--

    Structural Logic continued:
    Structure and Logic     (4/30/07):

    This entry cites
    Alice Devillers of Brussels--

    Alice Devillers

    "The aim of this thesis
    is to classify certain structures
    which
    are, from a certain
    point of view, as homogeneous
    as possible, that is
    which have
      as many symmetries as possible."

    "There is such a thing
    as a tesseract."

    -- Madeleine L'Engle