March 21, 2007

  • Sequel

    Finite Relativity
    continued

    This afternoon I added a paragraph to The Geometry of Logic that makes it, in a way, a sequel to the webpage Finite Relativity:

    "As noted previously, in Figure 2 viewed as a lattice the 16 digital labels 0000, 0001, etc., may be
    interpreted as naming the 16 subsets of a 4-set; in this case the
    partial ordering in the lattice is the
    structure preserved by the lattice's group of 24 automorphisms-- the same
    automorphism group as that of the 16 Boolean connectives.  If,
    however,
    these 16 digital labels are interpreted as naming the 16 functions from
    a 4-set to a 2-set  (of two truth values, of two colors, of two
    finite-field elements, and so forth), it is not obvious that the notion
    of partial order is relevant.  For such a set of 16 functions, the
    relevant group of automorphisms may be the affine group of A
    mentioned above.  One might argue that each Venn diagram in Fig. 3
    constitutes such a function-- specifically, a mapping of four
    nonoverlapping regions
    within a rectangle to a set of two colors-- and that the diagrams,
    considered simply as a set of two-color mappings, have an automorphism
    group of order larger than 24... in fact, of order 322,560. 
    Whether
    such a group can be regarded as forming part of a 'geometry of
    logic' is open to debate."

    The epigraph to "Finite Relativity" is:

    "This is the relativity problem: to fix objectively a class of
    equivalent coordinatizations and to ascertain the group of transformations S
    mediating between them."

    -- Hermann Weyl, The Classical Groups,
    Princeton University Press, 1946, p. 16

    The added paragraph seems to fit this description.