Series: Penn State Logic Seminar

Date: Tuesday, December 9, 2003

Time: 2:30 - 3:45 PM

Place: 113 McAllister Building

Speaker: Emily Grosholz, Penn State University, Philosophy

Title: Reduction and Representation (and Representation Theory)

  Representation theory studies physical and mathematical systems in
  terms of their symmetries. A system like a geometric figure (or
  e. g. a molecule, or the algebraic completion of the rationals)can
  be studied in terms of the group structure of the automorphisms that
  leave its shape (or e. g. its shape considered as a configuration of
  component atoms, or the rational numbers) invariant. In
  representation theory, these symmetry groups are mapped onto groups
  of matrices; these groups of matrices in turn are reduced to the
  canonical form of groups of block-factored matrices that exhibit the
  irreducible n x n representations, which can then be studied in
  terms of their characters (the product of their diagonal entries)
  and hence numbers. This is a striking reduction of complex, often
  infinitary objects to symbolic expressions that lend themselves well
  to computation.

  The lessons that I wish to elicit from the example of representation
  theory, used variously in chemistry and mathematics, in order to
  amplify current philosophy of mathematics, are the following:

  (1) Other important formal idioms besides predicate logic organize
  science and mathematics, reducing spatial configuration and dynamic
  processes to numerical computation; they make possible analyses that
  are quite different from those offered by predicate logic. When
  different formal languages are used to analyze intelligible objects,
  they reveal different kinds of conditions of intelligibility. 

  (2) The use of symbolic notation to investigate e. g. a chemical
  object typically makes use of iconic representations in tandem with
  the symbolic notation, and their conjunction is mediated and
  explained by natural language. 

  (3) While symbolic notations may in certain carefully defined
  situations be treated as uninterpreted, and manipulated in ^purely
  formal^ ways, their rational deployment in the sciences as in
  mathematics requires that their interpretations be present and
  reinstated into the problem context, and these presences and
  reinstatements are often, though not always, indicated by means of

  (4) Symbolic notations themselves have spatial and iconic dimensions
  that play important (and irreducible) roles in the knowledge they
  help to generate. This is evident in the chemical table and in
  certain printed nucleotide sequences of genes, where not only
  horizontal but also vertical correspondences in the representations
  exhibit important features of the gene; horizontal correspondences
  are typically rather more iconic (representing spatial
  side-by-sideness or addition of component parts), and vertical
  correspondences more esoteric.  

  (5) A representation that is symbolic with respect to one kind of
  thing may become iconic with respect to another kind of thing
  depending on context. The use of numerals in chemical applications
  of group theory, for example, are symbolic with respect to molecules
  but have iconic features with respect to numbers; and if a computer
  programmer is inspecting them as part of a formal language
  (considered as an object of mathematical study) then they are in a
  different way even more iconic. To be effective in the reduction of
  chemistry to group theory, those numerals must refer ambiguously to
  molecules, to numbers, and to ^themselves.^ 

  The tincture of iconicity that every symbolic representation has
  -^noted in both (4) and (5) ^- is important to my claim that there
  is no single correct symbolic representation of an intelligible
  thing. Icons, precisely because they are like the things they
  represent, are distortions, and distortions may take many forms;
  iconicity introduces style, and styles are manifold and
  changeable. Symbols, though they are relatively unlike the things
  they represent, must nonetheless share certain structural features
  with what they represent in order to count as representations at
  all, and just as there may be more than one set of structural
  features essential to the thing (hence different symbolizations that
  capture them), there may also be more than one way to represent a
  given set of structural features. Consider the representation of the
  natural numbers by Roman and Arabic notation, and by the notation 0,
  S (0), SS (0), ^ and so on. Moreover, as we have seen, in general
  symbolic representations must be supplemented by icons and natural
  language to function as symbols.  

  Another way to make the point is to observe that symbols are icons
  of idealized versions of themselves: the symbol (x) f (x) is also an
  icon of a certain wff of predicate logic, considered as a
  mathematical thing. G"odel shows us that every wff can also be
  represented (much more symbolically) by an integer whose prime
  decomposition encodes information about the wff, and that this
  representation yields important facts about the axiomatized systems
  of predicate logic, in particular, the results known as G"odel's
  Incompleteness Theorem. In many areas of human endeavor, discourse
  precipitates new intelligible things (laws, university charters,
  Hamlet, the sonnet) to add to the furniture of the world. What is
  distinctive about mathematics is that things precipitated by its
  discourse (like the wffs of predicate logic, considered as objects
  of study rather than just as modes of representation) are highly
  determinate, and come to stand in highly determinate relations with
  things that were already there, like the natural numbers, as happens
  in G"odel's proof. In sum, they may generate new mathematics. This
  doesn't mean that everything in mathematics is precipitated by
  discourse, as constructivism urges; indeed, even mathematical things
  that are precipitated by discourse turn out to have features that go
  beyond the original notation that gave rise to them.