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) Abstract: 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 icons. (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.