Tuesday, 1 April 2008
Mathematical Representation
Emily Grosholz, Penn State University
12:05 pm, 817R Cathedral of Learning

Abstract: The subject of my recent book Representation and Productive Ambiguity in Mathematics and the Sciences is demonstration. I begin with Galileo’s demonstration that projectile motion describes a parabolic trajectory, and make three kinds of arguments. First, the cogency of this demonstration cannot be merely a matter of logical syntax. Its importance and its durability are due to its contribution to the truth of mechanics in one sense and of geometry in another; to view the demonstration as a proof that is essentially a syntactic object cannot do justice to the fact that it explains why projectile motion describes a parabola. Moreover, the demonstration depends on the combination of the notation of proportions and a geometrical diagram (which ambiguously refers to the finite and the infinitesimal) by an argument in natural language that explains how we are to understand this novel, indeed unprecedented, conjunction. The heterogeneity of the modes of representation required for the demonstration, and the yoking of the finite and infinitesimal, the geometrical and the dynamical, insure that the reasoning cannot instantiate a deductive argument scheme. Second, the demonstration depends on the intelligible unity of a curve, in this case of a parabola. The iconic display of a parabola is essential to the proof; shape is irreducible to numbers, algebraic formulas, sets of points, or logical notation, though any of these may be useful in certain contexts for certain purposes as symbolic representations of shape. Mathematics proceeds by both symbolic and iconic notation, because it must both indicate what it is talking about and exhibit the conditions of intelligibility of what its discourse denotes. No single, homogeneous, unambiguous idiom can perform both functions at the same time, which is why real mathematical language is multi-vocal and polysemic. Moreover, the denotation must stop somewhere; it stops at intelligibly unified, existing things, not structures. So my argument is anti-structuralist. Third, the demonstration is a unified discourse; its ambiguity and heterogeneity cannot be eliminated by decomposing its reasoning into parts, because such decomposition also makes the force of that reasoning, as a demonstration, disappear. After four hundred years, Theorem I, Proposition I (Fourth Day) of the Discorsi still stands as a paradigm of rational persuasion.