acase@tulane.edu

One of the most interesting aspects of Descartes' philosophy is the notion that mathematical truths are, in some sense, contingent. In what has become known as the doctrine of the eternal truths, Descartes writes that mathematical truths are grounded in divinely created essences, and that, like all divine creations, these essences were freely chosen. Thus, Descartes concludes, God could have made two times four unequal to eight. My project is to understand what Descartes' metaphysics and epistemology must be like in order to accommodate this contingency I begin by arguing for the coherence of the claim that God contingently chose to make mathematical truths necessary. Mathematical essences are contingent creations, but they nonetheless ground the immutable truths of mathematics. Alvin Plantinga and Harry Frankfurt have both argued that admitting to such contingency makes all truth suspect; I argue instead that mathematical truths are logically distinct from both the laws of motion and from logical truths. While these truths are also eternal, I argue that the laws of motion stem directly from God's nature (and not from his will, as mathematical truths do), and logical truths have no metaphysical grounding at all As creations, mathematical essences must have some kind of being, but they do not neatly fit into any existing category of Cartesian ontology. By rejecting interpretations such as Jonathan Bennett's and Walter Edelberg's, I argue that they are not reducible to either of the two created substances that Descartes recognizes. I therefore conclude that the essences of mathematics have the same kind of being that the essence of material things would have if there were no material things. The essences of mathematics must be ontologically on a par with the essences of created substances, and they must also be particulars---e.g., the essence of triangularity must be distinct from the essence of circularity The ontological independence of particular mathematical essences from the essences of created substances does complicate the usual picture of Cartesian ontology, but ultimately it is the only way to accommodate all of Descartes' texts and retain the notion that mathematical truths are contingent / acase@tulane.edu

acase@tulane.edu

Recent work by S. Donaldson yields an invariant of the diffeomorphism type of four-dimensional manifolds. In the case of an algebraic surface this invariant can be expressed in terms of the fundamental class of the surface and the moduli space of stable SU(2) bundles V on the surface whose Chern classes are c(,1)(V) = 0 and c(,2)(V) = 1. R. Friedman and J. Morgan have calculated this invariant in the case of certain simply-connected elliptic surfaces. In this dissertation we study elliptic surfaces X with cyclic fundamental group, dim H('2)(X,(//R)) = 10 and dim H(,+)('2)(X,(//R)) = 1. We show that for every finite cyclic group G there is a topological four-dimensional manifold with fundamental group G that allows infinitely many different diffeomorphism types / acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

Let R be a commutative integral domain with 1 and Q its field of quotients. We show that if p.d.$\sb{\rm R}$ Q = 1, then Q/R is a direct sum of countably generated R-modules. We also show that any divisible torsion module of projective dimension one over R with p.d.$\sb{\rm R}$ Q = 1 is a direct sum of countably generated R-modules. These give us two characterizations of domains R with p.d.$\sb{\rm R}$ Q = 1. We find a classification theorem of divisible modules of projective dimension one over R with p.d.$\sb{\rm R}$ Q = 1. We show that countably generated torsion-free modules over valuation domains R with p.d.$\sb{\rm R}$ Q $>$ 1 can be embedded in free R-modules and generalize this to uncountably generated torsion-free modules / acase@tulane.edu

acase@tulane.edu

acase@tulane.edu