Kant’s post-1800 Disavowal of the Highest Good Argument for the Existence of God

I have two main goals in this paper. The first is to argue for the thesis that Kant gave up on his highest good argument for the existence of God around 1800. The second is to revive a dialogue about this thesis that died out in the 1960s. The paper is divided into three sections. In the first, I reconstruct Kant’s highest good argument. In the second, I turn to the post-1800 convolutes of Kant’s Opus postumum to discuss his repeated claim that there is only one way to argue for the existence of God, a way which resembles the highest good argument only in taking the moral law as its starting point. In the third, I explain why I do not find the counterarguments to my thesis introduced in the 1960s persuasive

A Dilemma for Mathematical Constructivism

In this paper I argue that constructivism in mathematics faces a dilemma. In particular, I maintain that constructivism is unable to explain (i) the application of mathematics to nature and (ii) the intersubjectivity of mathematics unless (iii) it is conjoined with two theses that reduce it to a form of mathematical Platonism. The paper is divided into five sections. In the first section of the paper, I explain the difference between mathematical constructivism and mathematical Platonism and I outline my argument. In the second, I argue that the best explanation of how mathematics applies to nature for a constructivist is a thesis I call Copernicanism. In the third, I argue that the best explanation of how mathematics can be intersubjective for a constructivist is a thesis I call Ideality. In the fourth, I argue that once constructivism is conjoined with these two theses, it collapses into a form of mathematical Platonism. In the fifth, I confront some objections

Somekawa's K-groups and Voevodsky's Hom groups (preliminary version)

We construct a surjective homomorphism from Somekawa's K-group associated to a finite collection of semi-abelian varieties over a perfect field to a corresponding Hom group in Voevodsky's triangulated category of effective motivic complexes.Comment: 15 page

Holomorphic Removability of Julia Sets

Let $f(z) = z^2 + c$ be a quadratic polynomial, with c in the Mandelbrot set. Assume further that both fixed points of f are repelling, and that f is not renormalizable. Then we prove that the Julia set J of f is holomorphically removable in the sense that every homeomorphism of the complex plane to itself that is conformal off of J is in fact conformal on the entire complex plane. As a corollary, we deduce that the Mandelbrot Set is locally connected at such c.Comment: 48 pages. 9 PostScript figure

A sheaf-theoretic reformulation of the Tate conjecture

Let p be a prime number. We give a conjecture of a sheaf-theoretic nature which is equivalent to the strong form of the Tate conjecture for smooth, projective varieties X over F_p: for all n>0, the order of pole of the Hasse-Weil zeta function of X at s=n equals the rank of the group of algebraic cycles of codimension n modulo numerical equivalence. Our main result is that this conjecture implies other well-known conjectures in characteristic p, among which: - The (weak) Tate conjecture for smooth, projective varieties X over any finitely generated field of characteristic p: given a prime l different from p, the geometric cycle map from algebraic cycles over X to the Galois invariants of the l-adic cohomology of the geometric fibre of X, tensored by Q_l, is surjective. - For X as above, the algebraicity of the Kunneth components of the diagonal and the hard Lefschetz theorem for cycles modulo numerical equivalence. - For X as above, the existence of a filtration conjectured by Beilinson on the Chow groups of X. - The rational Bass conjecture: for any smooth variety X over F_p, the algebraic K-groups of X have finite rank. - The Bass-Tate conjecture: for F a field of characteristic p, of absolute transcendence degree d, the i-th Milnor K-group of F is torsion for i>d. - Soule's conjecture: given a quasi-projective variety over F_p, the order of the zero of its Hasse-Weil zeta function at an integer n is given by the alternating sum of the ranks of the weight n part of its algebraic K'-groups