Wittgenstein and Hume on Miracles

In this paper, I intend to contrast the positions of Ludwig Wittgenstein and David Hume on miracles. While Hume holds that miracles are violations of laws of nature which can never be probable, Wittgenstein would reject this definition. Instead, he takes a broader stance on miracles and holds that many events which are not transgressions of laws of nature can be seen as miraculous. And the point of this is to highlight the vastly different events we call miracles. Contra Hume, Wittgenstein thinks that even some of our greatest certainties can call up in us a sense of absolute wonder and awe, and thus, be thought of as miracles

Global bifurcation for monotone fronts of elliptic equations

In this paper, we present two results on global continuation of monotone front-type solutions to elliptic PDEs posed on infinite cylinders. This is done under quite general assumptions, and in particular applies even to fully nonlinear equations as well as quasilinear problems with transmission boundary conditions. Our approach is rooted in the analytic global bifurcation theory of Dancer and Buffoni--Toland, but extending it to unbounded domains requires contending with new potential limiting behavior relating to loss of compactness. We obtain an exhaustive set of alternatives for the global behavior of the solution curve that is sharp, with each possibility having a direct analogue in the bifurcation theory of second-order ODEs. As a major application of the general theory, we construct global families of internal hydrodynamic bores. These are traveling front solutions of the full two-phase Euler equation in two dimensions. The fluids are confined to a channel that is bounded above and below by rigid walls, with incompressible and irrotational flow in each layer. Small-amplitude fronts for this system have been obtained by several authors. We give the first large-amplitude result in the form of continuous curves of elevation and depression bores. Following the elevation curve to its extreme, we find waves whose interfaces either overturn (develop a vertical tangent) or become exceptionally singular in that the flow in both layers degenerates at a single point on the boundary. For the curve of depression waves, we prove that either the interface overturns or it comes into contact with the upper wall.Comment: 60 pages, 6 figure

Wittgenstein on Miscalculation and the Foundations of Mathematics

In Remarks on the Foundations of Mathematics, Wittgenstein notes that he has \u27not yet made the role of miscalculating clear\u27 and that \u27the role of the proposition: I must have miscalculated ...is really the key to an understanding of the foundations of mathematics.\u27 In this paper, I hope to get clear on how this is the case. First, I will explain Wittgenstein\u27s understanding of a \u27foundation\u27 for mathematics. Then, by showing how the proposition \u27I must have miscalculated\u27 differentiates mathematics from the physical sciences, we will see how this proposition is the key to understanding the foundations of mathematics

Our American Artists. IV. William M. Chase

A biographical sketch of Indiana-born United States painter William M. Chase, with illustrations

Global Bifurcation of Anti-plane Shear Fronts

We consider anti-plane shear deformations of an incompressible elastic solid whose reference configuration is an infinite cylinder with a cross section that is unbounded in one direction. For a class of generalized neo-Hookean strain energy densities and live body forces, we construct unbounded curves of front-type solutions using global bifurcation theory. Some of these curves contain solutions with deformations of arbitrarily large magnitude.Comment: 21 pages, 2 figure