For the rest of the course, we will be looking at discussions of philosophical issues that arise concerning particular sciences. This week is the first of two concerned with physics, and it will focus on issues raised by the Einstein’s general theory of relativity, which were probably the central issues for philosophers of science of Carnap’s generation.

In fact, we won’t talk about general relativity until Thursday. In this assignment, Carnap introduces the idea of non-Euclidean geometry and questions it raises concerning the nature and status of our knowledge of geometry.

You should do three things to prepare for our discussion of it.

•  First, ask yourself whether you understand the points Carnap makes about geometry as well as he seems to intend (he doesn’t intend more than an introductory overview) or, more importantly, well enough to be able to think about the philosophical issues he suggests. One of the things we should do in discussion, then, is to sort points in his discussion of geometry that aren’t as clear as they need to be.

•  Second, think about the views of the nature of geometrical knowledge that are presented. The two views Carnap focuses on in these chapters are associated with Immanuel Kant (1724-1804) and Henri Poincaré (1854-1912), who lived on either end of the time when non-Euclidean geometry was developed. (In spite of the title of ch. 15, Carnap’s real discussion of the view he associates with Einstein comes in chs. 16-17.) You should try to come to an understanding of these views, and look for points about them that you think we should discuss. Also, think what can be said on behalf of each.

•  Third, I’d encourage you to ask yourself how specific these issues are to geometry. Could there be views analogous to those of Kant and Poincaré in the case of other parts of mathematics? Would they be more or less plausible in those cases? Could analogous things be said about fields outside of mathematics? (Newton’s three laws of motion is a good example to think about.)

People often think of Poincaré together with a contemporary of his, the physicist and philosopher and historian of science Pierre Duhem (1861-1916). Duhem had a view of science that could be thought of as falling between Carnap’s and Kuhn’s. He emphasized the impossibility of conclusively refuting any scientific theory (because of the possibility of attributing a failed test to the many auxiliary assumptions required to conduct it). He thought that theories would normally be revised and made gradually more complex in response to anomalous results rather than being abandoned. But he thought that it was part of the training of a scientist to acquire the “good sense” needed to recognize when a theory had finally become so complex that it should be abandoned in favor of an alternative approach.