This paper starts with an intuitive notion of representational spaces, which is intended to provide an improved version of Kuhn's concept of paradigms. It then proceeds to study the following topics in terms of this new notion: incommensurability, paradigm change, explanation of anomalies, explanation of regularities, explanation of irregularities, and physical necessity. In the course of the investigation, "representational space" gets clarified and defined. It is envisaged that this new concept should throw light on many issues in the philosophy of science.
In scientific explanations, the explanans theory is sometimes incommensurable with the explanandum empirical data. How is this possible, especially when the explanation is deductive in nature? This paper attempts to solve the puzzle without relying on any particular theory of reference. For us, it is rather obvious that the geometric idea of projection plays a key role in Kepler’s explanation of Tycho Brahe’s empirical data. We discover that a similar mechanism operates in theoretic explanations in general. In short, all theoretic explanations are “projective” explanations. If so, there should be no logical reason why explanans theories cannot be incommensurable with explanandum data. For illustration, we analyse Einstein’s explanation of the results of the Michelson–Morley experiment in some detail.
The existence of a winning strategy in the well-known Banach-Mazur game in a completely regular topological space X is proved to be equivalent to the generic existence of solutions of optimization problems generated by continuous functions in X