An early, very preliminary edition of this book was circulated in 1962 under the title Set-theoretical Structures in Science. There are many reasons for maintaining that such structures play a role in the philosophy of science. Perhaps the best is that they provide the right setting for investigating problems of representation and invariance in any systematic part of science, past or present. Examples are easy to cite. Sophisticated analysis of the nature of representation in perception is to be found already (...) in Plato and Aristotle. One of the great intellectual triumphs of the nineteenth century was the mechanical explanation of such familiar concepts as temperature and pressure by their representation in terms of the motion of particles. A more disturbing change of viewpoint was the realization at the beginning of the twentieth century that the separate invariant properties of space and time must be replaced by the space-time invariants of Einstein's special relativity. Another example, the focus of the longest chapter in this book, is controversy extending over several centuries on the proper representation of probability. The six major positions on this question are critically examined. Topics covered in other chapters include an unusually detailed treatment of theoretical and experimental work on visual space, the two senses of invariance represented by weak and strong reversibility of causal processes, and the representation of hidden variables in quantum mechanics. The final chapter concentrates on different kinds of representations of language, concluding with some empirical results on brain-wave representations of words and sentences. (shrink)

Modern Philosophy of Science has not yet recognized the significance of physical invariance principles for science and daily life. In this paper we investigate as a simple example, how time independence or time translational invariance of natural laws determines the time scale. We start with informal definitions of the invariance of concepts and laws. We then ask if time independence is an essential feature of natural laws or if time dependent laws are plausible without loss of predictive relevance. In section (...) 3 we investigate the question wether time independence of natural laws is a matter of convention. In sections 5-7 the determination of the time scale by time translational invariance is studied. In section 8 we study the conditions under which this determination is unique. (shrink)