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 intellectuel triumphs of the nineteenth century was the mechanical explanation of such familier 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 rime must be replaced by the spacetime invariants of Einstein's special relativity. Another exemple, 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, from automate to stimulus-sampling models ; it ends with some empirical results on brain-wave representations of words and sentences.