As science developed from antiquity to the present, it was recurrently faced with the question of whether nature is subject to chance to any extent, or evolves deterministically. For most of that time, the need to include chance was avoided by abstraction; the subject of study was so delimited as to allow its description as a deterministically evolving system. As the theory of probability developed in modern times, even chance came within the fold of predictability, and the degree of abstraction could be lowered. A great deal of the systematic description of deterministic systems still carries over, however, and it is indeed possible now to form a clearer idea of what exactly that ideal of determinism signified.
Two preambles are needed before we try to explicate the idea of a deterministic system. The first concerns symmetry, its associated strong but sometimes deceptive intuitions, and its proper role in the systematic description of mathematical spaces. The second concerns the use of such spaces to study physical systems, by representing their possible states and possible modes of evolution. Here we begin the first preamble.
When we read that certain conclusions were reached on the basis of considerations of symmetry, we will not always find a pure and a priori demonstration. Instead, there may just be an assumption of actual symmetry in the actual world. To say that a symmetry was broken means then only that the world did not after all go along with the assumption. But at other times it