In a Search for a Structure, Part 1: On Entropy

Mathematics is about ”interesting structures”. What make a structure interesting is an abundance of interesting problems; we study a structure by solving these problems. The worlds of science, as well as of mathematics itself, is abundant with gems (germs?) of simple beautiful ideas. When and how may such an idea direct you toward beautiful mathematics? I try to present in this talk a few suggestive examples. 1 States, Spaces, Crystals and Entropy. What is the ”number of states” of a (classical as opposed to quantum) system, S, e.g. of a crystal? A ”naive physicist’s” system is an infinite ensemble of infinitely small mutually equal ”states”, where you know nothing about what these states are but you believe they are equal because of observable symmetries of S. The number of these states, although infinite, can be assigned a specific numerical value by comparing with another system taken for a ”unit of entropy”; moreover, this number can be measured experimentally with a predictable error. The logarithm of this number is called (mean statistical Boltzmann) entropy of S, [14]. What is the ”space of states” of S? No such thing exists – would be our physicists answer (unless he/she is a Shroedinger’s cat). Even the ”number of states” – the value of entropy – may depend on accuracy of your data. This S is not a ”real thing”, nor is it a mathematician’s ”set”, it is ”something” that depends on a class of mutually equivalent imaginary experimental protocols. This, probably, appeared gibberish to mathematicians of the late 19th century, when Boltzmann developed his concept of entropy, and even of the early 20th century, when Lebesgue (1902) and Kolmogorov (1933) expressed the ideas of measure and probability in the language of the set theory of Kantor (1873).