His main conclusion seems to be (see his final paragraph in part two) that mathematics starts off with an empirical basis, but it "begins to live a peculiar lif

His main conclusion seems to be (see his final paragraph in part two) that mathematics starts off with an empirical basis, but it "begins to live a peculiar life of its own" which corresponds more to a creative discipline, whose primary criteria for evaluation are aesthetic. An excerpt from that final paragraph: But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up. It would be easy to give examples, to trace specific evolutions into the baroque and the very high baroque, but this, again, would be too technical. Seems very much along the lines of software projects: we lay down some framework that we work in which appears simple, elegant and well-suited to the conception of the product we're building. Then, as time goes one, it becomes complex as we 'patch' the imperfections of the initial system—then we refactor: a sort of paradigm shift occurs and we're working with new basic terms, hopefully restoring elegance to the system.