Mathematics

I discuss the implications of this book for the foundations of mathematics mainly on pages 772–821 and in the rather extensive corresponding notes. With a sufficiently general definition of mathematics, however, the whole core of the book can in fact be viewed as a work of experimental mathematics. And even with a more traditional definition, this is at least true of much of my discussion of systems based on numbers in Chapter 4. The notes to almost all chapters of the book contain a great many new mathematical results, mostly emerging from my analysis of some of the simpler behavior considered in the book. Pages 606–620 and 737–750 discuss in general the capabilities of mathematical analysis, while pages 588–597 address the foundations of statistics. Note that some ideas and results highly relevant to current frontiers in mathematics appear in some rather unexpected places in the book. Specific examples include the parameter space sets that I discuss in connection with shapes of plant leaves on page 407, and the minimal axioms for logic that I discuss on page 810. A more general example is the issue of smooth objects arising from combinatorial data that I discuss in Chapter 9 in connection with the nature of space in fundamental physics.