Equal (if and only if) is common in more mathematical settings, while Xor is widespread in discrete mathematics.

Yet while this means that there is in effect less new information in each theorem that is proved, it turns out that in most areas of mathematics these theorems are usually the ones that are considered interesting.

For a while it was thought that perhaps life might be defined by its ability for self-reproduction.

And this means that while the colors of these cells can be updated according to a wide range of different possible rules, the underlying number and organization of cells always stays the same.

And while from the point of view of modern intellectual thinking this may come as quite a shock, it is perhaps not so surprising at the level of everyday experience.

For while I believe that the basic science that I develop in this book provides a remarkably powerful new framework, coming up with an actual model requires all sorts of detailed work and analysis.

And so, for example, while the digits of π have a simple description in terms of traditional mathematics, the results in Chapter 4 suggest that outside of this framework they normally do not.

Leonhard Euler studied many continued fractions, while Joseph Lagrange seems to have thought that it might be possible to recognize any algebraic number from its continued fraction.

For large symmetric matrices with random entries following a distribution with mean 0 and bounded variance the density of normalized eigenvalues tends to Wigner's semicircle law 2Sqrt[1 - x 2 ] UnitStep[1 - x 2 ]/ π while the distribution of spacings between tends to 1/2( π x)Exp[1/4(- π )x 2 ] The distribution of largest eigenvalues can often be expressed in terms of Painlevé functions.

Pages 606 – 620 and 737 – 750 discuss in general the capabilities of mathematical analysis, while pages 588 – 597 address the foundations of statistics.