But while extending the instruction set can increase the speed of operations, it does not appear to yield a much larger density of machines with complex behavior.

Standard examples of recursive sequences that do not come from linear recurrence relations include factorial f[1] = 1; f[n_] := n f[n - 1] and Ackermann functions (see below ).

But this classification does not immediately provide a practical way to enumerate all possible groups.

But in fact the main point is that if the evolution of the whole system is to be reversible, then the demon must store enough information to reverse its own actions, and this limits how much the demon can do, preventing it, for example, from unscrambling a large system of gas molecules.

But while I believe that every feature of our universe does indeed come from an ultimate discrete model, I would be very surprised if the values of constants which happen to be easy for us to measure in the end turn out to be given by simple traditional mathematical formulas.

History of experimental mathematics The general idea of finding mathematical results by doing computational experiments has a distinguished, if not widely discussed, history. … What I do in this book—and started in the early 1980s—is, however, rather different: I use computer experiments to look at questions and systems that can be viewed as having a mathematical character, yet have never in the past been considered in any way by traditional mathematics.

Shells where successive whorls do not touch (as in the first picture on row (c) of page 415 ) appear to be significantly less common than others, perhaps because they have lower mechanical rigidity. They do however occur, though sometimes as internal rather than external shells.

Around 1621 Wilhelm Schickard probably built a machine based on gears for doing simplified multiplications involved in Johannes Kepler 's calculations of the orbit of the Moon. But much more widely known were the machines built in the 1640s by Blaise Pascal for doing addition on numbers with five or so digits and in the 1670s by Gottfried Leibniz for doing multiplication, division and square roots. … And with the idea of storing programs electronically, this became fairly easy to do, so that by 1950 more than ten stored-program computers had been built in the U.S. and in England.

When doing high-precision arithmetic, Mathematica follows the principle that it should only ever give digits that are known to be correct on the basis of the input that was provided. … What they do is to give a fixed number of digits as the result of every computation, whether or not all those digits are known to be correct. … It is important to realize however that this randomness has little to do with the details of the initial conditions.

(Least squares fits do this for models in which the data exhibits independent Gaussian variations.)