At first, one might think that each individual cell that comes into existence might use a different section of the underlying genetic program.

A certain range of patterns emerges—almost all of which turn out to be quite similar to patterns that one sees on actual animals.

There are certainly some tasks—such as playing chess or doing algebra—that at one time were considered indicative of human-like thinking, but which are now routinely done by computer.

For it potentially allows one to get a large number of different kinds of components without ever needing to increase the complexity of the underlying rules at all.

And indeed I suspect that by looking at issues such as how easy a given theorem makes it to get from one part of a network to another it will be possible to formalize many intuitive notions about the practice of mathematics—much as earlier in this book we were able to formalize notions of everyday experience such as complexity and randomness.

Chords Two pure tones played together exhibit beats at the difference of their frequencies—a consequence of the fact that Sin[ ω 1 t] + Sin[ ω 2 t]  2 Sin[1/2( ω 1 + ω 2 ) t] Cos[( ω 1 - ω 2 ) t] With ω ≃ 500 Hz , one can explicitly hear the time variation of the beats if their frequency is below about 15 Hz, and the result is quite pleasant.

In practical computing, one usually thinks of computational programs as being set up much like the register machines of page 896 and halting when they have finished executing their instructions.

But I strongly suspect that with appropriate definitions there is often undecidability in for example the three-body problem, so that the questions such as whether one of the bodies in a particular scattering process will ever escape to infinity are in general undecidable.

It appears to have always been a common view that communication somehow involves transferring thoughts from one mind to another. … In the 1600s the philosophical idea that the only way to get information with certainty is from the senses led to emphasis on observable aspects of communication, and to the conclusion that there is no way to tell whether an accurate transfer of abstract thoughts has occurred between one mind and another. … One can also imagine models of the historical development of languages which will tend to lead to redundancy at the level of Shannon information.)

(Even if one can do operations on all digits in parallel it still takes of order n steps in a system like a cellular automaton for the effects of different digits to mix together—though see also page 1149 .) … If one evaluates NIntegrate or NDSolve by effectively fitting functions to order s polynomials the difficulty of getting results with n -digit precision typically increases like 2 n/s .