(Page 0574) what happens if one divides an image into a collection of nested squares, but imposes a lower limit on the size of these squares. And what one sees is that as the ...

(Page 0595) what probability a given deviation from the average should occur for a sequence that is genuinely chosen at random. The result turns out to be quite consistent ...

(Page 0612) functions together with factorials and multinomial coefficients then it appears that there is not. But if one also allows higher mathematical functions then it ...

(Page 0682) idea is to have each of the various kinds of lines in the picture be emulated by some collection of localized structures in rule 110. But at the outset it is by ...

(Page 0755) dies out. But already in example (b) it is not so easy. One can go for 1000 steps and still not know what is going to happen. And only after 1017 steps does it ...

(Page 0762) So are there computations that take still longer to do? In Turing machine (f) the maximum number of steps increases exponentially with the length of the input. ...

(Page 0772) Implications for Mathematics and Its Foundations Much of what I have done in this book has been motivated by trying to understand phenomena in nature. But the ...

(Page 0784) reached -- corresponding to the fact that statements must exist that cannot be proved either true or false from a given set of axioms. But what does it take to ...

(Page 0792) unprovability. But the issue remains why such phenomena have not been much more obvious in everyday work in mathematics. At some level I suspect the reason is ...

(Page 0795) suggest that if one is ever going to study many important phenomena that occur in nature one will also inevitably run into them. But to traditional mathematics ...