But with all the first million initial conditions, only one other structure is produced, and this structure is again one that does not move.

And just as in the other systems we have discussed, what then emerges on average from all these small-scale random changes is overall behavior that again seems in many ways smooth and continuous.

But in all cases it turns out that the original angle between successive elements is almost exactly 137.5°.

All these rules preserve the planarity of a network.

Note that all the pictures are shown on the same scale; the last picture appears coarser because the structures it contains are larger.

So before too long it will no doubt be possible to implement all the processes of thinking that go on in a single human—or even in billions of humans—in a fairly small piece of material. … So given, say, an ordinary piece of rock in which there is all sorts of complicated electron motion this may in a fundamental sense be doing no less than some system of the future constructed with nanotechnology to implement operations of human thinking.

Predicate logic avoids this particular kind of case by implicitly assuming that what is meant is a general statement about all values of any variable—and avoids cases like the expression x + y by requiring all statements to be well-formed formulas (see page 1150 ).

Testing for reversibility [in cellular automata] To show that a cellular automaton is reversible it is sufficient to check that all configurations consisting of repetitions of different blocks have different successors. … But although these results suggest that in general it should suffice to test only up to n = k 2 r , all that has so far been rigorously proved is that n = k 2 r (k 2 r -1) + 2 r + 1 (or n = 15 for k = 2 , r = 1 ) is sufficient.

In the mid-1800s it became clear that despite their different origins most of these functions could be viewed as special cases of Hypergeometric2F1[a, b, c, z] , and that the functions covered the solutions to all linear differential equations of a certain type. ( Zeta and PolyLog are parametric derivatives of Hypergeometric2F1 ; elliptic modular functions are inverses.) … So, for example, if there are more parameters it becomes difficult to find continuous definitions that work for all complex values of these parameters. … (For rational f Paul Painlevé in the 1890s identified just 6 additional types of functions that are needed, but even now series expansions are not known for all of them.)

Starting in 1879 with his "formula language" ( Begriffsschrift ) Gottlob Frege followed a somewhat similar direction, suggesting that arithmetic and from there all of mathematics could be built up from predicate logic, and later an analog of set theory. … All of these were ultimately shown to be universal (see page 784 ) and thus in a sense capable of reproducing any mathematical system. … And following the work of the Bourbaki group beginning at the end of the 1930s it has become almost universally accepted that structures together with set theory are the appropriate framework for all of pure mathematics.