The results of this book, however, make it quite clear that this is not the case, and that in fact traditional mathematics has reached only a tiny fraction of all the kinds of abstract systems that can in principle be studied.

For example, to a distant observer, an object falling into a black hole will seem to take an infinite time to cross the event horizon—even though to the object itself only a finite time will seem to have passed.

But often an axiom system will not only be incomplete, but will also be what is called essentially incomplete.

Having made the definition Attributes[s] = Flat the state of a sequential substitution system at a particular step can be represented by a symbolic expression such as s[1, 0, 1, 0] . The rule on page 82 can then be given simply as s[1, 0]  s[0, 1, 0] while the rule on page 85 becomes {s[0, 1, 0]  s[0, 0, 1], s[0]  s[0, 1, 0]} The Flat attribute of s makes these rules apply not only for example to the whole sequence s[1, 0, 1, 0] but also to any subsequence such as s[1, 0] . (With s being Flat , s[s[1, 0], 1, s[0]] is equivalent to s[1, 0, 1, 0] and so on.

However, for additive rules the pattern of differences is just exactly the pattern that would be obtained by evolution from an initial condition consisting only of the changes made. … (Note that the pattern of differences between two initial conditions in a rule with k possible colors can always be reproduced by looking at the evolution from a single initial condition of a suitable rule with 2k colors.) In 2D class 3 cellular automata, the region of change usually ends up having a roughly circular shape—a result presumably related to the Central Limit Theorem (see page 976 ).

The suggestion is also sometimes made that we perceive a kind of harmony in nature because we see only a limited number of types of forms in it.

But when meiosis forms egg and sperm cells they get only one version of each.

(Some physicists might argue for a somewhat narrower definition that allows only discontinuities in the so-called partition function of equilibrium statistical mechanics, but for many of the most interesting applications, the definition I use is the appropriate one.)

More particles [in physics] To produce more massive particles requires higher-energy particle collisions, and today's accelerators only allow one to search up to masses of perhaps 200 GeV. … There is evidence against any more than the three known generations of quarks and leptons in that the decay process Z 0  ν ν has a rate that rather accurately agrees with what is expected from just three types of low-mass neutrinos.

But once the data has somehow been learned, only parts directly associated with handling it usually appear to be active.