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Attractors in systems based on numbers
Particularly for systems based on ordinary differential equations (see page 922 ) a geometrical classification of possible attractors exists.
Character of mathematics
Since at least the early 1900s several major schools of thought have existed:
• Formalism (e.g.
The pictures show all the distinct maximal cases that exist for a 7 ×7 grid, corresponding to possible circles with diameters Sqrt[m 2 +n 2 ] .
And so for example, in the case of cellular automata, it seems that all the essential ingredients needed to produce even the most complex behavior already exist in elementary rules.
Yet even so, there could still in principle exist in nature some other kind of system that would be able to do this.
Evaluation chains
The idea of building up computations like 1 + 1 + 1 + … from partial results has existed since Egyptian times.
But since the evolution of the system continually shifts digits to the left, it is inevitable that the differences that exist in later digits will eventually become important.
In rule 128, for example, the fact that regions of black shrink by one cell on each side at each step means that any region of black that exists after t steps must have at least t white cells on either side of it.
And as picture (d) begins to emphasize, one can think of these stripes as indicating what causal relationships or connections exist between updating events.
Yet with undecidability one believes that there is absolutely no construct that can explicitly exist in our universe that allows the problem to be solved in any finite way.