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He argued that the best scientific model is one that minimizes this complexity—which with probabilities 0 and 1 is equivalent to minimizing the number of nodes in the network.
Constructible reals
Instead of finding successive digits using systems like Turing machines, one can imagine constructing complete real numbers using idealizations of mechanical processes.
Computation of [recursive] sequences
It is straightforward to compute the various sequences given here, but to avoid a rapid increase in computer time, it is essential to store all the values of f[n] that one has already computed, rather than recomputing them every time they are needed.
[Computing] square roots
A standard way to compute √ n is Newton's method (actually used already in 2000 BC by the Babylonians), in which one takes an estimate of the value x and then successively applies the rule x 1/2 (x + n/x) .
In all cases the rules have been at least slightly more complicated than the ones I consider here, and behavior starting from simple initial conditions does not appear to have been studied before.
On page 160 the effect is much larger, and almost all the pictures would be completely wrong—with the notable exception of the one that shows localized structures.
One feature of it, however, is that it maps the equation for quantum mechanical time evolution into the equation for probabilities in statistical mechanics, with imaginary time corresponding to inverse temperature.
But now finally with the framework developed in this book one potentially has a meaningful foundation for doing this.
Until recently, the only kinds of shapes widely discussed in science and mathematics were ones that are regular or smooth.
in the majority of cases the best evidence that some particular set of effects are in fact the important ones ultimately comes just from the success of models that are based on these effects.