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And for example particularly in lower animals there are all sorts of cases where very simple and predictable responses to stimuli are seen. … And the crucial point is that this happens just through the intrinsic evolution of the system—without the need for any additional input from outside or from any sort of explicit source of randomness.
At some rather abstract level one can immediately recognize a basic similarity between nature and mathematics: for in nature one knows that fairly simple underlying laws somehow lead to the rich and complex behavior we see, while in mathematics the whole field is in a sense based on the notion that fairly simple axioms like those on the facing page can lead to all sorts of rich and complex results. … Yet at some level there is still all sorts of complexity in mathematics.
undecidability—this is no longer the case, and as I discussed above I suspect that it will actually be quite common for there to be all sorts of short theorems that have only extremely long proofs. … And so, for example, what are usually viewed as more successful areas of pure mathematics may have more compact networks, while areas that seem to involve all sorts of isolated facts—like elementary number theory or theory of specific cellular automata—may have sparser networks with more tendrils.
Adaptation to all sorts of complex situations also occurs in a great many systems. … My guess is that ultimately there is not, and that in fact any workable definition of what we normally think of as intelligence will end up having to be tied to all sorts of seemingly rather specific details of human intelligence.
And indeed it is sobering to notice that if one just listens even to bird songs and whale songs there is little that fundamentally seems to distinguish them from what can be generated by all sorts of processes in nature—say the motion of chimes blowing in the wind or of plasma in the Earth's magnetosphere. … But it is extremely common in all sorts of natural systems to see effects that propagate from one element to another.
Other examples [of minimal systems] Minimal systems achieving particular purposes are shown on page 619 for Boolean functions evaluated with Nand s, pages 759 and 889 for Turing machines, page 1142 for sorting networks, and page 1035 for firing squad synchronization.
Doing this certainly required experience in all sorts of different areas of science. … And the fact that I had managed to make this work so many times in Mathematica was part of what gave me the confidence to try doing something similar in all sorts of areas of science.
From the tradition of the existing sciences one might expect that its answer would depend on all sorts of details, and be quite different for different types of physical, biological and other systems. … And indeed in the later parts of this book I will show that even remarkably simple programs seem to capture the essential mechanisms responsible for all sorts of important phenomena that in the past have always seemed far too complex to allow any simple explanation.
But it turns out that there can still be other kinds of curvature—described for example by the so-called Riemann tensor—and these can in fact lead to all sorts of phenomena. … So this means that to know what will happen even in phenomena primarily associated with gravity one typically has to know all sorts of properties of matter.
And in the past there have certainly been several instances when new algorithms have suddenly allowed all sorts of computations to be done much more efficiently than had ever been thought possible before. … But what my discoveries have shown is that in fact even very small programs can be quite capable of doing all sorts of sophisticated computations.
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