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And quite probably the same is true of many nerve cells involved in the general process of doing the analog of producing hash codes.
But one of the central discoveries of this book is that this is far from true—and that actually it is rather common for rules that have extremely simple descriptions to give rise to data that is highly complex, and that has no regularities that can be recognized by any of our standard methods.
One way to see that this must be true is to note that any particular computer system or computer language can always be set up by appropriate programming to emulate any other one.
For even though a system itself may follow the Principle of Computational Equivalence, there is no guarantee that this will also be true of idealizations of the system.
The underlying rules for the Turing machine then define constraints on which sequences of such statements can be true.
For while simple infinite quantities like 1/0 or the total number of integers can readily be summarized in finite ways—often just by using symbols like ∞ and ℵ 0 —the same is not in general true of all infinite processes.
Yet surely, one might argue, there must be something fundamentally more to true intelligence of the kind that we as humans have.
The standard mathematical framework of relativity theory implies that any massless particle must propagate at c in a vacuum—so that not only light but also gravitational waves presumably go at this speed (and the same is at least approximately true of neutrinos).
To a very good approximation this appears to be true, but it turns out that in certain esoteric particle physics processes small deviations have been found. … Despite this, it was still assumed that CP and T would be true invariances.
For any m the same is true for sufficiently large n ; it is known that m = 4 requires n ≥ 35 and m = 5 requires n ≥ 178 .
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