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Real algebra [and axioms]
A notion of real numbers as measures of space or quantity has existed since antiquity.
It still seems likely however that some general inequalities should exist between the rate and quality of randomness that can be extracted from a system with particular thermodynamic properties.
Rarely is there a unique minimum that can be found, but the approach nevertheless seems to work fairly well whenever a good layout exists in a particular number of dimensions.
And since nothing like this seemed likely to exist otherwise, I decided to build it.
For our purposes more useful definitions tend to concentrate not so much on whether there exists in principle a simple way to generate a particular sequence, but rather on whether such a way can realistically be recognized by applying various kinds of analysis to the sequence.
But they have somehow implicitly assumed that no other kinds of exceptions can exist.
Undoubtedly, therefore, one of the main reasons that the discoveries I describe in this chapter were not made before the 1980s is just that computer technology did not yet exist powerful enough to do the kinds of exploratory experiments that were needed.
Should we really imagine that the complete spacetime history of the universe somehow always exists, and that as time progresses, we are merely exploring different parts of it?
But if one looks not just at individual strings but rather at the sequences of strings that exist along paths in the multiway system, then one finds that these can no longer be so arbitrary.
But what about an explicit particle of the kind we discussed in the previous section that exists as a structure in a network?