Chapter 9: Fundamental Physics

Section 7: Space as a Network

Definitions of distance

Any measure of distance—whether in ordinary continuous space or elsewhere—takes a pair of points and yields a number. Several properties are normally assumed. First, that if the points are identical the distance is zero, and if they are different, it is a positive number. Second, that the distance between points A and B is the same as between B and A. And third, that the so-called triangle inequality holds, so that the distance AC is no greater than the sum of the distances AB and BC. With distance on a network defined as the length of shortest path between nodes one immediately gets all three of these properties. And even though all distances defined this way will be integers, they still make any network formally correspond in mathematical terms to a metric space (or strictly a path metric space). If the connections on the underlying network are one-way (as in causal networks) then one no longer necessarily gets the second property, and when a continuum limit exists it can correspond to a (perhaps discontinuous) section through a fiber bundle rather than to a manifold. Note that as discussed on page 536 physical measures of distance will always end up being based not just on single paths in a network, but on the propagation of something like a particle, which typically in effect requires the presence of many paths. (See page 1048.)

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From Stephen Wolfram: A New Kind of Science [citation]