Page:Lawhead columbia 0054D 12326.pdf/186

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Fig. 5.2

While the specifics of this objection are only relevant to statistical mechanics, there is a more general lesson that we can draw: the track that we started down a few pages ago—of using formal features of chaos theory to put a straight-forward cap on the precision of our predictions on a given system after a certain amount of time—is not as smooth and straight as it may have initially seemed. In particular, we have to attend to the fact that simple distance across a state-space may not always be the best measure of the relative “similarity” between two different states; the case of thermodynamics and statistical mechanics provides an existence proof for this claim. Without an independent measure of how to group regions of a state space into qualitatively similar conditions—thermodynamic macroconditions in this case—we have no way of guaranteeing that just because some collection of states falls within the bounds of the region defined by 5(j) they are necessarily all similar to one another in the relevant respect. This account ignores the fact that two states might be very close together in state space, and yet differ