cw (07/31/82)
Gardner said that you can "use a Taylor series to extrapolate the behaviour (sic: Canadian) of a function everywhere from its behaviour (sic: consistency) at a single point" anabdalllah asks how this can be. I am sure Gardner meant that some functions have this property, not neccessarily all, as the statement was used as an analogy to suggest that some universe models are deterministic though not all are. Charles
ark (08/01/82)
It is indeed true that you can extrapolate the behavior of a function everywhere from its behavior at a single point, provided that (a) "behavior" means "value of the function and all its derivatives," and (b) the function and all its derivatives are continuous (and, by implication, differentiable) everywhere.
jagardner (08/03/82)
Allow me to clarify some of the things I said about cosmological models. Of course you can only use Taylor series to extrapolate functions if the functions satisfy certain conditions. On the real number line, they have to be analytic, which means (among other things) that they have to be infinitely differentiable. In analysis, infinite differentiability is really quite a strong assumption. In practical physics, since we can only measure to a certain degree of accuracy anyway, you can actually get away with a polynomial function to model all observations, and therefore you always get infinite differentiability. This is not particularly satisfying, but it does motivate many assumptions when constructing mathematical models. I indicated that there are mathematical models of the universe in which everything can be predicted from a single three-dimensional slice. There are also models in which this cannot be done. Since our knowledge of the universe is so meagre, we have no way as yet to choose between models. Maybe we never will. And even if our universe does follow a predictable model, it seems unlikely that we would ever be able to make predictions with it except on the largest of scales--even assuming that we could gather total information about a three-dimensional slice, the extrapolation process would be so computationally complex as to be impossible in any practical sense. ---Jim Gardner