From: utzoo!watmath!jagardner Newsgroups: net.misc Title: Re: Using Taylor Series To Extrapolate Article-I.D.: watmath.3183 Posted: Mon Aug 2 21:14:04 1982 Received: Tue Aug 3 02:22:53 1982 References: watmath.3155 eagle.435 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