[net.misc] Using Taylor Series To Extrapolate

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