Re: Evolutionist says you can't build a large wooden ship
- From: Rupert Morrish <rupert@xxxxxxxxxxx>
- Date: Fri, 05 Sep 2008 08:34:31 +1200
Michael Siemon wrote:
In article <1imqx22.4pjx8n343ujhN%j.wilkins1@xxxxxxxxx>,
j.wilkins1@xxxxxxxxx (John Wilkins) wrote:
...
Stick to teaching your grandmother how to do metaphysics instead :-)Please don't try to teach your grandmother to suck eggs.I admit that is obscure. But for ordinary spheres in ordinary (orientable) contexts, there will be two "poles".Sorry, it is obscure but you interpret it wrong. I already told you
how to do it. There are a very large number of math oriented web
sites describing the hairy ball theorem all of which say there must be
at least one and absolutely zero of them saying there must be two.
Look at Figure 2 in the following web site to see a mapping.
http://www.math.byu.edu/~jarvis/sperner.pdf
This is exactly the result of doing what I already suggested.
Umm, John? Did you _read_ the article you cited? It is a proof[*] by
contradiction that there cannot be _no_ zero point.
That's how I read it, too.
I think I can see that two poles can be arbitrarily close, but to have them coincide would require that some arbitrarily small area would always contain two tangents at 180 degrees to each other, which I think contradicts the definition of "smooth".
You can have a parting and a cowlick as close as you like, but they can't be in the same place.
That gives exactly no indication of how _many_ zero points will be present in vector fields on a sphere. The diagrams in your reference are all done _starting_ from
a "pole" on the sphere ("By the continuity assumption there exists an open disc P on S 2 about the north pole N within which all hairs essentially point in the same direction.") This is actually a bit odd,
as there is nothing "polar" about any of the argument -- but it is a
hint that something similar may be happening at the "south" pole :-).
Note also the persistent "hemispherical" context of the argument.
The article slides (without comment) from "popular" language (about some "hair" that has to "stick out" in a "cowlick") to a more accurate
specification about non-zero tangent vectors at each point on the
sphere. Frankly, I think the exposition is really terrible. I know
what they are talking about, and I find it hard to follow.
My grandmother was a strict atheist, and didn't believe in metaphysics.
---
[*] effectively, all proofs I have seen of this result are roughly
similar, in being proof by contradiction to the claim that there are
_no_ zeros of the tangent field. If you look at tangent fields in more detail, questions of orientable manifolds vs. non-orientable ones
begin to be very much relevant. Projective spaces _can_ have single
zero points in a tangent field. I can't baldly assert it to be true
(it's been 40 years since I actually knew this stuff :-)), but I do
not think that is the case for spheres.
.
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