On Thu, 09 Aug 2001 16:53:43 -0400 george murphy <gmurphy@raex.com>
writes:
> "D. F. Siemens, Jr." wrote:
> > But I suspect that one reason the applicability of mathematics is
> lauded
> > is that the dead ends are not pursued and are soon forgotten. This
> type
> > of selectivity certainly functions in other areas of human recall.
> So I'm
> > not too surprised that our mathematics fit the world.
>
> Dave -
> 3 comments, in (I think) ascending order of importance.
> 1) A possible source of confusion (not just in your post)
> is the
> ambiguity of the term "Riemannian geometry." This can mean either
> a. the result of choosing the alternative to the
> Euclidean
> parallel postulate which Dave describes, a geometry which can be
> realized on
> the surface of a sphere with antipodes identified, or
> b. a differential metric geometry - i.e., one in which
> a unique
> separation is defined between any two nearby points, but whose
> properties may
> vary from one point to another. It's the 4-D version of this that
> Einstein
> used for general relativity.
If my memory serves, the latter is usually described as a Riemannian
metric rather than a geometry. I ran across "differential geometry," one
of the extensions of geometry and other mathematical approaches which I
think applies. The abstract areas do not lend themselves to the
visualizations of finite planes, spherical and saddle surfaces of the
traditional variations on geometry. But some of the labels cling from the
connections.
> 2) From the date I suspect that the other theory of
> Einstein which
> you mention is his last attempt at a unified field theory which used
> a
> differential geometry more general than that of Riemann.
> (It's described in Appendix II of the last edition of Einstein's The
> Meaning
> of Relativity.) The notion that only 3 living persons had the
> background to
> understand it was nonsense - like "Only six men in the world
> understand
> relativity." The math is somewhat more complicated than that of
> general
> relativity but not qualitatively so. I did some work on it - or
> more
> precisely, on a closely related approach of Schroedinger's.
> Unfortunately
> I've come to be about 98% sure that it's a dead end as far as
> physics is
> concerned.
First, thanks for the clarification. Second, I think there is some
ambiguity in terms of understanding. Let me illustrate. I know enough
simple math to calculate either correlation (Pearson or Spearman). It was
somewhat tedious with paper and pencil, but clearly possible.
Unfortunately, the computation will work whether I distinguish rank from
ratio and use the correct formula or don't know the difference and
produce garbage. I ran across a text on statistics which computes the
foundations of these measures from mathematical principles. It was way
over my head, but many know enough calculus to follow the proofs.
However, there are very few who could start from scratch as I presume the
text's author did. I take him to be analogous to the three, and those who
can work with the given system as you did to be like those who can
understand the text.
> 3) Yes, a lot of theories which are mathematically
> consistent (at
> least as much as Goedel will allow), beautiful, &c are bad physics -
> i.e.,
> they don't correspond with the real world.
snip
> This is why Torrance,
> e.g.,
> has insisted on the idea of the contingent rationality of creation.
> I don't think that this makes what Wigner called the
> "unreasonable effectiveness" of math in describing the world any
> less
> remarkable. There is no a priori reason why our experience of the
> physical
> world should correspond closely to any math pattern.
>
From: "iain.strachan2" <iain.strachan2@ntlworld.com>
in small part
You would appear to differ from Polkinghorne on this one. It's not a
question necessarily of picking the maths to fit the world. When Abel
Galois laid down what was to become the foundations of group theory, he
had
no idea that it was going to have direct relevance to particle physics
and
quantum theory, because those fields of science had not even been
discovered. It really was just an "abstract free creation of the human
mind". Maybe God's mind as well? Maths is full of peculiar facts that
apparently have no relevance to nature. For example, the "Ramanujan
constant" e^(pi*sqrt(163)), which differs from a 40 digit integer by
around 10^-12. A coincidence? No, apparently there is a very deep
reason,
also, I understand, connected with group theory - who knows? maybe there
is
a physical relevance to this as well - but it started out as "just an
interesting fact" in the abstract realm. It is this aspect - of the
abstract ideas of the human mind turning out to fit the world, that
Polkinghorne finds so amazing, and that he relates to the Logos/Creator.
Iain.
I don't see that either George's or Iain's statements are responsive to
my point. We cannot prove mathematical systems consistent. For example,
the best we can do is demonstrate that if one of the 3 geometries is
consistent, the others are as well. But, after millennia of use, we can
presume consistency. The same holds, though with less confidence, for the
many variants mathematicians have produced.
Mathematicians play with systems. They put two together, as Descartes did
to produce analytical geometry. They change or add axioms to see what
happens. They come up with conjectures and try to prove them. These are
purely abstract structures. If Riemann and Galois (or someone) had not
produced their systems, they could not have become basic to a scientific
theory. What isn't available cannot be used.
All the consistent theories are true in all possible worlds. But, in this
sense, they are vacuous. However, when one of the systems is interpreted
in such a way that it fits a set of empirical observations, we can expect
it to yield predictions for additional observations. Not every system can
be so interpreted, of course, but mathematicians have produced some that
can be applied in various areas. This does not allow indiscriminate
interpretations. A Riemannian metric cannot be substituted for group
theory, or vice versa. Further, there is no warrant that mathematicians
will have produced a system which can be interpreted to match the
empirical data, or that any given scientist will see a connection.
There is another piece to my suggestion that there is no reason to wonder
that math gives us the basis for scientific theories. Poincare (I'm right
now too lazy to find the references) noted a paper by Koenigs about the
turn of the last century that proved that any set of data fitting the
least action principle has an infinite number of mechanical models. This
least me to the view that any consistent set of data has an infinite
number of logico-mathematical models. So I am not surprised that we
should encounter a connection between one of the systems giving
statements true in all possible worlds and the facts in our world.
Polkinghorne and others are amazed simply because they have not
considered the number of potential matches, if we but recognize them.
Dave
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