"D. F. Siemens, Jr." wrote:
> 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.
The terminology varies but "Riemannian Geometry" is used for b. E.g.,
that's the title of a classic (now dated) monograph on the subject by
Eisenhart. That's only one type of differential geometry. There are more
general types, like those Einstein et al. used in later attempts at unified
field theories.
> > 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.
It depends on what you mean "from scratch." It took Riemann's
genius - building on work of Gauss &c to come up with Riemannian geometry
(sense b) in 1854. It was relatively straightforward for Weyl & others to
generalize it. Nowadays all workers in general relativity can work out the
math & its application from some basic principles, having worked with the
stuff long enough. Same with other branches of math - a decent math
undergrad today understands the foundations of calculus better than did
Newton or Leibniz, but that doesn't mean he/she is smarter than they were!
> > 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.
To some extent this depends on how platonic one is. I'm enough of a
platonist (though with important qualifications because of both theology &
physics) to want to say, e.g., that Bolyai & Lobachevski, e.g., discovered
non-Euclidean geometry, & not simply that they invented it. But of course
"discovery" here means something different in pure math than it does in
physics. & as I noted before, what they discovered was not a unique "true"
geometry.
> 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.
But it's worth noting that group theory is used in differential
geometry. In fact, some of the most important recent work on gauge theories
&c has come about through the use of group theories in various abstract
spaces.
> 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.
Poincare (& I'm not looking stuff up either) argued for some form of
"conventionalism." E.g., you can simply declare at the start that Euclidean
geometry is correct & develop whatever auxiliary hypotheses you need in order
to explain observational data consistently with that basic convention. That
would be done by saying that gravitation affects clocks & measuring rods in
appropriate ways.
In the language of Lakatos' philosophy of science, Euclidean geometry
in that case is the "hard core" of the theory & the statements about clocks &
measuring rods are part of the "protective belt" which can be altered as need
be to keep the hard core intact. The question then is, can the theory
predict novel phenomena, or do you have to keep adding auxiliary hypotheses
as new observations are made? If the latter is the case, & that keeps up
long enough, the research program is "degenerating" & will be abandoned.
That's what happened, e.g., with steady state cosmology.
While new math tools (Riemannian geometry, matrices, group theory &c)
keep being introduced into physics, it's always been possible so far to show
that the new theories are covering theories of the old. There isn't one
realm of phenomena where you must use Euclidean geometry and another where
you must not. Rather, the areas where Euclidean geometry works are those in
which Riemannian geometry approximates Euclidean. Thus there really is a
sense in which there is a single math pattern for all physical phenomena. &
the fact that math has often been used to predict qualitatively new phenomena
suggests that we're not just making up this pattern as we go along. Both of
these facts together don't prove that there is a objective rational math
pattern to the world, but they indicate it strongly enough that the amazement
of Wigner, Polkinghorne, &c seems justified.
Shalom,
George
George L. Murphy
http://web.raex.com/~gmurphy/
"The Science-Theology Dialogue"
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