"D. F. Siemens, Jr." wrote:
> Sorry to break this off, but this part involves something on which a
> comment is warranted. First, math is a broad area in which all the
> statements are "true in all possible worlds." This applies to arithmetic
> and number theory, geometry, the calculus, complexity theory, and all the
> rest of the special areas. This does not mean that all the theorems apply
> equally to all situations. Both "one line parallel to a given straight
> line can be drawn through a point on the plane not on the given line" and
> "there are no parallel lines" cannot be true when applied within any
> world simultaneously. They hold only in Euclidean and Riemannian
> geometries. Something similar holds for ordinary and modular arithmetics.
>
> Since within a specific mathematical calculus or system all statements or
> theorems are proved "true," if a set of interpreted statements holds, one
> may interpret other theorems in the expectation that they will also yield
> physical truths. There is no guarantee that the process will yield an
> unending series of truths. Changes may be required at some point. This is
> seen most obviously in Newton's work, which may be viewed as an
> interpreted Euclidean geometry. Everything worked for a couple of
> centuries. But the next step was Einstein's interpreted Riemannian
> geometry (Special Theory) which provided for Newton's results as a
> limiting case. The math gets stickier with the General Theory.
>
> As I see it, the problem from the empirical side is collecting enough
> data on which a discipline can be based; on the theoretical, a system
> that fits the observations, so that predictions can be made. This leads
> to expansion and testing. I recall a report in the 50s that the
> mathematics of something Einstein produced was so difficult that only
> three persons living had the background to understand it. One of them had
> taken the time to work through the math (18 months ?) and announced that
> it was consistent. The report quoted him, "I'm only a mathematician. I
> can't comment on the physics." I'm only a philosopher, so I don't know
> what Einstein had presented, nor whether it was an advance or a dead end.
> 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.
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.
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. I think the significance of the
discovery of consistent non-Euclidean geometries by Bolyai & Lobachevski
isn't often enough noted in this regard. This showed that more than one
consistent mathematical pattern for a world is possible. Thus one can't, in
platonic fashion, regard the physical world as a representation of a unique
math. Even if one wants to think in semi-platonic fashion, the creator must
have had some choice of which pattern to use. This is why Torrance, e.g.,
has insisted on the idea of the contingent rationality of creation.
I don't think that this makes 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.
Shalom,
George
George L. Murphy
http://web.raex.com/~gmurphy/
"The Science-Theology Dialogue"
This archive was generated by hypermail 2b29 : Thu Aug 09 2001 - 16:53:32 EDT