Re: definition of science

From: D. F. Siemens, Jr. <dfsiemensjr@juno.com>
Date: Wed Apr 27 2005 - 15:53:23 EDT

I oversimplified the geometry. Sorry. However, the last I heard, the
universe is flat to some ridiculous decimal. For the rest, your quotation
from Einstein illustrates the principle underlying my analogy--a match
between mathematics (the model) and the measurements. However, the shift
to a Riemannian geometry because of the inclusion of time is not
necessarily that simple. There are, after all, multidimensional Euclidean
geometries.

As to why simplicity, the answer that immediately suggests itself is that
that is all the human intellect can grasp. There are, after all, an
infinite number of models for each set of data, but only a few fall
within human ken. It has been noted that scientists have been aware that
combining simple linear equations can produce nonlinear results, but,
faced with that problem, they substituted a linear approximation. That
they could handle. Only recently has complexity theory been developed and
applied to empirical matters. But this still seems to be in its infancy.
Dave

On Wed, 27 Apr 2005 05:26:09 -0700 "Don Winterstein"
<dfwinterstein@msn.com> writes:
I don't understand the relevance of your first three paragraphs. Perhaps
if you were to add a few words to explain...?

DS: "...A simple geometrical requirement gives us radiation falling off
according to the square of the distance. ... On the other hand, is the
mathematics of GUTs all that simple?"

DW: But note that the "geometrical requirement" is that of Euclidean
geometry. Euclid did not know enough to justifiably claim that real
space corresponded to his geometry except possibly over a range of small
distances. For distances either greater or smaller than he could have
conveniently measured, the geometry of space could have been, in
principle, anything. We know now that real space does not correspond
precisely to Euclidean geometry.

As for the concept of mathematical simplicity, I rely on A. Einstein: In
a lecture "On the method of theoretical physics" he stated, "Our
experience hitherto justifies us in believing that nature is the
realization of the simplest conceivable mathematical ideas." What can
that mean? The average college senior would not find the math of General
Relativity all that simple, either. From the rest of the essay I
conclude that by simple math he means math that is as simple as possible
while at the same time consistent with everything we know. From Special
Relativity we know space has four dimensions, so he assumes a Riemannian
metric. "If I [then] ... ask what are the simplest laws which such a
metric system can satisfy, I arrive at the relativist theory of
gravitation in empty space. If in that space I assume a vector-field or
an anti-symmetrical tensor-field which can be inferred from it, and ask
what are the simplest laws which such a field can satisfy, I arrive at
Clerk Maxwell's equations for empty space." And later on, "The important
point for us to observe is that all these constructions and the laws
connecting them can be arrived at by the principle of looking for the
mathematically simplest concepts and the link between them." Well,
speaking as an ex-physicist, the main thing these comments do for me is
make me envious of his grasp of the subject. But although I don't know
precisely how--given his assumptions--he determines what is
mathematically simplest, I assume he knows what he's talking about.

My ultimate point would be that there's no a priori reason to believe the
world should be mathematically simple; hence it comes as a surprise.

Don

----- Original Message -----
From: D. F. Siemens, Jr.
To: dfwinterstein@msn.com
Cc: asa@calvin.edu
Sent: Tuesday, April 26, 2005 3:53 PM
Subject: Re: definition of science

Let mr try a simple analogy. First, any mathematical of logical calculus
will produce logically true theorems. They are consistent for all
substitutions of variables, though practitioners will have fits about
substitutions, partly because they will not be true for all
substitutions. Thus Peano's postulates, beginning "Zero is a number," are
consistent and true (but limited), for integers. The parody, "Fido is a
dog," is equally consistent but not true. However, Peano's pattern
perfectly fits other numerical sequences, like 0, 2, 4, 6, ... and any
other sequences that can be set in a 1 to 1 relationship to the integers.
This last includes, as a matter of practice outside of mathematics,
partial matches.

Now consider the sequence 1111, 1110, 1101, ..., 0000, sixteen of them.
Imagine that they are part of a set of 4x4 matrices, 256 of them,
representing the entire range of mathematical calculi--though that's a
stretch. Now imagine that we make empirical measurements giving ?11?,
matched to the first row of the matrices. These are compatible with all
matrices containing the first, second, ninth and tenth sequence in that
row, that is, a quarter of the 256. But each of these 64 make different
"predictions" in the rest of the matrix. So the empirical task is to make
measurements corresponding to these predictions, in so far as possible.
Some will match. Others will not. Those that match are appropriate
theories.

What if a measurement turns up beyond the rows or columns of the
matrices? Either we look for a larger matrix, or we engage an auxiliary
hypothesis.

As to why we seem to get simple and elegant connections, I think it is
partly because we break up our problems into smallish pieces that bits of
math can fit. But some of it may spring from the crudity of our
measurement. A simple geometrical requirement gives us radiation falling
off according to the square of the distance. But there has been some
discussion of the factor being 2-n rather than 2, with the hope that more
accurate instruments would show this at great distances--that is, beyond
the Local Group. On the other hand, is the mathematics of GUTs all that
simple?
Dave
Received on Wed Apr 27 15:59:15 2005

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