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,
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
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
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
On Tue, 26 Apr 2005 14:12:24 -0700 "Don Winterstein"
Dave Siemens wrote:
"That mathematical calculi should apply to empirical matters is
surprising only if one does not recognize their nature, which is the same
in this regard as that of logical calculi. Both are empirically empty but
present necessary connections. If the logical or mathematical variables
(terms) match the empirical ones closely enough, then the formal
connections will represent the empirical ones reasonably. ...."
I assert the surprise remains, and goes deep. What connections are
necessary? Why should math variables have anything to do with empirical
ones? Group Theory was irrelevant to physics for a long time, and to its
discoverers its variables had nothing to do with the real world. Now
it's become basic to particle physics, molecular physics and
crystallography. (Maybe others.) Riemannian geometry at one time was
similarly irrelevant (as far as I know), as was n-dimensional space for
n>3. In general, all branches of math begin with axioms and deduce the
rest. There's no requirement that the axioms correspond to any physical
What is the necessary reason, for example, that potentials in simple
cases fall off as r^-2 rather than, say, as r^-pi out to a certain
distance, then r^-1.87 for another range of r, etc.? What is the reason
that physical relationships should have any necessary connection to
purely symbolic relationships that people dream up in ivory towers?
Whatever an actual physical relationship might be, it would always be
possible to fit mathematical functions to it piecemeal to get a match.
Lacking compelling reasons to think otherwise, we ought to expect to have
to do such arbitrary fitting all the time. The surprise is in how often
observed relationships for simple systems are mathematically simple and
elegant. That seems to be telling us something interesting.
Received on Tue Apr 26 18:59:34 2005
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