From: D. F. Siemens, Jr. <dfsiemensjr@juno.com>

Date: Tue Apr 26 2005 - 18:53:57 EDT

Date: Tue Apr 26 2005 - 18:53:57 EDT

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

On Tue, 26 Apr 2005 14:12:24 -0700 "Don Winterstein"

<dfwinterstein@msn.com> writes:

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

reality.

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.

Don

Received on Tue Apr 26 18:59:34 2005

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