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

Date: Wed Apr 27 2005 - 15:53:23 EDT

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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