From: Don Winterstein <dfwinterstein@msn.com>

Date: Thu Apr 28 2005 - 04:02:36 EDT

Date: Thu Apr 28 2005 - 04:02:36 EDT

"...The shift to a Riemannian geometry because of the inclusion of time is not necessarily that simple...."

You're right, Einstein needed a kind of 4-D space that could locally change shape.

"As to why simplicity, the answer that immediately suggests itself is that that is all the human intellect can grasp."

Perhaps we've beaten this almost to death; but what I hear Einstein saying is that the simplest math is not just what is comprehensible to human minds but is what the world precisely fits. Measurements support the precision of fit. He's making a statement about how the world is made.

Don

----- Original Message -----

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

To: dfwinterstein@msn.com<mailto:dfwinterstein@msn.com>

Cc: asa@calvin.edu<mailto:asa@calvin.edu>

Sent: Wednesday, April 27, 2005 12:53 PM

Subject: Re: definition of science

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<mailto: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.<mailto:dfsiemensjr@juno.com>

To: dfwinterstein@msn.com<mailto:dfwinterstein@msn.com>

Cc: asa@calvin.edu<mailto: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 Thu Apr 28 04:05:44 2005

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