Re: definition of science

From: Don Winterstein <dfwinterstein@msn.com>
Date: Tue Apr 26 2005 - 17:12:24 EDT

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

  ----- 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 11:51 AM
  Subject: Re: definition of science

  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. There is not an exact match, for the gapless continuum cannot exist in a "granular" universe of atoms, quarks, electrons and photons. But the numerical applies down to the physical limit.

  There is a restriction in the application of any formal calculus to "reality." There has to be a match. Aristotelian logic cannot be used for items that exhibit degrees, for example. Statistics can deal with matters where a simple true-false doesn't work. Still, correlation measures are often misapplied, usually measures appropriate to ratios are used for ranked data. The formulas are as easily calculated to produce nonsense as to retain relevance. I recall hearing, some 3 decades ago, that a stat prof required his students to present six misuses of statistical measures in peer-reviewed publications. Some universities require that experimental designs be checked by a statistician before being implemented.

  Obviously, not all calculi can be applied. Einstein required a Riemannian metric. Current measurements indicate that the universe is flat, i.e., Euclidean. I do not recall encountering an application of Lobachevskian geometry. Also, applications differ. I recall ignoring imaginary roots that emerged from word problems in high school algebra, but physicists find them relevant in electromagnetic theory. "Imaginary" popularly equates to "nonexistent," but not always. One may need to change outlooks radically to discern the "obvious."
  Dave

  On Tue, 26 Apr 2005 06:51:45 -0700 "Don Winterstein" <dfwinterstein@msn.com<mailto:dfwinterstein@msn.com>> writes:
    Dave Siemens wrote:

    "...Many mathematicians are persuaded that they
    discover rather than construct such relationships...."

    Many mathematicians indeed believe their results somehow exist independently of human minds and are discovered rather than invented, but they continue to recognize that these results involve symbols exclusively and not objects of the physical world. Some of the mathematicians I've talked with have been horrified to think that their beautiful relationships might be considered for application to real world problems! There is a lot of math that as yet has no application to the physical world.

    The surprising thing to many physicists, formerly also A. Einstein, is how well certain physical relationships are described by certain math relationships. So there seems to be some deep connection between the world of pure symbol (math) and the physical world. ("God is a mathematician.") This connection underlies our ability to understand our world, especially as we expand the math meaning of symbol to include word, concept or idea. That some physical relations are accurately described by abstract math relations tantalizingly supports the idea that human understanding is sometimes close to absolute understanding and hence more than just metaphor.

    Don
Received on Tue Apr 26 17:36:49 2005

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