Re: definition of science

On the infinite number of models, we go back to the turn of the past
century. French engineers were working on a project of designing devices
to produce various motions, with the aim of having a specific plan to
introduce to produce whatever motion their larger device needed. They
hoped for a simple menu to solve engineering problems. A country
engineer, Koenigs, presented a proof that any pattern of motion could be
arrived at by an infinite number of different devices. I ran the paper
down in /Comptes rendues/ some decades back, but do not remember the
issue. Poincare expanded the proof in one of his books to cover any set
of data that fall under the least action principle. Logico-mathematical
models are more flexible than mechanical devices, which finishes the
proof.

As a practical matter for supercession, theorists have to produce
theories that make different predictions. However, such replacement is
not necessary. It may be enough to produce a more convenient version. I
have noted that some recent proofs, Fermat's last theorem and the 4-color
map problem, are appallingly long. A new approach leading to an elegant
proof would not prove something new, but would be hailed. From a
different angle, experimentalists won't know what to look for unless a
theorist has produced a tentative map. Theorists can't know that they've
produced something relevant until experimentalists can test it.
Denigrating the alternate approach is quite silly, but very human.

As for QM, the earliest approach to quanta involved particles--Planck,
Einstein, Bohr, Sommerfeld. De Broglie devised a wave theory later, I
think about 1923. I believe Schrodinger showed the equivalence of the two
approaches a few years later. Since the historical development of theory
is not all that important to practitioners, but solutions and
applications are, you got the amalgamated approach.

When I need to check out with my groceries, the several clerks are
equivalent. Each one is equally skilled in scanning the items and taking
my money. They are, however, not identical. Leibnitz held the theory of
the identity of indiscernibles. But it does not seem to be a necessary
principle. I may not, in mathematics, be able to distinguish between 3+1
and 2+2, but there is a difference between a fried egg and a 3-egg omelet
and 2 2-egg omelets. Euclid's "Things equal to the same thing are equal
to each other" holds in geometry, but not necessarily for comestibles.
Dave

On Sun, 1 May 2005 06:23:15 -0700 "Don Winterstein"
<dfwinterstein@msn.com> writes:
DFS: I have noted that there are an infinite number of theories/models.
To expand, some of them will be exact matches (QM alternates).

My thesis chairman used to say, "Theorists can explain anything in a
variety of ways. It's up to experimenters to say who's right." He was
an experimenter (as I also became). To be relevant, theorists need to be
able to make predictions that distinguish their theories from others. I
suspect theories that are exact matches in that all their predictions are
identical to those of another theory are in fact identical theories even
though expressed differently. In other words, it should be possible to
show how two theories that predict the same things are alternate ways of
saying the same thing.

You say, "Philosophers insist that equivalence is not identity." I'd
need proof. In classical mechanics we have various formulations such as
those by D'Alembert, Lagrange and Hamilton, but I've never heard anyone
call these different theories; and in fact I'm sure it's possible to show
they're all different ways of saying the same thing.

Going back to your comments on string theory: No doubt the math is very
complicated (but I haven't looked at it, so I can't speak from
experience). Greater complications are to be expected when the scope of
the theory is greater. General Relativity is more complicated than
Newton's law of gravitation, but it explains more. String theory tries
to explain a lot more. But complicated as these theories are, in terms
of Einstein's previously cited essay they would be simple in Einstein's
usage.

(But string theory at this stage is still a branch of math. Not until it
makes testable predictions can it be considered physics. Science
requires a hard connection between the idea and the real world.)

QM as I understand it, however, would not be simple in that sense.
Robert Dicke (with James Wittke) was the author of my undergraduate QM
text, which lists seven "postulates of quantum mechanics." These are
essentially assumptions that relate the math to the physics. To give the
flavor, his postulate 1: "It is assumed for a system consisting of a
particle moving in a conservative field of force (produced by an external
potential) that there is an associated wave function, that this wave
function determines everything that can be known about the system, and
that it is a single-valued function of the coordinates of the particle
and of the time." So QM is not derived from basic principles but simply
assumes a lot of stuff at the outset. This is not significantly
different (I think) from the "piecemeal curve fitting" I mentioned
earlier (and possibly gave Einstein fits).

DFS: My impression from outside is that the "particle" theory and the
"wave" theory give identical results for all computations.

DFW: OK, I have to say I don't know where you're coming from here. I've
had several graduate-level courses in QM but have never heard of this.

Don

----- Original Message -----
From: D. F. Siemens, Jr.
To: dfwinterstein@msn.com
Cc: asa@calvin.edu
Sent: Saturday, April 30, 2005 12:34 PM
Subject: Re: definition of science

As I recall, I have noted that there are an infinite number of
theories/models. To expand, some of them will be exact matches (QM
alternates). Others will match up to a point (Whitehead and Einstein).
The impression I got is that some of the theories mentioned in the SciAm
article belonged in each of these categories--Dicke's in the latter.

Call it what you will, a particle is not a wave, and neither are
wavicles. A hole can be punched with a bullet or with a laser. But they
are not the same. My impression from outside is that the "particle"
theory and the "wave" theory give identical results for all computations.
However, some computations that are devilishly difficult in the one are
relatively simple in the other, but there are two theories for the same
results. The pair, because equivalent, are used as one by scientists.
Philosophers insist that equivalence is not identity. Scientists round
off and substitute linear approximations for nonlinear equations.
Mathematicians say it's wrong. This depends partly on the fact that
scientists have to measure, necessarily with limited accuracy, while
mathematicians submit to a perfect ideal. After measuring with limited
accuracy, scientists idealize their results. Empiricism has its
requirements, as do formalisms.
Dave

On Sat, 30 Apr 2005 01:41:02 -0700 "Don Winterstein"
<dfwinterstein@msn.com> writes:
DFS: Are you suggesting that only one mathematical model fits? ...

DFW: Einstein believed, as do I, that one is superior to all others.
Only experiment can decide. (Although, if you're Einstein, you can
apparently decide also on the basis of which theory has the simpler math.
 : ) ) RH Dicke also presented an alternative to Einstein's General
Relativity, as I recall. I think he wrote the SciAm article. But none
of the other versions have earned widespread acceptance.

DFS: Quanta may be approached either as particles or as waves, equivalent
theories. ....

DFW: There are different representations of the QM formalism (e.g.,
Dirac's bra & ket notation), but it's the same theory. Lots of people
would like a more "reasonable" theory, and I've heard of attempts, but no
one has yet improved on QM as we know it. As for waves and particles,
it's not that we're invoking different but equivalent theories but that
we're doing different kinds of experiments. Particles are also waves;
but when measuring them, some experiments detect particle-like behavior
and others wave-like behavior. It's called complementarity. Or don't I
understand what you're saying?

Don

----- Original Message -----
From: D. F. Siemens, Jr.
To: dfwinterstein@msn.com
Cc: asa@calvin.edu
Sent: Thursday, April 28, 2005 11:02 AM
Subject: Re: definition of science

Are you suggesting that only one mathematical model fits? After Einstein
presented his work, Whitehead came up with a different version. Not
liking Riemannian geometry, his was based on Euclidean. Eddington proved
that the two were equivalent on the four matters then recognized as
relevant. Later work disproved Whitehead's version of relativity because
of other matters. I recall an article in /Scientific American/ that
presented additional relativity theories, though it did not discuss the
calculus underlying them. Some apparently were equivalent to Einstein's
theories, while others were designed to be slightly different.

Quanta may be approached either as particles or as waves, equivalent
theories. I understand that two approaches to string theory were
demonstrated equivalent. In other words, the fit is multiple. Beyond
that, are four dimensions simple? What about 10 or 11? Is seeing a
matching pattern simple? Once seen, it's "obvious," of course. Then why
does it take brilliant people so long to see it? How many have an /annus
mirabilis/?
Dave
Received on Sun May 1 19:38:31 2005