rhersh at math.unm.edu
Sat Mar 21 18:53:03 EST 1998
Again hoping not to contribute to the flood of irrelevant postings,
I must reply to the argument that potential theory always existed,
at least while there was a solar system, Lagrange's theorem was
always true, at least while there were piles of pebbles, etc.
I beg leave to stick to astronomy, leaving aside pebble sorting
for another occasion.
Nowadays applied mathematicians and physicists talk about models
(not logic-theoretic models!) more often than about theories.
It is commonly accepted that most physical problems of interest
are subject to several different descriptions, of which more than
one may be useful and interesting.
For Newton , the planets were point masses. He proved that this
was mathematically equivalent to their being homogeneous spheres,
and he knew as well as we do that in fact they were neither point
masses nor homogeneous spheres, but "Approximate spheres" in some
sense, whose density distribution was (and is) unknown, but to
a first approximation homogeneous, or at least independent of longitude.
His theory said the motion of the planets is influenced by the mass
of *every* heavenly body, but he could solve his equations of motion
only for the case of n bodies, n = 2. Thus the motions of the planets
were calculated as if each of them successively were alone with the
sun; and the satellites, as if each were alone with its planet. The
result was a spectacular success, he could derive his dynamics from
Kepler's 3 laws, and vice versa, and in particular got elliptical orbits.
So, for instance, ellipses are facts of nature.???
Today, of course, we think of Newton's ellipses as only first
approximations. The "perturbing" influences of other planets may
have to be taken into account. Newton treated the interplanetary
space as a perfect vacuum, but it's imperfect, the friction of
interplanetary matter may need to be taken into account. And
everybody knows that Einstein in some sense superseded Newton;
we say that we may have to take "relativistic effects" into
Depending on what we do or don't take into account, we get
various "models." Some of the models require physical data which
we can only guess at; you might say each different guess at
interplanetary garbage and inhomogeneity and asphericity of
the planets gives a different model.
Some of the asteroids have such irregular shapes that as they
circle they sun they tumble and turn in such complex ways that
there is no hope of predicting their orientation more than
a few orbits ahead. In fact, such motions are cited as a
physical example of a fractal and of chaos. This example shows
that the relatively simple orbits of the 9 planets are a piece
of good luck for earthly science. A planet orbiting alternately
around the two members of a double star might have posed a
hopeless puzzle for some extragalactic Ptolemy or Copernicus.
As soon as we give up Newton's 2-body idealization, with perfect
vacuum and homogeneous spheres, we lose any hope of exact
explicit calculations or formulas. Must have a computer.
What does that mean? Hardware, software, machine language,
compilers, high level languages, general purpose differential
equation solvers, special purpose differential equation solvers,
a whole amorphous know-how of numerical analysis, algorithm
invention and analysis, and finally a computer code that has
been tested (not proved!) to be stable and sufficiently accurate.
Not that I have ever touched NASA, but I imagine that when
Foebe went to the moon, Robin tracked her constantly. I
imagine that from time to time Foebe's position and speed
were a bit different than had been calculated aforehand,
and either Robin from the ground or Foebe in space performed
some adjustment to keep the craft on track.
Once the flight was over, I suppose the records compiled at
Houston Space Headquarters were sent to some think tank,
where an applied mathematician, with help from computer jocks,
tried to analyze where and how the program that controlled
the flight could be corrected or improved, either in
the assumptions about physical data or in the numerical
Why did I put you through this long tedious bit of ugly real life?
Well, the flight of Foebe's spacecraft is not essentially
different from any other problem in space dynamics, except that
Foebe or Robin had the capacity to make corrections in flight.
The question is, what if any of this very large, complex
physical-mathematical-computational effort is independent
of human thought?
Well, the position and properties of earth and moon. If we modify
this story to make it about a lifeless asteroid rather than a
NASA spacecraft, we could also say the shape, weight, etc of the
asteroid. And if the motion of the asteroid is not interfered with
by human hands, we would also say its actual path or trajectory is
independent of human thought.
I suppose there's no argument about that.
What about the different mathematical models, (none of which
is perfect!) the different computational tricks to solve a
system of nonlinear o.d.e.s, the non-well-posed inverse problems
to try and guess about the density distribution of earth, moon ,
or any relevant planets, and all the codes, languages and software?
Do you say that's all out there in space?
Of course if you believe any and all math is celestial and
timeless, than that would apply to the whole big deal to
track and control a satellite.
But keep in mind this isn't math as you all talk about it.
Damn few definitions or theorems. A lot of guesswork, trial
and error, seat-of-the-pants know-how.
You could just say, that isn't math, that's plumbing. But
then how do you use astronomy to prove the divinity of math?
I understand that everything in the computer is really nothing
but a sequence of ons and offs, or ups and downs, or zeroes and
ones. If you say zeroes and ones, the whole thing is just
some binary number or other. Of course like all binary, ternary
etc numbers it has existed since FAther Time first yawned.
But this doesn't have much to do with astronomy.
If you know already, regardless of any IRRELEVANT information,
that math is transcendentally abstractly inhumanly timeless,
then you're all set. No problem.
If you think math as it's really done (both pure and applied)
might really be relevant, then you have to pay attention
to what's really going on.
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