FOM: lagrangian pebbles
rhersh at math.unm.edu
Wed Mar 18 18:26:10 EST 1998
I haven't studied number theory since my prelims at Courant,
so I'll take an easier example.
3 + 6 = 9
This is a theorem in PA and also in the 2d grade.
Say we know the solar system has 3 inner planets (mercury, venus,
earth) and 6 outer ones (mars, jupiter, saturn, neptune, uranus, pluto.)
Then we can conclude that there are 9 planets in the solar system.
Futhermore, since we are pretty sure the solar system is older than
homo sapiens, there were 9 planets in the solar system before anybody
knew about it.
No problem with any of that.
But now Prof. P. (for Platonist) says this shows that the number
9 (not to mention 6 and 3) "existed" before there were any people.
If there were 9 planets, doesn't that mean there was the number 9?
It seems to me that here we have an important confusion, between
an adjective and a noun. 9 as in nine planets is an adjective
(Watch out! Here comes V.P., "shocked" at an "elementary blunder.")
9 as in the number 9 is a noun.
9 as in nine planets is a counting number. It's part of
what Bunge calls "protophysics." Our language for talking
about physical reality. It does not take much reflection to
see that while there is no largest counting number, no sharp
upper bound, it is easy to write down a numerical expression
than which every actual act of counting that ever takes place
will be much smaller.
On the other hand, as we know, the noun numbers, as in PA or
in the 2d grade, are infinitely many, and have no upper bound
(no fair dragging in an aleph!)
So the counting numbers and the pure numbers (noun numbers)
are not the same thing. Of course they are closely
related, historically, psychologically, and in every act
of practical computation . Nevertheless, they ain't the
same. The noun numbers, or the pure numbers, are
our invention, as much as the reals, the complexes, the
quaternions, Clifford numbers, square and rectangular matrices,
semi-infinite and infinite matrices.....
The counting numbers are part of nature, in the same
sense that mass, volume, etc. are part of nature.
If there were a spherical homogeneous object in nature, no
doubt its mass would equal its density times its volume, which
would be an instance of a mathematical theorem. The theorem
as a general statement would not be stated, therefore would
not exist, therefore would not be true, until someone thought of
it. There's nothing obscure about an instance existing prior to
a generalization which includes it.
Why do I believe astronomers can "predict" the motions of the
planets far in the past? Or your cute thing with Lagrange's
pebbles? Because I believe nature is in some senses orderly,
behaving in the past in a way like it behaves today.
So if Newton's law seems good today, we can believe it was
good at least in the recent past (astronomically speaking.)
That means it was a good description of the behaviour of the
nine planets. It was an appropriate adjective.
But the mathematical subject of potential theory (the 3 dimensional
gravitational potential) as existing apart from, independently of
physical phenomena, was created by Newton and his successors. It
did not exist even in Neanderthal times, let alone in the Jurassic.
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