From: John K Clark (jonkc@att.net)
Date: Fri Oct 25 2002 - 22:48:41 MDT
> > The Identity of Indiscernibles is a principle of analytic ontology
> > first explicitly formulated by Wilhelm Gottfried Leibniz in his
> > Discourse on Metaphysics, Section 9 (Loemker 1969: 308). It states
> > that no two distinct substances exactly resemble each other. This is
> > often referred to as 'Leibniz's Law' and is typically understood to
> > mean that no two objects have exactly the same properties.
I once wrote a dialog called "Waiting for Zed" , it rambles on about a lot
of stuff but the part about The Identity of Indiscernibles is below and
shows how you can use it to work out exchange force and deduce the
laws of Chemistry.
=========
Bob: Ever hear of The Identity Of Indiscernibles? The philosopher who
discovered it was Leibniz about 1690.
Alf: Didn't he co-invent The Calculus along with Newton?
Bob: Yea, same fellow. He said that things that you can measure are what's
important, and if there is no way to find a difference between two
things then they are identical and switching the position of the objects
does not change the physical state of the system.
Alf: Big deal. I'm not talking about some un-provable idea in pure
Philosophy, I'm talking about practical questions, like if it's worth
paying extra for an original, or even more practical if a copy of you is
really you. Maybe Religion can help us with questions like that, but not
Science.
Bob: Actually Science can help us, and Leibniz's idea turned out to be very
practical, although until the 20th century nobody realized it, before that
his idea had no observable consequences because nobody could find two things
that were exactly alike. Things changed dramatically when it was discovered
that atoms have no scratches on them to tell them apart. By using The
Identity Of Indiscernibles you can deduce one of the foundations of modern
physics the fact that there must be two classes of particles, bosons like
photons and fermions like electrons, and from there you can deduce The Pauli
Exclusion Principle, and that is the basis of the periodic table of
elements, and that is the basis of chemistry, and that is the basis of life.
If The Identity Of Indiscernibles is wrong then this entire chain breaks
down and you can throw Science into the trash can.
Alf: That's an awful long chain of reasoning, if it has one weak link
perhaps you should put it in the trash can, how can you base it all on
The Identity Of Indiscernibles?
Bob: I wish Zed was here, he knows a lot more about this than I do, but
let's start with one of the first and greatest discoveries in Quantum
Mechanics, The Schrodinger Wave Equation. It proved to be enormously useful
in accurately predicting the results of experiments, and as the name implies
it's an equation describing the movement of a wave, but embarrassingly it
was not at all clear what it was talking about. Exactly what was waving?
Schrodinger thought it was a matter wave, but that didn't seem right
to Max Born. Born reasoned that matter is not smeared around, only the
probability of finding it is. Born was correct, whenever an electron is
detected it always acts like a particle, it makes a dot when it hit's a
phosphorus screen not a smudge, however the probability of finding that
electron does act like a wave so you can't be certain exactly where that dot
will be. Born showed that it's the square of the wave equation that
describes the probability, the wave equation itself is sort of a useful
mathematical fiction, like lines of longitude and latitude, because
experimentally we can't measure the quantum wave function F(x) of a
particle, we can only measure the intensity (square) of the wave function
[F(x)]^2 because that's a probability and probability we can measure.
Let's consider a very simple system with lots of space but only 2 particles
in it. P(x) is the probability of finding two particles x distance apart,
and we know that probability is the square of the wave function, so P(x)
=[F(x)]^2. Now let's exchange the position of the particles in the system,
the distance between them was x1 - x2 = x but is now x2 - x1 = -x. The
Identity Of Indiscernibles tells us that because the two particles are the
same, no measurable change has been made, no change in probability, so P(x)
= P(-x). Probability is just the square of the wave function so
[ F(x) ]^2 = [F(-x)]^2 . From this we can tell that the
Quantum wave function can be either an even function, F(x) = +F(-x),
or an odd function, F(x) = -F(-x). Either type of function would work in our
probability equation because the square of minus 1 is equal to the square of
plus 1. It turns out both solutions have physical significance, particles
with integer spin, bosons, have even wave functions,
particles with half integer spin, fermions, have odd wave functions.
Alf: Wait a minute. Are you saying that an electron, something that can not
be pinned down and doesn't even have a diameter in the usual sense of the
word, is spinning around like a child's top?
Bob: No, not really. It's called "spin" for historical reasons and it's true
you can make an analogy with the everyday meaning of "spin", but the
analogy is no better than mediocre. For example, it is possible to tip the
axis of spin of an electron with a magnetic field, you might think that
if you turn an electron by 360 degrees it would end up just as it was
before, after all, if you make one complete turn you end up looking in the
same direction, but that's not true for an electron. Turn an electron once
and it's different, you need to turn it twice, 720 degrees, before
it's the same as it was before.
Alf: So if I spin around twice, the world would look exactly the same to me
after one revolution or two, but for an electron it would look
different. Do you think that means we can only see half the universe that an
electron can see?
Bob: I don't know, when Zed gets here why don't you ask him, but I was
trying to show that we must assume that atoms are interchangeable or modern
Physics becomes incomprehensible. If we put two fermions like electrons in
the same place then the distance between them, x , is zero and because they
must follow the laws of odd wave functions, F(0) = -F(0), but the only
number that is it's own negative is zero so F(0) = 0 . What this means
is that the wave function F(x) goes to zero so of course [F(x)]^2 goes to
zero, thus the probability of finding two electrons in the same spot is
zero, and that is The Pauli Exclusion Principle.
Two identical bosons, like photons of light, can sit on top of each other
but not so for fermions, The Pauli Exclusion Principle tells
us that 2 identical electrons can not be in the same orbit in an atom.
If we didn't know that then we wouldn't understand Chemistry, we wouldn't
know why matter is rigid and not infinitely compressible,
and if we didn't know that atoms are interchangeable we wouldn't understand
any of that. Atoms have no individuality, If they can't even give
themselves this property I don't see how they can give it to us.
John K Clark jonkc@att.net
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