[extropy-chat] (no subject)

[John K Clark]

ME:
>> Leibniz's  Identity Of Indiscernibles is the idea that if you exchange
>> the
>> position of two things and there  is no change in the system then the
>> two things are the same. If I place you (the copy) and the original an
>> equal distance from the center of a symmetrical room so you see the
>> same things and then instantly swap your bodies position with the
>> original then neither you nor the original nor any outside observer
>> could detect the slightest change.  Then I'm afraid you don't really
>> understand Leibniz's law.

Heartland" <velvethum at hotmail.com> Wrote:

 > Regardless of the arrangement, you can always come up with
>*at least one arbitrary* property  (measurement of the distance
> from a non-equidistant point, for example)

No, you are quite wrong, if 2 identical objects are on either side of a
point you can not say one particular object is to the right or the left
because you have no way of knowing if the objects exchanged positions. If
you were right then the laws of chemistry would be quite different.
Curiously I once write a play and this fact was one of the plot elements:

Alf: Big deal. I'm not talking about some un-provable idea in pure
     philosophy,  I'm talking about practical questions, like whether
     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 Schrödinger 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? Schrödinger 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