[FOM] Choice of new axioms 1 (reply to Friedman)
joeshipman@aol.com
joeshipman at aol.com
Mon Feb 13 22:57:12 EST 2006
Friedman:
I don't see that you have made the connection. Again, I doubt any
intuition
behind RVM for people who are not confused, once one realizes that there
cannot be any translation invariant countably additive probability
measure
on [0,1]. This was known very early on.
So a countably additive probability measure on all subsets of [0,1]
can't be
translation invariant, and hence the idea that there is any intuition
that
there is a countably additive measure on all subsets of [0,1] is very
far
fetched.
Sihpman:
You again miss my point. I am talking about the time PRIOR to the
Vitali/Hausdorff/Banach-Tarski results which called into question
translational and rotational invariance (these were respectively
1905/1914/1924).
Again, I am not claiming that the validity of RVM depends in any way on
obsolete intuitions. All I am claiming is that it USED to be the case
that RVM was considered intuitively plausible, and that IF physics had
been developed in a different way so that "naive" physical intuitions
persisted for a while longer, mathematicians might have reacted to the
Vitali/Hausdorff/Banach-Tarski results by discarding the intuition that
space was invariant rather than discarding the intuition that matter
was infinitely divisible.
Eventually, BOTH intuitions would be discarded, but in the meantime the
intuition that matter was infinitely divisible would lend sufficient
plausiility to RVM that it would have a chance to be established as a
MATHEMATICAL axiom, once the work of Ulam and Godel showed that RVM
established Con(ZFC).
Friedman:
I do not know of any evidence concerning the number of mathematicians
back
then that were even considering the idea of expansion of the axioms. You
seem to think that this number was significant enough to speculate about
alternate historical paths.
Shipman:
No, I'm assuming that mathematics would have followed approximately the
SAME path, in the sense that there would have been the same crisis in
foundations brought on by the Russell paradox, and the same successful
response to the crisis involving the development of ZFC -- but IF some
one had formulated RVM 30 years before Ulam did, and it was regarded as
an interesting open question at a time when "intuition" still supported
it, the reaction to the discovery that it proved the Consistency of ZFC
would have been "we need to add this as an axiom" rather than "this
wasn't plausible anyway".
Of course, many mathematicians reacted to the
Vitali/Hausdorff/Banach-Tarski complex of results by becoming
suspicious that AC was false; so another path would be to deny AC and
assume there was an invariant countably additive measure on all subsets
of [0,1].
Here's another way of looking at it. Hilbert could have discovered in
1900, BEFORE Vitali, that RVM is inconsistent with CH; don't you think
he would have considered this very significant, and regarded it as
providing some evidence that CH might be false?
>
> Shipman:
>
> I mean that no reformulation of fundamental physics has been plausibly
> proposed that avoids using the real numbers and only talks about
finite
> objects (or even countable objects; all the fundamental theories
> involve ontologies going way beyond second-order arithmetic).
Friedman:
This is incorrect. All of the so called fundamental theories I know
anything
about are conveniently formulated in very very weak systems. Z_2
discusses
real numbers and complete separable metric spaces in the obvious way.
One
can use very weak fragments of ZF\P if one wishes to avoid some of the
standard coding involved. If one wishes to avoid coding entirely, then
one
can use convenient conservative extensions.
Shipman:
It is not incorrect. The systems they are formulated in may be weak
LOGICALLY, but as I said, they are not weak ONTOLOGICALLY. You may
choose to code the functions, Hilbert space operators, and other
objects of higher type in second-order arithmetic, but the resulting
theory would be unnatural and unsuitable for actually doing physics.
When I said "no reformulation...has been plausibly proposed" I meant a
reformulation that a PHYSICIST would find plausible.
This was discussed here a couple of years ago. The OBJECTS of the
fundamental physical theories, the things that physicists actually
refer to, would be completely unintelligible if the theories were coded
into Z2, and even Z2 still involves real numbers in an essential way.
Friedman:
I am referring to the position taken that general relativity is not a
fundamental theory because advanced quantum theory must be taken into
account in any truly fundamental theory.
shipman:
Don't let the perfect be the enemy of the good. General Relativity is
the best we've got -- a theory can be "fundamental" without having to
be a "theory of everything"!
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