[FOM] Choice of new axioms 1 (reply to Friedman)

[joeshipman@aol.com]

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"!