[FOM] Reverse Mathematics and underdetermination

[Monroe Eskew]

What is a "part" of mathematics?  Could it be a finite set of theorems?  If so, then any finite part can be given a good axiomatization.  Could it be the set of all logical consequences of a given statement?  The obviously that statement is an axiomatization of this part.  So these cases do not seem relevant to the philosopher's worry.  The philosopher must be referring to mathematics as a whole, or something more complex or nebulous.

There is a large number of, but finitely many, mathematical facts which are known to be equivalent to WKL_0 over RCA_0.  Therefore RCA_0 + WKL_0 is a good, succinct axiomatization of these facts.  Furthermore it is an axiomatization of the set of all its consequences.  As argued, this kind of case does not seem relevant to the philosopher's worry.

It could be that I don't understand the philosopher's worry.

Best,
Monroe


On Jul 19, 2012, at 2:40 AM, Sam Sanders <sasander at cage.ugent.be> wrote:

> Dear members of the FOM-list,
> 
> During a recent conference, a number of philosophers presented the usual 
> problem of underdetermination (in the context of FOM).
> 
> 1) There are (potentially) infinitely many mathematical theorems.
> 
> 2) We can only know a limited number of mathematical theorems.
> 
> 3) Given 1) and 2), how can we know that an axiomatization (of a part of) mathematics is a good one,
> as we only know it to agree with a limited number of theorems?
> 
> The obvious slogan of underdetermination is that "any model is underdetermined by the data".
> 
> 
> I was wondering what your opinions are on underdetermination in light of Reverse Mathematics?  For instance:
> 
> Could we not say that e.g. WKL_0 is OVERdetermined (even if you only agree with a weak version of the Main Theme of Reverse Mathematics)?
> In particular, are the large number of equivalences over RCA_0 with WKL_0 not a mountain of evidence in support of the 
> claim that WKL_0 corresponds very tightly to a certain part of mathematics?
> 
> Furthermore, the claim has been made that Reverse Mathematics is too dependent on e.g. the choice of base theory (RCA_0).
> Would this dependence claim nullify the overdetermination claim?
> 
> 
> With kindest regards,
> 
> Sam Sanders
> 
> ps: Please note that I did not claim that RM is too dependent on choice of the base theory RCA_0.  
> I believe RM is not (too dependent on …).  
> 
> 
> 
> 
> 
> 
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