[FOM] Reverse Mathematics and underdetermination

Sam Sanders sasander at cage.ugent.be
Thu Jul 19 05:40:30 EDT 2012

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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