[FOM] Reverse Mathematics and underdetermination

Aatu Koskensilta Aatu.Koskensilta at uta.fi
Fri Jul 20 14:39:32 EDT 2012

Quoting Sam Sanders <sasander at cage.ugent.be>:

> 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".

   Well, what is the model here and what is the data? You say, for  
instance, that  there are (potentially) infinitely many mathematical  
theorems. Just what is meant by mathematical theorem? Ditto for "good  
axiomatization", "part of mathematics", and so on.

> 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?

   It is impossible to say without first examining in some detail just  
what is involved in the over or underdetermination claim.

Aatu Koskensilta (aatu.koskensilta at uta.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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