[FOM] Reverse Mathematics and underdetermination

[Marcin Mostowski]

YOUR PROBLEM, FORMULATED AS FOLLOWS

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? 

IS BASED ON MISUNDERSTANDING.

Our mathematical experience is not based on theorems, but on mathematical reality!

Having two elements you have potentially infinte set of truths about them.

Having more elements you have less determined reality.

Nevertheless huge universes, like those of size 2^1000, are still determined, because we can experience them.

The problem is that we imagine that there are very huge numbers which are outside of our experience. This is a mistake. We can experience greater numbers than we experienced previously. However it does not mean that mathematics is about thouse 'unexperienced numbers'. Just opposite, it says only about thouse experienced!

Logic is about mathematical theorems, but not mathematics itself!

Marcin Mostowski