[FOM] extramathematical notions and the CH

Aatu Koskensilta Aatu.Koskensilta at uta.fi
Tue Feb 5 19:11:07 EST 2013

Quoting joeshipman at aol.com:

> Any constant which can be approximated arbitrarily closely by an  
> algorithm is by definition arithmetical.

   Sure. But let me be even more explicit about the hypothetical  
scenario I have in mind. Suppose we have a physical theory that  
predicts that, say, the mass of a particle is equal to 0 if there are  
non-measurable sets of reals and 1 otherwise. It then follows that if  
we manage to measure the mass of the relevant particle, and find it to  
be 1, we're faced with the decision of either rejecting the physical  
theory or the axiom of choice. As you say, this situation can't arise  
in case of the sort of physical theories we currently consider even  
remotely plausible (or conceivable). But it can't be ruled out solely  
on basis of considerations involving "non-absolute" and "absolute"  
mathematical notions and principles. Any such conclusion must be based  
on an examination of actual physical ideas, theories, principles, e.g.  
the holographic principle, the Bekenstein bound, quantum information  
theory, philosophical musings about physics and measurement, the  
logical and mathematical properties shared by mathematical  
formulations of physical theories so far proposed, and so on, and  
inescapably must involve non-logical and mathematical ingredients. It  
appears we in fact entirely agree on this, so please consider my  
harping on it just a somewhat pointless reminder it's a good idea to  
be clear on what is based on purely logical considerations and what on  
observations about and of physics as we currently know it.

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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