[FOM] inconsistency of P

Sam Sanders sasander at cage.ugent.be
Tue Oct 4 23:14:55 EDT 2011

I agree with what Tim said on the relation between reality and our representation thereof.
However, there seems to be an acceptable way of qualifying whether our representations are any good.

For instance, the numerous independent ways of deriving Avogadro’s constant (with negligible errors) are 
taken by Ian Hacking to be sufficient evidence for the real-world existence of molecules and atoms.
(See 'Representing and Intervening: Introductory Topics in the Philosophy of Natural Science' Cambridge University Press, 1983.)

The central idea here is robustness, which is stability under (small) variations.  Our representation of reality will always contain
*some* errors, but if our model does not collapse under small variations of assumptions, then we can believe it to be close to the reality,
even though it contains some errors.  Put in another way: for robust scientific models, the errors present do not add up to anything significant.

It is an interesting question what robustness means in the foundations of mathematics.  For instance, can we use a notion of robustness to claim
that a certain logical theory is close to the 'reality' of mathematics (If you believe such to exist).

> In a first course in quantum mechanics, we're taught to represent various 
> subatomic quantities as wave functions.  Is an electron *really* a wave 
> function?  Everett might say yes, but others woud say no: it's a useful 
> fiction that is theoretically tractable and helps us make experimental 
> predictions.  Similarly, in general relativity, we represent spacetime as 
> a pseudo-Riemannian manifold.  Is spacetime *really* a pseudo-Riemannian 
> manifold?  Is the continuum hypothesis a physical question whose truth 
> depends on whether there is really an uncountable set of spacetime points 
> out there not in one-to-one correspondence with the set of all spacetime 
> points?  Most would say no.  The physical world is a messy place and if we 
> want as *accurate* as possible a description of it, then our beautiful 
> theories are not up to the task.  This is not to say that we shouldn't 
> create and study beautiful theories, but we shouldn't conflate our 
> beautiful fictional stories with the ugly reality that they are trying to 
> approximate.

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