2 ms·
Physical theories can't be "proven" in the same way as mathematical conjectures, so we have to rely on agreement with experiment, self-consistency, "simplicity"
by chriswarbo 10y ago
Physical theories can't be "proven" in the same way as mathematical conjectures, so we have to rely on agreement with experiment, self-consistency, "simplicity" (e.g. Kolmogorov complexity or some approximation), lack of counterexamples/disproof, etc.
In this sense, I agree with Robert Harper (e.g. Practical Foundations for Programming Languages, section 22.2) that unprovable claims regarding computability may be merely "theses" to logicians, but to scientists they should be considered "laws" just as much as Newton's law of universal gravitation.
In fact, other than purported claims of "hypercomputation" ( https://en.wikipedia.org/wiki/Hypercomputation#Criticism https://en.wikipedia.org/wiki/Hypercomputation#Criticism ) or the anthropocentric "microtubule"-type theories popularised by Penrose ( https://en.wikipedia.org/wiki/Orchestrated_objective_reduction#Criticism https://en.wikipedia.org/wiki/Orchestrated_objective_reducti... ), it seems that the computability of the universe is commonly accepted 'background knowledge' by most researchers in the field.
Even those making extreme claims in this area, like those refuted in http://www.scottaaronson.com/papers/npcomplete.pdf http://www.scottaaronson.com/papers/npcomplete.pdf are presumably assuming that the universe is computable; there doesn't seem to be much point debating computational efficiency of soap bubbles if I thought (for example) that halting oracles existed!
- wyager 10y agoUnless I'm mistaken, the computable numbers are not complete in the Cauchy Sequence sense. Therefore, there can be no hilbert space based on the computable numbers, because a hilbert space has to be complete in the Cauchy sense. It would only be a pre-hilbert space. I believe this means one can no longer correctly use limits in QM; are there other consequences of this?
- chriswarbo 10y agoVery interesting, and the kind of technical aspect I tried to avoid by not refuting "face-on" since I don't feel qualified to answer this without looking into it further :) Still, if there are reasons to allow uncomputable numbers as elements of a quantum theory, that still doesn't invalidate the larger point that any predictions of those theories are computable; it's just a constraint on which axiom scheme we follow when manipulating our equations. Again, even if a theory involves uncountable sets, it may still have countable models (following either Skolem's "paradox", or a similar argument; since QM operates at much higher level than formal systems, I'm not sure if it's first-order, second-order, higher-order, etc.).