[FOM] Extramathematical notions and CH

[Joe Shipman]

Chow said:
**** 
Why couldn't someone come up with a physical theory that we somehow come to accept as correct, that predicts that some experimentally measurable value tells us the truth value of CH?  Once we leave the well-defined territory of finite verification and allow ourselves to "know" physical *theories*, how do we prevent the physicists from working CH into their theories? Granted, there's no plausible proposal on the horizon, but if you are urging us not to draw arbitrary lines, then I don't see why we should draw an arbitrary line between absolute and non-absolute statements.
****

I reply:

See below; the short answer is that I am not leaving the territory of finite verification, because we can interpret experimental measurements as verifying not only pi^_0 statements but also, with appropriate confidence levels, pi^0_1 and sigma^0_2 statements.

Chow said:
****In 4b, there's a giant finite object that I have good, albeit fallible, reasons to believe has been fully explored by someone or something or some collection of people or things.  In 4a, I *know* that the finite object hasn't been fully explored.  While I'm not going to reject 4a as useless, I have no difficulty making a principled distinction between knowledge of type 4a and knowledge of type 4b.
****

I reply:

You seem to think I am trying to erase the distinction between 4a and 4b despite my making that very distinction; if you accept that 4a represents a VALID but DIFFERENT TYPE OF knowledge of mathematical statements, I have made my point, because I never claimed case 5 (Quantum computation) was like cases 1 (Wiles proof), 2 (Classification of finite simple groups), 3 (probabilistically checkable proofs), and 4b (4-color theorem), only that it was like case 4a (random polynomial time or BPP). I merely claimed cases 4a and 5 were useful, which you now agree with.

Chow said:
****
I do think it's possible to draw another line between 5 and 6.  To summarize:
1, 2, 3, 4b: Finite and completely verified, as far as we can tell.4a, 5: Finite but definitely not completely verified.6. Infinite.
Or maybe I should add a seventh category, and say instead:
6. Infinite, but absolute.7. Infinite and non-absolute.I'd prefer to stick to 1, 2, 3, 4b when using the term "mathematical knowledge."  If you are happy slipping down the slope to 6 then I don't see why we might not slip down to 7 some day.
****

I reply: 

The problem with the distinction you draw between cases 4a/5 and 6 (the theory entails that a nonrecursive bit sequence might be experimentally measurable) is that the "finite object" in cases 4a and  5 is too big to be contained in this universe, rather than merely being too big to be humanly feasible but manageable by a computer, as in cases 3 and 4b. This is an important distinction between 3/4b and 4a/5, which renders the further step to case 6 of no additional practical importance. 

The difference between case 6 and case 7 (which I thank you for including because I forgot to go that final step) is that non-absolute mathematics cannot be settled by physical experiment because, as Paul Elliott perceptively pointed out, we could be living in a Skolem hull. In more straightforward terms, our physical theories can't be pushed beyond arithmetical statements and still be experimentally testable, because other theories with the same arithmetical consequences will be indistinguishable from them. 

For all the theories we currently contemplate, which involve measurable quantities that are summations of countably infinitely many computable quantities, the statement that a measurable physical quantity has a particular real value is Pi^0_3, the statement that its real value lies in an interval is Sigma^0_2, and if there are computable moduli of convergence we can get down to Pi^0_1. Experimental measurements will be of the form "with confidence level 10 sigma, the value of the quantity lies in the interval [a,b]", which will provide practical certainty if not metaphysical certainty that if our physical theory is correct, then the Sigma^0_2 mathematical statement that the quantity is in [a,b] is true, and we will have previously gained confidence in the physical theory through other experiments.

 I cannot see how theories with the same arithmetical consequences could possibly be distinguished from each other by experiments that end up with numerical data, which is why I think that case 7 is essentially different from case 6. This also answers Aatu Koskensilta's concern, because CH and V=L and similar non-absolute axioms cannot decide any arithmetical statements that were not already decidable in ZF.

 -- JS 
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