Do physicists know about mathematical undecidability? And do they believe it might play a part in Nature? Foundations of The Quantum Logic: http://steviefaulkner.wordpress.com/
Do I understand your idea correctly? You want to equate quantum indeterminancy with mathematical undecidability - particularly the example of the square root of -1? On the face of it, these seem to have quite different kinds of uncertainty despite both employing a 2D number plane in their modelling. With QM, as you zoom in to pin a value to the x axis, you lose control over value in the y axis (it could be anything out to infinity). So there is this yo-yo balance where increasing certainty in one direction leads to increasing uncertainty in the other, with planck scale setting the ultimate limits. But with mathematical undecidability of square root of -1, surely the 2D plane then allows complex values to be exactly specified in relation to x and y axis? So a different behaviour?
Thank you for your reply apeiron, Undecidability is not at all the same as uncertainty. I urge you to read further into my work. If you would like the full paper when it is finished, leave a comment on my blog with your email address. SteveFaulkner.
I would prefer a quick justification as - as you say - uncertainty is not undecidability. There would be two kinds of undecidable I think. The first where you flip-flop between binary choices (trapped in self-reference as in the liar's paradox) and then second where you have asymptotic approach to a limit, so like when calculating pi or e. Trapped now in a non-completing process (but one always headed in a certain direction). To me, QM indeterminancy is something else again. And perhaps there is not a good mathematical analogue? I would liken it to symmetry and symmetry breaking. But that does seem to give us the extra aspect of yo-yoing around a fixed point value like the planckscale. So why not say a few words as to how you see square root of -1 as an analog of QM indeterminancy. You may be right, but I did skim your preliminary arguments and could not spot an answer in language that I could understand.
Thank you apeiron for your interest. I'll try and give a brief overview. I see QM as an applied mathematics which is a semantic theory. I set formalism of Wave Mechanics within Mathematical Logic by making it a first-order theory [nothing to do with approximation methods]. This places the Field Axioms as axiomatic over all scalar components of all mathematical objects in the theory. Due to Soundness and Completeness, theorems of Model Theory, there is an excluded middle of propositions that are mathematically undecidable with respect to the Field Axioms. If the square root of minus one is logically independent of the Field Axioms, the proposition of this square root's existence falls into this excluded middle. In re-writing formalism of Wave Mechanics as a first order theory, introduction of the imaginary unit can be postponed by replacing it by a bound variable until a certain step in the derivation of the theory. That point is normalisation; and the reason that the imaginary unit is needed there is that the theory requires products between vectors in an orthogonal space. In fact, it is these products that assume existence of the square root of minus one. [ Baylis, W.E., Huschilt, J. and Jiansu Wei (1991) Why i? American Journal of Physics 1992 60/9, pp788-797 ] Baylis et.al demonstrate this for 3space but do not seem to realise it applies to infinite dimensional spaces. Requiring the introduction of the square root's existence at this stage establishes its logical independence of the Field Axioms. The fundamental mathematical undecidability, here, is the existence of the products of orthogonal vectors, rather than existence of the imaginary unit. Such undecidabilty is within any theory of quantum physics that relies on such products, not only Wave Mechanics.
Still sounds apples and oranges to me. Square root of -1 was an exact entity - the result of a natural operation - that was found to have been excluded from the 1D numberline. And its position was recovered when the numberline was relaxed to reveal the numberplane (so to speak). So it was the exclusion of something definite - the definite outcome of a definite operation. But QM is more complicated and seems quite different - apart from fact that we need at least two dimensions of measurement to begin to capture one thing. As I say, you are performing operations in a complex plane - which is already a constraint from Hilbert infinity because we have constrained naked QM indeterminancy to a two dimensional question - where are you and when are you? And then the answers oscillate around the planckian fixed point. So with sqrt -1, you are expanding dimensionality to recover definite things that were excluded - and finding not just this particular point on a numberplane but the whole numberplane itself. And with QM, you are modelling the limits of the ability to exclude middles via dimensional constraint. You are trying to bottle up duality, in effect - the complementary qualities of scale and heat (location and momentum, energy and time, etc) - and finding that imaginary numbers are necessary because the duality proves irreducible in the limit. To sum up, a case of something mistakenly thought excluded vs a case of a dichotomy that cannot actually be collapsed?
Unlike the numbers 1, 2, 3, 4,...or any rational numbers, existence of the square root of minus one cannot be proved as a theorem of the Field Axioms. Existence of every rational number can be proved as a theorem. Nevertheless, existence of the square root of minus one cannot be disproved by the the Field Axioms. Since it's existence can be neither proved nor disproved, it has undecidable existence under the Field Axioms. This does not stop it from having a definite, exact value. Foundations of The Quantum Logic: http://steviefaulkner.wordpress.com/ You will find that all irrational numbers are undecidable by the same logic.
In our science forums, we don't accept solicitation for review of non-peer reviewed works (except in some rare cases in our independent research forum, where very specific criteria must be met), but our guidelines are not specific on philosophy submissions. Overall, we don't wish to this site to be a vehicle for promoting personal websites, blogs, papers, and theories, so I am going to ask for consultation from the other mentors. I'll get back to you. Thanks! Update: resolved via PM.