Are physicists aware that, as opposed to the classical logic of true and false, arithmetic as a formal axiomatised system, consists of formulae that are either: provable from axioms, or, negatable by axioms, or, logically independent of axioms (neither provable nor negatable)? And that arithmetic, as an axiomatised system, conveys its logic through quantum theory?
Only conflating formal provability with truth and formal disprovability with falsity can be said that formal arithmetic involves a 3-valued logic. In addition, the questiion seems to conflate as well formal provability with logical consequence, which coincide only in first order logic: in second order arithmetic there are sentences that are neither formally provable nor formally disprovable but no sentence that is logically independent of the axioms; this happens because second order arithmetic is categorical though incomplete.
Only conflating formal provability with truth and formal disprovability with falsity can be said that formal arithmetic involves a 3-valued logic. In addition, the questiion seems to conflate as well formal provability with logical consequence, which coincide only in first order logic: in second order arithmetic there are sentences that are neither formally provable nor formally disprovable but no sentence that is logically independent of the axioms; this happens because second order arithmetic is categorical though incomplete.
This is Godel's Incompleteness Theorem. Any axiomatic logical system which contains the operation of addition must be either incomplete (there are true theorems which cannot be proved in the system) or inconsistent (within the system there are theorems, A and not A, which can both be proven). I don't see the connection to Quantum Mechanics, though. It is important to be aware that none of the mathematical theorems that are used in Quantum Mechanics, or any other field of physics for that matter, are either among those that satisfy the incompleteness or inconsistency conditions. If at some point in the future we find that a new theory which has superior predictive power relies on mathematics derived from a theorem which does fit one of these criteria, well then that would be very interesting.
Lawrence,
Thanks for your contribution. I am writing a paper now in which I think you will be interested. And I shall be pleased for your comments when its ready. I will send you a copy. It concerns Arithmetic as a formal axiomatised system and the interaction of this system with quantum theory. I show logical independence in quantum theory associated with existence of the square root of minus one. I do not go as far as to show this logical independence results from incompleteness of Arithmetic. Nevertheless, the circumstances would seem to suggest this rather strongly.
Steve,
I'd be very interested to read it. So complex numbers are to blame? Sounds interesting.
Lawrence,
Presburger arithmetic contains the addtion operation and is complete, consistent and even decidable.
Well, it so happens that I am a physicist (and a logician) and simultaneosuly, I am also a co-author of the paper: T. J. Stepien, L. T. Stepien, "The Formalization of The Arithmetic System on The Ground of The Atomic Logic", published in Journal of Mathematics and System Science (T. J. Stepien and L. T. Stepien, J. Math. Syst. Sci. 5, No.9, 364-368 (2015)): http://www.davidpublisher.org/index.php/Home/Article/index?id=18008.html . In this paper, it has been proved that the classical Arithmetic can be based on the Atomic Logic. The Atomic Logic is three-valued logic.
I encourage also to read the papers:
The following paper could be relevant to this discussion:
Solomon Feferman. Why a Little Bit Goes a Long Way: Logical Foundations of Scientifically Applicable Mathematics
PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association Vol. 1992, Volume Two: Symposia and Invited Papers (1992), pp. 442-455
Abstract
Does science justify any part of mathematics and, if so, what part? These questions are related to the so-called indispensability arguments propounded, among others, by Quine and Putnam; moreover, both were led to accept significant portions of set theory on that basis. However, set theory rests on a strong form of Platonic realism which has been variously criticized as a foundation of mathematics and is at odds with scientific realism. Recent logical results show that it is possible to directly formalize almost all, if not all, scientifically applicable mathematics in a formal system that is justified simply by Peano Arithmetic (via a proof-theoretical reduction). It is argued that this substantially vitiates the indispensability arguments.
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