Rebuilding Mathematics on a Quantum Logical Foundation

We construct a rich first-order quantum logic which generalizes the standard classical predicate logic used in the development of virtually all of modern mathematics, and we use this quantum logic to build the foundations of a new quantum mathematics. First, we prove both soundness and completeness for the quantum logic we develop, and also prove a powerful new completeness result which heretofore had been known to hold for classical, but not quantum, first-order logic. We then use our quantum logic to develop multiple areas of mathematics, including abstract algebra, axiomatic set theory, and arithmetic. In some preliminary investigations into quantum mathematics, Dunn found that the Peano axioms for arithmetic yield the same theorems using either classical or quantum logic. We prove a similar result for certain classes of abstract algebras, and then show that Dunn’s result is not generic by presenting examples of quantum monoids, groups, lattices, vector spaces, and operator algebras, all which differ from their classical counterparts. Moreover, we find natural classes of quantum lattices, vector spaces, and operator algebras which all have a beautiful inter-relationship, and make some preliminary investigations into using these structures as a basis for a new mathematical formulation of quantum mechanics. We also develop a quantum set theory (equivalent to ZFC under classical logic) which is far more tractable than quantum set theory previously developed. We then use this set theory to construct a quantum version of the natural numbers, and develop an arithmetic of these numbers based upon an alternative to Peano’s axioms (which avoids Dunn’s theorem). Surprisingly, we find that these “quantum natural numbers” satisfy our arithmetical axioms if and only if the underlying truth values form a modular lattice, giving a new arithmetical characterization of this important lattice-theoretic property. Finally, we show that these numbers have a natural interpretation as quantum observables with whole number eigenvalues, and find that our quantum arithmetic of these numbers yields a new sum and product for such observables with some truly remarkable properties.