The paper is a continuation of another paper (https://philpapers.org/rec/PENFLT-2) published as Part I.
Now, the case of “n=3” is inferred as a corollary from the Kochen and Specker
theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would
correspond to an admissible disjunctive division of qubit into two absolutely independent parts
therefore versus the contextuality of any qubit, implied by the Kochen –
Specker theorem. Incommensurability (implied by the absence of hidden
variables) is considered as dual to quantum contextuality. The relevant
mathematical structure is Hilbert arithmetic in a wide sense (https://dx.doi.org/10.2139/ssrn.3656179), in the framework of which Hilbert arithmetic in a narrow
sense and the qubit Hilbert space are dual to each other. A few cases involving
set theory are possible: (1) only within the case “n=3” and implicitly, within
any next level of “n” in Fermat’s
equation; (2) the identification of the case “n=3” and the general case
utilizing the axiom of choice rather than the axiom of induction. If the former
is the case, the application of set theory and arithmetic can remain
disjunctively divided: set theory, “locally”, within any level; and arithmetic,
“globally”, to all levels. If the latter is the case, the proof is thoroughly
within set theory. Thus, the relevance of Yablo’s paradox to the statement of
Fermat’s last theorem is avoided in both cases. The idea of “arithmetic
mechanics” is sketched: it might deduce the basic physical dimensions of
mechanics (mass, time, distance) from the axioms of arithmetic after a relevant
generalization, Furthermore, a future Part
III of the paper is suggested: FLT by mediation of Hilbert arithmetic in a
wide sense can be considered as another expression of Gleason’s theorem in
quantum mechanics: the exclusions about (n = 1, 2) in both theorems as well as
the validity for all the rest values of “n” can be unified after the theory of
quantum information. The availability (respectively, non-availability) of
solutions of Fermat’s equation can be proved as equivalent to the non-availability
(respectively, availability) of a single probabilistic measure as to Gleason’s
theorem.
Keywords: arithmetic mechanics, Gleason’s
theorem, Fermat’s last theorem (FLT), Hilbert arithmetic, Kochen and Specker’s
theorem, Peano arithmetic, quantum information
The paper as a PDF file or @ repositories: @ SSRN, @ PhilPapers, @ HAL, @ EasyChair, @ CambridgeOpenEngage, @ Preprints, @ PhilSci Archive (Pittsburgh)
