The previous two parts of the paper (correspondingly, https://philpapers.org/rec/PENFLT-2 and https://philpapers.org/rec/PENFLT-3) demonstrate that the interpretation of Fermat’s last theorem (FLT) in Hilbert arithmetic meant both in a narrow sense and in a wide sense can suggest a proof by induction in Part I and by means of the Kochen - Specker theorem in Part II. The same interpretation can serve also for a proof FLT based on Gleason’s theorem and partly similar to that in Part II. The concept of (probabilistic) measure of a subspace of Hilbert space and especially its uniqueness can be unambiguously linked to that of partial algebra or incommensurability, or interpreted as a relation of the two dual branches of Hilbert arithmetic in a wide sense. The investigation of the last relation allows for FLT and Gleason’s theorem to be equated in a sense, as two dual counterparts, and the former to be inferred from the latter, as well as vice versa under an additional condition relevant to the Gödel incompleteness of arithmetic to set theory. The qubit Hilbert space itself in turn can be interpreted by the unity of FLT and Gleason’s theorem. The proof of such a fundamental result in number theory as FLT by means of Hilbert arithmetic in a wide sense can be generalized to an idea about “quantum number theory”. It is able to research mathematically the origin of Peano arithmetic from Hilbert arithmetic by mediation of the “nonstandard bijection” and its two dual branches inherently linking it to information theory. Then, infinitesimal analysis and its revolutionary application to physics can be also re-realized in that wider context, for example, as an exploration of the way for physical quantity of time (respectively, for time derivative in any temporal process considered in physics) to appear at all. Finally, the result admits a philosophical reflection of how any hierarchy arises or changes itself only thanks to its dual and idempotent counterpart.
In a previous paper (https://dx.doi.org/10.2139/ssrn.3648127 ), I have demonstrated an elementary and thoroughly arithmetical proof of Fermat’s last theorem by induction if the case for “n = 3” is granted as proved only arithmetically (a fact a long time ago). Now the same case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1968): the eventual validity of Fermat’s equation would correspond to an admissible disjunctive division of a qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen –Specker theorem. The proof involves set theory, but only within the case “n=3” and implicitly, within any next levels of “n”. Thus the application of set theory and arithmetic can remain disjunctively divided as well: set theory, “locally”, within any level; and arithmetic, “globally”, to all levels. Thus, the relevance of Yablo’s paradox to the statement of Fermat’s last theorem is avoided. However, the mathematical structure relevant to the proof cannot be restricted to Peano arithmetic but to a generalized structure called “Hilbert arithmetic” and discussed in detail in another previous paper (https://dx.doi.org/10.2139/ssrn.3656179 ). Furthermore, Fermat’s last theorem in virtue of that kind of proof can be considered as another expression of Gleason’s theorem in quantum mechanics (or vice versa): 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 the non-availability (respectively, availability) of a single probabilistic measure as to Gleason’s theorem.
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.