On the essential logical structure of inter-universal Teichmüller theory in terms of logical AND “∧”/logical OR “∨” relations: Report on the occasion of the publication of the four main papers on inter-universal Teichmüller theory

Abstract

The main goal of the present paper is to give a detailed exposition of the essential logical structure of inter-universal Teichmüller theory from the point of view of the Boolean operators --such as the logical AND “∧”logical OR “∨” operators-- of propositional calculus. This essential logical structure of inter-universal Teichmüller theory may be summarized symbolically as follows: A ∧ B = A ∧ (B₁ ∨˙ B₂ ∨˙...) ⇒ A ∧ (B₁∨˙B₂∨˙...∨˙ B́₁ ∨˙ B́₂ ∨˙...) -- where · the “∨˙” denotes the Boolean operator exclusive-OR, i.e., “XOR”; · A, B, B₁, B₂, B́₁, B́₂, denote various propositions; · the logical AND “∧'s” correspond to the Θ-link of inter-universal Teichmüller theory and are closely related to the multiplicative structures of the rings that appear in the domain and codomain of the Θ-link; · the logical XOR “∨˙'s” correspond to various indeterminacies that arise mainly from the log-Kummer-correspondence, i.e., from sequences of iterates of the log-link of inter-universal Teichmüller theory, which may be thought of as a device for constructing additive log-shells. This sort of concatenation of logical AND “∧'s” and logical XOR “∨˙ 's” is reminiscent of the well-known description of the “carry-addition” operation on Teichmüller representatives of the truncated Witt ring ℤ/4ℤ in terms of Boolean addition “∨˙” and Boolean multiplication “∧” in the field F₂ and may be regarded as a sort of “Boolean intertwining” that mirrors, in a remarkable fashion, the “arithmetic intertwining” between addition and multiplication in number fields and local fields, which is, in some sense, the main object of study in inter-universal Teichmüller theory. One important topic in this exposition is the issue of “redundant copies”, i.e., the issue of how the arbitrary identification of copies of isomorphic mathematical objects that appear in the various constructions of inter-universal Teichmüller theory impacts-- and indeed invalidates-- the essential logical structure of inter-universal Teichmüller theory. This issue has been a focal point of fundamental misunderstandings and entirely unnecessary confusion concerning inter-universal Teichmüller theory in certain sectors of the mathematical community. The exposition of the topic of “redundant copies” makes use of many interesting elementary examples from the history of mathematics.