What is the simplest (at the lowest level feasible) explanation of the approach of “forgetting the history” of a mathematical object, as used in Inter-universal Teichmüller Theory (IUTT)? Please explain:
- What are these operations and their history? (Without that we get nowhere!)
- How can an exposition or proof forbid the reader to use previous known information about an object? (This just sounds unreasonable to me.)
- What is a really simple example of a history with re-initialisation playing a useful rôle? (This would help one to understand and accept the method.)
- Is this method broadly accepted? (Scholze and Stix do not mention it in their reports², though it is evidently very important to Mochizuki.)
- I am not asking what is the status of the purported proof of the abc conjecture, though that is obviously relevant.
- I admit to not having so much as started to try to study the IUTT papers³, but do not think that amounts to an unreasonable lack of effort to solve my problem myself, given that I understand that many professional mathematicians have shied away from them. I am hoping that someone else has at least got far enough to answer.
Shinichi Mochizuki, initiator of IUTT, says in §5 of his report¹ of discussions with critics of IUTT that:
The logical origin of the differences in viewpoint and so of misunderstandings by critics might be the different approaches to histories of operations on mathematical objects (e.g. structures):
Conventional: Conventionally, one regards all operations as parts of a single history, wherein they are all accessible.
IUTT: IUTT frequently uses re-initialisation operations — i.e. one “forgets” the previous history of an object, regarding it as inaccessible in subsequent discussions. Re-initialisation necessitates labels for “before and after versions” of an object and explicit specification of the types (in IUTT: “species”) of objects, particularly before and after re-initialisation (e.g. “automorphism groups of fields” / “abstract topological groups”).
¹ http://www.kurims.kyoto-u.ac.jp/~motizuki/Rpt2018.pdf – Report by Shinichi Mochizuki of discussions in March 2018 between himself and Yuichiro Hoshi (expounding IUTT) and Peter Scholze and Jakob Stix (questioning its methods).
- (see also other documents at http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-discussions-2018-03.html)
² http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf (August 2018), critique by Scholze and Stix, one of the other documents referred to in ¹
- (first draft: http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-05.pdf (May 2018)).
³ S. Mochizuki, Inter-universal Teichmüller Theory (August 2012):
I: Construction of Hodge Theaters, RIMS Preprint 1756
II: Hodge-Arakelov-theoretic Evaluation, RIMS Preprint 1757
III: Canonical Splittings of the Log-theta-lattice, RIMS Preprint 1758
IV: Log-volume Computations and Set-theoretic Foundations, RIMS Preprint 1759