ON 31st July 1974 André Joyal gave a talk with the title Arithmetisation dans le topos. Anders Kock took notes, and kindly shared them with me a few years back. He gave me permission to host them publicly, so here they are: pdf link.
I see the benefit here as at least simplifying what seems like the sound part of Mochizuki’s work (even if dependent on something still regarded as conjectural) to ordinary mathematics; no dismantling alien ring structures or odd metaphors about how school students get confused by logarithms. Not only that, but the statement of Corollary 3.12 is rendered in ordinary mathematics, rather than in the language of Frobenioids and their ilk.
Just a short post to point out the possibilities of writing cardinals in a more modern typeface. Pick your favourite solution from this list. The following compares the Euler math font version from the AMS with the FDSymbol version at the TeX.SE answer.
I’ve been reading Bunge, Gago and San Luis’s book Synthetic Differential Topology to review for MathReviews. I’ve finally sent off the written review, to appear here within short order (requires subscription).
On the whole, the book is ok, pretty much self-contained, but could have done with a bit more copy-editing. I think of this as more a fault of the publisher, but given that it’s in a lecture notes series, they probably didn’t give it as thorough a going-over. I’m very much intrigued by the link between Penon’s intrinsic topology on objects of a topos, which plays a big rôle in SDT, and Mike Shulman’s real cohesion, as hinted at in these slides by David Jaz Meyers.
(My thanks to the AMS for the free reviewers copy of the book).