# Groups are numbers (3). Axiom (convex)

This post will be updated as long as the em draft will progress. Don’t just look, ask and contribute.

UPDATE 3: Released: arXiv:1807.02058.

UPDATE 2: Soon to release. I found something so beautiful that I took two days off, just to cool down. Wish I release this first em-convex article in a week, because it does not modify the story told in that article. Another useful side-effect of writing this is that I found a wrong proof in arXiv:0804.0135 so I’ll update that too.

UPDATE: Don’t mind too much my rants, I have this problem, that what I am talking about is in the future with respect to what I show. For example I tried to say it several times, badly! that chemlambda may be indeed related to linear logic, because both are too commutative. Chemlambda is as commutative as linear logic because in chemlambda we can do the shuffle. Or the shuffle is equivalent with commutativity, that’s what I tried to explain last time in Groups are numbers (1). There is another, more elaborate point of view, a non-commutative version of chemlambda, in the making. In the process though, I went “oh, shiny thing, what’s that” several times and now I (humbly try to) retrace the correct steps, again, in a form which can be communicated easily. So don’t mind my bad manners, I don’t do it to look smart./

The axiom (convex) is the key of the Groups are numbers (1) (2) thread. Look at this (as it unfolds) as if this is a combination of:

• the construction of the field of numbers in projective geometry and
• the Gleason and Montgomery-Zippin solution to the Hilbert 5th problem

I think I’ll leave the (sym) axiom and the construction of coherent projections for another article.

Not in the draft available there are about 20 pages about the category of conical groups, why it is not compact symmetric monoidal (so goodbye linear logic) but it has as a sub-category Hilb. Probably will make another article.

I sincerely doubt that the article form will be enough. I can already imagine anonymous peer reviews where clueless people will ask me (again and again) why I don’t do linear logic or categorical logic (not that it is useless, but it is in the present form heavily influenced by a commutative point of view, is a fake generalization from a too particular particular case).

A validation tool would be great. Will the chemlambda story repeat, i.e. will I have to make, alone, some mesmerizing programs to prove what I say works? I hope not.

But who knows? Very few people deserve to be part of the Invisible College. People who have the programming skills (the easy part) and the lack of prejudices needed to question linear logic (the hard, hacker part).