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A recent email from my colleague Jean Gallier explains why the new edition of Discrete Mathematics is available on his website:

A bit more that three years ago, Springer suggested that I revise my
“Discrete Mathematics” (published in 2010).
Unfortunately, Jocelyn and I
waited too long and now that we are done
Springer no longer wants it.

I added a chapter on probability theory and made some
significant changes (improvements!). I also changed the title.
There is even an intro to the simply-typed lambda-calculus!
In any case, temporarily I am falling back on the little known publisher
“Kurt Reillag and House of Cats and Dogs”.

Interested readers may observe a certain relationship between "Reillag" and "Gallier", and in any case web search will turn up Reillag's home page — which provides this post's main linguistic relevance.

From that page, and without further comment from me, here's the abstract for a 1992  talk that Jean presented on behalf of Reillag:

 

Linear Logic is nonregressively productive and not self-realizable, Nonmonotonic Logic is regressively nonproductive and self satisfied.

 

Kurt W.A.J.H.Y Reillag
Hospices de Beaune

ABSTRACT: The amazing results quoted in the title are proved. They have unprecedented and unpredictable consequences such as:

1. Linear logic being nonregressively productive, no matter how good a theorem you produce, a harder and better one is waiting out there.

2. Linear logic is never satisfied with itself, which means that it is an infinite source of new results.

In contrast,

3. Nonmonotonic logic being regressively nonproductive, no matter how bad a result you produce, there is a worse one waiting out there.

4. This does not matter anyway, since nonmonotonic logic is self satisfied.

There are other incredible consequences of these results. For example, the discontinuum hypothesis is independent of NMZF (nonmonotonic ZF), and Riemann's Hypothesis gives in.

The proof techniques are the most revolutionary aspect of this work. The main new concept is that of an FBI sheaf. To define an FBI sheaf, it was necessary to define a variant of Grothendieck topologies, named undercover agents. Then, an undertopos is like a topos, except that it has a subobject declassifier. An FBI site is a category with undercover agents. FBI sheaves cannot be glued, but they can be shredded. By iterating sheaf shredding long enough, we get an FBI sheaf sufficiently small that, it splits (et oui!). This is why the French say: "Voila le topo(s)". The most amazing of all is that in fact, using Freyd undercovers and the fact that A => A is intuitionistically provable, we get that every FBI sheaf over a big site is nonregressive. This is because all undercover agents eventually split. It it then an easy matter to get our fundamental results. They explain a lot, in particular about funding.

Unfortunately, I learned this morning that Kurt W.A.J.H.Y Reillag went for his annual retreat at some unknown mountain winery. This means that someone here at Penn will have to give the talk if we want to hear it. I did receive a large stack of wine bottle labels containing the entire paper scribbled on them.

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You may also enjoy the slides for a presentation by Jean Gallier and Kurt Reillag on the topic "What is a proof?"

And Discrete Mathematics is also well worth your time — especially the new Chapter 1.

 

March 15, 2022 @ 7:07 am · Filed by Mark Liberman under Language and mathematics

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