Saturday, March 31, 2012

Erdös number update

My previous post requires an update, since I received a note reminding me that Ed Keenan also published with Jonathan Stavi, ["A semantic characterization of natural language determiners," Linguistics and Philosophy (1986) 9:253-326]. Stavi has an Erdös number of 2, in virtue of two different collaborations with Erdös coauthors Menachem Magidor and Marcel Herzog. This makes Keenan's number a rare 3 (very low for anyone primarily in linguistics), and my own a more interesting 4. This type of game is what passes for a hobby among mathematicians.

Thursday, March 29, 2012

Erdös number

Scholars with even a passing interest in mathematics usually know what an Erdös number is; it is the number of degrees of separation between a scholar and Paul Erdös, calculated by stepping through collaborative publications. Paul Erdös was a sort of enigmatic freelance mathematician who was very good at proving things, and very prolific with the aid of something like 500 different collaborators. It has become something of a sport in modern times for mathematical scholars to compute their Erdös number, since basically everyone who has published a mathematics article with a collaborator ends up having an Erdös number. I've read that most real mathematicians have an Erdös number of 8 or less.

A more interesting thing is when scholars in neighboring fields, such as mathematical linguistics, end up with Erdös numbers due to cross-fertilization. It turns out that one of my professors, Ed Keenan, appears to have an Erdös number of 4, which is really very low for a scholar outside mathematics. It's so unusual, it bears a stated proof. One chain of collaborative work connecting Keenan with Erdös is the following:

Keenan, E. and Westerståhl, Dag (2011) “Generalized quantifiers in linguistics and logic,” in J. van Benthem and A. ter Meulen (eds.), Handbook of Logic and Language. Second Edition, Amsterdam: Elsevier.

Hella, Lauri; Väänänen, Jouko; Westerståhl, Dag (1997) "Definability of polyadic lifts of generalized quantifiers." J. Logic Lang. Inform. 6:305–335.

Magidor, Menachem; Väänänen, Jouko (2011) "On Löwenheim-Skolem-Tarski numbers for extensions of first order logic." J. Math. Log. 11:87–113.

Erdős, P.; Magidor, M. (1976) "A note on regular methods of summability and the Banach-Saks property." Proc. Amer. Math. Soc. 59:232–234.

And the wonderful conclusion to all this is the theorem that my own Erdös number is 5, thanks to a paper I published with Ed Keenan in a rather obscure special volume of Linguistische Berichte in 2002. Does this increase my mathematical credibility? Not really. But it's nice to know I am somehow closer to the inner circle than I thought. A special bonus will soon come from my upcoming collaboration with Nick Chater, who also has an Erdös number of 4. So then I'll have earned my number 5 in two different ways.

Thursday, March 22, 2012

Survey of proof nets

This isn't much to be proud of, but I seem to have written an unpublishable paper. It is "A survey of proof nets and matrices for substructural logics," which I intended for the Bulletin of Symbolic Logic. However, it seems that they are not much interested in surveys, or if they are, they require them to be pitched to a highly technical audience already versed in the subject. At least that's what I could glean from the short note I got which rejected the paper at a preliminary stage. The "outside expert" who the editor found to make a brief assessment said that the paper was nearly devoid of all content. I don't believe I managed to fill 25 pages with precisely nothing (quite an achievement that would be), so I posted it on arXiv anyway. I believe it will benefit anyone wanting to find out about proof nets and how they relate to the logics which underpin type-logical grammar. Fortunately I already have tenure and am not too proud, so I posted the link to my unpublished masterpiece. All comments here will be appreciated---unless you feel it is devoid of all content. That particular comment has been used up.