The Nature of Mathematical Proof

In his talk this morning, Grisha Mints referred to a paper by Paul Cohen. He didn’t have the reference handy, so I tracked it down:

Paul J. Cohen, Skolem and pessimism about proof in mathematics. Phil. Trans. R. Soc. A (2005) 363, 2407–2418.

The entire issue, on meeting on “The nature of mathmatical proof” organized by Alan Bundy in 2004, is of interest:

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences (Volume 363, Number 1835 / October 15, 2005)

LaTeX trick: rising diagonal dots

Just in case you ever need it: \ddots going the other direction:

\makeatletter
\def\Ddots{\mathinner{\mkern1mu\raise\p@
\vbox{\kern7\p@\hbox{.}}\mkern2mu
\raise4\p@\hbox{.}\mkern2mu\raise7\p@\hbox{.}\mkern1mu}}
\makeatother

Then you can say: \varepsilon_0= \omega^{\omega^{\Ddots}}

Modality Morning

This morning has two talks on modal logic: first up was Marcus Kracht with a survey on the development of modal logic; now Steve Awodey is reporting on joint work with Kishida on topological semantics of first-order modal logic. Marcus talked about some interesting results in the mathematics of modal logic, especially general semantics for first-order modal logic. Steve’s talk is old-skool chalkboard math, with pretty drawings. I just learned what sheaf is. He has a beautiful model theory for first-order S4 in terms of sheaves.

Why the Faculty Scholarly Productivity Index doesn’t mean anything in philosophy

Ok. Brit posted about it. Apparently some people claim that the Faculty Scholarly Productivity Index (FSP) shows something about the rankings produced by the Philosophical Gourmet Report (PGR) (e.g., that they’re off). But it doesn’t.

That is not because the PGR is actually the best possible way to measure program or even faculty quality. It is also not because the FSP is a bad way to measure program or faculty quality. In fact, I think if you’re going to measure faculty quality other than by doing a reputation survey, the methodology of the FSP is the way to go: (a) figure out who’s in what department (b) count books and journal articles of those people (c) count citations. What the FSP is aiming to do is to measure the productivity of individual people and then produce an aggregate ranking. There are some things that are questionable. For instance, why exactly should you count a book the same as 5 journal articles? In general: the specific weighting of parameters is questionable–what would probably be more indicative of faculty quality would be to publish individual and mean rankings on each dimension (articles, books, citations).

But what really sinks the FSP in its current form is the quality of the data: (1) books from Amazon, i.e., any book regardless of publisher is included. (2) Journal articles from scopus.com. Scopus’s selection of philosophy journals is, well, atrocious. I could find about half a dozen philosophy journals that I recognized the title of. As far as I can tell, it only includes a couple of top journals, notably not: PPR, Phil Q, JPhil, Phil Review, Synthese. So the only reason that some good departments show up on FSP’s top 10 is that they have faculty who publish books and many papers in Ethics, BJPS, and Linguistics and Philosophy. As far as I can tell, there isn’t even a bias for analytic philosophy and against continental: Kant Studien is not in scopus, and neither is any other continental journal I know of. There are no logic journals at all.

The short of it: if you’re going to measure faculty productivity, but only count journal articles in an arbitrary selection of journals only a few of which are generally of high quality, you’re going to get unreliable results.

Strict Conditional in LaTeX

I just had occasion to have to typeset Lewis’s strict conditional symbol <img src="http://plato.stanford.edu/archives/win2004/symbols/fishhook.gif" alt="- in LaTeX. It turns out it isn’t in the standard AMS fonts. Peter Smith’s LaTeX for Logicians to the rescue! There I found:

  1. that the strict conditional symbol is in the fonts that are part of the txfonts and pxfonts packages, and
  2. that there is a wonderful 110 page/3 MB comprehensive listing of all LaTeX symbols (by Scott Pakin).

Now it turns out that the point of txfonts and pxfonts is to give you output in Times Roman and Palatino fonts, respectively, with matching math and symbol fonts. That’s useful in itself–but if you happen to not want your document to be in Times or Palatino, you can still get \strictif by putting this in the preamble:

\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\DeclareMathSymbol{\strictif}{\mathrel}{symbolsC}{74}

Philosophy Genealogy

I just noticed that Josh Dever’s Philosophy Family Tree now comes with a little Java applet that gives you a list of your philosophical ancestors (easier to use than the PDF list).

Josh, any plans to make the tree capable of dealing with more than one advisor?

A complete first-order temporal logic of time with gaps

Source

Theoretical Computer Science 160 (1996) 241-270
(with Matthias Baaz and Alexander Leitsch)

The first-order temporal logics with \(\Box\) and \(\bigcirc\) of time structures isomorphic to \(\omega\) (discrete linear time) and trees of \(\omega\)-segments (linear time with branching gaps) and some of its fragments are compared: The first is not recursively axiomatizable. For the second, a cut-free complete sequent calculus is given, and from this, a resolution system is derived by the method of Maslov.

DOI: 10.1016/0304-3975(95)00107-7

Preprint

First-order Gödel logics

Annals of Pure and Applied Logic 147 (2007) 23-47 (with Matthias Baaz and Norbert Preining)

First-order Gödel logics are a family of finite- or infinite-valued logics where the sets of truth values V are closed subsets of [0,1] containing both 0 and 1. Different such sets V in general determine different Gödel logics GV (sets of those formulas which evaluate to 1 in every interpretation into V). It is shown that GV is axiomatizable iff V is finite, V is uncountable with 0 isolated in V, or every neighborhood of 0 in V is uncountable. Complete axiomatizations for each of these cases are given. The r.e. prenex, negation-free, and existential fragments of all first-order Gödel logics are also characterized.

DOI: 10.1016/j.apal.2007.03.001

Preprint

Logic Degree Programs?

I’m giving a talk on Friday for a general audience, and I thought it would be cute to claim in the little blurb about me that I am “probably the only person in the world who holds both undergraduate and doctoral degrees in logic.” Several people have asked me if that’s really true, and now I wonder myself. There aren’t many programs that offer degrees in logic (and by this I mean: degrees that have “logic” in the name of the degree but not as a specialization of, say, mathematics, and also not a degree in mathematics with a thesis that happened to be in logic). There’s an undergrad degree given out by Carnegie Mellon. The University of Vienna used to offer degrees in “Logistik,” but now mathematical logic is part of the mathematics major there, as far as I can tell. The Computational Logic degree administered by the Universities of Dresden / Bolzano / Madrid / Vienna (TU) / Lisbon and the MA in Logic at the University of Amsterdam are graduate (MSc) degrees. The only straight PhD program in logic I know of is the one at Berkeley. There are also programs in Amsterdam, Carnegie Mellon, Irvine, and Munich, but these, to the best of my knowledge, grant PhDs in Computer Science, Mathematics, or Philosophy with some kind of specialization in logic. I don’t know exactly what the deal is with degrees from Paris 7. Do you know of other programs? Do you know of anyone who got both their undergrad and PhD degrees from logic programs?

Help Studia Logica get into ISI

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Stanley’s Erdős Number is 5

Over dinner yesterday, Jason and I got to talking about Erdős numbers of various people. He didn’t know his, so I looked it up–the Mathematical Reviews database MathSciNet has a “compute collaboration distance” function in the author search. It produces output like this:

Jason Stanley coauthored with Richard G. Heck, Jr. MR1234144 (94k:03006)
Richard G. Heck, Jr. coauthored with George Stephen Boolos MR1701948 (2000k:03003)
George Stephen Boolos coauthored with John P. Burgess MR1898463 (2003a:03001)
John P. Burgess coauthored with R. Daniel Mauldin MR0628885 (82j:28002)
R. Daniel Mauldin coauthored with Paul Erdös MR0412390 (54 #516)

It’s not perfect, since it only computes distance based on papers in the MR database, i.e., mathematical papers. But, you can compute the distance not just to Erdős, but to anyone in the database: my Stanley number, for instance, is ≤6.

The Mexican Multiplier Trounces Dr. Evil in Large Number Duel

Agustín Rayo wins over Adam Elga, uses “googol” and a variant of Berry’s paradox in knock-out punch: see here.

Dr. Evil clutched his heart as though it had been pierced by an arrow. Trembling, he fell to his knees on the floor of the crowded stuffy room, all eyes watching him. The Mexican Multiplier threw up his hands in victory, smiling, as Dr. Evil whispered, “I’ve been crushed.” The battle was finally over.

HT: Henri Galinon/Kai von Fintel

Gödel quote

Varol Akman kindly sent a link to this picture of the poster advertising the Gödel exhibition in Vienna, with the nice quote: “Today philosophy has arrived, at best, at the point mathematics was at in Babylonian times.”