Kripke in NY Times

This was recently posted on Leiter Reports, but is probably worth repeating. You can find the article about Kripke here. You can certainly trust journalists to value understatements. I've included my favourite passage underneath:

In many circles, Mr. Kripke, who in 2001 was awarded the Schock Prize, philosophy's equivalent of the Nobel, is thought to be the world's greatest living philosopher, perhaps the greatest since Wittgenstein. Mr. Kripke is actually superior to Wittgenstein in at least two respects. Wittgenstein did not accomplish some of his most important work while still in high school. And unlike Wittgenstein, who was small, slender and hawklike, Mr. Kripke looks the way a philosopher ought to look: pink-faced, white-bearded, rumpled, squinty.

"Not one of the riff-raff" - GM Magnus Carlsen & Corus 2006

Last time we left the young and promising Carlsen (portrayed above together with Kasparov (left) before his retirement) was just before his encounter with Kamsky in the FIDE Knock-out Championship . For those who were not updated somewhere else, he won the first game brilliantly, but later succumbed to the older American and settled for 11th place. Note however, that his impressive performance ensured him a place in the candidate matches, more specifically against no. 1 in the same Siberian tournament, namely Aronian (2752).

While waiting for this spectacular match however, we have yet again been impressed by "the Mozart of chess" (I'm not responsible for this nickname). In the Corus tournament in Wijk aan Zee, Carlsen played in the B group, competing for a place in the A group next year. The expectations were anything but modest, seeing that his archnemesis, the 15 year-old Ukranian GM Karjakin, won the B group last year, and thus gets to swim with the big fish this year.

Carlsen started out with characteristic strength, soon taking the lead in the B group. At the most he had a whole point against his fiercest rivals, Motylev and Almasi, but as the tournament came to an end, he lost a decisive game against Motylev. This cost him the lead and left more than enough excitement for the last two tournament days. But when Motylev lost his game and Carlsen drew on the day before the last, Almasi caught the 1st place with only one game to go. Consequently, my day yesterday was spent watching computer chess for hours on, waiting for the results.

Luckily, Almasi lost his game, leaving both Carlsen and Motylev with winning chances. But the hope of Carlsen taking the 1st place alone soon vanished when Motylev played an easy win without even reaching the 30th move. According to Chessbase, then, with the score even, Motylev wins the tournament on tie-break rules (as far I can see, it is due to the fact that Motylev won the only game they played against eachother). Yesterday, after all the results were in, the speculations were rampant on the Chessbase forum: Will Carlsen still be invited to play in the A group next year? Yes, he will. This morning Chessbase News could report that Carlsen AND Motylev will indeed be playing in the A group in Corus 2007.

Postscript: In the A group, an even race between Anand and Topalov (current FIDE Champion) ended with equal score.

Random Remarks on Triviality

After reading Andreas' post about the Nottingham conference and Lowe's use of the word 'triviality', I've decided to post something about it myself. The post is not a direct comment on Lowe, but rather a comment on the more general 'What do we mean when we say that something is trivial?'.

Who hasn't had the pleasure of being beaten over the head with 'triviality' in introductory logic courses? Indeed, it seems that in logic more or less anything can become trivial given the appropriate range of time and depth of analysis. When my logic teacher called theorems taking up three to four whiteboards trivial, it struck me that this lovely pet phrase of mathematicians (" 'Tis trivial") is in need of some scrutiny.

First of all: 'Trivial' is not synonymous with 'obvious'. In the following, I will take 'O(p)' (read 'p is obvious' or 'it is obvious that p') to have a subjective ring, whereas 'Tr(p)' (read 'p is trivial' or 'it is trivial that p') should be something independent of whoever inquires whether p is the case. Of course, I perfectly allow that some would prefer to conflate these and say that 'Tr(p)' is equally subjective, but the point is rather to investigate what sense can be made of triviality understood in the former way. Can we say anything intersting about the Tr-operator apart from what can be said about the O-operator in a psychological vein? Could the Tr-operator operator be subject to formalization?

Second: There is a sense of the word 'trivial' which I'm not interested in, namely in the sense of 'they had a conversation about trivial things', that is "everyday things". In this sense, I'm sure the weather in St. Andrews is quite trivial, but this is not what I intend to capture with the Tr-operator. Rather, as alluded to above, I'm interested in the use common among mathematicians, logicians and, to a certain degree, philosophers.

Third: A final consideration has already been anticipated by the above stipulations. A natural question to ask is what we ascribe triviality to? In the logico-mathematical context (the one we focus on here), triviality is frequently predicated to proofs themselves, e.g., 'The deduction is trivial' or 'There is a trivial proof for p'. However, as indicated by the Tr-operator, I prefer ascribing triviality to propositions. But that is not to say that I do not appreciate the the relationship between proofs and triviality, it is just that the relationship, I believe, is a bit different.

Obviously (sorry about that), Tr(p) -> Tp, i.e., if a proposition is trivial, then it's true. Furthermore, Tr(p) -> Pr(p), i.e., if a proposition is trivial, then it's provable. And, finally, Pr(p) -> Tp, i.e., if p is provable, then p is true. It goes without saying that we do not have the converses: p can be true without being provable, p can be true without being trivial, and p can be provable without being trivial. Now, although I want to keep psychology out of the analysis, I do agree with Andreas that triviality is connected to cognitive work, though we also seem to agree that triviality cannot be equated with a degree of cognitive work. My idea, rather, is that both cognitive work (in this context) and triviality are intimately connected with complexity of proofs. Thus, the answer to the question 'What makes p trivial', lies with the proof(s) of p, even though it is not the proofs themselves which are trivial or non-trivial.

To give an easy example, we could equate the complexity of proofs simply with the length of the proofs, say n. Thus, a bit arbitrarily, we can say that Tr(p) iff p has a proof of no more than n lines. In particular, then, if n = 2, any two-line proof of p (mostly just stating an axiom) would render p trivial. Of course, the particular example given here is not satisfactory, due to the over-simplification of the notion of complexity involved. We can perfectly well imagine a system where there are two-line proofs which are highly non-trivial, which demand substantial work to reach. This can, for instance, be a result of an extremely complex set of rules. Consequently, the rules of the system must also have an impact on the involved notion of complexity.

If we grant that we can fix this notion of complexity, what is achieved? As mentioned above, we get a non-psychological account of triviality (i.e., it can be separated from O(p)), one which is a candidate for further formalization. Moreover, the account does seem to fit with at least some important aspects of the logico-mathematical use. (1) triviality will be a result of complexity; (2) triviality and non-triviality will therefore typically be related to less and more cognitive work (though not necessarily); (3) it is possible for a proposition p (say in number theory) to be non-trivial when first discovered (due to complexity of proof, of course), but later gain status as trivial (by simplification of the proof).

Now, one of the chief challenges is to give an adequate approximation of complexity of proofs. This task, again, rests heavily on our ability to render the idea of proof as transparent as possible. Since complexity is at best vague in respect to informal proofs, the best way to go, it seems, is to study formal proof systems and relate complexity to these. But then we quickly encounter another difficulty: Which proof system? It goes without saying that Tr(p) might hold in proof system S1, while not-Tr(p) for S2. Just take a formula like 'p -> p' for instance - in most natural deduction systems, this is a straight-forward proof, easily dubbed trivial; in a Hilbert-style axiomatic system, however, the same formula would demand for some work. (Note that it is still tempting to say that O(p -> p)). Even more, for any p we can construct a proof system such that Tr(p) or a proof system such that not-Tr(p). But considering that Tr(p) -> T(p), there are additional constraints on the proof systems. However, these constraints will not be enough to single out one particular system, they will only eliminate the extreme cases (such as an inconsistent system). In want of a better reply, then, it's possible to simply admit that triviality, at least in the sense studied here, is relative to proof systems.

A final problem: How do we account for nested Tr-operators? The expression 'Tr(Tr(p))' says something about the complexity of the proofs of the complexity of proofs of p.

Epistemic Logic

An introduction to epistemic logic has recently been added to Stanford Encyclopedia of Philosophy. The introduction is written by Vincent Hendricks and John Symons. It contains a short summary of the basics of epistemic logic, including some more contemporary themes. Although quite short, the authors have been able to provide a few interesting words on the philosophical motivation for epistemic logic.

If you're interested in further reading, check out Hintikka's Knowledge and Belief: An Introduction to the Logic of the Two Notions (1962), recently reissued.

Truth and Proof: Kurt Gödel and the Foundations of Mathematics

A brief statement in the middle of my exam period: 25-26th of March 2006 there will be a conference at the University of Edinburgh titled 'Truth and Proof: Kurt Gödel and the Foundations of Mathematics'. Details can be found here. Am I going? Yes, I am. Among the speakers are Prof. Shapiro (Ohio State\St. Andrews) and Richard Zach (University of Calgary). The latter runs a very interesting blog called LogBlog. Check it out.

Analytic vs. continental philosophy

The stubborn old debate about whether it is any point in distinguishing between analytic and continental philosophy has recently loomed large on Leiter Reports. Since I've been educated at a more or less predominantly continental department (University of Bergen), and have now shifted to a department with a clear overweight of analytic philosophers (University of St. Andrews), I thought I might make a brief remark on the issue.

Some analytic philosophers have written metaphilosophical papers on the nature of analytic philosophy. Among the more well-known is Michael Dummett's Origins of Analytical Philosophy, where the linguistic turn is the chief characteristic of analytical philosophy, thus excluding some familiar names in the anglo-american philosophy (I believe Dummett's example is Gareth Evans). Another, and I guess, not so frequently read attempt at capturing the essence of analytic philosophy, can be found in Dagfinn Føllesdal's 'What is analytic philosophy and why should one engage in it?', published in The Rise of Analytic Philosophy by H-J. Glock. I will say nothing else than that the argument of this paper is as designed for getting people to shout and scream and wave their fists at you. I quite enjoyed it myself, but of course mostly as entertainment.