New APPS: Art, Politics, Philosophy, Science

Non-monotonicity is a fever

(Cross-posted at M-Phi)

In his 2008 paper ‘Logical dynamics meets logical pluralism?’, Johan van Benthem writes (p.185):

… Many observations in terms of structural rules address mere symptoms of some more basic underlying phenomenon. For instance, non-monotonicity is like ‘fever’: it does not tell you which disease causes it.

I’ve always been puzzled by this observation – among other reasons because I’m a non-monotonicity enthusiast, so it seemed odd to me to claim that non-monotonicity would be like the symptom of some disease! But beyond the disease metaphor, it was also not clear to me why Johan saw non-monotonicity as this unspecified, possibly multifaceted phenomenon. After all, there should be nothing esoteric about non-monotonicity: a non-monotonic consequence relation is one where addition of new premises/information may turn a valid consequence into an invalid one. The classical notion of validity has monotonicity as one of its defining features: once a consequence, always a consequence, come what may. This is why a mathematical proof, if indeed valid/correct, remains indefeasible for ever and ever, come what may.

Racism in mathematics

And now, something much more serious from The Guardian: an opinion piece by African-American mathematician Jonathan Farley on racism in mathematics.

[T]here are no black winners of the Fields medal, the "Nobel prize of mathematics". [...] In reality, black mathematicians face career-retarding racism that white Fields medallists never encounter. Three stories will suffice to make this point.

He then goes on to narrate three very depressing stories, the last one about himself. It makes for sobering reading, and it does resonate with the stories we've been hearing about what it's like to be a member of a racial minority in the philosophy profession as well.

UPDATE: On a positive note, it occurred to me that, in this context, it would also be fitting to highlight the Infinite Possibilities series of conferences, whose aim is to celebrate and promote diversity in the mathematical sciences both on the gender and the ethnic/racial dimension. It is a conference "designed to promote, educate, encourage and support minority women interested in mathematics and statistics." The latest installment took place a few weeks ago, and had my fellow country-woman Valeria de Paiva among the keynote speakers. A wonderful initiative!

Diagonalization and Différance: a mismatch of scope

(This post is part of the NewAPPS symposium on Paul Livingston's 'Derrida and Formal Logic: Formalizing the Undecidable')

Paul Livingston’s paper presents a comparative analysis of Gödel’s incompleteness results, Priest on diagonalization, and Derrida on différance. One of the goals seems to be to show that there are significant analogies between these different concepts, which would be at odds with the fact that Derrida’s ideas encounter great hostility among most philosophers in the analytic tradition. In particular, Derrida’s différance and the concept/technique of diagonalization are presented as being each other’s counterpart (a view earlier defended by Graham Priest in Beyond the Limits of Thought).

But crucially, while différance is presented as cropping up in any discourse whatsoever, for a particular language/formal system to have the kind of imcompleteness identified by Gödel specifically with respect to arithmetic, certain conditions must hold of the system. So a fundamental disanalogy between what could be described as the ‘Gödel phenomenon’ (incompleteness arising from diagonalization and the formulation of a so-called Gödel sentence) and Derrida’s différance concerns the scope of each of them: the latter is presented as a perfectly general phenomenon, while the former is provably restricted to a specific (albeit significant) class of languages/formal systems. Although Livingston does not fail to mention that a system must have certain expressive characteristics for the Gödel phenomenon to emerge, it seems to me that he downplays this aspect in order to establish the comparison between différance and diagonalization. (There is much more that I could say on Livingstone’s thought-provoking piece, but for reasons of space I will focus on this particular aspect.)

Summer school: Formal methods in philosophy

I just sent out the announcement for the summer school on formal methods in philosophy that I am organizing. It seems to me that more sustained methodological discussions of applications of formal methods in philosophy are at this point much needed. The summer school is an attempt to foster such debates and motivate students and young researchers to be attentive to the to methodological issues underlying their work. See below for the official announcement, and check the webpage of the summer school for further details.

The Fibonacci sequence, plant growth, and Vi Hart

(Cross-posted at M-Phi)

The Fibonacci numbers are those in the following sequence of integers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 etc. By definition, the first two numbers are 0 and 1, and each subsequent number is the sum of the previous two. The sequence is named after Fibonacci, aka Leonardo of Pisa, who introduced the sequence (known already in Indian mathematics) to Western audiences in his famous book Liber Abaci (1202) – which, by the way, is also one of the main sources for the dissemination of Hindu-Arabic numerals in Europe, no less. (Fibonacci had learned ‘Eastern’ mathematics while studying to become a merchant in North Africa -- see an earlier post on the importation of Indian and Arabic mathematics into Europe through a sub-scientific, merchant tradition.)

Explaining the gender gap in mathematics, chess and philosophy

What do mathematics, chess and philosophy have in common? Among many other things, they have a glaring gap between men and women.  And the reason in all three cases may be cultural, rather than biological. 

Ten to the sixth

NewAPPS will soon reach its millionth page view. How many is that? The image below contains a million dots.

To see them all, download it (right-click or control-click on the image and choose one of the “Download” options), view it in Preview or something similar and blow it up five or six times. The file’s name is “OybTz.png”, and even though it contains a million dots it is very small.

Just for you compulsive types, I’ve made one dot red.

On Mathematicians and Philosophers.

In the exchange over Dennis' wonderful post on infinitesimals, Dennis writes, "Casting doubt on someone’s pronouncements is very far from devising a consistent theory to show them false (consistent because it has models in the category of sets). A mathematician would regard that as the difference between gilt and gold." I call Dennis' move here (and it is one that Russell also was frequently attracted to), an instance of "Newton's Challenge to Philosophy." That is, a philosopher appeals to natural science (or mathematics) to settle a dispute within philosophy. Let me grant Dennis' claim about the "mathematician." But within philosophy burden-shifting is no small achievement. Note, in particular, that Russell appeals to mathematics in order to condemn Leibniz's wrong turn to "speculation." That is to say, it is one thing to get the math wrong or to be unable to provide a mathematical proof for a claim within mathematics. It is another thing to make a claim to the effect that metaphysics of mathematics has been settled. I suspect it was inevitable that Russell would be wrong about the latter.

The (in)consistency of PA and consensus in mathematics

Last week, the foundations of mathematics community was shaken by yet another claim of the inconsistency of Peano Arithmetic (PA). This time, it was put forward by Edward Nelson, professor of mathematics in Princeton, who claimed to have found a proof of the inconsistency of PA. A few months ago, quite some stir had been caused when Fields medalist V. Voevodsky seemed to be saying that the consistency of PA was an open question; but Nelson’s claim was much more radical; he claimed to have proved that PA was outright inconsistent! (Here is a great post by Jeff Ketland with a crash-course on PA and a discussion of ways in which it might be inconsistent.)

Nelson announced his results on the FOM mailing list on September 26^th 2011, providing links to two formulations of the proof: one in book form and one in short-summary form. Very quickly, a few math-oriented blogs had posts up on the topic; we all wanted to understand the outlines of Nelson’s purported proof, and most of us bet all our money on the possibility that there must be a mistake in the proof. External evidence strongly suggests that PA is consistent, in particular in that so many robust mathematical results would have to be revised if PA were inconsistent (not to mention several proofs of the consistency of arithmetic in alternative systems, such as Gentzen’s proofs -- see here).

Indeed, it did not take long for someone to find an apparent loophole in Nelson’s purported proof, and not just someone: math prodigy and Fields medalist Terence Tao (UCLA), who is considered by many as the most brilliant mathematician currently in activity. The venue in which Tao first formulated his reservations was somewhat original: on the G+ thread opened by John Baez on the topic. (So those who dismiss social networks as a pure waste of time have at least one occurrence of actual top-notch science being done in a social network to worry about!)

The 2011 Royal Society Winton Prize for Science Books

Today the shortlist for the 2011 Royal Society Winton Prize for Science Books was announced. As it turns out, I’m a big fan of popular science books; when they are good, they are not only entertaining to read, but I often find insights and ideas that I then go on to use in my academic philosophical work. Of course, they are mostly a starting point, as you still need to do your homework and check the actual scientific articles/sources, but comprehensive overviews can be a valuable source of insight and inspiration.

The nominated books are:

· Alex’s Adventures in Numberland by Alex Bellos (Bloomsbury)

· Through the Language Glass: How Words Colour Your World by Guy Deutscher (William Heinemann)

· The Disappearing Spoon by Sam Kean (Doubleday (UK); Little, Brown and Company (USA) )

· The Wavewatcher’s Companion by Gavin Pretor-Pinney (Bloomsbury)

· Massive: The Missing Particle That Sparked the Greatest Hunt in Science by Ian Sample (Basic Books (USA); Virgin Books (UK))

· The Rough Guide to The Future by Jon Turney (Rough Guides)

In the official announcement you can read short descriptions of each of them, and download the first chapters of each. So far, I’ve only read Alex’s Adventures in Numberland, which I very much enjoyed (as reported on this M-Phi post), but pretty much all the others look like they could be really interesting. 

And speaking of math, a few months ago I reported on Fields medalist Voevodsky apparently questioning the consistency of Peano Arithmetic. Well, now somebody else has gone even wilder: Edward Nelson, professor of mathematics in Princeton, claims to have a proof of the inconsistency of Peano Arithmetic! Details can be found here.

 

Representing the hyperbolic plane with crochet

I just posted something on M-Phi about models of the hyperbolic plane construed with crochet. I won't cross-post it here this time, but for readers interested in either crochet or non-Euclidean geometry (or both), it might be something you will enjoy reading. Just to give you an idea, here's a picture of such a model:

       

Beautiful, isn't it? It looks like a coral.

Mathematical reasoning and external symbolic systems

(Cross-posted at M-Phi)

A few weeks ago I wrote a post on blind mathematicians, discussing the case of Bernard Morin and the eversion of the sphere in particular. I had been thinking about blind mathematicians then because I was working on a paper on the role of external symbolic systems (written systems such as notations in particular) for mathematical reasoning and mathematical practice. I have now completed a first, preliminary draft of the paper, and uploaded it on my academia website (it's on top of the list under 'Papers'). Should anyone be interested in taking a look, comments would be most welcome! I discuss the case of Bernard Morin all the way at the end of the paper, as well as the case of Jason Padgett, the man with acquired savant syndrome who sees shapes as fractals and can hand-draw fractals of pretty much any image you can think of. Here is the abstract:

What is it like to be a blind mathematician?

(Cross-posted at M-Phi, and in the spirit of the "Count me in" campaign.)

I am now working on a paper on the role of manipulating notations in mathematical reasoning, which reminded me of an issue that interested me a few years ago: how do blind mathematicians do their work, given that the usual manipulation of notations (which obviously appeals crucially to vision) is not an option to them?

I then sent a query on this topic to the FOM list, and got some very interesting replies. In particular, many people mentioned this fascinating notice published by the American Mathematical Association on blind mathematicians, which I highly recommend to anyone interested in how mathematics 'comes about'. An excerpt:

A sighted mathematician generally works by sitting around scribbling on paper: According to one legend, the maid of a famous mathematician, when asked what her employer did all day, reported that he wrote on pieces of paper, crumpled them up, and threw them into the wastebasket. So how do blind mathematicians work?

Latest news on the 'inconsistency of PA' affair

UPDATE: I'm adding a few small changes to the original post, which were suggested by Steve Awodey.

(Cross-posted at M-Phi.)

As previously reported (here and here), over the last few weeks there has been a heated debate within the foundations of mathematics community on what appeared to be a controversial statement by Fields medalist V. Voevodsky: the inconsistency of PA is an open problem. But what seemed at first instance a rather astonishing statement has become more transparent over the last couple of days, in particular thanks to posts by Bill Tait and Steve Awodey over at the FOM list. (Moreover, I’ve had the pleasure of discussing the matter with Steve Awodey in person here in Munich, in conversations which have been most informative.)

Voevodsky and the (in)consistency of Peano Arithmetic

For those of you who might want to take a break from the Synthese affair :), there is an interesting discussion going on at the FOM mailing list (Foundations of Mathematics); you can read the messages here. It was prompted by videos of lectures by Fields medalist Vladimir Voevodsky (here and here), where (at least according to some) he seems to display a complete lack of understanding of Goedel's incompleteness results.

I wrote a slightly more technical post on this topic over at M-Phi, but let me try to summarize the main points here, assuming that people who do not have a background on the foundations of mathematics may also be interested. As is well known, Goedel's incompleteness results prove the impossibility of proof-in-S of a sentence expressing the consistency of S, for any consistent and sufficiently strong system S (containing Peano arithmetic, PA). But Goedel's results do not prove that the consistency of PA cannot be proved simpliciter, i.e. in absolute terms. And so Gentzen spent a good chunk of the 1930s formulating different proofs of the consistency of PA -- obviously not in PA itself or a system containing it, which Goedel had proved to be impossible, but rather in a different system, namely the theory obtained by adding quantifier free transfinite induction to primitive recursive arithmetic. The details don't matter much (for more, see von Plato's SEP entry on the development of proof-theory), but one important thing is that this system is not strictly stronger than PA, as there are theorems that PA can prove and this system cannot. So it was not just a matter of switching to a stronger theory to prove the consistency of PA. Gentzen's proofs are widely held to be cogent proofs of the consistency of PA (relative, of course, to the perceived reliability of the theory in which they are formulated).