New APPS: Art, Politics, Philosophy, Science

What's wrong with Mochizuki's 'proof' of the ABC conjecture?

(Cross-posted at M-Phi)

A few days ago Eric had a post about an insightful text that has been making the rounds on the internet, which narrates the story of a mathematical ‘proof’ that is for now sitting somewhere in a limbo between the world of proofs and the world of non-proofs. The ‘proof’ in question purports to establish the famous ABC conjecture, one of the (thus far) main open questions in number theory. (Luckily, a while back Dennis posted an extremely helpful and precise exposition of the ABC conjecture, so I need not rehearse the details here.) It has been proposed by the Japanese mathematician Shinichi Mochizuki, who is widely regarded as an extremely talented mathematician. This is important, as crackpot ‘proofs’ are proposed on a daily basis, but in many cases nobody bothers to check them; a modicum of credibility is required to get your peers to spend time checking your purported proof. (Whether this is fair or not is beside the point; it is a sociological fact about the practice of mathematics.) Now, Mochizuki most certainly does not lack credibility, but his ‘proof’ has been made public quite a few months ago, and yet so far there is no verdict as to whether it is indeed a proof of the ABC conjecture or not. How could this be?

As it turns out, Mochizuki has been working pretty much on his own for the last 10 years, developing new concepts and techniques by mixing-and-matching elements from different areas of mathematics. The result is that he created his own private mathematical world, so to speak, which no one else seems able (or willing) to venture into for now. So effectively, as it stands his ‘proof’ is not communicable, and thus cannot be surveyed by his peers.

What is logic?

A few weeks ago Jeff Bell had a post with the suggestive title ‘What is philosophy?’, referring to the eponymous work by Deleuze and Guattari. Now, as it turns out, a similar question, ‘What is logic?’, has preoccupied David Marans, a lecturer in logic at St. Thomas University in Miami, for quite some time. He thus created the Logic Gallery, a compilation of statements by philosophers and logicians across the centuries (starting with Aristotle and ending with Tim Williamson) on the nature of logic (or else comments from which one can infer the author’s conception of logic). It is a useful glimpse at the different views on logic held across time, and as such well worth spending some time on. In particular, it illustrates the early influence of the conception of logic as pertaining to debating and discussing, and the later predominance of the conception of logic as pertaining to thought and mental processes.

Marans welcomes suggestions for improvement and additional submissions. (I should perhaps add the proviso that there are no exact references to the original location of the quotes, and also that there may be some issues of translation.)

Desiderata for formal theories of truth

I have a fairly technical post over at M-Phi on desiderata for formal/axiomatic theories of truth, picking up on some ideas I discussed in a NewAPPS blog post of two years ago. My feeling is that it is a bit too technical/specific for cross-posting here at NewAPPS, but do check it out if formal theories of truth is what you like to read about on a Sunday afternoon!

Indirect proofs in the Prior Analytics

(Cross-posted at M-Phi)

A few days ago I wrote a post on a dialogical conceptualization of indirect proofs. Not coincidentally, much of my thinking on this topic at the moment is prompted by the Prior Analytics, as we are currently holding a reading group of the text in Groningen. We are still making our way through the text, but here are some potentially interesting preliminary findings.

I am deeply convinced that the emergence of the technique of indirect proofs marks the very birth of the deductive method, as it is a significant departure from more ‘mundane’ forms of argumentation (as I argued before). So it is perhaps not surprising that the first fully-fledged logical text in history, the Prior Analytics, offers a sophisticated account of indirect proofs. 

A dialogical conception of indirect proofs

(Cross-posted at M-Phi)

In his commentary on Euclid, the 5^th century Greek philosopher Proclus defines indirect proofs, or ‘reductions to impossibility’, in the following way (I owe this passage to W. Hodges, from this paper): 

Every reduction to impossibility takes the contradictory of what it intends to prove and from this as a hypothesis proceeds until it encounters something admitted to be absurd and, by thus destroying its hypothesis, confirms the proposition it set out to establish.

Schematically, a proof by reduction is often represented as follows:

[~A]

.

.

.

⊥

------

A

 It is well know that indirect proofs pose interesting philosophical questions. What does it mean to assert something with the precise goal of then showing it to be false, i.e. because it leads to absurd conclusions? Why assert it in the first place? What kind of speech act is that? It has been pointed out that the initial statement is not an assertion, but rather an assumption, a supposition. But while we may, and in fact do, suppose things that we know are not true in everyday life (say, in the kind of counterfactual reasoning involved in planning), to suppose something precisely with the goal of demonstrating its falsity is a somewhat awkward move, both cognitively and pragmatically.

The complexity of poker

(A second in a series, drawn from joint work with K. Joseph Mourad.)
How do we measure the complexity of decision procedures in poker?  This is a question that is both complex and subtle, and seems to me interesting in thinking about the interplay between formal modeling of epistemological situations and more concrete strategic epistemic thinking.

impredicativity in finite (empirical) contexts

(This will be the first in a series of posts designed to suggest that the mathematics of impredicativity - especially methods of definition that make use of revision-theoretic procedures - are relevant to empirical contexts.  Everything I say in these grows out of joint work with my math colleague Joe Mourad.)

Two basic points about the notion of impredicativity: first, it is much broader than what non-expert philosophers tend to think of under the rubric of paradoxes, vicious circularity, and the like.  Second, it is a property of definitions - or, more generally, procedures - not of concepts or sets, in the first instance.  Given an appreciation of these points, it is not hard to see that the general phenomenon can pose important epistemological issues in contexts in which there are no infinite totalities in play, indeed, in the context of various empirical discussions.  

The Modus Ponens-Modus Tollens asymmetry and non-monotonic logics

(Cross-posted at M-Phi)

A well-known phenomenon in the empirical study of human reasoning is the so-called Modus Ponens-Modus Tollens asymmetry. In reasoning experiments, participants almost invariably ‘do well’ with MP (or at least something that looks like MP – see below), but the rate for MT success drops considerably (from almost 100% for MP to around 70% for MT – Schroyens and Schaeken 2003). As a result, any theory purporting to describe human reasoning accurately must account for this asymmetry. Now, given that for classical logic (and other non-classical systems) MP and MT are equally valid, plain vanilla classical logic fails rather miserably in this respect.

As noted by Oaksford and Chater (‘Probability logic and the Modus Ponens-Modus Tollens asymmetry in conditional inference’, in this 2008 book), some theories of human reasoning (mental rules, mental models) explain the asymmetry at what is known as the algorithmic level (a terminology proposed by Marr (1982)) – that is, in terms of the mental process that (purportedly) implement deductive reasoning in a human mind. So according to these theories, performing MT is harder than performing MP (for a variety of reasons), which is why reasoners, while still trying to reason deductively, have difficulties with MT. Other theorists defend that participants are not in fact trying to reason deductively at all, so the asymmetry is not related to some presumed competence-performance gap. (Marr’s term to refer to the general goal of the processes, rather than the processes themselves, is ‘computational level’ – the terminology is somewhat unnatural, but it has now become standard.) Oaksford and Chater are among those favoring an analysis at the computational level, in their case proposing a Bayesian, probabilistic account of human reasoning as a normative theory not only explaining but also justifying the asymmetry.

The phylogeny and ontogeny of deductive reasoning

(Cross-posted at M-Phi)

In a recent paper, the eminent psychologist of reasoning P. Johnson-Laird says the following:

[T]he claim that naïve individuals can make deductions is controversial, because some logicians and some psychologists argue to the contrary (e.g., Oaksford & Chater, 2007). These arguments, however, make it much harder to understand how human beings were able to devise logic and mathematics if they were incapable of deductive reasoning beforehand.

This last claim strikes me as very odd, or at the very least as poorly formulated. (To be clear, I side with those, such as Oaksford and Chater, who think that deductive reasoning must be learned to be mastered and competently practiced by reasoners.) It looks like a doubtful inference to the best explanation: humans have in fact devised logic and mathematics, which are crucially based on the deductive method, so they must have been capable of deductive reasoning before that. Something like: birds had to have fully formed wings before they could fly – hum, I don’t think so… Instead, the wing analogy suggests that there must be some precursors to deductive reasoning skills in untrained reasoners, but the phylogeny of the deductive method (and to be clear, I’m speaking of cultural evolution here) would have been a gradual, self-feeding process.

Logical Properties of 'It Seems'

I am currently finishing a paper on the semantic and logical properties of 'seem'. As 'seem' is a subject-raising verb, we can treat 'it seems' as a sentential operator. This raises the question of how this operator behaves logically. Is it hyperintensional? Does it distribute over conjunction? Over disjunction? Over conditionals? Does it commute with negation?

I think it's fairly obvious that 'it seems' is hyperintensional. It seems to Lois Lane that Superman is not Clark Kent but it doesn't seem to her that Superman is not Superman. The other questions are harder.

Suppes on neuroscience as foundations of mathematics

(OK, so it looks like I’m over-posting a bit today… Just one more!)

Between today and tomorrow, the workshop ‘Groundedness in Semantics and Beyond’ is taking place at MCMP in Munich, co-organized with the the ERC project Plurals, Predicates, and Paradox led by Øystein Linnebo. The workshop’s program seems excellent across the board, but the opening talk is what really caught my attention: Patrick Suppes on ‘A neuroscience perspective on the foundations of mathematics’. The abstract:

I mainly ask and partially answer three questions. First, what is a number? Second, how does the brain process numbers? Third, what are the brain processes by which mathematicians discover new theorems about numbers? Of course, these three questions generalize immediately to mathematical objects and processes of a more general nature. Typical examples are abstract groups, high dimensional spaces or probability structures. But my emphasis is not on these mathematical structures as such, but how we think about them. For the grounding of mathematics, I argue that understanding how we think about mathematics and discover new results is as important as foundations of mathematics in the traditional sense.

The best medieval solution to the Liar ever

I posted this on M-Phi yesterday, but now it occurs to me that it might be of interest to some NewAPPS readers as well (the interest sparked by Brit's recent post suggests this much!). It is in fact a review of Stephen Read's edition and translation of Bradwardine's treatise on insolubles, which I just wrote for Speculum. 

------------------------------------------------------

Thomas Bradwardine (first half of the 14^th century) is well known for his decisive contributions to physics (he was one of the founders of the Merton School of Calculators) as well as for his theological work, in particular his defense of Augustinianism in De Causa Dei. He also led an eventful life, accompanying Edward III to the battlefield as his confessor, and dying of the Black Death in 1349 one week after a hasty return to England to take up his new appointment as the Archbishop of Canterbury.

What is thus far less well known about Bradwardine is that, prior to these adventures, in the early to mid-1320s, he worked extensively on logical topics. In this period, he composed his logical tour de force: his treatise on insolubles. Insolubles were logical puzzles to which Latin medieval authors devoted a considerable amount of attention (Spade & Read 2009). What is special about insolubles is that they often involve some kind of self-reference or self-reflection. The paradigmatic insoluble is what is now known (not a term used by the medieval authors themselves) as the liar paradox: ‘This sentence is not true’. If it is true, then it is not true; but if it is not true, then what it says about itself is correct, namely that it is not true, and thus it is true after all. Hence, we are forced to conclude that the sentence is both true and false, which violates the principle of bivalence. It is interesting to note that, in the hands of Tarski, Kripke and other towering figures, the liar and similar paradoxes re-emerged in the 20^th century as one of the main topics within philosophy of logic and philosophical logic, and remain to this day a much discussed topic.

Liars and Truth-Tellers

This work is licensed under a Creative Commons Attribution-NonCommercial 2.5 License. This means you're free to copy and share the comic (but not to sell it). Thanks to Kevin Lepore for the reference.

A Few Puzzles for a Rainy Sunday

Five old puzzles to brush up on your logic skills before the GREs:

(1)
Yesterday I ran into my colleagues Eric Wiland and John Brunero, who were trying to sneak out to watch the baseball game. I asked Eric:
"Is any of you a truth-teller?"
Eric said something that sufficed for me to know the answer to my question. Are Eric and John liars or truth-tellers?

Domains of quantification - part 2: does science quantify over reals?

Short version: 
Science is often said to be committed to reals, because physics, for example, essentially makes use of sentences with real-quantifiers.  But we have perfectly good countable, well-founded, constructive models of full second order arithmetic.  So why can't physics, for example, simply explicitly embrace one of these as what they are working over and thereby radically simplify their alleged ontological commitments?

Domains of quantification - part 1

It seems to me that there is an issue with the epistemology of domains of quantification that has important implications for the epistemology and semantics of math generally, and which has received less attention than it deserves.  In quick outline, the point is this:

A quantificational sentence has a determinate meaning only if there is some determinate fact of the matter as to what its domain of quantification is.

So one knows what one is saying with such a sentence only if one knows what domain one is quantifying over.  If we are discussing anything as complex as the reals - equivalently second order arithmetic - and mean to quantify over the "intended model" - that is, do not specify some constructable model as our domain - then we do not know what we are quantifying over.

Thus, we do not know what we are saying when we make claims with second order arithmetic quantifiers.

Stebbing Vindicated: on technical expertise as a criterion in being the useful subject of attention

Mohan wrote a useful follow up to my post on the unjustly forgotten Susan Stebbing. Before I get to the core of my response to Mohan's post I want to prevent a potential confusion. Mohan suggests  "it is a pity that historians of analytic philosophy don’t give her more airtime." Well, as some commentators on my post noted, historians of analytic philosophy DO write about Stebbing (and this is how I explained how I came to know of her work). My claim is, rather, that even if we leave aside Stebbing's fascinating treatment of Spinoza, reading Stebbing can be instructive to us (by which I mean professional philosophers today): (i) by reminding us that not all early analytic philosophy was fundamentally anti-Metaphysical; following up the work of Moore and Wisdom (etc), Stebbing is developing and theorizing the tools of analysis in order to offer a distinctly analytic account of metaphysics; (ii) I call it distinctly analytic because it anticipates in rudimentary and non-trivial ways the orientation of the program of David Lewis.  (Of course, I don't want to overstate the claim--modality is not driving Stebbing's approach.) This matters even if there is no line of influence. (iii) Stebbing's understanding of the continuity between Bradley and Russell is a useful corrective to a lot of narratives within the disipline. 

By contrast, Mohan claims that Stebbing's "lack of technical expertise in the key tropes of analytic philosophy (Russell's theory of descriptions, no less) meant that it was not very useful for later generations to read her work." At the end of this post I say something about the criterion that Mohan applies here. Now Mohan offers a bunch of examples of Stebbing's lack of "technical expertise." Dan Kervick has usefully challenged Mohan's examples. Now while I have a healthy admiration for Mohan's technical chops (around NewAPPS I am among the techno-ignoramuses), I want to develop Kervick's points a bit. I do so in reverse order (because all my comments turn, in part, on what Stebbing means by "refer.") Mohan writes:

Susan Stebbing: The History of Metaphysics at the Dawn of Analytic Philosophy

From this point of view metaphysicians are apt to seek for an indubitable datum. Descartes, for example, saw clearly the need for such a datum. Whatever one may think of the success of his efforts, it will probably be admitted that his procedure was correct provided that the aim of metaphysics be to provide reasons for our commonsense beliefs. Curiously enough, it is Descartes, rather than Spinoza who has shown most clearly the futility of a constructed system. Spinoza, so it seems to me, used the form of a deductive system in order to exhibit his vision of the universe. I see no reason why he should be dismayed by the charge of circularity; there is no reason why he should not wrap up in his definitions and axioms all that he desired to bring forth from them. For Descartes, however, such a charge, if it could be substantiated, would be fatal.--Susan Stebbing (69).

One of the great scandals of analytic philosophy is our utter ignorance of our history. Long after my  graduate education I encountered Susan Stebbing´s name for the first time in Mike Beany´s writings on early analytic. (But don't try looking for Stebbing in Soames.) Stebbing (1885 – 1943) was a crucial organizational figure in the movement (she helped found Analysis, brought Carnap to Cambridge for the first time, etc) and important popularizer. (Stebbing is useful to those who wish to tell an alternative to my stories (and here) about Ernest Nagel as Prophet of Analytic Philosophy.) Even so, here I focus on her because she was arguably the first significant theorist of analytic philosophy and the method proper to it. In particular, it is astonishing that Stebbing's (1932) master-piece, "The Method of Analysis in Metaphysics," was (I believe) never properly anthologized in the early readers of analytic philosophy.

Now, Stebbing (and her critic, Max Black) worked hard at explicating what the technique and method of philosophical analysis amounts to. What is crucial is that long before Strawson, Benardete, and Kripke-Lewis, a pre-eminent analytic philosopher had articulated (on the basis of deep reflection on Russell and Moore) how one should think about the marriage of analytic philosophy and metaphysics, which according to Stebbing "is a systematic study concerned to show what is the structure of the facts in the world to which reference is made, with varying degrees of indirectness, whenever a true statement is made." (65) For Stebbing metaphysical analysis had a direction toward the precise understanding of absolutely simple elements (that are taken to exist). Now Stebbing insisted on a distinction between logical analysis and metaphysical analysis. Black denied the distinction and offered a deflationary model for philosophy, which is primarily interested in the "structure of sentences rather than with fact." (258) Black seems to have won the day initially (although I think he misunderstands Stebbings position as uncovering facts rather than the structure of facts), but either way from our post-Lewisian vantage point, Stebbing seems most prescient. (Black ridicules the very idea of an analysist being able to describe basic facts.) We can understand Lewis' natural properties as giving Stebbing her cake (Lewis and Stebbing share a deep respect for common sense and natural science) with little loss of...ahum...simplicity. 

Interview with Keith Stenning in The Reasoner

(Cross-posted at M-Phi)

The latest issue of The Reasoner is out, and among other interesting features, it has an interview with the cognitive scientist Keith Stenning, conducted by me. Keith is (in my opinion) the sharpest, most interesting researcher currently working on reasoning, so everyone should be thoroughly familiar with his work! He has two fairly recent can't-miss books, Seeing Reason and (with Michiel van Lambalgen) Human Reasoning and Cognitive Science. In the interview, he talks about the status of cognitive science and psychology as disciplines, his interdisciplinary professional trajectory, and his recent collaboration with Michiel van Lambalgen tracking down the inherent component of defeasibility in human reasoning.

Do check it out if reasoning is one of your hobby horses!

Deductive proofs: transferability and the 'built-in opponent'

(Cross-posted at M-Phi)

This week I read an extremely interesting paper by Kenny Easwaran, ‘Probabilistic proofs and transferability’, which appeared in Philosophia Mathematica in 2009. Kenny had heard me speak at the Formal Epistemology Workshop in Munich a few weeks ago, and thought (correctly!) that there were interesting connections between the concept of transferability that he develops in the paper and my ‘built-in opponent’ conception of logic and deductive proofs; so he kindly drew my attention to his paper. Because I believe Kenny is really on to something deep about mathematics in his paper, I thought it would be a good idea to elaborate a bit on these connections in a blog post, hoping that it will be of interest to a number of people besides the two of us!

Paradoxes give him heartburn

Via Language Log, an irresistable remix of the Liar Paradox, from Ed McBain’s Blood Relatives (1975). The setup includes Police Detective Carella, a memo from the Commissioner about memos, and a rubber stamp. Go read the rest…

The Spinoza-Clarke dispute in Boole (the origins of analytic philosophy, reconsidered)

A few weeks ago, I quoted this line from Russell: "The ethical work of Spinoza, for example, appears to me of the very highest significance, but what is valuable in such work is not any metaphysical theory as to the nature of the world to which it may give rise, nor indeed anything which can be proved or disproved by argument." (On Scientific Method In Philosophy) In commenting on the full passage, I remarked, "Russell systematically defends the beauty and nobility of Spinoza's ethics, but denies it is philosophical for three reasons: (a) Spinoza's metaphysics of substance has been disproved by science. I call such appeal to the authority of science to settle philosophic argument within philosophy, "Newton's Challenge to Philosophy." Russell here echoes eighteenth century Newtonian criticism of Spinoza. Once Spinoza's metaphysics is discredited, (b) Spinoza's ethics is not really founded on or supported by argument, but by feeling (and this is indeed also Russell's view of Bergson) and "maxims." I actually think this reading of Spinoza has a lot going for it (and the significance of maxims is a way in which Spinoza and Smith are connected), but about that some other time. (c) The feeling that Spinoza's ethics promotes is oriented toward practice, and (c*) Russell denies that philosophy is fundamentally concerned with practice, but rather with understanding." In private I wondered why Russell would have thought (b); had he done an analysis of Spinoza's arguments in the Ethics, which (we learn from Monk's biography) he admired greatly, and found them wanting? I doubt he did.

Rather, I suspect Russell remembered chapter 13 (and 14) of Boole's masterpiece, An Investigation of the Laws of Thought (1854). Elsewhere, Russell credits Boole with having discovered "pure mathematics," or formal logic. (Oddly, Boole is not mentioned by Monk; this neglect is emblematic for a more general problem in the way "early analytic" is treated, but about that some other time more.) Boole writes, "The analysis of its [Spinoza's Ethics'--ES] main argument is extremely difficult, owing not to the complexity of the separate propositions which it involves, but to the use of vague definitions, and of axioms which, through a like defect of clearness, it is perplexing to determine whether we ought to accept or to reject. While the reasoning of Dr. Samuel Clarke is in part verbal, that of Spinoza is so in a much greater degree; and perhaps this is the reason why, to some minds, it has appeared to possess a formal cogency, to which in reality it possesses no just claim." (p. 145) What are these remarks doing in a book that regardless of its limitations from a more recent point of view, invents "pure mathematics"?

The 'logic-as-umpire' view of logic: a Kantian relic

(Cross-posted at M-Phi)

As widely circulated on the internet, Tim Williamson has a recent The Stone post, on the purported ‘neutrality’ of logic. He writes:

Here’s an idea many philosophers and logicians have about the function of logic in our cognitive life, our inquiries and debates. It isn’t a player. Rather, it’s an umpire, a neutral arbitrator between opposing theories, imposing some basic rules on all sides in a dispute. The picture is that logic has no substantive content, for otherwise the correctness of that content could itself be debated, which would impugn the neutrality of logic. One way to develop this idea is by saying that logic supplies no information of its own, because the point of information is to rule out possibilities, whereas logic only rules out inconsistencies, which are not genuine possibilities. On this view, logic in itself is totally uninformative, although it may help us extract and handle non-logical information from other sources.

The idea that logic is uninformative strikes me as deeply mistaken, and I’m going to explain why. […]

Is non-monotonicity a symptom?

This post is a follow-up on Catarina's excellent post "non-monotonicity is a fever".  I want to defend one interpretation of Van Benthem's comment that Catarina says "always puzzled" her: "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 have not read the relevant book, so this may not be what he had in mind, but I think it is right.

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.

Women in logic -- Historical figures

Prompted by the changes to the ‘Women in Logic’ list announced earlier this week,* Erich Reck writes to inquire on historical figures of female logicians. The ‘Women in Logic’ list focuses on researchers currently active, but Erich rightly points out that it would be very important to pay more attention to female logicians from the past, in particular but not exclusively in the context of logic courses.

A similar work of ‘rediscovery’ of the importance of female philosophers for the history of philosophy has been undertaken by a number of people (see here); it is a well-known phenomenon that history tends to be rather ‘forgetful’ of these important female figures (as well as of other groups – see my post on the neglect of Arabic Philosophy in standard curricula). But I am not aware of any list of historical female figures specifically for logic, so I’d like to ask readers to add suggestions here in comments (another list in the making…).

Bergson says Spinoza, Russell says Leibniz

 In his nice post on the origins of analytic philosophy, a theme that has been addressed many times at this blog, Eric points out that Bergson was among Russell's principle targets. It is perhaps understandable that this part of the story has been forgotten. The Frege-Russell narrative concerning the origins and legitimacy of analytic philosophy has largely maintained its hegemonic status. Within this narrative, Bergson emerges not only as a  marginal “continental” figure safely ignored by analytic philosophers but moreover he is a philosopher whose influence on the continent was itself surpassed by the phenomenologists (who, such as Merleau-Ponty, rejected Bergson’s vitalism). However, following up on a comment I made on the thread to John Protevi’s post about Adrian Moore’s book, I would push back the genealogy of an important aspect of the continental/analytic divide to the choice between Spinoza or Leibniz. Understood in this way, Bergson, à la the song "Let's call the whole thing off," says Spinoza and Russell says Leibniz. In targeting Bergson, therefore, Russell is ultimately one in a line of many who are seeking to exorcise the specter of Spinoza.

A tutorial on diagonalization

Related to the recent NewAPPS symposium on Livingston’s paper on Derrida and diagonalization, my M-Phi co-blogger Jeff Ketland posted a really neat tutorial on diagonalization and the diagonal lemma. One of the things that emerged in discussions of Livingston’s paper is that there are quite a few different conceptions of diagonalization floating around, so in this context Jeff’s post may be of interest to many readers – something like ‘everything you’ve always wanted to know about diagonalization but were afraid to ask’. Enjoy!

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.