How good are Bayesian models of cognition?

The current issue of Behavioral and Brain Sciences may be of interest to empirically-minded Bayesian M-Phi'ers. The target article is 'Bayesian Fundamentalism or Enlightenment? On the explanatory status and theoretical contributions of Bayesian models of cognition', by Matt Jones and Bradley C. Love. Here is the abstract:

The prominence of Bayesian modeling of cognition has increased recently largely because of mathematical advances in specifying and deriving predictions from complex probabilistic models. Much of this research aims to demonstrate that cognitive behavior can be explained from rational principles alone, without recourse to psychological or neurological processes and representations. We note commonalities between this rational approach and other movements in psychology – namely, Behaviorism and evolutionary psychology – that set aside mechanistic explanations or make use of optimality assumptions. Through these comparisons, we identify a number of challenges that limit the rational program's potential contribution to psychological theory. Specifically, rational Bayesian models are significantly unconstrained, both because they are uninformed by a wide range of process-level data and because their assumptions about the environment are generally not grounded in empirical measurement. The psychological implications of most Bayesian models are also unclear. Bayesian inference itself is conceptually trivial, but strong assumptions are often embedded in the hypothesis sets and the approximation algorithms used to derive model predictions, without a clear delineation between psychological commitments and implementational details. Comparing multiple Bayesian models of the same task is rare, as is the realization that many Bayesian models recapitulate existing (mechanistic level) theories. Despite the expressive power of current Bayesian models, we argue they must be developed in conjunction with mechanistic considerations to offer substantive explanations of cognition. We lay out several means for such an integration, which take into account the representations on which Bayesian inference operates, as well as the algorithms and heuristics that carry it out. We argue this unification will better facilitate lasting contributions to psychological theory, avoiding the pitfalls that have plagued previous theoretical movements.

Among the commentators (BBS works with the formula of one target article and a number of peer-commentaries per issue), some familiar names such as Lawrence Barsalou, Nick Chater, Mike Oaksford, Clark Glymour, and my esteemed colleague Jan-Willem Romeijn. I haven't had the chance to check it out yet, but it looks promising!

Here we go again: "PA is inconsistent" (Edward Nelson)

UPDATE October 1st: Nelson withdraws his claim.

Readers probably recall the 'Voevodsky affair' of a few months ago (as reported here and in subsequent posts), prompted by Voevodsky's claim (or suggestion?) that the consistency of PA is an open problem. This week, an even more daring claim has circulated at the FOM list: PA is downright inconsistent. Its author is Edward Nelson, professor of mathematics at Princeton, known for his work on Internal Set Theory and Robinson arithmetic. In his words:

I am writing up a proof that Peano arithmetic (P), and even a small fragment of primitive-recursive arithmetic (PRA), are inconsistent.

He refers to a draft of the book he is working on, available here, and to a short outline of the book, available here. I've skimmed through the outline, which focuses mostly on a critique of finitism for making tacit infinitary assumptions, and towards the end there are some interesting considerations on the methodology he has been working with. In particular, he has devised a automated theorem-checker, qea:

Qea. If this were normal science, the proof that P is inconsistent could be written up rather quickly. But since this work calls for a paradigm shift in mathematics, it is essential that all details be developed fully. At present, I have written just over 100 pages beginning this. The current version is posted as a work in progress at http://www.math.princeton.edu/~nelson/books.html, and the book will be updated from time to time. The proofs are automatically checked by a program I devised called qea (for quod est absurdum, since all the proofs are indirect). Most proof checkers require one to trust that the program is correct, something that is notoriously diffi cult to verify. But qea, from a very concise input, prints out full proofs that a mathematician can quickly check simply by inspection. To date there are 733 axioms, de nitions, and theorems, and qea checked the work in 93 seconds of user time, writing to les 23 megabytes of full proofs that are available from hyperlinks in the book.

At this point, I really do not know what to think of Nelson's claims, and I doubt that I would be able to make much sense of his proofs anyway. So for now I'm just acting as a 'reporter', but I'd be curious to hear what others think!

UPDATE: Over at The n-Category Cafe', John Baez has a much more detailed post on Nelson's claims, including a suggestion by Terry Tao on a G+ thread as to what seems to be wrong with the proof. (Yay for G+!) Check also Tao's comment at 5:29.

ANOTHER UPDATE: Edward Nelson has replied at FOM to some of the queries that had been put forward. You can read his message here. He replies in particular to Tao's observations:

So far as I know, the concept of the "Kolmogorov complexity of a theory", as opposed to the Kolmogorov complexity of a number, is undefined. Certainly it does not occur in Chaitin's theorem or the Kritchman-Raz proof.  I work in a fixed theory Q_0^*. As Tao remarks, this theory cannot prove its own consistency, by the second incompleteness theorem. But this is not necessary. The virtue of the Kritchman-Raz proof of that theorem is that one needs only consider proofs of fixed rank and level, and finitary reasoning leads to a contradiction.

UPDATE AGAIN: I really encourage everybody to go check the comments at the The n-Category Cafe' post; the explanations of what is wrong with Nelson's purported proof are really very clear and accessible. I'm wondering if it would be worth writing a separate post on this? (Anyone?) At any rate, as an anonymous commentator says below, there's much to be commended in a purported proof whose loop-hole(s) can be identified fairly quickly; at least it was well formulated to start with.

PhD and post-doc positions in Groningen

In the context of my new research project The Roots of Deduction, two positions have just been advertised, one for a PhD student and one for a post-doc. Details on the two positions can be found here. The PhD candidate will reassess the literature on psychology of reasoning and mathematical cognition from the point of view of the dialogical, multi-agent conception of deduction underpinning the project, and the post-doc will work on the historical development of the deductive method in ancient Greece, again from a dialogical point of view.

Please help me spread the word, and do get in touch if you think you know of suitable candidates. Thanks!

Everything you always wanted to know about epistemic arguments for Bayesianism (but were afraid to ask)

It's in Richard Pettigrew's latest entry in the Stanford Encyclopedia of Philosophy.

(It occurred to me that a couple of friends and M-PHIers in Munich will find this much more useful than my awkward attempts to meet their queries a few nights ago ;-)

Formal epistemology at its best

(Cross-posted at NewAPPS.)

So far, I have not been following developments in formal epistemology very closely, even though the general project has always been in the back of my mind as a possible case-study for my ideas on the methodology of using formal tools in philosophy (and elsewhere). Well, last week I attended two terrific talks in formal epistemology, one by Branden Fitelson (joint work with Kenny Easwaran) in Munich, and one by Jeanne Peijnenburg (joint work with David Atkinson) in Amsterdam. (Full disclosure: Branden is a good friend, and Jeanne is my boss in Groningen! But I’m sure everybody will agree they are among the very best people working on formal epistemology these days.) These two talks illustrate two different ways in which the application of formal methods can be illuminating for the analysis of epistemological concepts and theories, and thus confirmed my hunch that formal epistemology can be a good case study for a more general reflection on formal methodology.

Let me start with Branden’s talk, 'An 'evidentialist' worry about Joyce's argument for probabilism'. The starting point was the preface paradox, and how (in its ‘bad’ versions) it seems to represent a conflict between evidential norms and coherence/accuracy norms. We all seem to agree that both coherence/accuracy norms and evidential norms have a normative grip over our concept of knowledge, but if they are in conflict with one another (as made patent by preface-like cases), then it looks like we are in trouble: either our notion of knowledge is somewhat incoherent, or there can’t be such thing as knowledge satisfying these different, conflicting constraints. Now, according to Branden (and Kenny), Jim Joyce’s move towards a probabilistic account of knowledge is to a large extent motivated by the belief that the probabilistic framework allows for the dissolution of the tension/conflict between the different kinds of epistemic norms, and thus restores peace in the kingdom.

However, through an ingenious but not particularly complicated argument (relying on some ‘toy examples’), Branden and Kenny show that, while Joyce’s accuracy-dominance approach to grounding a probabilistic coherence norm for credences is able to resist the old ‘evidentialist’ threats of the preface-kind, new evidentialist challenges can be formulated within the Joycian framework itself. (I refer the reader to the paper and the handout of the presentation for details.) At Q&A, I mentioned to Branden that this looks a lot like what we’ve had with respect to the Liar paradox in recent decades: as is well known, with classical logic and a naïve theory of truth, paradox is just around the corner, which has motivated a number of people to develop ‘fancy’ formal frameworks in which paradox could be avoided (Kripke’s gappy approach, Priest’s glutty approach, supervaluationism, what have you). But then, virtually all of these frameworks then see the emergence of new and even more deadly forms of paradox – what is referred to as the ‘revenge’ phenomenon. What Branden and Kenny’s work seemed to be illustrating is that the Joycean probabilistic framework is not immune to revenge-like phenomena; the preface paradox strikes again, in new clothes. Branden seemed to agree with my assessment of the situation, and concluded that one of the upshots of these results is that there seems to be something fishy with how the different kinds of epistemic norms interact on a conceptual level, which cannot be addressed simply by switching to a clever, fancy formalism. In other words, probabilism is great, but it will not make this very problem go away.

This might seem like a negative conclusion with respect to the fruitfulness of applying formal methods in epistemology, but in fact the main thing to notice is that Branden and Kenny’s results emerge precisely from the formal machinery they deploy. Indeed, one of the most fascinating features of formal methods generally speaking is that they seem to be able to probe and explore their own limitations: Gödel’s incompleteness results, Arrow’s impossibility theorem, and so many other revealing examples. It is precisely by deploying these formal methods that Branden and Kenny can then conclude that more conceptual discussion on how the different kinds of epistemic norms interact is required.

Three days later, I attended Jeanne’s talk at the DIP-colloquium in Amsterdam (the colloquium I used to run when I was still working there). The title of the talk is great, ‘Turtle epistemology’, which of course refers to the famous anecdote ‘it’s turtles all the way down!’. Jeanne and her co-author David are interested in all kinds of regress phenomena in epistemology, in particular in the foundationalist claim that infinite regress makes any justification impossible. I quote from the abstract:

The regress problem in epistemology traditionally takes the form of a one-dimensional epistemic chain, in which (a belief in) a proposition p1 is epistemically justified by (a belief in) p2, which in turn is justified by (a belief in) p3, and so on. Because the chain does not have a final link from which the justification springs, it seems that there can be no justification for p1 at all. In this talk we will explain that the problem can be solved if we take seriously what is nowadays routinely assumed, namely that epistemic justification is probabilistic in character. In probabilistic epistemology, turtles can go all the way down.

They start with a formulation of justification in probabilistic terms, more specifically in terms conditional probabilities: proposition E_n+1 probabilistically supports E_n if and only if E_n is more probable if E_n+1 is true than if it is false.

P (E_n | E_n+1) > P (E_n | ~E_n+1)

The rule of total probability then becomes:

P (E_n) = P (E_n | E_n+1) P (E_n+1) + P (E_n | ~E_n+1) P (~E_n+1)

Again through an ingenious and very elegant argument, Jeanne and David then formulate infinite chains of conditional probabilities, but show that it is simply not true that they do not yield a determinate probability to the proposition in question. This is because, the longer the chain, and thus the further away the ‘ur-proposition’ is (the one we cannot get to because the chain is infinite), the smaller its influence on the total probability of E₀. At the limit, it gets cancelled out, as it is multiplied by a number that tends to 0 (for details, check their paper here, which appeared in the Notre Dame Journal of Formal Logic).

The moral I drew from their results is that, contrary to the classic, foundational axiomatic conception of knowledge and science, the firmness of our beliefs is in fact not primarily grounded in the very basic beliefs all the way down in the chain, i.e. the ‘first truths’ (Aristotle’s Arché). Rather, their influence becomes smaller and smaller as we go up the chain. At this point, there seem to be two basic options: either we must accept that the classical foundationalist picture is wrong, or we reject the probabilistic analysis of justification as in fact capturing our fundamental concept of knowledge. Either way, this particular formal analysis was able to unpack the consequences of adopting a probabilistic framework, and to show not only that in this setting, infinite regress need not be an insurmountable problem, but also that the epistemic weight of ‘basic truths’ may be much less significant than is usually thought. In a sense, this seems to me to be an example of Carnapian explication, where the deployment of formal methods can in fact unravel aspects of our concept of knowledge that we were not aware of.

Thus, these two talks seemed to me to illustrate the strength of formal methodologies at their best: in investigating their own limits, and in unpacking features of some of our concepts that are nevertheless ‘hidden’, buried under some of their more superficial layers. I guess I’m starting to like formal epistemology…

Book draft: Formal Languages in Logic

(Apologies for shameless self-promotion!)

In several of my posts, I mentioned the book on formal languages that I've been working on for the last few years. I now have a draft of the book ready for (moderate!) public consumption, which is now available here. The two final chapters are still missing, but the draft is already something of a coherent whole, or so I hope.

Many people have kindly expressed their interest in checking out the material, hence my decision to make it available online at this point, despite the fact that it is still a somewhat rough draft (references are still a mess). Needless to say, comments are always welcome :)

Roy's Fortnightly Puzzle: Volume 9

OK, so not so much a puzzle as a question this time.

I am currently co-teaching a graduate seminar on philosophy of mathematics this semester (structuralism versus logicism, to be more specific). We did a pretty good job of advertising the seminar, and as a result have a number of mathematicians sitting in the class (both faculty and graduate students).

The issue is this: As we talk about the philosophical questions and their possible solutions (for example, last week we read Benecerraf's "What Sets Could Not Be" and "Mathematical Truth", since these set up the issues at issue between modal structuralism and Scottish logicism quite nicely), the mathematicians kept coming back to the fact that none of these issues seem to have any bearing on what mathematicians actually do.

At one level I agree with this - when actually doing mathematics, mathematicians need not, and probably ought not, be thinking about whether their quantifiers range over abstract objects or something else. Rather, they should be worrying about what follows from what (to put it in an overly simplistic way).

There might be an exception to the above paragraph in moments of mathematical crisis - for example, if one were a nineteenth-century mathematician working in real analysis. But in general the point seems, on a certain level, right.

On the other hand, however, it seems obvious to me that mathematicians will benefit from thinking about philosophical issues (and benefit qua mathematician). But it is somewhat difficult to articulate why they would benefit.

So, any thoughts? In short, what should we say to mathematicians regarding why they ought to care about what philosophers say?

Normal science in a nutshell

On the largest mathematical proof -- aka the Enormous Theorem

I came across an interesting short note about the largest existing mathematical proof, composed of 15.000 pages; it involved more than 100 mathematicians in its formulation. I wonder if there's an entry for it in the Guinness Book of Records? There should be!

The Enormous Theorem concerns groups, which in mathematics can refer to a collection of symmetries, such as the rotations of a square that produce the original shape. Some groups can be built from others but, rather like prime numbers or the chemical elements, "finite simple" groups are elemental.

There are an infinite number of finite simple groups but a finite number of families to which they belong. Mathematicians have been studying groups since the 19th century, but the Enormous Theorem wasn't proposed until around 1971, when mathematician Daniel Gorenstein of Rutgers University in New Jersey devised a plan to identify all the finite simple groups, divide them into families and prove that no others could exist.

Hit parade

It'd better not escape notice that two M-PHIers, Hannes Leitgeb and Richard Pettigrew, made it to the top ten 2010 papers in philosophy with "An Objective Justification of Bayesianism II: The Consequences of Minimizing Inaccuracy," Philosophy of Science, 77 (2010), 236-272.

Congratulations, Hannes and Richard!

Van Benthem on system imprisonment

(Cross-posted at NewAPPS.)

Johan van Benthem is one of my favorite philosophers of logic (and not just because I'm ultimately an Amsterdam child!). He is completely idiosyncratic as a philosopher of logic, as he refuses to 'waste his time' with classical topics such as truth, consequence, paradoxes etc. But this is exactly what I like about what he has to say: he looks at the practices of logicians (being one himself!) and tries to make sense of what it is that we are doing when we 'do logic' in the currently established ways -- at times, adopting a rather critical stance as well. True enough, his observations are very much connected with his own research agenda, and yet they are also surprisingly general.

One of the concepts he's been talking about -- not so much in his 'official' papers, but mostly at talks, personal communication and interviews -- is the concept of system-imprisonment. (It is, however, mentioned in his 1999 'Wider Still and Wider: resetting the bounds of logic', in A. Varzi, ed., The European Review of Philosophy, CSLI Publications, Stanford, 21–44.) Here are some interesting passages:

But how good is the model of natural language provided by first-order logic? There is always a danger of substituting a model for the original reality, because of the former’s neatness and simplicity. I have written several papers over the years pointing at the insidious attractions and mind-forming habits of logical systems. Let me just mention one. The standard emphasis in formal logical systems is ‘bottom up’. We need to design a fully specified vocabulary and set of construction rules, and then produce complete constructions of formulas, their evaluation, and inferential behavior. This feature makes for explicitness and rigor, but it also leads to system imprisonment. The notions that we define are relative to formal systems. This is one of the reasons why outsiders have so much difficulty grasping logical results: there is usually some parameter relativizing the statement to some formal system, whether first-order logic or some other system. But mathematicians want results about ‘arithmetic’, not about the first-order Peano system for arithmetic, and linguists want results about ‘language’, not about formal systems that model language.

(I can't disclose the source for this quotation for now, as it is from a paper for a project I'm involved with which must remain a secret for a few more months... Anyway, the remark on mathematicians wanting results about 'arithmetic' also reminds me of the series of posts on Voevodsky and the incompleteness of arithmetic that we had a while ago.)

Nevertheless, I am worried by what I call the ‘system imprisonment’ of modern logic. It clutters up the philosophy of logic and mathematics, replacing real issues by system-generated ones, and it isolates us from the surrounding world. I do think that formal languages and formal systems are important, and at some extreme level, they are also useful, e.g., in using computers for theorem proving or natural language processing. But I think there is a whole further area that we need to understand, viz. the interaction between formal systems and natural practice.

(This is from an interview at the occasion of the Chinese translation of one of his books.)

I submit that the notions of system imprisonment and system generated problems must be taken seriously when we are using formal methods to investigate a given external target phenomenon. Oftentimes, a whole cottage industry becomes established to tackle what is taken to be a real issue, which is in fact an issue emerging from the formalism being used, not an issue pertaining to the target phenomenon itself. My favorite example here is the issue of 'free variables' in de re modal sentences, which then became seen as a real, deep metaphysical issue. In truth, it is simply an upshot of the formalism used, in particular the role of variables and the notions of bound or free variables. By adopting a different framework (as I did in a paper on Ockham's modal logic of many years ago, in the LOGICA Yearbook 2003 - pre-print version here) which does not treat quantification by means of variables, the 'issue' simply vanishes.

More generally, system imprisonment points in the direction of the epistemic limits of formal methods. Ultimately, what we prove is always relative to a given formal system, and the result lives or perishes with the epistemic reliability of the formal system itself. This does not mean that we should resign ourselves to some form of skepticism and/or relativism (Johan clearly does not!), but simply that we must bear in mind that the formal models are exactly that: models, not the real thing.

Rational psychos?

All too often, the choice of a benchmark of rational agency critically affects the consequences drawn from the empirical study of behavior in cognitive science. Fascinating examples arise from classical studies in the psychology of reasoning (here's a older post touching upon this). A recent Cognition paper by Daniel Bartels and David Pizarro provides challenging evidence concerning normative standards of moral judgment. The abstract goes as follows:

"Researchers have recently argued that utilitarianism is the appropriate framework by which to evaluate moral judgment, and that individuals who endorse non-utilitarian solutions to moral dilemmas (involving active vs. passive harm) are committing an error. We report a study in which participants responded to a battery of personality assessments and a set of dilemmas that pit utilitarian and non-utilitarian options against each other. Participants who indicated greater endorsement of utilitarian solutions had higher scores on measures of Psychopathy, machiavellianism, and life meaninglessness. These results question the widely-used methods by which lay moral judgments are evaluated, as these approaches lead to the counterintuitive conclusion that those individuals who are least prone to moral errors also possess a set of psychological characteristics that many would consider prototypically immoral." (Bartels, D. & Pizarro, D., "The mismeasure of morals: Antisocial personality traits predict utilitarian responses to moral dilemmas", Cognition, 121, 2011, pp. 154-161.)

(I owe the hint to Thoughts on Thoughts.)

New book: Set Theory, Arithmetic, and Foundations of Mathematics

Yesterday the announcement of a new book was sent around at the FOM list, and as it looks like a very interesting book, I thought I'd put a notice of it here at M-Phi too. It is edited by Juliette Kennedy and Roman Kossak, and the (somewhat vague) title is Set Theory, Arithmetic and Foundations of Mathematics. Here is the table of contents:

1. Introduction - Juliette Kennedy and Roman Kossak;
2. Historical remarks on Suslin's problem - Akihiro Kanamori;
3. The continuum hypothesis, the generic-multiverse of sets, and the Ω conjecture - W. Hugh Woodin;
4. ω-Models of finite set theory - Ali Enayat, James H. Schmerl and Albert Visser;
5. Tennenbaum's theorem for models of arithmetic - Richard Kaye
6. Hierarchies of subsystems of weak arithmetic - Shahram Mohsenipour;
7. Diophantine correct open induction - Sidney Raffer;
8. Tennenbaum's theorem and recursive reducts - James H. Schmerl;
9. History of constructivism in the 20th century - A. S. Troelstra;
10. A very short history of ultrafinitism - Rose M. Cherubin and Mirco A. Mannucci;
11. Sue Toledo's notes of her conversations with Gödel in 1972–1975 - Sue Toledo;
12. Stanley Tennenbaum's Socrates - Curtis Franks;

13. Tennenbaum's proof of the irrationality of √2.

I'm not sure what the idea is behind grouping this particular collection of papers (I have not had the chance to check it out, there's probably something on this at the introduction), but it does look like many of these papers are a must-read. I'm particularly interested in the papers concerning non-standard models of arithmetic and Tennenbaum's theorem (full disclosure: Juliette Kennedy and I had a very interesting correspondence on the topic a few years ago), but the set-theory section is also high-power stuff for sure!