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.
Friday, 30 September 2011
Tuesday, 27 September 2011
I am writing up a proof that Peano arithmetic (P), and even a small fragment of primitive-recursive arithmetic (PRA), are inconsistent.
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.
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.
Monday, 26 September 2011
Sunday, 25 September 2011
Everything you always wanted to know about epistemic arguments for Bayesianism (but were afraid to ask)
(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 ;-)
Tuesday, 20 September 2011
(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
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
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 En+1 probabilistically supports En if and only if En is more probable if En+1 is true than if it is false.
P (En | En+1) > P (En | ~En+1)
The rule of total probability then becomes:
P (En) = P (En | En+1) P (En+1) + P (En | ~En+1) P (~En+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…
Monday, 19 September 2011
Saturday, 17 September 2011
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?
Thursday, 15 September 2011
Friday, 9 September 2011
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.
Congratulations, Hannes and Richard!
Monday, 5 September 2011
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.
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.)
Friday, 2 September 2011
"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.)
Thursday, 1 September 2011
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;
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!