Sound and Fury

I’ve just got back from a philosophy of probability conference in Oxford. It was very interesting.

Here are some thoughts I had about Hilary Greaves’ talk. The project is to flesh out an idea of epistemic rationality by analogy to practical rationality and practical decision theory. The idea is that in the “Cognitive Decision Theory” the sort of acts you are interested in are various beliefs you could adopt. Or various belief functions you could adopt. There is an idea of cognitive utility which is supposed to be a measure of how epistemically good you think a certain belief state is. Greaves and Wallace 2006 show that it is rational to update by conditionalisation when some conditions are satisfied.

The thing is, their theory starts by assuming that belief is represented by a probability function. This is a fairly standard, but perhaps too restrictive assumption. To paraphrase from Greaves’ talk yesterday: there are various intuitions we have about epistemic rationality. If we can derive lots of these from the theory, without building them in to the theory, then that’s a good thing. The less we have to start with and still get back all (or most) of our intuitions, the better.

So let’s apply that to the issue of presupposing that belief is probabilistically coherent. Why not start with some more general belief function set up and see under what circumstances probability measures are the uniquely rational way to structure your beliefs? The answer to this question is, I imagine, “because it’s really hard to prove anything about any more general framework”. True enough. But what about the case where belief is represented by a set of probability measures? How much of the argument of Greaves and Wallace goes through in the case where you just replace every instance of “probability measure, ” with “set of probability measures ” and replace every with .

A large spread of probability measures in an agent’s representor can be seen as indicating that agent’s desire to withhold judgment pending more conclusive evidence. I think that’s a valuable thing to want to model formally. The probability measure representation of belief requires that the agent has to be maximally probabilistically opinionated, which I don’t think is epistemically rational…

Presumably in this broader context it is no longer possible to prove that conditionalise is the best updating rule, but perhaps some analogue of conditionalise which incorporates a method for conditionalising on sets of probability measures still works.

Here’s a suggestion as to when it would be rational to have probabilistic credences. If your epistemic utility function were close to the scoring rules of Joyce 1998 then perhaps probabilism would be uniquely rational.

In general however, I think that as well as accuracy (which moves you towards probabilism), there is another source of epistemic utility. Or rather, a source of epistemic disutility which you want to avoid. Accuracy means you derive utility from being close to the truth. You derive disutility from being far from the truth.

Here’s a kind of geometrical analogy to illustrate the sources of utility in contention. Alice’s credence in an event is represented by some subinterval while Bob’s credence is some one point in the unit interval. The event is actually some value . While the vagueness of Alice’s credence is itself a source of epistemic disutility, the fact that is a good thing and gives Alice some positive utility. Bob, on the other hand, gets no “vagueness penalty”, but he does get some disutility proportional to how far wrong he was, i.e. proportional to . So, if an agent does not have enough information to pin down the actual value, it might be epistemically rational to take the hit from the “vagueness penalty” in order to avoid Bob’s “wrongness penalty”. In the same way, paucity of information might make vague interval valued credences preferable to probabilistic point values.

When I went for an interview at Oxford for my undergrad degree one of the things we had to do was a kind of logic test. It had things like “All men wear hats, some men wear ties” and you were asked what it was possible to conclude from that. Presumably they were looking for something like “Some men wear ties and hats”. One of the questions was “Most men wear ties, most men wear hats”. I said that it wasn’t possible to conclude anything from this. But it was then explained to me that if hat-wearers and tie-wearers are both in a majority, then there must be some who are both: the proper conclusion is that “Some men wear ties and hats”.

“All men” is standing in for the universal quantifier (x) and “Some men” stands for the existential quantifier (Ex). But “Most men” is awkward to put in similar terms without resorting to some sort of logic enriched with predicates relating to numerical proportions. But that sort of “statistical logic” is too complex. The fact that some predicate applies not to all but to a majority of some domain is an easily understood fact that does not need a whole logic of proportions behind it. Let’s make “Most of” a new kind of quantifier: (Mx). What properties does (Mx) have?

This is the property my interviewer at Oxford was exploiting. If most x’s are P’s and most x’s are Q’s, then some x’s must be both.

These two properties simply say that if all men wear hats then most men wear hats and if most men wear hats then some men wear hats.

Do we really need a third quantifier here? Is there some way to express “most of” in terms of universal and existential quantifiers? I’m not sure there is. Here’s a first stab. When you say “(Mx)Px” what you are really saying is:

But that’s no good, because that expression still contains an “Mx”. We are kind of going beyond standard predicate logic, but only a little bit. Perhaps Mx shouldn’t be a quantifier, but a property of predicates? But I don’t think that is a particularly satisfying suggestion. A property of predicates would be a third-order entity, and that seems extravagant given the modest goal of formalising the easily understood concept of a majority.

Soon I’ll post again and discuss the “dual” of (Mx): “Only a few men wear hats” “Hat-wearers are in a minority”.

A note on the symbolism: I wanted to use and but I couldn’t think of a way to “invert” the letter “M” in a similar fashion. It shares the vertical symmetry of “A” but when reflected around a horizontal axis as is, you get a “W”, which is no good if the idea is to come up with a new symbol. So I reverted to old-style quantifiers. Suggestions regarding how to “symbolise” majority are most welcome.

I was sad to read that I.J. Good died recently. I only heard about him a couple of weeks ago. Even from the fairly academic work of his I was reading you got a sense that he was a really interesting character with a unique sense of humour. His obituary confirms that impression.

In other news, I’m desperately trying to finish various papers and so on that need doing and I’m not really progressing particularly fast. Not that that is really news. That seems to be my default status. In the coming weeks I have to finish up short papers on the following topics:

• Frege on the definition of numbers
• Why probability is not always the best way to represent uncertainty
• Quantum Mechanics and Structural Realism
• Similarity relations and the theory-world link

Interesting topics all (apart from the first one), but it’s frustrating to have all these little bits and pieces to get done when what I really want to do is get down to a Big Project like my literature review… Although the second and fourth topics at least have some bearing on the topic of the lit rev, so that’s a bonus.

Two upcoming conferences that I will be attending:

• Scientific Realism Revisited
• Graduate Conference: Philosophy of Probability

Exciting stuff, no?

I’ve been thinking about the Dutch Book Argument (DBA) recently. I think the constraints on rational betting preference that underpin the force of the argument are unreasonable. At least they are not always reasonable. I think a better way to think of the argument is as a conditional argument: “Given these rational betting preference conditions, this is how rational degrees of belief should be constrained.” Then you can have a whole series of different DBAs with different betting preference conditions leading to different constraints on credence. It would be interesting to see how one would have to constrain betting preferences in order to have you beliefs behave like upper and lower probabilities or Dempster-Shafer belief functions…

I think this isn’t to undermine the force of the DBA, but to reinforce it. With this wider framework we can understand why people often fail to reason probabilistically. We can understand what aspects of rational betting preference are “non-probabilistic”.

That’s not to say that the DBA isn’t without its flaws. Some elements that concern me are:

• Using betting behaviour as a proxy for belief
• Existence of exactly specific numerical credence (and utility)
• Reasoning as calculating expected utilities
• The idealisations involved in discussing “ideal rational agents”: utility maximising, purely self interested, perfect calculating agents…

This New Scientist article made me wonder about time. The latest super-clocks are sensitive to differences in height (actually differences in gravitational field) small enough that their accuracy would be affected if they were placed on a slightly lower table, for example. This is madness. Maybe we’ve reached a point where time doesn’t really mean all that much any more. On those kind of scales, at that sort of precision, maybe time just stops being a useful concept. Like the length of a coastline stops being a useful concept when you measure it too accurately. (Because it’s a fractal). Or temperature; if you keep zooming in, you reach a point where the “temperature” of the volume becomes meaningless – if you zoom in enough that the volume contains maybe one atom or even no atoms, then surely temperature is unhelpful. Perhaps the same thing happens to time when we start trying to subdivide it into 10^-18ths of a second or whatever it is they are doing…

Here’s another article on time where some people seem to be reaching the same conclusions… I’m not sure these articles will be available – I’m on the campus network and they might have some sort of subscription to the magazine…

Following Benacerraf’s rightly famous paper “What numbers could not be” a variety of authors have written papers entitled “What * could not be”. Here is a list of the ones I’ve come across:

• What numbers could not be, P. Benacerraf
• What conditional probabilities could not be, A. Hajek
• What structures could not be, J. Busch
• What possible worlds could not be, R. Stalnaker
• What chances could not be, J. Ismael
• What justification could not be, M.T. Nelson
• What mathematical truth could not be, P. Benacerraf
• What unarticulated constituents could not be, L. Clapp
• What equality of oppurtunity could not be, M. Risse

OK, I haven’t read most of these – I just did a google scholar search for “What * could not be”, but it’s interesting to see how a good title is “remixed”… (I’ve only read the top three or four) Also worthy of mention is “Numbers can be just what they have to” by Colin McLarty, another way to refer obliquely to Benacerraf. Another good title is “what is it like to be a bat?” My favourite title to play on this classic paper is “What is it like to be boring and myopic?”

I read this article which is in the Reuters “Oddly Enough” section, where they put their quirky, weird stories. It was just a little too bleak for my liking. A conman has been exectued in China. And one of the people he conned has commited suicide… Wahey! Funny old world, eh?

Synonymy must be one of very few words to have 3 “y”s in.

There was recently an article in the Guardian saying that some watchdog or thinktank or something had put the UK in the group that is likely to be worst hit by the credit crunch. Along with the Netherlands, Spain, Hungary and Luxembourg… All I could think was “Wow cool, Luxembourg got mentioned in the paper. Wait a minute. Oh shit…” (P.S. I’m not sure those were the right countries, but it was something like that…)

I’m writng an essay on Bertrand’s paradox and similar problems with the principle of indifference, but I’ve changed my mind so many times in the course of writing it… It’s going to need some serious tidying up once I decide what my actual position is. My thinking on the problem over the last year-ish has gone: Ooh Bertrand’s paradox, It’s OK, Jaynes has solved it. Oh no he hasn’t. It’s OK van Fraassen solves problems with indifference, Ah. he hasn’t. Actually, the wine/water problem is fairly conclusive. Actually, Mikkelson has solved it. Ah no he probably hasn’t… Since writing the essay, I’m vacillating between thinking that there is no problem, there’s only a problem in infinite cases, there’s only a problem in uncountable cases or the real problem is only in finiite cases. Now I’m not sure what to think…

The solution to the Monty Hall problem (switching wins you 2/3 of a car) depends for its answer on the fact that you know how Monty will act. Other host behaviours are possible. So my question is this: what is the best strategy if you don’t know what Monty’s behaviour is? Is it different in single case vs long run scenarios? In the latter case, what about a strategy that allows you to alter your behaviour depending on Monty’s behaviour? I don’t really know how to answer these questions; I have enough trouble convincing myself of the solution to the original problem!

In other news, a couple of books by D.H. Mellor are available for free online! Matters of Metaphysics and The Matter of Chance. And more philosophy gubbins- Philosophy Bites: Bitesize philosophy podcasts. Wonderful.

One last thing. Tim Minchin and Duke Special look quite similar. They both play piano type music. But Tim Minchin is from Australia and does comedy songs and Mr. Special is from Northern Ireland and plays “proper music.”

Tim Minchin

Duke Special

I have been reading some craaa-aaazy stuff today. So this one paper suggested that the fundamental metaphysical nature of the world is that of a graph. An unlabelled asymmetric graph-theoretic entity. OK. It kind of fits in with a lot of stuff about ontic structural realism. Kind of.

Now I’m reading something about the bird-eye view versus the frog-eye view of space and time. The “bird” sees the whole spacetime structure from the outside. What looks like a particle moving with constant speed to the frog looks like a strand of spaghetti to the bird. Two particles orbiting each other, to the bird, looks like two strands of pasta entwined like a double helix. Like the weird blob thing in Donnie Darko. So the bird sees the frog as an ensemble of worldlines for the frogs particles. The frog looks like a tube of pasta strands to the bird. I did not make up any of this up. (Except the Donnie Darko reference). It’s all there in Max Tegmark’s The Mathematical Universe. The weak point in his argument is that he claims that any “theory of everything” will be entirely mathematical. This simply cannot be the case. We have plenty of theories that are entirely mathematical; go ask your local maths professor. If it is to be a theory of the physical world, the theory is going to have to involve some kind of pointers as to how to apply its results to the world. So we had a theory of Riemannian manifolds before Einstein came along, but that didn’t make the (mathematical) theory a scientific theory. Not until Einstein started showing how the manifold could relate to our conception of space. Tegmark pretty much agrees with this point, but then says that that isn’t fundamental to the theory. He says we have a mathematical theory and then the interpretation is done afterwards and isn’t necessarily part of that theory. This is both methodologically backwards and I think just plain wrong. The interpretation is central to that mathematical theory qua scientific theory.

I am sympathetic to the (ontic) structural realist flavour of what Tegmark is getting at, but I don’t think his “derivation” of his “MUH: Mathematical Universe Hypothesis” really works. I have to say I gave up after the first 10 or so pages because it was getting near to dinner time and the two column layout is a pain in the arse to read on the computer.

Tegmark also has that annoying scientist’s habit of not putting the names of the articles in his bibliography. So in the text he will cite “[14]” which isn’t helpful. Then if I scroll to the bibliography I will see that [14] is “J. Ladyman Studies in History and Philosophy of Science 29 409-424 (1998)” God dammit. If he just wrote Ladyman (1998) in the text instead of [14] I’d know immediately that he meant What is structural realism? It would mean much less hopping back and forth. And what about if the author and name wasn’t enough for me to identify the paper? I’d have to bloody well look it up on the internet. I appreciate the practice makes sense in science where knowing the actual paper under discussion isn’t important to the argumentation, and that titles of scientific papers are long and would lead to bloated bibliographies, but come on! It isn’t even as if it would be a lot of work to change it. How hard is it to add the line “\usepackage{natbib209}” to the preamble of your LaTeX document and change your bibliography style to chicago, or similar? I bet the names of the articles are already in the bibtex file…

I did promise to post something that wasn’t a rant. And this started out as a light-hearted look at some of the dangerously bonkers stuff I take seriously every day. But it turned into a rant about bibliography formatting, of all things. I appear to be incapable of not ending up complaining about something. I guess that means I am just a mean spirited cynical rantophile.

I didn’t know google had a “blog search” thing. So I tried it out. I searched for “philosophy of science” (obviously) One of the results was about experimental philosophy. I read it and decided I don’t like “X-Phi” (which is the annoying name that these people have chosen to give to their approach). Bristol university has its own x-phi page here. I mention this out of some kind of misplaced loyalty, not out of any approval for such an endeavour.

Here is some experimental philosophy in action. And here is an NYTimes article about it. It contains the following passage which I think is marvelous:

Colleagues in biology have P.C.R. machines to run and microscope slides to dye; political scientists have demographic trends to crunch; psychologists have their rats and mazes. We philosophers wave them on with kindly looks. We know the experimental sciences are terribly important, but the role we prefer is that of the Catholic priest presiding at a wedding, confident that his support for the practice carries all the more weight for being entirely theoretical.

I don’t take issue with what these people are doing. I just wonder whether the label philosophy really applies… I mean what makes this new experimental philosophy philosophy, rather than, say anthropology or psychology or political science? Obviously philosophers should take empirical findings into consideration. Kant’s view that Euclidean geometry is necessarily the only possible geometry, and therefore necessarily the geometry of the world is cleary untenable in our post-elliptic geometry, post-Minkowski spacetime world. Any number of examples could be given here. But philosophy is still somehow separated off from empirical science. Of course you could just criticise me for having an old-fashioned view of the nature of philosophy. Armchair theorising is out of favour.

OK, here is the problem; what can the results of these polls tell us? If they tell us that some notion, X, is actually pretty much universal, then  philosophers are allowed to say things like “it is clear that X” or ” most people would agree that X.” But if everyone agrees with X, it’s already clear that we’re allowed to assert it. No experiment is needed to demonstrate that everyone thinks X is true. Everybody already knows that: X is pretty damn obvious, right? If the poll gives more ambiguous results, it is unlikely that the notion under scrutiny is the kind of thing philosophers would go about asserting without justification. So this Experimental Philosophy might take its questions from philosophical literature, but its methods don’t seem to fit with philosophical practice.

I’m stressing the philosophical part. Obviously it is interesting to see what people think about the truth value of sentences like “the king of France is bald” or “the king of France is not on a state visit to Prussia” but is it really philosophy to go around asking people?