Continuous Motion?

In the new issue of Analysis that Brian just mentioned, there’s an article by Hud Hudson called “How to Part Ways Smoothly”. He describes two point-objects that are colocated at every time before 100, but then are at a different location at that time, although both move continuously. The way this works is that both rotate around a clock face doubling their speed after each rotation, so that they go around infinitely many times before 100, and that at 100 one of them is at the 3 on the clock face and the other is at the 9.

I don’t understand why he is justified to claim that “neither character ever moves discontinuously”. It’s true as he says that no matter how small an interval you look at before time 100, there are time-slices of each character that are arbitrarily close to their destination. However, I normally think of continuity as being characterized by a different set of quantifiers – for every spatial distance, there is some temporal duration such that all time-slices within that duration are within that distance of the destination.

Because their trajectories satisfy Hudson’s definition of continuity but not mine, the space-time trajectories are said to be “connected, but not path-connected”, and curves like this are standard counterexamples in topology. But is there any reason why metaphysicians might adopt Hudson’s account of continuous motion and not mine? If not, then an example like his could be constructed whereby a particle traces out successive approximations to a Peano space-filling curve with constantly doubling speed, so that in the limit every point in space could be the result of “continuous” motion.

Bleg and Banff

First a bleg. Those of you who read over there may have noticed that I haven’t posted at my other blog in a few weeks. At first it was just the usual not having the right combination of time and ideas, but recently the software has given me trouble again, and now doesn’t believe that there’s a blog there for me to edit. (Fortunately it’s still displaying all the old posts.) So I’m planning on moving away from MovableType. But since I can’t log in to export the old posts, I was wondering if anyone knows of a way to extract them efficiently so I can host them on different software, possibly on a different site. (I lost control of about a year ago when I switched hosting companies, and they seem to have accidentally renewed the registration, so I may have to change domain names again.)
Anyway, I was just in Banff for probably the most blogged-about conference I’ve ever been to. I count nine posts between , Gillian, and Richard, as well as two mentions on Certain Doubts. Also, I believe Aldo Antonelli got a lot of good pictures, including one of Gillian and me that should appear in the upper-left corner here at some point after I get a copy from either him or Richard Zach.
I found it to be a great workshop, and I’m very glad that Richard invited me! Despite the very broad “focus” on mathematical methods in any area of philosophy, there were some very interesting series of talks that gave some coherent threads. For instance, theories of truth were the focus of the survey by JC Beall and Michael Glanzberg, as well as more specific talks by Volker Halbach, Jeff Ketland, Sol Feferman, and Greg Restall. And a variety of different semantics for modal logics (especially quantified and non-classical) were discussed by Marcus Kracht, Steve Awodey, Eric Pacuit, Greg Restall, and Graham Priest. There were also two interesting proposals, by Delia Graff Fara and Kai Wehmeier, suggesting that identity as a relation doesn’t play quite the role we think it does, the former pointing out that for a lot of purposes (especially modal and temporal ones) we can’t use strict identity, but rather some “same F as” relation; the latter arguing that identity shouldn’t even be treated as a relation in logic at all!

Given my experiences at FEW (abstract deadline today if you want to submit!) I was surprised by a relative lack of talks on probability (basically only Branden Fitelson and Tim Williamson, and one fourth of Hannes Leitgeb’s virtuoso four talks in one hour performance). But there’s always got to be some trade-off. Anyway, I had lots of productive conversations (some while cross-country skiing) that will shape my dissertation, some papers I’ve been working on, and almost certainly some future blogposts either here or wherever I get my other blog working once I can transfer the old posts.

FEW 2007

Fourth Annual Formal Epistemology Workshops, FEW 2007

We are in the process of organizing the fourth annual formal epistemology workshop. The purpose of these workshops will be to bring together individuals, both faculty and graduate students, using mathematical methods in epistemology in small focused meetings. Topics treated will include but are not limited to:

  • Ampliative inference (including inductive logic);
  • Game theory and decision theory;
  • Formal learning theory;
  • Formal theories of coherence:
  • Foundations of probability and statistics;
  • Formal approaches to paradoxes of belief and/or action;
  • Belief revision;
  • Causal discovery.

Besides papers with respondents, each workshop will typically include short introductory tutorials (three or four topically related presentations) on formal methods. These tutorials will be oriented particularly to graduate students.

The fourth workshop is scheduled for 31 May – 3 June 2007 and will be held at Carnegie Mellon University in Pittsburgh.

FEW 2007 SUBMISSION DEADLINES: – Paper submissions (at least a 1/2 page abstract): By Monday, Feb. 26th. – Notification: By Monday, March 30th.

Please send submissions by email to Branden Fitelson <>.

Those interested in participating, either by presenting papers, responding, or providing tutorials, or in helping with organization, should contact one of the organizers listed below. Limited funds are available for graduate student travel. (A workshop web-site will shortly be set up. It will appear at the usual location:

Richard Scheines <> CMU

Sahotra Sarkar <> UT-Austin

Branden Fitelson <> UC-Berkeley

Physics and Philosophy

I just saw an interesting post on the Bohm interpretation of quantum mechanics, arguing that it doesn’t really help explain any of the interesting facts of quantum computing. (I think the many-worlds interpretation might help, but I’m no expert.)

However, I think the discussion was particularly interesting because it shows what physicists thinking about philosophical questions think “philosophical” means:

The unfortunate word “interpretation” suggests that foundational work should never impact the ring-fenced portion of the theory and should be confined to something like “finding the right language” for talking about quantum theory in order to avoid the difficulties with measurement, etc. Indeed, many philosophers actually like this distinction, because it makes “interpretation” a purely philosophical question that they can go off and study on their own, safe in the knowledge that they won’t find themselves proved wrong by physicists.

For my own part, I would have thought that this is a distinction that philosophers would want to reject. Isn’t it more appealing to the ambitious philosopher to say that the problem of interpretation is a philosophical question that is important to the physics. Thus, physicists will have to pay attention to us, even if it does mean that on some level we run the risk of being proved wrong by them.

Now, you may regard the “ontology problem” as a pure philosophy problem, to which physics can never supply a unique answer.

Calling a question philosophical is sometimes about the same for me as calling it moot. I agree that physics answers often have philosophical strength. In that sense, the philosophical aesthetic is valuable. But if you call a question philosophical, then you may be saying that if you don’t know if there really is a question.

Again, this seems to be a physicist with misconceptions about what philosophy is. (Though perhaps the fact that I think that just means that I have misconceptions about what philosophy is too.) I would have thought that once the Quine-Duhem problem was noted, it would be clear that “physics can never supply a unique answer” doesn’t mean that a question isn’t part of physics. In fact, the existence of this sort of question, which physicists do seem to find interesting once in a while (never mind their accusations at string theorists and others of engaging in pseudo-science) seems to me to lend strong support for a naturalist position on which there is no solid distinction between philosophical and physical questions, and the answers to either can have impacts on the other.

Anyway, it does seem strange to me that there’s so much discussion of the philosophical aspects of physics on the blogosphere, and so little input from actual philosophers of physics in these debates. It would seem that many of the physicists involved don’t even know that philosophers of physics exist! Is this the sort of case where our profession needs to assert itself a little more, or is it not even worth the attempt to win the hearts and minds of physicists?

Gettier Cases in Literature

I’ve been idly interested in Gettier cases in unusual locations for a while, at least in part to convince myself that they’re actually important, rather than just marginal philosophical cases.  Recently, I realized that there may be some that play important roles in literature.  However, the only one that came to mind is something I only very hazily recollect.  When I was young, I read several mystery novels by Agatha Christie.  In one, I seem to remember that the murderer had killed the victim in a very clever way and concealed the evidence extremely well, but used one more extra twist to protect herself.  (I believe it was a female murderer.)  She placed a lot of misleading evidence, that pointed to her having killed the person, but in a way different from how the victim actually died.  By framing herself, she hoped that the police would at first end up in a Gettier situation, with a justified, true belief that she killed the victim, but that once they saw through the flimsy framing evidence, that would throw them off the trail, so that she could get off without getting caught.  (Of course, in the end it didn’t work out for her.)

If anyone recognizes this story, it would be useful to have some more specific details, so that it can be used as an example.  And if anyone knows of any other cases that appear in literature, that would also be nice.  This one is nice also for illustrating why knowledge is much better than being Gettierized – as Timothy Williamson points out several times in Knowledge and its Limits, someone with knowledge is much less likely to get led astray than someone who merely has true belief, or for whom misleading evidence is available.  But I also think it may put some pressure on his conception of misleading evidence being only evidence for a false proposition, rather than Gettierizing evidence for a true one.

Sorites Arguments in Mathematics

First of all, I’d like to thank Brian for inviting me to post here (like Gillian and Carrie, and some others that haven’t decided to say anything yet).

The topic I’m interested in is a parallel between the sorites arguments typical in discussions of vagueness and certain arguments for the adoption of strong new axioms in set theory. (I discussed some of those arguments in this post on my other blog.)

Basically, the idea is based on Gödel’s second incompleteness theorem. For every nice enough theory T (basically, T needs to be strong enough to represent basic arithmetic, and orderly enough that you can tell whether or not a given statement is an axiom), there is an arithmetical statement called Con(T) that says that T is consistent. However, for any such T, the statement Con(T) is neither provable nor disprovable from T itself. But if T has only true statements as axioms, then it must clearly be consistent, so T is incomplete. In particular, there is a true theory T’ that proves all the consequences of T, but also proves Con(T), and we should adopt T’ instead of T.

Because this argument then applies to T’ as well, we seem to either have to withhold judgement on our initial theory, or adopt a theory far stronger than what we started out with. (A recent post of mine discusses a parallel argument by Roger Penrose that claims to show that our mathematical knowledge is given instead by a non-computable theory.) Set theorists often use this argument to show that mathematicians who accept the axioms of ZFC (the standard framework most mathematicians tacitly accept as the foundations of what they do) must therefore accept much stronger principles as well, despite the fact that they can’t be proven.

I’d like to agree with the set theorists, but this argument reminds me of some fallacious reasoning in cases of vagueness. The idea is that for each theory T, if T is true, then T+Con(T) must be as well. This is similar to the claim that for every n, if n grains of sand don’t make a heap, then n+1 don’t either. The argument that shows mathematicians must accept every large cardinal claim that set theorists come up with is parallel to the argument that there are no heaps of sand. It seems plausible to me that whatever account of vagueness one has to block the heap argument could be adopted to block the set-theoretic argument. On the other hand, if a solution to the problem of vagueness doesn’t apply to the mathematical case (perhaps because it seems implausible to assign intermediate truth-values to mathematical claims or something), then one might see this solution as somehow lacking.

Some people might also run the sorites argument as a modus tollens instead of a modus ponens, saying that any number of grains of sand form a heap, and similarly that ZFC is not true (because adopting it as true forces them towards further claims about large sets that they have trouble believing).

A position like this is adopted by finitists, who accept claims about various finitary mathematical objects (like natural numbers, rational numbers, and the like) but only accept “potential infinities” (like a list that one can keep adding to) rather than “completed infinities” (like the actual precise value of some irrational number, or some non-computable set of natural numbers). However, most finitists accept Peano Arithmetic as a set of axioms, and a similar argument works starting with PA to drive one seemingly inexorably towards ZFC, and thus to the higher infinite.

The even more drastic solution accepted by some is known as ultrafinitism, on which one doubts even some “finite” numbers. In practice, these doubts arise about extremely large numbers, like a googolplex, which are believed to be larger than the number of subatomic particles in the entire universe. However, a similar sorites argument is going to cause trouble for the ultrafinitist – if the ultrafinitist accepts that the natural number n makes sense mathematically, then it would also seem that she should accept that the natural number n+1 does as well. (After all, one can just take the set of n things that one already has, and add this set itself as an element to create a new set!) If this is right, then once one accepts that even a single natural number makes sense, this successor principle pushes one inexorably towards at least full finitism, if not towards the infinite. The ultrafinitist has to reject the claim that every natural number has a successor, but it seems that she shouldn’t point to any number as “the last one”.

I’d like to use these arguments to show that once one adopts any amount of mathematics, one basically has to go “all the way to the top” in terms of the scales of infinity. However, these arguments seem to share troubling features with sorites arguments that we do want to block, so I’m interested in seeing what accounts of vagueness might do to them.

(The idea for this post originated in a class I co-taught last week with Mike Shulman to mathematically talented high school students at the Canada/USA Mathcamp.)