Five Days of Formal Philosophy, and Uniform Solutions

22 05 2007

I just finished quite a streak of formal talks in philosophy. From Thursday night until Sunday, I (like Marc Moffett) was in Vancouver for the Society for Exact Philosophy, which was quite a fun little conference with a lot of interesting talks. Then on Monday, those of us at Berkeley working with Branden Fitelson got together with the people at Stanford working with Johan van Benthem for an informal workshop, which like last year had a lot of talks on probability from the Berkeley side and dynamic epistemic logic from the Stanford side, and again helped reveal a lot of interesting connections between these two rather different formal approaches to epistemological questions. And then today we had our quasi-monthly formal epistemology reading group meeting at Berkeley, with Jonathan Vogel from UC Davis.

There was a lot of interesting stuff discussed at all these places, but I’m glad there’s a bit of a break before FEW 2007 in Pittsburgh. Anyway, it’s also very nice to know that there is all this work going on relating formal and traditional issues, both in epistemology and other areas of philosophy.

Anyway, among the many interesting talks, the one that’s got me thinking the most about things I wasn’t already thinking about was the one by Mark Colyvan, on what he calls the “principle of uniform solution”. The basic idea is that if two paradoxes are “basically the same”, then the correct resolution to both should also be “basically the same”. So for instance, it would be very strange for someone to claim that the correct approach to Curry’s Paradox is that certain types of circularity make sentences ill-formed, while the correct approach to the Liar Paradox is to adopt a paraconsistent logic. Mark pointed out that there are some problems with properly formulating the principle though – do we decide when paradoxes are “basically the same” in terms of their formal properties, the sorts of solutions they respond to, or the role they’ve played in various arguments? For instance, Yablo’s Paradox was explicitly introduced in order to point out that self-reference is not the key issue in the Liar, Curry, and Russell paradoxes – which suggests either that the relevant formal property they share is something else, or that the proper way to think of paradoxes is something else.

In hearing this, I started to wonder just why we should believe anything like this principle of uniform solution anyway. The strongest cases of the relevant form of argument seem to me like the appeal in Tim Williamson’s Knowledge and its Limits to various different forms of the Surprise Examination Paradox – he points out that some traditional resolutions only solve the most traditional version, but that a slightly modified version gets through, and that his proposed solution to that version blocks the traditional version as well. Since both cases seem problematic, and one “covers” the other, it seems that we only need to worry about solving the covering case. I take it that something like this is at work as well when Graham Priest uses the Liar paradox to argue for dialetheism, and then suggests a return to Frege’s inconsistent axiomatization of mathematics rather than using the much more complex system of ZFC.

If this is the form of argument, then we shouldn’t always expect the principle of uniform solution to be worth following. If I (like most philosophers that don’t work directly on this sort of stuff) think that something like ZFC is the right approach to Russell’s Paradox, and something like Tarski’s syntactically typed notion of truth is the right approach to the Liar Paradox, then both get solved, but neither approach would work for both. Their formal similarities are interesting, but there’s no reason they should have the same solution, since there isn’t an obvious solution that works for both (unless you go for something as extreme as Priest’s approach). Formal or other similarities in paradoxes often help show that resolving one will automatically resolve the other, so that the above argument will work, but there’s no reason to think that this will always (or even normally) be the case.

But at the same time, something like this principle seems to work much more generally than in the case of paradoxes. There are certain similarities between the notion of objective chance, and the notion of subjective uncertainty, so it makes sense that we use a single mathematical formalism (probability theory) to address both. Alan Hájek has suggested that these analogies continue even to the case of conditionalizing on events of probability zero, though I think that this case isn’t as strong. (Though that might just be because I’m skeptical about objective chances.) There’s a general heuristic here, that similar issues should be dealt with similarly. In some sense, it seems very natural to suggest that differences in approaches to different issues should somehow line up with the differences between the issues. But we don’t expect it to always work out terribly nicely.

Anyway, there’s a lot of interesting methodological stuff here to think about, for paradoxes in particular, and for philosophy in general (as well as mathematics and the sciences).



8 05 2007

Jason Stanley has a post on the Leiter Reports discussing the problem of “Chmess”. As he puts it,

Chmess, for those of you who haven’t heard, is just like chess – except the king can move two squares in either direction. As Daniel Dennett has pointed out, Chmess provides a rich source of a priori truths to explore. However, the a priori truths of chmess are not particularly worthy of exploration. Dennett’s challenge of Chmess is to explain the difference between describing the a priori truths of Chmess and practicing philosophy.

He says that the sciences don’t face the problem of Chmess because they are clearly investigating an independently existing body of truths. He suggests that for the humanities, the problem doesn’t even arise, because they don’t see themselves as investigating a body of truths, but rather as changing people’s lives. I think he suggests that something like the latter is the right way to avoid the problem for philosophy as well:

In philosophy, as much as any other discipline, one engages in the practice of investigating alternative conceptual structures, be they systems of value or systems of belief. Such investigations may seem like theorising about Chmess, if we conceive of philosophy as a science. But [if] we regard philosophy instead as an activity intended to help those who learn it acquire the resources to lead a dignified life, it quite clearly does not.

However, I’m not convinced that this is the right way for philosophers to avoid the problem. (Admittedly, I don’t know what Dennett’s preferred solution is – he must disagree with Stanley here because of his scientific picture of philosophy.) To see that another solution must clearly be available, consider the case of mathematics. Working on Chmess problems bears only a very distant resemblance to working on science or the humanities, and a somewhat closer relationship to philosophy (especially when characterized as the study of a priori truths). However, many mathematicians explicitly claim that mathematics is just the study of what can be proven in arbitrary abstract formal systems, and many further claim (echoing G. H. Hardy’s famous A Mathematician’s Apology) that there is absolutely no use to any of their work and they only do it for their own amusement.

If that were really the case about mathematics, then the problem of Chmess would be extreme. Why shouldn’t a top university hire someone who was an incredible genius and had made all sorts of deep and subtle discoveries about Chmess?

My thought is that all mathematics is applied – if not to the physical world, then at least to other mathematics. There is a deep sort of interconnectedness between all the disciplines of mathematics. Chmess is unlikely to have these connections, just like almost any other arbitrary formal system, including Chess (though Go may have helped lead Conway to the surreal numbers). Mathematicians only work on problems that are relevant to other problems that already have established interest. If a certain formal system arises in multiple contexts, then that system will become of interest in its own right.

In philosophy, if one wants to take a “scientific” sort of view, one can see the point similarly. Unlike the other sciences, philosophy encourages discussion of non-actual systems in order to uncover their structure. However, these systems will ideally have some connection to actual systems, and hopefully will illuminate some of the relevant issues. And although philosophy very often deals with non-actual circumstances (twin earth, Gettier, trolley problems, etc.) the only cases I can think of where people really deal with formal systems that are not intended to describe something actual are in analytic metaphysics, or when they are specifically trying to disprove someone else’s theories.

So I think Jason Stanley’s argument is far from conclusive in establishing that philosophy should not be thought of as a science, but rather as something like a therapeutic activity. (I know that’s not what he said, but I can’t resist comparing him to Wittgenstein.)

Mathematical Existence

2 05 2007

Last night, I mentioned to my roommate (who’s a mathematician) the fact that some realist philosophers of mathematics have (perhaps in a tongue-in-cheek kind of way) used the infinitude of primes to argue for the existence of mathematical entities. After all, if there are infinitely many primes, then there are primes, and primes are numbers, so there are numbers.

As might be expected, he was unimpressed, and suggested there was some sort of ambiguity in these existence claims. He wanted to push the idea that mathematically speaking, all that it means for a statement to be true (in particular, an existence statement) is that it be provable from the relevant axioms. I was eventually able to argue him out of this position, but it was surprisingly harder than expected, and made major use of the Gödel and Löwenheim-Skolem-Tarski theorems.

The picture that is appealing to the mathematician is that we’re just talking about any old structure that satisfies the axioms, just as is the case when we talk about groups, graphs, or topological spaces. However, I pointed out that the relevant axioms for the natural numbers can’t just be Peano Arithmetic, because we’ll all also accept the arithmetical sentence Con(PA), once we’ve figured out what it’s saying. This generalizes to whatever set of axioms we accept for the natural numbers. As a result, Gödel’s theorem says that we don’t have any particular recursive set of axioms in mind.

To make things worse for the person who wants to think of the natural numbers as just given by some set of axioms, there’s also the fact that we only mean to count the members of the smallest model of PA, not any model. By the LST theorem, any first-order set of axioms that could possibly characterize the natural numbers will also have uncountable models. In order to pick out the ones we really mean, we either have to accept some mathematical class of pre-existing entities (the models of PA, so we can pick out the smallest one), or accept a second-order axiomatization, or give up any distinction between standard and non-standard natural numbers.

Of course, the same argument can be run for set theory, and at this point, accepting a second-order axiomatization causes trouble. Some philosophers might argue a way around this using plural quantification, but at least on its face, it seems that second-order quantification requires a prior commitment to sets, so it can’t underlie set theory. And we really don’t want to admit non-standard models of set theory either, because they give rise to non-standard models of arithmetic, which include certain infinite-sized members as “natural numbers”.

So it seems, for number theory and set theory at least, the mathematician has to admit that something other than just axioms underlies what they’re talking about. I’ve argued that the existence of and agreement on these axioms means that mathematicians don’t often have to think about what’s going on there. And from what I can see so far, the mathematician could take a platonist, fictionalist, or some other standpoint without any trouble. And in fact, most of the reasoning is going on at a high enough level that even if there’s some problem with the axioms (say, ZFC turns out to be inconsistent) this will have almost no effect on most other areas of mathematics, just as a discovery that quantum mechanics doesn’t correctly describe the behavior of subatomic particles will have relatively little effect on biology. But the mathematician can’t insist that existence in mathematics just means provability from the axioms – she can insist on some distinction between mathematical existence and ordinary physical existence, but it will be something like the minor distinction that the platonist recognizes, or possibly something like the distinction between real and fictional existence.