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David Chalmers has an interesting post on the differences between his and Frank Jackson's versions of two-dimensionalism. It turns out that my reading of a certain passage in "Why we need A-intensions" was right: Jackson believes that truth at a world considered as actual is somehow reducible via de-rigidification to truth at a world considered as counterfactual.
The difference between linguistic ersatzism, where possible worlds are replaced by sets of sentences, and modal fictionalism, where the pluriverse of all worlds is replaced by a large set of sentences describing all worlds at once, appears to be small. Nevertheless, I (still) think the analytic power of fictionalism is greatly diminished compared to that of linguistic ersatzism.
One of the great advantages of possibilia is that they provide a unified framework to reduce lots of kinds of things: properties can be identified with sets of possibilia, propositions with sets of worlds, meanings with functions from worlds to extensions, events with functions from worlds to regions, and so on. But suppose possibilia don't really exist, but exist only according to some fiction. Then properties can't be sets of possibilia. By the usual rule of interpreting statements about fictional entities, it will at most be true that according to the fiction, properties are sets of possibilia. But that doesn't help us if we're looking for a unified ontology. We'd like to know what properties really are, not what they are according to some fiction. If as fictionalists we think that properties really are sets of possibilia, then we have to conclude that properties don't really exist, just as the (other-worldly) possibilia don't really exist.
According to the Principle of Recombination,
for any things at any worlds there is a world containing a duplicate of each of these things and nothing else (that is, nothing that is not a part of the fusion of the duplicates).
Applied to the mereological fusion of David Hume and David Lewis, this says that there is a world containing nothing but a duplicate of the fusion of Hume and Lewis. This duplicate presumably has a part that is a duplicate of Hume and another that is a duplicate of Lewis. How are these parts spatiotemporally related?
Brian points to Gabriel Uzquiano's Cardinality Puzzle about Mereology and Set Theory (PDF), which he (Gabriel) introduced a while ago in the now-deceased Philosophy from the 617 weblog. I still don't know enough set theory and mereology to competently discuss the matter, but anyway, it seems to me that perhaps the puzzle can be strengthened, as follows.
The whole four-dimensional universe, including past, present and future times, does not change; it will not be different tomorrow; it remains the same at all times.
If the whole four-dimensional universe remains the same at all times, then presumably no part of it will ever fail to exist or has ever failed to exist.
So for example, the apple I'm just about to eat will never fail to exist. It will exist forevermore. As will I, and you, and this weblog.
There is but one totality of worlds; it is not a world; it could not have been different. (Lewis, Plurality, p.80)
If the totality of worlds could not have been different, then presumably no possible world could have failed to exist.
Then in particular, the actual world, @, could not have failed to exist.
So there is an actually existing thing, namely @, that could not have failed to exist.
Even worse, arguably @ has some of its parts essentially. So there are some actually existing things besides @ that could not have failed to exist.
One might even say that all worlds have all their parts essentially, simply because worlds do not exist at other worlds. Then it follows that no actually existing thing could have failed to exist.
Conservatism as a methodological principle says that we should prefer new theories that resemble our old theories. (I don't mean the principle that a new theory should be at least as good as its predecessors, nor the principle that it should explain the success and failures of its predecessors. Very non-conservative theories can do that.)
What is the status of conservatism? Is it a primitive rule telling us that even if we know that some revisionary theory is as good as a conservative one -- that both explain roughly the same data, make roughly the same predictions, are equally simple, etc. --, we should prefer the conservative theory? (An otherwise good theory according to which there are no birds, but only bird-halluzinations, say, just seems incredible, in particular if a more credible alternative is available.) In this case, conservatism would resemble the simplicity principle that tells us to always prefer the simpler of otherwise equal theories.
When I prepared my talk at Heidelberg, I noticed some errors and oddities in the paper I had written. There were also a few interesting points raised in the discussion which I wanted to address. So in Switzerland, I almost completely rewrote the paper. Here is the new version: "Emperors, dragons and other mathematicalia".
[Update 2004-12-31: I've corrected another mistake: condition (1) on singleton relations should say that they are injective functions, not just that they are functions.]
I'm back. The conference was good; and Switzerland was quiet, white and beautiful.
The mountain whose rather flat slope you can see in the foreground is called "Um Su". It is a mountain without a summit.
I'm on my way to the German-Italian philosophy conference in Heidelberg. After that, I'll spend some days in the Alps. I won't have internet access there. (Also, my computer didn't want to boot this morning. Hopefully it was just too cold.)
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