[FOM] Possible worlds
Timothy Y. Chow
tchow at alum.mit.edu
Mon Dec 18 10:34:51 EST 2006
A popular way of dealing with the notions of possibility and necessity is
to appeal to the concept of possible worlds. Here is an observation about
possible worlds that has long vaguely bothered me, and I was wondering if
anyone has thought this through.
In 1990, if you had said, "Fermat's Last Theorem might be false," then
most people would have agreed with you. Today, most people will not. The
sense of possibility involved here does not seem to me to be easily
handled in the standard framework of possible worlds. Standardly, since
we have a proof of Fermat's Last Theorem, we know that it must be true in
all possible worlds. (If you have qualms about FLT being a "mathematical"
statement that might not be "logically" true, then replace FLT with some
logical validity that was unknown to be a validity at time t and known to
be a validity at some later time t'. I'll continue to use FLT as an
example.) On this reading, it would seem that the folks back in 1990 were
simply asserting a false statement. But intuitively, this doesn't seem to
a satisfactory analysis of the situation.
We could take the attitude, "So much the worse for the possible worlds
theory," and leave it at that. However, is there a more productive point
of view? Is there perhaps a way of formulating a version of the possible
worlds theory, or a version of modal logic, that satisfactorily accounts
for "FLT might be false"? There seem to be overtones of the debate
between intuitionism and platonism here. Is the standard approach to
possible worlds laden with platonistic assumptions? David Lewis has
defended the "reality" of possible worlds; while this seems to be a
minority view, the mere fact that he would take such a view and not be
immediately shot down does perhaps suggest that there are tacit
platonistic assumptions in the background somewhere.
It is not clear to me, though, that intuitionism furnishes an immediate
answer. In 1990, we might have vaguely imagined a "possible world" in
which someone finds a counterexample to FLT. But today, it is hard to
imagine fleshing out such a scenario in detail. Just which digits would
be written down on the page containing the counterexample to FLT? Our
inability to flesh out this alleged "possible world" by specifying the
digits does not seem to be merely a lack of mathematical ingenuity or
computational power; it seems to be, in a strong sense, *impossible* to
specify such a "possible world." This would seem to be a difficulty for
intuitionists and platonists alike.
Perhaps the notion of "possibility" implicit in "FLT might be false"
simply cannot be handled by the possible worlds theory? Even if this is
so, is there any interesting way in which the relevant sense of
possibility can be axiomatized?
Tim
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