[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?


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