[FOM] Possible worlds
4mjmu at rogers.com
Mon Dec 18 13:58:57 EST 2006
Timothy Chow wrote:
> 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?
We could certainly imagine "mathematicians" in some possible world producing
a counterexample to FLT if in that possible world they used some law of
mathematical reasoning not considered valid in the actual world.
Presumably, this wouldn't be the right sort of possible world for you.
That aside, do you not have the identical situation when you discover that
Water = H20? Kripke would say you did, I think. So in this case,
possibility in math isn't really different from physical possibility.
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