Brent Meeker writes:
> > Even if you say that, there is still a sense in which arithmetic is independent of the
> > real world. The same can be said of Euclidian geometry: it follows from Euclid's axioms
> > *despite* the fact that real space is not Euclidian. The fact that real space is not
> > Euclidian means that Euclidian geometry does not describe the real world, not that
> > it is false or non-existent.
> >
> > Stathis Papaioannou
>
> But the fact that a theorem is true relative to some axioms doesn't make it true
> or existent. Some mathematicians I know regard it as a game. Is true that a
> bishop can only move diagonally? It is relative to chess. Does chess exist?
> It does in our heads. But without us it wouldn't.
What more could we possibly ask of a theorem other than that it be true relative to some
axioms? That a theorem should describe some aspect of the real world, or that it should
be discovered by some mathematician, is contingent on the nature of the real world, but that
it is true is not.
Stathis Papaioannou
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Received on Sat Aug 19 2006 - 00:55:33 PDT