Stathis Papaioannou wrote:
> 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.
That it is a true description of the real world, or that it is a true theorem
relative to the axioms. It is a mistake to conflate the two, which I suspect is
done by people claiming mathematical theorems are true.
Brent Meeker
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Received on Sat Aug 19 2006 - 02:50:38 PDT