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 
>>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 
done by people claiming mathematical theorems are true.

Brent Meeker

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