Bruno Marchal wrote:
>
> Le 29-août-07, à 02:59, marc.geddes.domain.name.hidden a écrit :
>
>> I *don't* think that mathematical
>> properties are properties of our *descriptions* of the things. I
>> think they are properties *of the thing itself*.
>
>
> I agree with you. If you identify "mathematical theories" with
> "descriptions", then the study of the description themselves is
> metamathematics or mathematical logic, and that is just a tiny part of
> mathematics.
That seems to be a purely semantic argument. You could as well say arithmetic is metacounting.
> After Godel, even formalists are obliged to take that distinction into
> account. We know for sure, today, that arithmetical truth cannot be
> described by a complete theory, only tiny parts of it can, and this
> despite the fact that we can have a pretty good intuition of what
> arithmetical truth is.
But one would not expect completeness of descriptions. So the incompleteness of mathematics should count against the existence of mathematical Truth - as opposed to individual propositions being true.
Doesn't it strike you as strange that arithmetic is defined by formal procedures, but when those procedures show it to be incomplete, mathematicians resort to intuition justify the existence of some whole? Theology indeed!
Brent Meeker
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Received on Wed Aug 29 2007 - 17:11:47 PDT