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Re: [ontolog-forum] Search engine for the ontology

To: "John F. Sowa" <sowa@xxxxxxxxxxx>
Cc: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Pat Hayes <phayes@xxxxxxx>
Date: Fri, 29 Feb 2008 16:23:29 -0600
Message-id: <p06230906c3ee3505bf0b@[]>
At 4:18 PM -0500 2/28/08, John F. Sowa wrote:

As soon as you go from one to two, you have adopted an inadequate
approximation to infinity:

PH> Well, OK, there are TWO foundations. I agree, CT [category theory]
> is widely treated as a real mathematical foundation in practice.

But if you use "widely treated", you make it a popularity contest.

Just a quick remark: exactly, it IS a popularity contest (among mathematicians). I'm happy with that (as I suspect you are also), but I just wanted to dispel the impression that there are hundreds of competing 'foundations' for mathematics, or that everyone agrees that it is fine to just choose your own foundation for your own theory. So for example, Axcel invents a new set theory, and one of his main results is that it is consistent relative to ZFC, ie if ZFC is consistent, it is. And - and this is the point - this is widely and immediately accepted as having established its credentials as a coherent alternative set theory. If he had shown it was consistent relative to some random axiomatic theory, this would not have cut much ice: the choice of ZFC is important. It is a kind of 'standard meter' for foundational consistency.

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