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

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Pat Hayes <phayes@xxxxxxx>
Date: Thu, 28 Feb 2008 15:02:13 -0600
Message-id: <p0623090bc3ecd06319e5@[]>
At 3:55 PM -0500 2/28/08, John F. Sowa wrote:
>PH> But more seriously: in fact, in actual practice, there is a
>  > privileged universal foundation for virtually all of mathematics,
>  > to wit, ZFC.
>But the main question is "Whose practice?"
>If you mean logicians who like set theory, then indeed ZFC is
>a commonly used foundation.
>But if you mean mathematicians who prefer category theory,
>they will tell you that category theory is better than
>any version of set theory as a foundation.    (01)

Well, OK, there are TWO foundations. I agree, CT is widely treated as 
a real mathematical foundation in practice.    (02)

>But if you talk to people who follow Lesniewski,    (03)

There aren't any mathematicians who follow Lesniewski :-)    (04)

>  they will tell
>you that Lesniewski's program based on mereology is better.
>But if you look at what Goedel did, he took arithmetic as
>the foundation and mapped logic to arithmetic because most
>mathematicians at that time (and I believe even at present)
>have more faith in arithmetic than they have in logic.    (05)

I think you are just historically wrong about this. Goedel didn't map 
logic to arithmetic for foundational reasons: he did it to show that 
arithmetic couldn't be given a foundation, in a sense. The reason for 
choosing arithmetic was (1) it was rich enough to support the 
mappings he needed and (2) it was - and still is - regarded as the 
most 'obvious' and simple part of mathematics, so his result was all 
the more shocking. If he had shown that, say, transfinite group 
theory theory was undecideable, we wouldn't have had all this fuss 
about it.    (06)

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