Pat, (01)
PH> But more seriously: in fact, in actual practice, there is a
> privileged universal foundation for virtually all of mathematics,
> to wit, ZFC. (02)
But the main question is "Whose practice?" (03)
If you mean logicians who like set theory, then indeed ZFC is
a commonly used foundation. (04)
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. (05)
But if you talk to people who follow Lesniewski, they will tell
you that Lesniewski's program based on mereology is better. (06)
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. (07)
And if you talk to most mathematicians who *use* mathematics
to solve actual problems (as opposed to logicians who use logic
to talk about mathematics), the most frequent comment is what
Chris's friend said: (08)
Friend of CM> "Most working mathematicians would rather not think
> about foundational issues, just as most meat-eaters would rather
> not think about how the animals are slaughtered." (09)
My personal preference: Every axiom system is the foundation for
a particular mathematical theory. There is no reason why one axiom
system should be considered more fundamental than all others. If
I had to pick one, I would agree with Kronecker that natural numbers
are preferable. But mostly, I sympathize with Chris's friend. (010)
John (011)
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