Re: Isomorphism between Mereology and Boolean algebra without least element
Michel,
I agree with your remarks.
The point I was trying to emphasize is the need
to keep all options open. I think that the idea
of picking one single foundation for mathematics
would be a disaster.
Set theory is just one popular way of describing
collections, and there is no reason why it should be
considered more fundamental than any other. Mereology
is another way (actually, a large family of related
ways -- see Peter Simons' book for a lot more). But
there are undoubtedly other ways that could be used
for different purposes.
Category theory is a very general way of characterizing
structures by looking at what is preserved when you
map from one domain to another. I agree that most
working mathematicians don't use it, but there are
many computational tools that are beginning to take
advantage of its power and flexibility.
Topos theory happens to be used with category theory,
but I'm sure that there could be many other variations
that could also be used, some of which may be more
compatible with more popular versions of logic.
I think that the word "foundation" is confusing
people. It suggests that work on "foundations" is
somehow more fundamental and has to be done first
before you build anything on top.
But that's not how foundational work is used at all.
Peano's axioms, for example, are self-contained.
There is absolutely no need to interpret them in terms
of sets or categories or anything else before you start
to prove theorems with them.
Instead of talking about the "foundations" of mathematics,
a better term is "metamathematics": all the work that
Hilbert, Frege, Russell, Brouwer, etc., were doing is
mathematics *about* mathematics. It is orthogonal to
what ordinary "object-level" mathematicians do, and there
is absolutely no reason why working mathematicians should
wait for (or even care about) what the metamathematicians
are doing.
That is not to say that metamathematics is unimportant.
More precisely, it is misleading to call it "foundational".
John