Re: Isomorphism between Mereology and Boolean algebra without least element
Mathematics has been enjoying foundational crises since at least the
invention of zero. Someone in each generation finds fault with something in
the previous work. What Peano and the others did was to recognize the need
for formal axioms of that sort. Some foundational questions never do get
entirely settled to the satisfaction of all, but progress is made, and
clarity added.
Topos theory, supporting intuitionistic logic, has less overall support than
FOL. Set theory, by some measure, is the most widely used foundation, but
the devil is in the details.
Metamathematics (metalogic) is the best tool for studying the relationships
between the various competitors among formal foundations, but the same or
related foundational issues are likely to arise in some form at the meta
level also, as assumptions in the metalanguage/logic. 'Working'
mathematicians should worry, though, about consistency in some way or
another, if they are to have any notion of proof (and of communication) at
all. And, in fact, they do worry about definitions, principles, and so
forth. Some worry more precisely than others.
-Jay
----- Original Message -----
From: "John F. Sowa" <sowa@BESTWEB.NET>
To: "Michel Eytan" <eytan@umb.u-strasbg.fr>
Cc: "Avril Styrman" <Avril.Styrman@helsinki.fi>;
<standard-upper-ontology@listserv.ieee.org>; <cg@CS.UAH.EDU>;
<alexander.heussner@gmx.net>; <aapo.halko@helsinki.fi>
Sent: Thursday, July 07, 2005 06:34
Subject: 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