Re: Isomorphism between Mereology and Boolean algebra without least element
Jay,
Those are important issues that are certainly worth
studying, discussing, and analyzing in detail:
JH> Well, there is still theoretical debate about
> mathematical validity, and its nature. It's true enough
> that core engineering and physics maths aren't practically
> disputed. But there remain plenty of areas where theoretical
> issues impinge on the possibilities of practical results.
> Even in Peano Arithmetic there are still simply stated
> theorems to be sought, and questions about undecidability,
> and essential undecidability.
I wasn't questioning the importance of these studies, but
the aptness of the term "foundation". I realize that word
has been in use for some time, but another word, which is
sometimes used and which I believe is less misleading, is
the word "metamathematics".
To use your example, there are three kinds of studies that
should be distinguished:
1. Finding new "simply stated theorems" is no different
in principle from finding complicated theorems. That
is ordinary "object-level" mathematics.
2. Questions "about undecidability and essential
undecidability" are definitely metamathematical, since
they address the method of proving rather than doing
the actual proving.
3. The third kind would be to take an undefined primitive,
such as "natural number" in Peano's theory, and replacing
it with something else. One example would be Frege's
idea of defining the integer 5 as the set of all sets
whose cardinality is equal to 5.
I have no quarrel with anyone who wants to work on any of
these three kinds of problems. They're very useful and
respectable things to do.
Activities of the third kind would be the best candidates for
being "foundational", since they replace primitives in one
theory with structures that are defined in another theory.
But the reason why I believe that the word "foundational"
is a misleading metaphor is that its connotations are wrong.
When you build a house, the foundations are the first
thing you construct, there is one and only one foundation,
and the house can never be more solid than the foundation.
But in mathematics, the so-called foundational work is
almost never done at the beginning, it is not unique,
and many theories, such as arithmetic and Euclidean
geometry, are far more solid than any proposed foundation.
Brouwer, for example, pointed out that the integers are
usually taught to children by the process of counting, and
for that reason, Peano's successor function is a far more
"natural" foundation than Frege's sets of sets.
And as I mentioned earlier, arithmetic and Euclidean geometry
are usually the first systems that logicians use as models
to ensure that their logical "foundations" are sound and
consistent relative to arithmetic or geometry.
ME>> ... that thanks to Bourbaki who have been geniuses as
>> far as ads go ;-)
JH> Very good, no doubt :)
Yes, but as Pierre Cartier, one of the later members of the
Bourbaki, pointed out, many of them had very serious blind
spots -- especially in their attitude towards diagrams,
category theory, and physics:
http://www.ega-math.narod.ru/Bbaki/Cartier.htm
Bottom line: You can *never* consider any single proposed
foundation for math to be unique or best or completed.
John