SUO: Foundations of mathematics
As usual, when Pat makes a factual statement about math or
logic, I agree. I just wanted to make a few more observations.
PH>This is just one brick in the wall of ingenious reconstructions of
>mathematics in set theory which was built on the foundation provided
>by Russell & Whitehead. The upshot of all this ingenuity is that
>almost anything that can be said in mathematics can, if challenged as
>to its underlying consistency, refer to the challenge to the question
>of the consistency of set theory. ("Almost" because some aspects of
>category theory don't fit onto set theory very easily, and category
>theory has itself been proposed as a more appropriate foundation for
>mathematical consistency. There has been a kind of religious struggle
>for the soul of mathematics, with Saunders MacLane as its Martin
>Luther.)
There is nothing magical about set theory, or category theory,
or any other proposed foundation for "all of mathematics."
During the 19th century, an even more heated argument broke
out about whether set theory was an appropriate foundation for
such things as integers, which seem to many (including me) as
more solidly established than set theory with all its paradoxes
(or antinomies, for those who want to be more charitable).
In the early 20th century, the intuitionists, led by Brouwer
challenged set theory, at least the nonfinitary parts of it:
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Brouwer.html
http://virgo.bibl.u-szeged.hu/202Library/Brouwer%20Cambridge%20Lectures%20on%20Intuitionism%20(1951)%20.htm
In Poland, Stanislaw Lesniewski established an alternative
foundation for mathematics based on mereology, but didn't
publish it before WW II came and destroyed most of his
manuscripts. In England, Whitehead had promised to write
a fourth volume on geometry for the Principia Mathematica,
which would be based on his version of mereology. He never
finished that project, but he used his version (called
extensive abstraction) in several of his other publications.
>Apart from providing this consistency proof, however, the
>set-theoretic reconstruction isn't all that interesting from a
>mathematical point of view. In particular, it would be a mistake to
>assume that ordered pairs *really are* sets of the form {{a}{a b}},
>or that integers *really are* sets of the form {},{{}},{{{}}},... or
>of the form {},{{}},{{}{{}}},{{}{{}}{{}{{}}}},... or whatever. They
>might be those things; but they might also be something else
>altogether, and the mathematics works just fine without committing
>itself to the ultimate nature of the things it is talking about.
I agree completely. If all you want is a theory of the
integers, it is much, much easier to start with Peano's axioms
than to start with set theory. The same is true for almost any
other traditional branch of mathematics. (You usually need
some version of "collection", but you don't need to go beyond
countable infinities, and you don't need sets of sets. For
such purposes, a very simple version of mereology is adequate.)
For anyone who may be interested in the wild and wooly issues
of the foundations of mathematics and what, if anything, they
might mean, I recommend the papers by Greg Chaitin:
http://www.umcs.maine.edu/~chaitin/
Following is an article about him in the March 2001 issue
of New Scientist:
http://www.newscientist.com/features/features.jsp?id=ns22811
Following are a couple of paragraphs excerpted from the middle
of that article.
John Sowa
_______________________________________________________________
Mathematics has always been considered free of
uncertainty and able to provide a pure foundation
for other, messier fields of science. But maths is
just as messy, Chaitin says: mathematicians are
simply acting on intuition and experimenting with
ideas, just like everyone else. Zoologists think
there might be something new swinging from
branch to branch in the unexplored forests of
Madagascar, and mathematicians have hunches
about which part of the mathematical landscape
to explore. The subject is no more profound than
that.
The reason for Chaitin's provocative statements is
that he has found that the core of mathematics is
riddled with holes. Chaitin has shown that there
are an infinite number of mathematical facts but,
for the most part, they are unrelated to each
other and impossible to tie together with unifying
theorems. If mathematicians find any connections
between these facts, they do so by luck. "Most of
mathematics is true for no particular reason,"
Chaitin says. "Maths is true by accident."