Re: Isomorphism between Mereology and Boolean algebra without least element
Jay,
Those aren't foundational crises. Those are just
new theories that extend, but do not invalidate
any of the old ones:
JH> Mathematics has been enjoying foundational crises
> since at least the invention of zero.
The word "enjoying" is a good choice, because new
additions make math more interesting. But they
don't create crises -- except for people who don't
like to be bothered with newfangled ideas.
> What Peano and the others did was to recognize
> the need for formal axioms of that sort.
The 19th century did a better job of stating axioms
more precisely, but that can still be considered
normal math. Even Peano's axioms are about integers
-- they don't reduce integers to set-theoretic
construction.
Dedekind cuts for defining real numbers, however,
are different. They do replace a real number with
something else -- the set of all rational numbers
less than or equal to the cut.
The difference is that Peano's axioms are actually
used by working mathematicians while doing number
theory, etc. Dedekind cuts, however, are *never*
used by working mathematicians. Nobody uses a
Dedekind cut to calculate a square root, a cosine,
or a logarithm or to prove a theorem about them.
So Peano's axioms are part of "object-level" math,
but Dedekind cuts are part of metamathematics.
The following is true, but working mathematicians are
content to worry at the object level, not the metalevel:
> '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.
First of all, consistency is generally accomplished by
finding a model -- and the two most important sources
of models are the integers and Euclidean geometry --
the two subjects that are far more secure (i.e., free
from contradiction) than any work on metamathematics.
Remember that Goedel used integer arithmetic to model his
famous theorem about logic. That's because mathematicians
have more far faith in the consistency of arithmetic than
they have in the consistency of higher-order logic (and
with very good reason -- despite the fact that nobody has
proved or could ever prove the consistency of arithmetic).
Precision is important at any level, not just the
metalevel. Defining a real number as a Dedekind cut
does *not* make any object-level math more precise.
It does not improve the qualify a proof about logarithms,
cosines, or Bessel functions.
Bottom line: Arithmetic and Euclidean geometry got along
quite well for millennia without any need for "foundations".
And whenever anybody tries to prove that their notion of
foundation is consistent, they use arithmetic or geometry
as the standard of security.
John