Re: Isomorphism between Mereology and Boolean algebra without least element
Avril,
Thanks for the reference:
http://people.imise.uni-leipzig.de/alexander.heussner/files/Mereology.pdf
That paper is a good exploration of the parallels, but
the opening sentence is not quite accurate:
> Mereology has been considered an important feature
> of philosophy and mathematics for nearly one hundred
> years, its origins lie in set theory and logic calculus.
Actually, mereology predates set theory, or at least
Cantor's version. Boole developed his algebra with three
different interpretations of the same symbols:
1. Propositions: p+q is exclusive or, p*q is logical and,
and -p is not.
2. Predicates: the same operators applied to monadic
predicates.
3. Sets: the same operators applied to sets.
Many logicians, such as De Morgan, objected to interpreting
p+q as exclusive or and preferred the inclusive or. That
option leads to "De Morgan's laws" for propositions,
predicates, and sets. But those laws were very familiar
to the medieval logicians (who also preferred inclusive or).
Following is Ockham statement: "It should be noted that
the contradictory opposite of a disjunctive proposition is
a conjunctive proposition composed of the contradictories
of the parts of the disjunctive proposition." (Note that
Ockham treated this as reminder, not a novel discovery.)
Peirce introduced the "claw symbol" for propositions, predicates,
and sets: "p -< q" was his way of writing "less than or equal",
which for propositions would mean "the truth value of p is less
than or equal to the truth value of q"; for sets, it would mean
"p is a subset of q". This is a convenient notation, because
you can write Cat -< Animal with the interpretation that every
cat is an animal or the set of cats is a subset of the set of
Animals. With Peano's notation, students get very confused
about reversing the implication symbol to get subset.
Before Cantor, there was a lot of dispute about how to treat
individuals. Some treated an individual as a one-member set,
but others wanted separate operators. Cantor insisted on
distinguishing "member-of" from "is a one-member subset of".
As soon as you introduce two separate operators, it is possible
to talk about sets as members of sets. With only one operator,
it is not possible have sets of sets of sets. That is the option
that enabled Cantor to build up his infinite hierarchy of sets,
and that is also the most controversial issue, which Lukasiewicz
wanted to eliminate by getting rid of the member-of operator.
That is a very important point: If you don't have "member-of"
you can never go beyond countable sets, you can't create the
paradox of "sets which are not members of themselves", etc.
In my opinion, Lukasiewicz's did an excellent job of showing
that it is possible to build up all of arithmetic without
assuming Cantor's hierarchy. Unfortunately, World War II
created a lot of trouble in that part of the world. After
the war, most people lost interest in the foundations of
arithmetic (except for a tiny minority who taught their
students how to churn out dissertations and papers for the
_Journal of Symbolic Logic_).
Summary: I agree with Kronecker: "Die ganzen Zahlen hat der
liebe Gott gemacht, alles andere ist Menschenwerk." Peano's
axioms are sufficient to define the integers, and mereology
plus Dedekind cuts is sufficient to define the real numbers.
It is foolish to think that set theory, with all its paradoxes,
is necessary to make arithmetic more secure.
Hilbert called Cantor's hierarchy of infinities a "paradise",
but Wittgenstein called it a "dismal swamp". That's a religious
dispute, which I believe is irrelevant to any mathematics that
is relevant to applications in physics or computer science.
John Sowa