Re: Isomorphism between Mereology and Boolean algebra without least element
Avril,
I'll start with your concluding sentence:
> I get the picture from your summary that you
> suggest that we should have many tiny and generic
> formalisms instead of one large formalism.
What I am actually advocating is pluralism: there
is no reason to edict a single foundation for all
of mathematics, and there are many good reasons for
looking at a multiplicity of alternate foundations.
And for that matter, working mathematicians (i.e. the
people who prove theorems about differential equations
or Fermat's last theorem) ignore all the foundational
work because it's irrelevant to what they do in their
daily work. It doesn't clarify anything.
Set theory is just one among an infinite number of
possible foundations for mathematics, and there is
no reason to make it *the* official foundation.
> In Kripke-Platek set theory with ur-element, the
> ur-elements a,b,c,d,e can be taken as indivisible atoms.
> .... but atom is not part of the Solar system on the
> same abstraction level as it is a part of a proton.
That's an argument for abstraction layers. You could,
if you wish, reuse the member-of operator of set theory
to build layers, or you could introduce a separate
layer-of operator. Instead of privileging set operators
over any others, you could introduce multiple operators,
and use category theory as the overall framework for
relating them.
Category theory is an alternative foundation for
mathematics, and many mathematicians say that category
theory is much more powerful as a foundation for math
than any version of set theory. (In fact, category
theory has a great deal of usefulness for working
mathematicians, and it would therefore qualify as
a better foundation than any version of set theory.)
And for that matter, there is a great deal of
confusion about the relationship between logic and
mathematics. Peirce, Hartley Rogers, and many other
mathematicians said that it's better to consider
mathematical logic as a branch of applied mathematics:
it is the application of mathematical techniques to
the study of reasoning.
I have a great deal of sympathy for that approach
because there are many different ways of axiomatizing
logic. Instead of treating logic as a privileged
foundation for all of mathematics, it would be better
to think of it as one tool among many.
I think it's a very good exercise to try to develop
alternative foundations and to show that they are
consistent relative to one another. That's important
because it's impossible to prove the absolute consistency
of any complex system. Therefore, the best we can do
is to show that they are consistent relative to one
another.
And for that matter, there are only two major systems
that mathematicians are really confident in their
reliability: the integers and Euclidean geometry.
Whenever you have a new system, it's a good idea check
it's consistency by defining a model of that system
in terms of either the integers (as Goedel did for his
famous theorem) or Euclidean geometry (as mathematicians
do for every other version of geometry).
Kronecker, Peirce, and many others said that it is
foolish to think that arithmetic needs set theory
in order to make it more reliable. I certainly agree.
John