Re: Fwd: SUO Quo Vadis
Avril,
I really don't see what issues you consider unfinished:
> let's finish this before Christmas. Please
> address the issues.
JS>> There's nothing
>> wrong with writing an autobiography to describe yourself.
AS> Yes, but then you would be writing about yourself, not
> about _how_ you write about yourself. And if you write about
> how you write about yourself, you don't write about how you
> write about how you write about yourself. It is plain that
> self reference does create an infinite regress.
It's not infinite. You continue that as far as you like.
And when you get tired, you stop. At every step, it's finite.
That's an example of Tarski's multiple metalevels, which are
always finite, but could be extended if desired. For further
discussion and references to Tarski's levels, see my article
on Laws, Facts, and Contexts:
http://www.jfsowa.com/pubs/laws.htm
> The axioms were created for set theories in order to avoid
> paradoxes in the first place...
Yes, Frege's version had the paradox, but Zermelo avoided it.
There are several ways of avoiding it.
> ... I'm suspicious about your antipathy towards set
> theory and sympathy towards Lesniewski's mereology.
Sometimes I use set theory, and sometimes I use mereology.
I agree with many linguists that mereology is better suited
to representing plurals in natural languages. But that doesn't
mean that set theory is inconsistent or bad for what it does.
> Boole lived 1815-1864. Lesniewski was born in 1886, and created
> mereology that is isomorphic with Boolean algebra, that was
> created by Boole before Lesniewski was even born!
There are various axioms for mereology. Some of them have an
empty or null element (which is necessary for any version that
is isomorphic to Boolean algebra). But there are others that
don't have a null element.
> Now, you also have a theory called 'theory' that describes,
> suggests, or explains something. I ask, what does that theory
> stand for, what does it explain or describe? It cannot be a
> tautology, since only TOP contains tautologies.
Three points:
1. A theory describes those things that the variables in its
propositions refer to. Those would normally be the elements
of any model M for which the theory T is true (i.e., any
M such that M|=T). See Section 1 of the theories.htm paper.
2. Every theory contains all the propositions in every theory
above it (i.e., every generalization). Therefore, every
theory contains the tautologies.
3. If you have a theory T about theories, it would be true of
models whose elements are theories (and T could very well
be an element in one of those models).
> If there are things outside of the scope of the present
> languages, these things must be excluded from the ontology
> because it is impossible to describe them.
What do you mean "present languages"? If it's not describable
in one language, you can move to another language. There are
infinitely many possible languages -- for example, Tarski's
hierarchy of metalanguages, all of which have exactly the same
grammar, but language N+1 includes the domain D of everything
that N can describe plus all the syntactic features of the
language N.
However, I agree that there are many things that cannot be
described in any finite sentence in any language that can be
written in a finite alphabet. Only a countable subset of
the real numbers, for example, are describable.
> A person realizes a category...
In mathematics, it's common practice to avoid introducing the notion
of the mind of any particular agent. Many mathematicians adopt a
Platonic stance: all possible mathematical structures that can be
consistently axiomatized exist in a Platonic heaven.
But I believe that it's possible to do good mathematics with any
of these conceptions. If you like, you can talk about what some
person thinks or what God thinks. Or you can take a formalist
stance: if a theory T is consistent and T contains an existential
quantifier (Ex), then it's grammatically OK to say that x exists.
To me, this seems like a discussion of metalevel questions that
had been answered many years ago. I don't see what is left to be
finished "before Christmas".
John