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fom

Posts: 1,098
Registered: 12/4/12

Re: when indecomposability is decomposable
Posted: Feb 20, 2013 10:03 AM

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On 2/17/2013 10:03 AM, Shmuel (Seymour J.) Metz wrote:
> In <88qdnU1ZNsB9j4LMnZ2dnUVZ_uudnZ2d@giganews.com>, on 02/15/2013
> at 11:02 PM, fom <fomJUNK@nyms.net> said:
>
>> When one invokes the axiom,
>
>> Ax(x=x)
>
>> by
>
>> a=a
>
>> there is an ontological interpretation of the
>> sign of equality corresponding with the sense
>> of indecomposability.
>
> I don't see how it is either omtological or indecomposable. The
> inference is valid regardless of how you model "=".
>

The validity of the inference being justified by
what stance?

A number of years ago, there was a John Correy on these
newsgroups who formulated a set of axioms that
could be syntactically manipulated to "prove" the
existence of non-self-identicals. He had been
hammered on the basis of the ontological import
of x=x.

In the first edition of "Principia Mathematica" distinction
was made between bare quantification ("real variables") and
scoped quantification ("apparent variables"). Then, in the
second edition, the position changed.

I agree that the step above, within a derivation,
is merely a transformation rule. Originally, Frege treated
universal quantification as arbitrary substitution
with all senseless substitutions being interpreted as
false. That was found to be objectionable by those
such as Russell.

Outside of a derivation, there is more ambiguity.

>> Of course, mathematicians generally do not know
>> of description theory.
>
> Is that true? I'd buy the claim that for most mathematicians it is not
> relevant to their sphere of interest.

That would have been a better way of completing that
phrase.

>
>> That is, when one presupposes
>> the ontological interpretation that gives
>> rise to the necessity of
>> |- (x=y -> Az(zex <-> zey))
>
> Isn't that a special case of a more general axiom schema? For any
> propositional function P of two variables, |- (x=y -> Az(P)z,x) >->
> P(z,y))
>

Correct. That schema is generally referred to
as Leibniz' law. This is Leibniz' actual
statement when introducing the sense of it:

"What St. Thomas affirms on this point
about angels or intelligences ("that here
every individual is a lowest species")
is true of all substances, provided one
takes the specific difference in the way
that geometers take it with regard to their
figures."

Although the value of investigations into formal
grammars cannot be denied, to call that schema
Leibniz' law is historically inaccurate.

The quote above is made with respect to an
intensional syllogistic logic and geometric
rigidity. So, one is looking at a nesting
of part-whole relations relative to a
metric. Cantor's intersection theorem is
closer to Leibniz' law than the schema
above.

It had been Frege who gave Leibniz' law
its logical form. Even if one disregards
the fact that he retracted his logicism
at the end of his career in favor of a
geometric foundation for mathematics,
the naive use of that schema is, at
least, questionable for set theory.
Frege, himself, made the distinction,

"But although the relation of equality
can only be thought of as holding for
objects, there is an analogous relation
for concepts."

It is simplest to express the difference
in his statements which follow with
the formulas:

AxAy(x=y <-> Az(xez <-> yez))

AxAy(x=y <-> Az(zex <-> zey))

The first is object identity and the
second is concept identity.

Moreover, Cantor certainly objected to
the treatment of sets as extensions.
He insisted that sets had a prior
criterion of identity. In his critique
of Frege he wrote:

"... fails utterly to see that quantitatively
the "extension of a concept" is something
wholly indeterminate; only in certain cases
is the "extension of a concept" quantitatively
determined; and then, to be sure, if it is
finite it has a determinate number; if it is
infinite, a determinate power. But for such
a quantitative determination of the "extension
of a concept" the concepts "number" and
"power" must previously be given from the
other side. It is a twisting of procedure
if one attempts to ground the latter
concepts on the concept of the "extension
of a concept".

One should additionally note that Cantorian
set theory had been predicated on a theory
of ones, or units, that had been argued
against by Frege.

The two formulas given above may be
paraphrased as:

"A set is a collection taken as an
object"

"A set is determined by its elements"

The first is asserting a prior identity
criterion whereas, for pure set theory,
the second is not.

What the situation comes down to is
this. The predicate symbols 'c', 'e',
and '=' are like the fundamental constants
in physics. The history of "undefined
language primitives" has led to treating
a system of relations whose relationship
to one another can be made clear as if
it cannot.

The real distinction of the situation is
found in epistemic logic. For the practical
purposes of mathematics, it comes down to
the construction of language through
definition, because a definition is not
the same as a stipulation. A stipulation
is impermanent and used to parametrize
a problem domain. A definition is expected
to have a structural relation to the
theoretical coherence of the science.