Date: Feb 20, 2013 10:03 AM Author: fom Subject: Re: when indecomposability is decomposable 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.