Math Forum Discussions

fom

Posts: 1,035
Registered: 12/4/12

Re: distinguishability - in context, according to definitions
Posted: Feb 19, 2013 12:16 PM

Plain Text

On 2/19/2013 6:29 AM, Shmuel (Seymour J.) Metz wrote:
> In <x_-dnZNsYePggrzMnZ2dnUVZ_rydnZ2d@giganews.com>, on 02/17/2013
> at 12:20 PM, fom <fomJUNK@nyms.net> said:
>
>> This is not how I understand mathematics.
>
> There are two very different issues; structure and presentation. It is
> cumbersome to always write proofs out in full, so working
> mathematicians use a set of informal shorthand notations.
>

Of course. And, I would not generally insist that any of it be
altered (except for one course in recursive function theory
whose professor worked without a book and filled the blackboards
faster than anyone could transcribe the material...).

The problem arises elsewhere.

>> Almost every reputable mathematics department is
>> giving courses in "mathematical logic," presumably
>> based on this received paradigm.
>
> I believe that such courses fo into more than just the logical
> machinery needed in Mathematics, and that they cover, e.g.,
> independence, consistency, models.
>

They do. But, the nature of identity is treated casually.
It is that subject matter over which I stumbled. The
first hint I had that what I had been struggling with was, in
fact, legitimate came when I found this paper:

http://arxiv.org/pdf/quant-ph/9906101.pdf

I did not discover that paper until 2003 or 2004. I had already
formulated an alternative set of axioms for the treatment of
identity in Zermelo-Fraenkel set theory. My first attempt at
writing a paper in 1994 had been dashed. I had written it in a
format called WordPerfect because I did not know about TeX.
Then, I taught myself to administer a Linux platform (minimally)
so that I could get a LaTeX installation. Of course, while doing
that, the MiTeX installation became available for Microsoft
platforms.

The second paper was submitted to "The Journal of Symbolic Logic".
I do not believe it had been reviewed at all. The day after
submission, I received a confirmation email. It addressed me as
"professor." I had been polite enough to inform them of their
error. Within 30 minutes I received an angry rejection about
wasting their time.

There had been an oversight in that paper (I hesitate to call it
an error because it remains a different modal possibility.). It
is corrected now because of someone who had been kind enough to
respond several years ago. It had not been a friendly response.
However, it identified the error. The only supportive comment I
ever received from anyone came from Wilfred Hodges who suggested
that I should look at the work from the Polish school.

While I am not an expert on the pursuit of "quantum logic" in
the sense of what those lattice theorists who wrote the paper
at the link above are pursuing, any computational model or
syntactic model with truth-functional capability will require
more than what they have accomplished.

In the manipulation of syntax in those other posts I had linked
is a characterization of truth-functional behavior not based on
specific representation by truth tables. It is not based on
Boolean algebra. In fact, if successful in its intent, it
characterizes semantics relative to a lattice mapping from the
free DeMorgan lattice on one generator into the ortholattice
O_6 mentioned in the paper above.

>> But, in the "logical" sense,
>> 1.000... = 0.999...
>> is merely a stipulation of syntactic equality
>> between distinct inscriptions that is prior
>> to any mathematical discourse.
>
> There is no syntactic identity there. There is, instead, an identity
> based on a specific[1] definition of the notation and a specific set
> of axioms.
>

Correct. My "stipulation of syntactic equality" is the same as
your "definition of notation".

The phrase "syntactic equality" is being taken directly from Carnap.

When I had been badly flamed quite some time ago, the flamer would
keep insisting that there was no problem with identity -- that, in
fact, any apparent problem is trivially addressed with term models
where the "objects" of the model are merely the equivalence classes
of terms assigned equality by prior stipulation.

The difficulty here is that "mere" has nothing to do with it.

One has a set of symbols. One has a partition on that set of
symbols. And, if one is claiming a homomorphism, then one is
looking at the uniqueness of an induced quotient topology on the
identification space/decomposition space/quotient space.
Individuals who are trained outside of mathematics do not
recognize the situation in this manner. In fact, when they
hear mathematicians use the word "identify" they often respond
with Russell's critical reference of Dedekind concerning
"honest toil."

Along the same lines, a partition of the domain into equivalence
classes has a lattice representation. Technically, such lattices
are atomistic lattices having the atomic covering property:

if p and q are atoms and if b/\p=0 then
p<=b\/q implies q<=b\/p

They are more commonly referred to as matroid lattices and have
a principal origin in the investigation of linearly dependent
and linearly independent sets of vectors where the numerical
coefficients have been ignored.

Needless to say, the axiom of choice and the generalized
continuum hypothesis come into play here.

Wittgenstein insisted on a form of identity wherein every
distinct symbol corresponded to a distinct object. We might
refer to that as canonical naming. If, in the formation of
the quotient space/quotient model above, one chooses instead
to use a map to a canonical name within each equivalence class,
then separation properties of the discrete topology that
characterize the bottom of the partition lattice are retained
in moving up a chain through the lattice. This is a natural
way to retain the syntactic distinction of symbol shape.

Now within any equivalence class of symbols, distinguishing
one at the expense of the others may be taken as fixing a
direction on the edges of a complete graph such that the
symbol chosen as a canonical name is the only symbol for
which every edge is outwardly directed.

Next, if one takes this as a prior constraint in relation
to well-orderings, then the choice functions for the set
are partitioned into those for which the chosen element
is first and those for which it is not. Among those for
which it is first, one may now consider the labeling problem
whereby the ordinals correspond to the labels and the
the directed edges of the complete graph emanating from
the canonical name are taken to be constraints.

In finite circumstances, path consistent labelings for a
complete graph reduce to path consistency along every
triple. Of course, to overlap triples sequentially
is to overlap the pairs. This is, perhaps, one interpretation
of Peano's axiom

(ae|N /\ be|N) -> (a=b <-> (a+1=b+1))

In addition, along the lines of the complete graph mentioned
earlier, one must organize the triples so that they are
overlapping. Among all of my senseless syntax, you will
find the use of Steiner Quadruple Systems. Such block
designs are configurations wherein every triple is uniquely
represented in blocks of size 4. Because the 2-dimensional
flats of every n-dimensional affine space over the Galois
field of order 2 form an SQS, an SQS can be formed for
every set of symbols with order 2^n, n>2.

For a long time, I did not know how to get the triples
topologically. Primarily, I had been focused on definite
descriptions and how Cantor's intersection theorem
reflected Leibniz' actual statements better than Russell's
logicist representation. But, my attention turned to
Hausdorff spaces when I realized that unary negation
facilitated organizing the formulas of first order logic
into a minimal Hausdorff topology, provided one also
augmented the set with the Fregean notions of "the True"
and "the False".

While looking at the connectedness properties of Peano
spaces two days ago, I stumbled on Problem 28A in
"General Topology" by Willard. The example discusses
how to form an indecomposable continua on 3 points.
I am now looking at what I had been doing with the
intersection theorem in terms of compact, connected
Hausdorff spaces.

No. There is nothing "mere" about my understanding
of these matters. And, you are correct. There are
axioms involved:

http://en.wikipedia.org/wiki/Axiom_of_symmetry

You will find the discussion of graph theory at the
end of the page.

> [1] Well, some of the posters seem to be unable to formulate what
> they mean by, e.g., "0.999...".
>

:-)

Infinity is... hard.