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:
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 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: