"mitch" wrote:
: Thank you for this post, Galathaea. I have run a number of searches
: from the text within and have found numerous papers of interest.
:
: galathaea wrote:
:
: > There is this very interesting paper by Richard Greechie
: > about a non-standard quantum logic (L44, M22), which can be
: > represented (fixed width):
: >
: > s v
: >
: > / \ / \
: >
: > r t u w
: >
: > / \ / \
: > \ /
: > p ----------- \ --- / ----------- n -------- m
: > \ /
: > | \ / |
: > | X |
: > | / \ |
: > | / \ |
: > | |
: > | g ----- f -------- e |
: > | |
: > | |
: > q | | k
: > | |
: > \ /
: > \ h d /
: > \ /
: > \ | | /
: > \ | | /
: > \ | | /
: > \ | | /
: > \ | | /
: > \ | | /
: >
: > i c
: >
: > \ /
: >
: > j b
: >
: > \ /
: >
: > a
: >
: > in what is known as Greetchie notation (Greetchie's own
: > papers describe it, or Josef Tkadlec has a paper on general
: > "Representations of Orthomodular Structures" and there are
: > other online resources).
:
: I need to learn about this notation. If what I have read is
: correct, it reflects certain relationship with maximal
: Boolean sublattice blocks.
Yeah, that's the basic interpretation. I tend to look at it as a
decomposition into "local" boolean connectives, in accord with my ideas on
orthomodular structures consisting of a notion of localisation of boolean
logic in accord with a metaphor on global versus local symmetry groups.
: As a diagram, however, it has an interesting symmetry. And, it
: is not unlike the kind of thing that I think about with the
: column vectors. One quick schematic for it, perhaps, is given
: through the specification for protons and neutrons in terms of
: their constituent quarks:
:
: up up down
:
: up down down
Exactly! I was hoping you would notice this, as this is exactly the point I
wanted to stress about this diagram. There is this 3 / 2 versus 2 / 3
symmetry (?? antisymmetry or dualisation ??) very apparent in much of the
study of this particular logic. Its not just diagrammatic either. The
violation of the ortho-arguesian law seems related to such.. uh...
symmetries (it's hard to find the appropriate term).
: The two vertexes at the top of the graph have paths that cross.
: So, you get a piece something like,
:
: left right
:
: f n
: | |
: p e h m
: | | | |
: q d g k
:
:
:
: But, if I use my "wizard" nonsense, the overloading of this 2-3
: relationship can be captured
:
:
: A B B A
: | | | |
: w i z a r d
: 4 9 1 1 9 4
: | |
: C C
:
:
: So, I am visualizing a very complicated way of juxtaposing two
: characters. Maybe I should have used something like
:
:
: A B C
: | | |
: w i z a r d
: 4 9 1 1 9 4
: | | |
: C B A
:
:
: to match the "ups" and "downs" lifted from the physical context.
It's funny because I had never noticed the symmetry / whatchamacallit in the
word wizard until I had read some posts by you about this example. Even
listening to Black Sabbath never showed me the kabbalah here...
=)
: I guess I cannot quite understand what people mean by an
: "unordered pair." I can understand not knowing the order of a
: pair and I can understand superposition of all possible orders.
: But, the connectivity of a pair without order is
: incomprehensible to me.
Yes!!! Yes! Yes and yes!!!
I once got into a discussion with my topology teacher about which was more
primary. He argued that ordered pairs required more definition, whereas my
point was that on all conceptual levels I could identify (visual, auditory,
etc.), the ordering seemed to follow most naturally from the input, and the
act of "unordering" seemed a latter abstraction.
a -> b is much more useful evolutionarily than, say, a = b.
: The diagram above makes me think of this question.
:
: Involving commutation is not far-fetched. Schein's paper on
: pseudosemilattices and pseudolattices is motivated by
: characterizing the identity,
:
: zxyz = zyxz
:
: And the notation I have for Jordan triple systems expresses
: something similar
:
: {xyz} = L(x,y)z = P(x,z)y = {zyx}
I would say that questions of commutativity are some of the things in which
orthomodularity is specially capable in answering. In particular, my
approach to orthomodular logics was originally introduced through the notion
of non-commutative spaces and the construction of what are now known as
quantum groups (ie. quasitriangular Hopf bialgebras). I haven't figured out
exactly what it is about my psychology, but noncommutative geometry always
intrigued me in much the same way that non-Euclidean geometry did, and
understanding the nature of the relationship between geometry and logic was
one of the first driving researche programmes I took on long ago.
Your mention of Jordan triple systems brings back some work I did where I
was looking for a semantic analysis of Jordan algebras, in particular in
relation to quantum logic. I only reached a modal view myself, but I did
have to confront the well known problems of Jordan algebras (more precisely
all C* and particularly all von Neumann algebras) in the face of dispersion
and entropic calculations.
: And, what seems to be going on is the relationship between
: a line segment (Hilbert's definition is the same as an
: unordered pair in set theory) and the use of a line segment
: for doing the calculus. What I mean by the latter assertion
: is the use of three points to characterize convexity:
:
: F(t)= (x - y) + y(1-t) for 0<=t<=1
:
: or something like that.
:
: In any case, no convexity, no (useful) calculus. The fact that
: sets are archetypical objects in mathematics does not mean that
: axiom systems convey the essence of mathematics. I guess this
: pairing thing just gets me. But, after a hundred years of this
: stuff you think we would have figured out that one just doesn't
: get away from the geometry (who is that? Wittgenstein?)
There was a point in my life when I admired Wittgenstein immensely, much I
would dare to say, comparable to your admiration of Kant. Althoug I would
now differ with both because of my own tangents I have wandered, it is so
blatantly obvious that these folk deeply pondered the nature of reality and
mathematical constructs far beyond most modern mathematicians.
Convexity is so fundamental in defining properties. The intersection of
convex regions is convex. Unions of properties do not distinguish. There
is something very fundamentally geometric in all of that.
: I found a used book on design theory. It begins with
:
: "An incidence structure is a triple
:
: D=(V,B,I)
:
: where V and B are any two disjoint
: sets and I is a binary relation between
: V and B"
:
: And, of course, in topological model theory you are working with
: a monadic second order language that extends a first-order
: similarity type with the membership relation and set variables.
: So, what you really have is
:
: M=(D,T,e)
:
: where D is a domain of objects, T is a domain interpreted as
: subsets from P(D), and e is an incidence relation interpreted
: as membership.
:
: So, we get a great connection between formal language and
: mathematics without need for huge debates.
Yes, and I've also posted in some far away land my suspicions that
Mitchell-Benabou provides a fairly comprehensive conversion of certain
natural and formalised language constructs in the language of categories and
topoi. This connection between topology, subobject classifiers, and logic
has always seemed to be much more fundamental than a purely rigorous
translation. It has seemed to be cognitive to me as well.
: Anyway, thank you very much for this post. I have saved it
: to a file and have been back to read it several times so far.
It has been only a partial response to your posts. I have worked hard to
track your thoughts and study up on some of the connections you make, but I
have such a hard time following all of the connections in a reasonable time.
I understand that some of my points have similarly been scattered across the
literature, so please understand that I don't expect any large scale
conclusions from my comments any more than I can give large scale responses
to yours (at least such jumps would surprise me!).
I did notice that my post failed to make the connection between the name L2
in the classification diagram and your work in which I reconstructed the
diagram. There were other connections I know did not expand upon as clearly
as I should have (I get lost in my thoughts way too easily to be as
expressive to others as I would hope). Also I made no effort to explain the
connection with quantales in sufficient depth. However, my point was less
to introduce new ideas and more to stress a context that I have been
following, so it is hard to choose what to state and what to leave to
literature findings.
There is, however, a thread of thought which I wanted to share that may
explain somewhat my approach to all of this. I have been interested in
"pre-geometric" approaches to quantum mechanics for most of my theoretical
life. One of my very first actual mathematical theories was work on the
characterisation of inertial dynamics (in my early work, this equals lines)
purely in terms of object-to-object distance relationships on the space. I
felt in my early theoretical life that there must certainly be some method
of consolidating quantum mechanics with topology if metric spaces were
generalised in a way to allow inconsistent geometries between particles as
long as every particle saw its own consistent geometry. Distance was my
first geometrical connective, though I later moved more general notions of
uniform spaces. My goal was to construct the quantum geometry on local
notions of consistency (which brings us back again on the Greechie notation
and other, more esoteric, beliefs). When I first read Mac Lane and
Moerdijk's "Sheaves in geometry and logic: a first introduction to topos
theory" and similar books and papers in the literature, I was stunned at the
eloquence in wich the theories were phrased. They were far more precise
definitions of what I had instinctively felt, yet they precisely expressed
my intuitions (that kind of confidence I have always felt in the works of
Poincare and Tarski).
I was able in a first theory to relate first differentials on the space-time
manifold, making bundle connectives primary. However, I have since moved to
more general differential topological notions in which to frame my
approaches. Such is my whacked out view of theory. I want to construct
space time from more local notions of connectivity, in the meanwhile
relaxing global notions of consistency to allow for a quantum description.
My interpretation of Mach does not require global consistency.
I could give a few equations for review, but my point is not to overload on
the connections (and I am quite far from even showing such an approach to be
viable on several fronts). I only wanted to show that along the way, I
noticed a few connections with other research. Speculation is rampant,
though it is well supported mathematically. That seems to be the way these
days with observation underfunded in respect to theory. It makes tolerance
a very interesting solution.
You once wrote, mitch:
: "The Earth is becoming smaller by the minute. We built
: modern civilizations out of admiration for ancient
: civilizations. Perhaps I should suggest that President
: Bush forget about Mars and start funding for the Corps
: of Pyramid Engineers. I do not really want to be a
: fatalist on issues like these, but it seems like there
: are various little hints in almost all branches of
: mathematics. I don't see expanding economies without
: expanding frontiers and those frontiers are becoming
: prohibitively expensive or carnivorously cannibalistic."
When you add what seems to me a very dual notion that controllable choices
should always be made to optimise health, and that competition is often more
healthy a decision than singleton action schemas, I think a much more
natural understanding forms on the frontiers of choice theory. I think it
forms the basis for a "scientific activism", where cellular propositions are
abstracted to multicellular and superorganismic models and quantifiability
of various aspects of health are modelled for entire societies.
When you look at education these days, it is still a fact that those with
the greatest education have the greatest opportunity for high economic and
social placement in a community. And one of the biggest barriers to
education is economics, leaving a wedge in class stratification which
negatively affects quite a few countries, the most prominent being the U. S.
Many underclass have to fight huge battles, work extreme workloads, actually
work towards unhealthy lifestyles, whereas the priveleged live lives of
privelege because of social structures they erected by force (I have read
you mention your struggles and I too had to work 48+ hour weeks in addition
to the 20-30 hour weeks for my education to meet ends). There are no easy
paths for the theorists, for the high road has been polluted by snake oil
and confidence games.
Every battle is critical.
Sun Tzu is the textbook of choice.
And it's so very hard to accept how, as the art of war puts it so well,
"I:18. All warfare is based on deception."
Ontologies are attacked at their foundations because those are the places
where one has to stand up for oneself, where confidence must be most secure.
Confidence is never secure. One can lie to oneself.
Sometimes in my fits of fancy, I like to view the entire scientific
hierarchy as a combination of chest beating ritual with observational
partial orderings. It is the way I explain the existence of so much
controversy in such experimentally indistinguishable frameworks.
Other times, I am much more forgiving.
--
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galathaea: prankster, fablist, magician, liar
