Some background on Stone Algebras
mitch
I found some material that might be of interest. There will be more,
including information on de Morgan algebras and Lukasiewicz algebras.
But the latter are related to Post algebras In turn, Post algebras
relate to Stone algebras.
"The study of pseudocomplemented distributive
lattices commenced with a paper by V. Gilvenko
in 1929. Although there is a reference to what
we now call Stone algebras in the 1937 paper of
M. H. Stone, it was G. Gratzer and E. T. Schmidt
who first solved Problem 70 of B. Birkhoff and
thereby generated widespread interest in the topic.
This, in fact, was the first paper in which the terms
Stone lattice and relative Stone lattice were used.
"A pseudocomplemented distributive lattice is a
distributive lattice L with 0, 1 such that for each
(a e L) there is a greatest element ~a which is disjoint
with a. The problem referred to above is then: What
is the most general pseudocomplemented distributive
lattice in which
~a + ~~a = 1 ?
"Another problem which attracted interest was posed
by O. Frink in 1962: Is every Stone algebra isomorphic
to a subalgebra of the algebra of all ideals of a complete
atomic Boolean algebra? This problem was solved by
G. Gratzer in 1963. More direct proofs were later
given by G. Bruns and G. Gratzer. More recently,
H. Lakser has shown that every pseudocomplemented
distributive lattice can be embedded (so as to preserve
+, *, 0, and ~) in the lattice of ideals of an atomic
Boolean algebra.
"A monograph devoted to Stone algebras by J. Varlet
appeared in 1963. In 1969 C. C. Chen and G. Gratzer
presented two papers in which Stone algebras were
represented as a triple--consisting of two 'simple structures'
and a 'connecting map'. The free Stone algebras on
finitely many free generators were determined in
R. Balbes and G. Gratzer, and the injective hulls of
Stone algebras were described using the 'triple method'
by H Lakser. In 1970, K. B. Lee gave a characterization
of all of the equational subclasses of pseudocomplemented
distributive lattices. This was followed by three significant
papers: H. Lakser (1) and G. Gratzer/H Lakser (2) on the
structure of pseudocomplemented distributive lattices,
dealing in particular with subdirect products, the congruence
extension property, amalgamation, and injectivity. A
generalization to Stone algebras of order n has been
investigated by T. Katrinak and A. Mitschke. Because
of their relationship to Post algebras we will cover this topic
in [...]
"Besides the papers we have mentioned, many others
have been published on pseudocomplemented distributive
lattices and closely related topics. We mention just a
few: [...]
"[...]
"The equational subclasses form a chain
B_(-1) < B_0 < B_1 < ... < B_omega
"In this chain B_(-1) is the trivial class, B_0 is the
class of Boolean algebras, B_1 is the class of Stone
algebras, and B_omega is the whole class of
pseudocomplemented distributive lattices."
--Balbes and Dwinger, "Distributive
Lattices"
George once mentioned that the things I wanted to talk about could be
reduced to Boolean algbras. I'm not so sure. But, we shall see.
:-)
mitch
mitch
There are two examples of chains from Balbes and Dwinger:
As an example of a Heyting algebra:
| 1 if a <= b
Chains with 0, 1 : a -> b = |
| b if b < a
...and as a pseudocomplemented distributive lattice with pseudocomplement
a*
| 0 if a <> 0
Chains with 0, 1 : a* = |
| 1 if a = 0
The first sentence in my system had been an order relation.
AxAy( x c y <-> (Az( y c z -> x c z ) /\ Ez( x c z /\ ~(y c z) ) ) )
As I told George, the structure of the sentence is such that is simply
translates the transitivity of the zero-order implication into a
first-order context. I certainly had no recognition of any relation to
Heyting algebras when I did this.
The order was purposely devised as a strict order because of what I had
been doing with the membership relation and identity. Once those things
were established, I added the sentences,
ExAy( ~( y c x <-> x = y ) )
ExAy( ~( x c y <-> y = x ) )
to give the system a '0' and a '1.'
When I had been requesting a little 'freedom' from strict first-order
logic, what I meant without knowing at the time was that my quantifiers
were to be treated intentionally. There would have been no problem
extending the axiom set with something analogous to a foundation axiom.
However, I needed to treat the maximal term on its own. This led to the
axiom,
Ax( Ey(x c y) -> Ey(x e y) )
This is an assertion of almost universality. So, the system (apparently)
declares a second-order semantics. Extensionality followed from this
(along with axioms not shown) in the theorem,
ExAy( ~(y e x <-> x = y) )
And this had been based on treating sets in the sense of descriptive set
theory whose axioms are not properly a part of Zermelo-Fraenkel set
theory. So, extensionality in the system is a proven property rather than
an assumption of the logic. By this I mean that the descriptive sense of
algebraic semantics governed interpretation rather than the
syncategorematic interpretation of standard first-order model theory.
In any case, it clearly distinguished '1' from '0' thus prioritizing
directionality and a truth condition for otherwise dual extrema.
If I had been aware of the relationship of lattice theory and logics to
begin with, I might have been able to avoid the arguments that happened.
But, there are models of set theory that are simply linear orders, and, I
had been thinking about the problem in terms of models for set theory.
Oh well.
:-)
mitch
mitch
mitch wrote:
> I found some material that might be of interest. There will be more,
> including information on de Morgan algebras and Lukasiewicz algebras.
> But the latter are related to Post algebras In turn, Post algebras
> relate to Stone algebras.
>
Here is the scoop on the de Morgan algebras.
As a matter of historical record, I had tried to talk about symmetries
associated with classical truth-table semantics when you obfuscated 'T' and
'F' Balbes and Dwinger illustrate the de Morgan algebra on 2^2 with the
following remarks,
"Unlike any of the classes we have considered,
order-isomorphisms and M-isomorphisms are
not the same in M. Indeed, the Boolean algebra
2^2 is not M-isomorphic with the de Morgan
algebra obtained from 2^2 by redefining
~ (0,0) = (1,1)
~ (0,1) = (0,1)
~ (1,0) = (1,0)
~ (1,1) = (0,0)
So, in terms of truth table semantics, what one would be looking at is a
transition from
A B |
------------
T T |
T F |
F T |
F F |
to
~B ~A |
------------
F F |
T F |
F T |
T T |
Needless to say, I had become interested in representing connective
invariance with respect to transformations like this. Unfortunately, I did
not know about de Morgan algebras. Enough history.
The definition is as follows:
Definition 1. A de Morgan algebra is an algebra of
the form
(L, (+, *, ~, 0, 1))
where + and * are binary operations, ~ is a unary
operation and 0, 1 are nullary operations, satisfying:
M_1 : A set of identities in +, *, and with a 0 and 1 which
define distributive lattices with 0 and 1
M_2 : ~(x+y) = ~x * ~y
~(x*y) = ~x + ~y
M_3 : ~~x = x
What is going on here has to do with involutions. Balbes and Dwinger give
the usual definition of an involution,
"If X is a set, then a function
phi: X ---> X
which satisfies
phi o phi = 1_x
is called an involution of X."
So, an involution is a function that yields the identity mapping when
applied successively.
Here are some set theoretic remarks,
"Definition 2: Let L be a de Morgan algebra. If L
consists of all the subsets of a set X with cup and
cap as binary operations and 0 and X as nullary
operations, then L is called a complete de Morgan
field of sets. If L is a subalgebra of a complete
de Morgan field of sets, then L is called a de Morgan
field of sets."
"The reader should note that the operation ~ is, in
general, not set theoretic complementation.
"... there is a one-to-one correspondence between
all complete de Morgan fields of subsets of a set
X and the involutions on X."
Balbes and Dwinger then offer some more facts on how de Morgan algebras fit
in among other types of algebras. In what follows, any mention of an
equational class is simply the class closed under homomorphic images,
subalgebras, and direct products. I will use an ampersand, &, to indicate
these closures.
One of the things I had tried to talk about before was labeling. It is my
understanding that an algorithmic approach to the labeling problem is based
on overlapping triples. So, I find the discussion of subdirectly
irreducible de Morgan algebras interesting.
"We are now able to characterize the subdirectly
irreducibles in &M. We introduce the following
notation. Let M_0, M_1, and M_2 be the de Morgan
algebras defined by
M_0 = 2 (standard ordinal, {0,{0}})
M_1 = 3 (standard ordinal, {0,{0},{0,{0}}}) with (~1)=(1)
M_2 =
1
/ \
/ \
a b
\ /
\ /
0
with (~a)=(a) and (~b)=(b)
[When I had been talking about invariance with respect to de Morgan
conjugation in sentential logic, this would have been the relationship
being preserved. Since standard negation existed in that system of
connectives as well, you get the familiar "the projection connectives and
their negations are preserved under de Morgan conjugation" whining I had
been doing.]
"M_0 is isomorphic to a subalgebra of M_1,
and, M_1 is isomorphic to a subalgebra of
M_2."
In fact, there is a nice relation among the equational subclasses:
"We will denote the trivial equational subclass
of &M, consisting of all one element algebras by
&M_(-1). &M_0, &M_1, and &M_2 will denote
the equational subclasses of &M generated by
M_0, M_1, and M_2 respectively
"The equational subclasses of &M are
&M_(-1) subclass &M_0 subclass &M_1 subclass &M_2 = &M
"Moreover, for (L e &M), (L e &M_0) if and only
if
x*(~x) = 0
is an identity in L
and (L e &M_1) if and only if
((x + ~x) * (y * ~y)) + ((y + ~y) * ( x * ~x)) = (y * ~y) + (x * ~x)
is an identity in L."
So, with regard to the labeling question,
"If (L e &M) and L is a chain, then (L e &M_1)"
So, maybe sci.logic will forgive me for arguing about "rigid designators,"
"labeling," or otherwise having so strongly emphasized linear orders. The
question of a foundation for mathematics has to reduce according to the
structures defined in mathematics. Clearly, logic cannot be excepted.
But, the one structure that coincides for these algebras is a chain.
It might, perhaps, be possible that &M_2 is the fundamental class here.
But, one must question how that relates to quantum mechanics where
experiments dictate a different approach.
Balbes and Dwinger indicate that the free de Morgan algebra on one free
generator is given by the diagram,
1
|
|
*
/ \
/ \
a ~a
\ /
\ /
*
|
|
0
They denote the standard distributive lattice obtained from a de Morgan
algebra by ignoring its unary operation with an superscript. Thus, the
free de Morgan algebra on one operator is equivalent to the standard
distributive lattice on two generators,
(F_M(1))^* = F_D01(2)
That is, F_D(2) is given by
x+y
/ \
/ \
x y
\ /
\ /
x*y
F_D01(2) is given by
1
|
|
x+y
/ \
/ \
x y
\ /
\ /
x*y
|
|
0
Thus F_M(1) treats (a) and (~a) as separate symbols for free generation,
thereby giving it the representation of F_D01(2) when the unary operation
is ignored.
But, naturally, I still will be unable to get anyone on sci.logic to think
in terms of duality... sigh
In any case, the cardinalities associated with the free algebras in general
satisfy
| F_M(n) | = | F_D01(2n) |
which probably explains why the hierarchy of orthomodular logics seem to
follow a similar rule.
Anyway, I am not so convinced that the things I think about in the
foundations of mathematics are properly described by Boolean algebras.
There is more to this than what can be represented. There is a question of
faithful representation as well.
:-)
mitch
galathaea
: George once mentioned that the things I wanted to talk about could be
: reduced to Boolean algbras. I'm not so sure. But, we shall see.
I point to the theorem that there does not exist a partial algebra
homomorphism for quantum logics on most Hilbert spaces in my recent piece on
quantum logic. I had been meaning to mention it here for several days, but
"so many things, so little time"...
Anyways, thats one of those theorems to keep in the pocket against the
universal Boolean claim.
--
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
galathaea
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). Now, an orthomodular lattice is
called a standard quantum logic if it is ortho-isomorphic
to some sub-orthomodular poset of the ortholattice
associated to some Hilbert space. The interesting thing
about this particular nonstandard orthomodular lattice is
that the ordering on the lattice is still strongly order
determining, and instead differs from a standard quantum
logic in that a variant of Desargues' theorem is violated.
Desargues' theorem is interesting because it relates the
three point perspectives to a point to the corresponding
three point perspectives to a line. It is an interesting
relation that can be translated somewhat into a logical
form called the ortho-Arguesian law.
Given a pair of three-point collections ai, bi and defining
ci = (aj \/ ak) /\ (bj \/ bk) (with i,j,k a cyclic permutation of 0,1,2)
x = (a0 \/ b0) /\ (a1 \/ b1) /\ (a2 \/ b2)
y = c2 /\ (c0 \/ c1)
z = ((a0 /\ (a1 \/ y)) \/ b0
A lattice is ortho-Arguesian if whenever ai # bi (where #
is the relation signifying relatively orthocomplemented)
for all i, we have implied x -> z.
(L44, M22) is not ortho-Arguesian, but all Hilbert spaces are.
The study of these types of spaces shows a path to
classifying possible generalisations of quantum mechanics to
spaces that do not obey certain classical geometric
equational theorems, in particular altering notions of
perspective here.
Looking at equational theorems to classify has been a
topic deeply studied by Gudrun Kalmbach (who discovered (L44,
M22)). In her paper "Omologic as a Hilbert type calculus",
she explores rules of inference over various orthomodular
logics, where she gives the classification MOn for the
omolattices:
T
/ / ... \
/ / \
/ / \
x1 x2 ... xn
\ \ /
\ \ /
\ \ ... /
_|_
M is used for the class of all finite modular ortholattices.
She calls L1 the lattice embeddable in (L44, M22) given by
.
/ \
. .
/ \
. .
| |
. .
| |
. ----- . ----- .
and describes a lattice given by mitch at one point:
========================================================
[begin fixed width]
| T |
| | \
/ | T | \
/ | | \
/ / | T | \
/ / | | \ \
/ / / | T | \ \
/ / / \ \
/ / / \ \
/ / / \ \
/ / / \ \
/ / / \ \
/ / / \ \
/ / / \ \
/ / / \ \
\ \
| T | | T | | F | \ \
| | | | | | \ \
| T | | F | | T | \ \
| | | | | | \ \
| F | | F | | F | \ \
| | | | | | \ \
| F | | T | | T | \ \
\ \
| | |
\ \ | | / / | F | | F |
\ \ / \ / / | | | |
\ x x / | T | | F |
x X x | | | |
/ x x \ | T | | F |
/ / \ / \ \ | | | |
/ / | | \ \ | T | | T |
| | | / /
| F | | F | | T | / /
| | | | | | / /
| F | | T | | F | / /
| | | | | | / /
| T | | T | | T | / /
| | | | | | / /
| T | | F | | F | / /
/ /
\ \ \ / /
\ \ \ / /
\ \ \ / /
\ \ \ / /
\ \ \ / /
\ \ \ / /
\ \ \ / /
\ \ \ / /
\ \ \ | F | / /
\ \ | | / /
\ \ | F | /
\ | | /
\ | F | /
| | /
| F |
====================================================
but from a different direction (this one gets messed up without
width set to more than 96).
Including MO (class of all modular ortholattices), B (boolean
lattices), and C the class of omolattices generated from
Hilbert spaces, she gives the rough sketch of equational
subclasses of the orthomodular lattices as:
OM
/ \
----------------/-------\--------------
| / \
|
| MO C
|
| | |
------------|------- | ???
| |
| ...
M |
|-------------------
|
...
|
MOn
| .........
...
| | | |
MO3 L0 L1 L2
\ / / /
\ / / /
\ / / / ...
MO2
|
B
|
0
This is the beginning of a map of the pantheon of orthomodular
logics, guideposts to further investigations. One could, for
example, go from here and look at MO3 and more generally MOn
in relation to partition logics and computation, as for example
done by Karl Svozil's studies of quantum mechanics and automata.
Instead, however, I want to make the connection with computation
a different way. I want to point out the Heyting algebra
connection. Bob Coecke has shown in "Complete lattices
represent complete Heyting algebras (or: quantum logic with an
intuitionistic implication)" that there exists a construction
that embeds the projection lattice into a complete Heyting
lattice with true operational disjunction. Although Piron showed
that the meet was a true operational (semantic) conjunction, it
is not true for the join on the projection lattice to necessarily
be an operational (semantic) disjunction, and this construction
builds the minimum completion of disjunction on the states
(showing in the process that posets may generally have such
completions in Heyting algebras). The construction introduces
an operational resolution which has been used to demonstrate
dualities between quantales; in fact each operational resolution
factors as a closure operator and a lattice isomorphism and
generalises the duality between states and properties. The
operational view is also known as the convex approach in quantum
logic because of the stress on the subspace structure of physical
states, and is an order theoretic interpretation of what Haag and
others viewed only in terms of linear orders.
There is a structure known as a petri net which describes a model
for linear logics. A petri net also generates a quantale.
Linear logics are used to describe computation in dynamically
changing environments (cf. Samson Abramsky's "Computational
interpretations of linear logic").
I would love at this point to make some connections of this with
Joyal's work as mentioned in John Baez's week 202 posting of
"This week's finds in mathematical physics" on sci.physics, but
I have some other things to work on right now. But I think some
of this already start to cast some of the other points made
elsewhere about the fascinating logical connections found in
quantum mechanics. Certainly, there is something very
"intuitive" hidden in here, and it has a lot to do with
information.
mitch
found numerous papers of interest.
galathaea wrote:
I need to learn about this notation. If what I have read is correct, it reflects certain
relationship with maximal Boolean sublattice blocks.
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
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.
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.
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}
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?)
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.
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.
:-)
mitch
galathaea
: Thank you for this post, Galathaea. I have run a number of searches
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.
mitch
galathaea wrote:
I think this is in all logic one way or another. It took a lot of work to
discern its presentation in classical logic. Of course, no one else knows what
I am talking about. :-)
> 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).
Reciprocalities?
Since the people who disagree with Russell's logicism are often familiar with
Kant's distinction between mathematics and logic, then one might look at his
categories of relation. Everyone looks at counting and tries to make that the
basis of arithmetic. But, if 2 and 3 "exist," then they exist independent of a
counting order. This is a reciprocal relation.
Quantum mechanics captures this by representing a partition across the first two
quantum numbers. Without regard for physical intutions, the number 3 would be
understood in relation to its partitions,
(0,3)
(1,2)
(2,1)
(3,0)
These are the geometric intuitions that are lost in logicism.
>
> : 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...
>
> =)
>
Go Ozzie. Black Sabbath was one of the more memorable concerts of my youth.
And, I never saw any rock-n-roll artist manipulate an audience so skillfully.
I learned about the word wizard from a Mathematical Games article published in
Scientific American sometime in the early 1970's. But, there was more in that
article than just the symmetry in wizard.
There were two "shifts" discussed in the article:
IBM ---> HAL
NY ---> OZ
The IBM context had been taken from Arthur C. Clarke's "2001:A Space Odyssey."
The NY context came from a curious fact about the "Wizard of Oz." Apparently,
the author had been from New York.
If you look at these you will see that they each shift the alphabet by one
letter in opposite directions. They are opposite in the sense that the mappings
are rigid presentations with respect to an actuality and a fiction.
The mappings are really of the form
n ---> n-1
n ---> n+1
But, of course, they are in a finite list.
If you look closely, you can see that there is another curiosity here.
B-M N-Y
| |
| |
A-L O-Z
If we ignore the 'I' from 'IBM,' then two mappings partition the alphabet down
the middle.
Somehow, I think I see this 2-3/3-2 reciprocality in the alphabet. Information
theory is becoming far more real for me than physics. :-)
The sad thing is that the alphabet somehow seems to be involved in completely
inexplicable ways. On alt.philosophy I offered immortalist some mathematical
facts about 26,
-----
There are 26 sporadic finite groups.
There are 26 Heyting algebras generated by finite projective formulas for
2-universal models.
There are 26 dimensions associated with the unimodular Lorentzian lattice in
which string theorists are interested.
-----
And given that my stated interest has always been the foundations of
mathematics, I went to my Bible the other day and looked at The Book of Numbers:
"The Book of Numbers derives its name from
the account of the two censuses of the Hebrew
people taken, one near the beginning and one
near the end of the journey in the desert (chapters
1 and 26)."
So, whatever influence that article had on me as a twelve-year old, I am
strongly of the opinion that the apparent idiocy of my presentation is a
manifestation of information-theoretic statistical entropy. But, for the
record, I did start out with formal axioms. And, from my perspective, a lot of
people are talking in circles but claiming otherwise.
>
> : 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.
>
I don't really like this notation because I immediately see inclusion and
identity. But, yes.
Your professor's statement reflects some measure of symbolic complexity. These
days, I might begin to ask for a characterization of the inclusive disjunction
connective. Clearly, it is not a complete connective. So, it is derivative.
Now, given its representation in terms of a complete connective, it is
ambiguously defined. Indeed, both NAND and NOR are complete and present
different representations of inclusive disjunction. So, now I would ask which
complete connective presents the *canonical* representation of inclusive
disjunction.
There is nothing simple about the definition of an unordered pair except the
presuppositions on which it is founded.
>
> : 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.
>
That is beyond me.
>
> : 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.
My admiration of Kant comes primarily from the fact that he asserted mathematics
to be different from logic. This had been something I concluded on my own
(although undoubtedly influenced by an earlier reading of "Critique of Pure
Reason" that I had almost entirely forgotten). Since I knew I was considering
things outside the usual paradigm of first-order logic, I had been searching for
philosophical justification. Had I understood just how much his work has been
misunderstood and misrepresented, I probably would have looked a little
further. But, the arguments would have been the same. At least I can still
claim that nothing about my mathematical intuitions derives from
intuitionism--yet, I have similar conclusions. :-)
> 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.
Yes.
>
> 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.
Yes.
Now, what is the psychology of that? There is a real sense that our approach to
education is reducing our knowledge base to simple variations on a handful of
geometric forms.
>
> : 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.
>
I did not get many hits on a Google search here. But, I am going to guess that
one can piece together some relation to the work done by Halmos in algebraic
logic. The existential quantifier and constants are associated with mappings.
The one paper that had some explanation of Mitchell-Benabou suggested that the
morphisms were interpreted as typed terms.
>
> : 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 probably cannot fix that. If you have questions, please ask. Maybe I can
explain why I jump around so quickly and carelessly.
If you look at that paper discussing the non-Boolean model for classical logic,
you will see that it is couched in a discussion of equivalences. Moreover,
these are in a lattice--an algebraic structure.
In 1986, I walked into a professor's office and tried to explain that the
continuum hypothesis was independent because of a problem with the identity
relation. I tried to claim that "an equivalence attachment" was needed. I now
realize that I was talking about Frege's identity puzzles. I was resolving them
by declaring that distinctness was mathematically understood relative to the
separation axioms in point-set topology. My reasoning was based on descriptive
set theory. The axiom of determinacy is not compatible with the axiom of
choice.
Of course, I did not know any of those issues at the time. Within two weeks I
was in the hospital as my "rigidly designated vocabulary" (all those
definitions!!!) started reorganizing itself. So, yes, I am not just a newsgroup
crackpot. I am a certified crackpot.
But, my subsequent work indicates that I had the right idea. It took me three
years to formalize a set of sentences that expressed my ideas. Central to the
construction was a distinction between "the characteristic equivalence of the
language" (also known as "the notion of well definition") and "the notion of
identity." In fact, I was claiming that identity in the sense of mathematics
was determined by "algebraic semantics" whereas identity in the sense of a
first-order language--that is, identity as a formal symbol of the language--was
a grammatical device needed for introducing constants and functions of the
language. When I finally learned TeX and wrote a paper, I called it "The Formal
Description of Identity."
So, I have eighteen years thinking about mathematics from a non-standard point
of view. In fact, it would probably be accurate to say that I do not even
understand mathematics the same way as other people.
I knew my ideas were non-standard and I knew I would have to understand how it
related to set theory. So, I spent many years studying set theory and comparing
structures in set theory with structures in other branches of mathematics. I
was not interested in set models since I recognized the infinite regress they
implied. So, I studied Goedel's constructible universe and Cohen forcing
carefully.
I had become convinced of a topological interpretation early on. Kuratowski's
closure axioms do not refer to a universal class and the transitive classes
(among which are the transitive closures uniquely associated with each element)
satisfy those axioms. So, every model of set theory is a topology.
From my perspective, the failure to investigate topological invariance as it
applies to models of set theory is a sad testament to the state of affairs in
foundational mathematics. It is like your professor who said that the
definition of an unordered pair is necessarily simpler.
Formulating a coherent picture of all of this has not been easy. Unfortunately,
it seem that this has resulted in a "private language."
Right. Good direction.
> 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).
Well, I don't know anything about quantum geometry. I have considered something
along the lines of localized topological features associated with a particle,
however.
Suppose I have a standard three-dimensional coordinate system. I want to take
the xz-plane as a complex plane with the positive z-axis as the positive
imaginary axis. I want to take the yz-plane as a complex plane with the
negative z-axis as the positive imaginary axis. The "points" in this space are
the directed line segments joining two complex coordinates.
That's how it starts, at least. I have not thought about the details for a long
time. The problem, of course, is that it would never lead to numerical
results....
We can only hope. One of the benefits of democratic republics is the general
health that accrues to the entire population.
One of my concerns with this, however, is that our economic theories are
directed toward minimizing risk and business models are directed toward
optimizations. As we reduce our choices, we evolve toward singleton action
schemas while proclaiming competition.
>
> 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.
I suppose that explains why I am beginning to consider belief systems and game
theory more central to questions of free will than physics these days. :-)
>
> 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.
>
For the most part, I am not too hostile. Of course, being flamed for a year did
nothing to make me less so. But, except for one or two particular weeks, I kept
my composure. The reality is that most scientists are not in decision-making
positions. So, they are much more worried about the kids and the mortgage.
:-)
mitch
mitch
galathaea wrote:
> 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 am not so sure it is worth the effort, Galathaea. As George once noted, the
"ability" to discern meaningful patterns among large data sets is associated
with mental illness. I have a diagnosis, and, prudence would dictate that that
is the relevant fact here.
:-)
mitch
George Greene
: : I guess I cannot quite understand what people mean by an
: : "unordered pair."
What utter idiocy.
Consider your parents.
Which of them is "first"?
mitch
George Greene wrote:
George,
That was me who wrote that. :-)
You really do not want to get into a discussion about haploid females
here, do you?
:-)
mitch
George Greene
That's not the point. The point is that YOU don't want to
get into that discussion. That discussion is NOT relevant
to the question. Who is or isn't haploid has nothing to do
with who is or isn't FIRST.
One could go on.
Consider a pair of shoes. Which of them is FIRST?
In math this problem goes away for pairs with roles;
you could always say that the left shoe is first
or that your father is first.
Therefore, consider a pair, NOT of shoes, but of SOCKS.
Which one is FIRST?
My point is simply that claiming that you don't know
what an unordered pair is IS STUPID. OF COURSE you
know what an unordered pair is. The fact that strings
get presented linearly does NOT change that. Indeed,
it is arguably a major FLAW of the string paradigm that
it IMPOSES order, ARBITRARILY.
--
--- The history of our nation has demonstrated that separate is seldom, if ever, equal.
--- (Feb.3,2004) Supreme Judicial Court of Massachusetts (4-3), adv.Sen.#2175
mitch
George Greene wrote:
I like the new signature quote.
I will agree with you, however. I could have chosen a much more complicated way of
expressing the ambiguity.
:-)
mitch
galathaea
: galathaea wrote:
: > 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).
:
: Reciprocalities?
:
: Since the people who disagree with Russell's logicism
: are often familiar with Kant's distinction between
: mathematics and logic, then one might look at his
: categories of relation. Everyone looks at counting
: and tries to make that the basis of arithmetic. But,
: if 2 and 3 "exist," then they exist independent of a
: counting order. This is a reciprocal relation.
:
: Quantum mechanics captures this by representing a
: partition across the first two quantum numbers. Without
: regard for physical intutions, the number 3 would be
: understood in relation to its partitions,
:
: (0,3)
: (1,2)
: (2,1)
: (3,0)
:
: These are the geometric intuitions that are lost in logicism.
Its funny how you explain that so similarly to the way I would have... =)
Physicists tend to divide weights in their Lie algebra representations by 2
because that gives them something closer to the idea of spin they had
developed prior to fixing the mathematical study. However, the 3rd weight,
which is 2 (from 0, 1, 2) and corresponds to a spin 1 wavefunction whose 3
spin component possibilities are (-1, 0, 1), seems to capture something
important about counting and collecting in quantum spaces. One arrives at
the same component possibilities when one takes 2 spin 1/2 particles
(electrons, say) and couples them. The collection process just on its own
is degenerate or ambiguous to a singlet and triplet, as:
|1, 1> = ^^
|1, 0> = 1 / sqrt(2) (^v + v^)
|1, -1> = vv
|0, 0> = 1 / sqrt(2) (^v - v^)
This is general to spin in all directions and corresponds to the
discretising perspective properties of quantum spaces. This can be
generalised to many collections of spin properties through Clebsch-Gordan
coefficients, where all you really are studying is the decomposition of
tensor products of modules, but it all starts with the simplest collection
identities found in the 2 spin 1/2. This gives the algebraic collection of
these degeneracies a purely geometrical structure and show how a strange
relationship forms between the numbers 3 and 2 in the most fundamental step
of the collection process: going from 1 thing to 2 of them.
The relationships between 3s and 2s, like between 1s and 4s, is also very
common in many folk tales, mythologies, etc. Dualities and trinities
abound. I would guess that has to do with 5 fingers on a hand and the
natural patterns we can make breaking that apart, but also some of the same
patterns arise. Take for instance that geometrical figure the pentagram
(the complete graph K5), which has been a source of human fascination
throughout history. There are two common methods of displaying this figure
along the plane of verticle symmetry in our vision: what one mythology calls
"one horn ascending" and "two horns ascending" (which are mirror images of
each other through the visual horizon, splitting the figure into a side with
2 and a side with 3). Again the same pattern between 2 and 3 and 5, and you
can also find a (1, 4) in there (vertices are / not on a symmetry plane.
And if you look reeel hard, you'll start to notice that the ideas of two
spin 1/2 (two state -- up or down) particles being collected can be
graphically represented on this figure. The use of v and ^ in the above to
indicate individual spin states even visually connect to the patterns of the
horns ascending / descending, but there are games you can play as well...
Shiva has three eyes, but only two kundulas in the ears. Yet the two
kundulas have names -- Alakshya ("without any possible sign or symbol") and
Niranjan ("unseen") -- which refer to three eyes through the duality of the
masculine / feminine. Then again, shiva has a five headed appearance and
has five purposes or acts to progress the cycle of existence.
2, 3, 5, 3, 5, 2, 1, 4, ...
People get fascinated with these relationships as they play out. We see
connectives, forming patterns. Some of them are related to the arithmetical
relationships of collecting 1 + 4 = 2 + 3 = 5. Some of them are graph
theoretical. All of them are topological.
: > : 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.
And they show up in the strangest places! Patterns everywhere. Connections
everywhere.
And we humans stare.
Because we know we'll see those patterns when we stare enough. That's what
we do so well. Not all patterns give us very good information on which to
make predictions because not all patterns have anything to do with
predictions and the future. Some patterns are only appreciated for
aesthetic reasons, but ther is almost always a level of abstraction where a
pattern becomes useful.
40,000 years ago, humans started drawing things like
/\/\/\/
/\/\/\/
/\/\/\/
and
oO O
O Oo
o
all over the place.
We began to project all of our entoptic phenomena back out to the world,
sharing our visual patterns.
Although we likely had mathematics from very early on in the birth of our
utterance languages, it seems likely that it was limited by memory to very
humble amounts of arithmetic and what could be conveyed by pebbles and the
like (das glasperlenspiel). However, around the time the visual designs
started appearing, we also find a growth in tally markings and computational
games.
[...]
: I learned about the word wizard from a Mathematical
And thats the "secret". The patterns _are_ real. In fact, they _are_ the
reality in experience as we can identify it.
Only the things we pattern can we name.
Physics asks for a different thing. They ask for models of causation. They
want dynamics.
They want logic.
Patterns are prior to that. They are the abstractions and connections we
use to build the structure of science. The justification in patterns is
that we see them, but science requires that we see them repeat.
: The sad thing is that the alphabet somehow seems
: to be involved in completely inexplicable ways.
: On alt.philosophy I offered immortalist some
: mathematical facts about 26,
:
: -----
: There are 26 sporadic finite groups.
:
: There are 26 Heyting algebras generated by finite
: projective formulas for 2-universal models.
:
: There are 26 dimensions associated with the
: unimodular Lorentzian lattice in which string
: theorists are interested.
: -----
:
: And given that my stated interest has always been
: the foundations of mathematics, I went to my Bible
: the other day and looked at The Book of Numbers:
:
: "The Book of Numbers derives its name from
: the account of the two censuses of the Hebrew
: people taken, one near the beginning and one
: near the end of the journey in the desert (chapters
: 1 and 26)."
William Burroughs was fascinated with the number 23. It started with this
incidence with boats and planes and crashes, linking all of these strange
coincidences about the names of the captains and the number 23 attached to
the various voyages. He noticed it one day passing through his information
context and noted it. As he noted more occurances of the strange number 23
(humans have 23 pairs of chromosomes, for example). Others, like the
Discordians and Robert Anton Wilson have continued his tradition.
: So, whatever influence that article had on me as a
: twelve-year old, I am strongly of the opinion that
: the apparent idiocy of my presentation is a
: manifestation of information-theoretic statistical
: entropy. But, for the record, I did start out with
: formal axioms. And, from my perspective, a lot of
: people are talking in circles but claiming otherwise.
The idiocy is that most people pretend so loudly they do not see the
patterns. I find it very rude for the patterns, who have done so much for
us. Our behavior towards them seems one of fear or snobbish avoidance.
They have never hurt anyone, and it is clear they are our betters.
: > : I guess I cannot quite understand what people
So it might boil down to the fact that distinction requires asymmetry, and
you need a distinction to pair?
=)
[... snipped stuff I need more time on ...]
: > It has been only a partial response to your
[and from elsewhere...]
: I am not so sure it is worth the effort, Galathaea. As
: George once noted, the "ability" to discern meaningful
: patterns among large data sets is associated with mental
: illness. I have a diagnosis, and, prudence would dictate
: that that is the relevant fact here.
I had a schizogenic experience when I was 12 due to abuse from my alcoholic
stepfather (also a Vietnam vet and a hunter with many guns in the house). I
was walking around for several weeks seeing auras around people and hearing
voices. It flared one night so badly that I thought it would be my very
last night, and there came this point where I felt the only choice I had,
what little freedom I had against the determinism of this chaos, had
narrowed down to some unspeakble question of paths. Of ways to look at the
world. I felt absolutely certain at that point that if I made the wrong
decision, I would dissolve into the chaos and was dead as an individual.
That was never diagnosed, but the years of depression afterwards eventually
were, and I was treated for severe depression much later.
So if were setting mental illness as our cautionary tales, it's probably
important that I point out the need to keep an eye on me as well. But then
again, I have also spent a lot of my obsessive attention on trying to
understand my psychology, and have found a lot of commonalities in these
attention disorders with characteristics of many of the people I admire.
People I consider some of the most creative, most intelligent, most
in_sight_ful, presented very similar symptoms in their life. So I just see
it as one of those sad, bittersweet things about life that seeing many
patterns is often a response to trauma.
: But, my subsequent work indicates that I had the right
: idea. It took me three years to formalize a set of
: sentences that expressed my ideas. Central to the
: construction was a distinction between "the
: characteristic equivalence of the language" (also known
: as "the notion of well definition") and "the notion of
: identity." In fact, I was claiming that identity in the
: sense of mathematics was determined by "algebraic
: semantics" whereas identity in the sense of a first-order
: language--that is, identity as a formal symbol of the
: language--was a grammatical device needed for introducing
: constants and functions of the language. When I finally
: learned TeX and wrote a paper, I called it "The Formal
: Description of Identity."
Yeah. In case I never mentioned it before, I did read it. It was my first
fuller-length introduction to the way you think...
: So, I have eighteen years thinking about mathematics
: from a non-standard point of view. In fact, it would
: probably be accurate to say that I do not even
: understand mathematics the same way as other people.
Or at least the way other people admit. I still see an avoidance reaction
there. An unacknowledged timidity.
: I knew my ideas were non-standard and I knew I would
: have to understand how it related to set theory. So,
: I spent many years studying set theory and comparing
: structures in set theory with structures in other
: branches of mathematics. I was not interested in set
: models since I recognized the infinite regress they
: implied. So, I studied Goedel's constructible
: universe and Cohen forcing carefully.
:
: I had become convinced of a topological interpretation
: early on. Kuratowski's closure axioms do not refer to
: a universal class and the transitive classes (among
: which are the transitive closures uniquely associated
: with each element) satisfy those axioms. So, every
: model of set theory is a topology.
:
: From my perspective, the failure to investigate
: topological invariance as it applies to models of set
: theory is a sad testament to the state of affairs in
: foundational mathematics. It is like your professor
: who said that the definition of an unordered pair is
: necessarily simpler.
:
: Formulating a coherent picture of all of this has not
: been easy. Unfortunately, it seem that this has resulted
: in a "private language."
[...other stuff for another day...]
When I see people on some of the newsgroups attacking (like Franz was doing
about quantum groups), actually pretending like no such patterns exist even
when they are spelled out rigorously in our common formalised language of
mathematics, I cannot fathom blaming those who have taken the time to try an
explanation of the patterns they have seen. I think of any mental disorder
that I would call a sickness, it would be this attacking psychology of
avoidance.
mitch
galathaea wrote:
> "mitch" wrote:
>
> : But, my subsequent work indicates that I had the right
> : idea. It took me three years to formalize a set of
> : sentences that expressed my ideas. Central to the
> : construction was a distinction between "the
> : characteristic equivalence of the language" (also known
> : as "the notion of well definition") and "the notion of
> : identity." In fact, I was claiming that identity in the
> : sense of mathematics was determined by "algebraic
> : semantics" whereas identity in the sense of a first-order
> : language--that is, identity as a formal symbol of the
> : language--was a grammatical device needed for introducing
> : constants and functions of the language. When I finally
> : learned TeX and wrote a paper, I called it "The Formal
> : Description of Identity."
>
> Yeah. In case I never mentioned it before, I did read it. It was my first
> fuller-length introduction to the way you think...
>
Or lack thereof. :-)
The real tie in to what I wrote there was he "visual" interpretation of the
circularly-defined predicates. I can now say that this is an interpretation
treating membership as a generalized incidence relation. That is why the design
theory pops up.
Moreover, it is related to the concept of determinacy via what is called
descriptive set theory.
And, the logic collapses to a linear order because it is probably more closely
to the category-theoretic dynamical truth object than to anything in our usual
notions of set theory.
You should tack on the axioms,
Ea Ab(
~( b proper_subset a <-> a = b ) )
Ea Ab(
~( a proper_subset b <-> b = a ) )
Aa(
Eb(a proper_subset b )
->
Eb( a in b ) )
to give you extremal terms (use part/subset to interpret the universal
quantifier--that is how the system was designed). Then the three statements
above give you an extensional domain as a consequence
Ea Ab(
~(b in a <-> a = b) )
So, what might this be relevant to? Well, someone in set theory might ask why
they should accept the possibility of models that are not hereditarily
ordinal-definable since the "boundary" between intensional and extensional
interpretation appears to require linear ordering of referents.
But the question is stupid since one would have to define a whole new
mathematics....
Keith Ramsay made some excellent remarks on sci.math today. You might look at
them. Thanks for reading the paper.
:-)
mitch