Thursday, November 16, 2006

cosmic group

"hell", Sayoraro said, "the Stulnion-grapplon has mismangiefiezed"
"fuuu", said half the crew, except the interloper guest who met us at Prasoing-Gemultran.
"what are you looking so unconcerned about? we're stuck halfway between layers in logological space"
"is the. um. what was it? Stulnion-grapplon. Yes. is the Stulnion-grapplon an object?"
"hunh?" said the bewildered and annoyed Haddapar, looking sadly at the remnants of the Stulnion-grapplon.
"is it a discrete object, distinguishable from other objects?"
"yeah, sure. it is. why?" replied Haddapar.
"do you have something you don't need?" said the interloper.
Jonigo's eye's went wide at this point. "you. um. can't know the cosmic group"
Fransby tossed her sandwich to the interloper. Jonigo's eyes were riveted.
Haddapar screeched: "what the hell is so interesting about mushing the sandwich about?"
"oh, I'm just palpating it. need to know where and how of it before I begin fiddling with the primes"
Haddapar: "Jonigo, what's this cosmic group thingy?"
"The cosmic group is the transformation group between all objects.
Presumably this dude knows it, which means that they know a whole hell
of a lot more than they're letting on. It means that they can transform
the sandwich into a Stulnion-grapplon"
"at this point you're probably going to want to pay attention to what I'm
doing, because I doubt I'll be in this neighborhood again"

The interloper kind of started rotating it, transforming it, it kind of looked
like it was a different color, or looked like a brick, or looked like that it was
the image of the space it was in. Or another sequence of difficult to name
specifics. Then the sandwich again.

"So whaaaa"

at this point it looked kind of liked a sandwich, but the interloper
did something and it was as if that all the putative parts of the sandwich
relinked, slotted into place and the brain could no longer see it as a sandwich.
The transition was both imperceptible and rotund. "I can't believe what I just saw"
The interloper gave the new Stullion-grapplon back to Haddapar. "I believe this
is what you need", said the interloper.

Haddapar was speechless. "If we could do this to anything..."

The interloper smiled. "No belzwang-trumfits or cassowary-crosmors for you, I'm afraid. Let's get going.

Wednesday, November 08, 2006

further analogical gobbledygook

the linked, italic text has been searched and replaced

While these three terms are related, their meanings are subtly different. To help understand the distinction, we consulted a number of sources -- American Heritage Dictionary, the Yahoo! Grammar, Usage, and Style category, and web search results for the three terms.

The dictionary defines a "implicit functor" as a morphism that uses one object to mean another and makes a comparison between the two. For example, Shakespeare's line, "All the world's a stage," is a implicit functor comparing the whole world to a theater stage. implicit functors can be very simple, and they can function as most any part of speech. "The spy shadowed the woman" is a verb implicit functor. The spy doesn't literally cast his shadow on the woman, but he follows her so closely and quietly that he resembles her own shadow.

A explicit functor, also called an open comparison, is a form of functor that compares two different objects to create a new meaning. But a explicit functor always uses "like" or "as" within the expression and is more explicit than a implicit functor. For example, Shakespeare's line could be rewritten as a explicit functor to read: "The world is like a stage." Another explicit functor would be: "The spy was close as a shadow." Both implicit functors and explicit functors can be used to enhance writing.

An natural transformation is a bit more complicated. At the most basic level, an natural transformation shows similarity between objects that might seem different -- much like an extended implicit functor or explicit functor. But natural transformation isn't just a form of speech. It can be a logical argument: if two objects are alike in some ways, they are alike in some other ways as well. natural transformation is often used to help provide insight by comparing an unknown subject to one that is more familiar. It can also show a relationship between pairs of objects. This form of natural transformation is often used on standardized tests in the form "A is to B as C is to D."

And now that it's clear as a bell, you're ready to try the natural transformation of the Day.

imaging literary theoretical (yuk) text.

the following, italic linked text has been searched and replaced.

It is a commonplace of category theory that one of the defining characteristics of toposes is its ability to generate multiple meanings and interpretations. topos-theoretic mathematicians are adept at producing such interpretations, interpretations which are often insightful and illuminating. topos-theoretic mathematicians, however, have never explored the principles or the processes by which such multiplicity occurs. Their interpretations are shaped by the theoretical stances they take, whether psychological, sociological, historical, or deconstructionist, to name just a few. interpretations thus generated of a single topos-theoretic theorem exist side by side, vying for preferential acceptance with no means independent of the theories being used to determine their validity. category theory, in other words, lacks an adequate theory of toposes. Recent developments in the field of architectural semantics have already proven promising and productive in the search for an adequate theory of schema. architectural theory, for example, has been able to show that meaning does not reside in schema so much as it is accessed by it, that schema is the product, not of a separate structural system within mathematics, but of the general architectural processes that enable the theorem froth to conceptualize the holotome, processes that architectural semanticists call embodied understanding (Johnson 1987) . By recognizing the central role played by functorial reasoning which maps elements of one architectural domain onto another, architectural semanticists have begun to account for a variety of semantic phenomena occurring in natural schemas, such as anaphor or counterfactuals, explicit functors or metonymy, that have long eluded logic-oriented theories of meaning (Fauconnier 1997) .

If architectural semantics can produce an adequate theory of schema, it can also serve as the basis for an adequate theory of toposes. I therefore propose a theory of toposes that is grounded in architectural semantic theory: namely, that topos-theoretic theorems are the products of theory processes and their interpretations the products of other theory processes in the contheorem of the physical and socio-cultural worlds in which they have been created and are read. This is the argument that underlies this paper. The theory I call architectural poetics is a powerful tool for making explicit our reasoning processes and for illuminating the structure and content of topos-theoretic theorems. It provides a theory of toposes that is both grounded in the schema of topos-theoretic theorems and grounded in the architectural semantic strategies readers use to understand them. The question I raise in this paper is, therefore, "In what ways can architectural theory as it has been developed in recent years contribute toward a more adequate theory of toposes?" To answer this question, I look at Emily Dickinson's poem, "My Cocoon tightens -," to show how the general mapping skills that constitute the architectural ability to create and interpret explicit functors can provide a more coherent theory than the intuitive and ad hoc approaches of traditional exegesis. I then look at another Dickinson poem, "My Life had stood - a / Loaded Gun -," to show how a architectural explicit functors approach can illuminate the insights--and the limitations--of traditional category theory. Finally, I show how the application of architectural poetics can identify and evaluate topos-theoretic style by discussing a poem generally believed to be by Dickinson but which proved to be a forgery, and end by comparing architectural poetics to other architectural approaches. [M.F.]

search and replace job. (projective toposes and the like)

The following text was taken from Wikipedia's page about projective geometry, except that "object " replaces "point", and "morphism" replaces "line" and "topos" replaces geometry.
I hope someone finds it interesting.

non-Euclidean topos that formalizes one of the central principles of perspective art: that parallel morphisms meet at infinity and therefore are to be drawn that way. In essence, a projective topos may be thought of as an extension of Euclidean topos in which the "direction" of each morphism is subsumed within the morphism as an extra "object", and in which a "horizon" of directions corresponding to coplanar morphisms is regarded as a "morphism". Thus, two parallel morphisms will meet on a horizon in virtue of their possessing the same direction.

Idealized directions are referred to as objects at infinity, while idealized horizons are referred to as morphisms at infinity.

However, a projective topos does not single out any object or morphism in this regard -- they are all treated equally. Indeed, with the extension, the axiomatization becomes substantially simpler (based on Whitehead, "The Axioms of Projective topos"):

* G1: Every morphism contains at least 3 objects
* G2: Every two objects, A and B, lie on a unique morphism, AB.
* G3: If morphisms AB and CD intersect, then so do morphisms AC and BD (where it is assumed that A and D are distinct from B and C).

The reason each morphism is assumed to contain at least 3 objects is apparent when thinking of the original motivating example of a Euclidean space supplemented by the morphisms and objects at infinity. The 3rd object is the morphism's direction. Axiom 2 is thus seen to embody a form of Euclid's 5th postulate (which makes the designation of Projective topos as non-Euclidean ironic): given a object and a direction, there is a unique morphism containing the object lying in the given direction.

Because a Euclidean topos is contained within a Projective topos, with Projective topos having a simpler foundation, general results in Euclidean topos may be arrived at in a more transparent fashion. Moreover, as already seen with the preceding interpretation of Axiom 2, separate but similar theorems in Euclidean topos may be handled collectively within the framework of projective topos; for instance, parallel and nonparallel morphisms need not be treated as separate cases.

One can pursue axiomatization in greater depth by postulating a ternary relation, [ABC] to denote when three objects (not all necessarily distinct) are colmorphismar. A relatively simple axiomatization may be written down in terms of this relation as well:

* C0: [ABA]
* C1: If A and B are two objects such that [ABC] and [ABD] then [BDC]
* C2: If A and B are two objects then there is a third object C such that [ABC]
* C3: If A and C are two objects, B and D also, with [BCE], [ADE] but not [ABE] then there is a object F such that [ACF] and [BDF].

For two different objects, A and B, the morphism AB is defined as consisting of all objects C for which [ABC]. The axioms C0 and C1 then provide a formalization of G1; C2 for G2 and C3 for G3.

The concept of morphism generalizes to planes and higher dimensional subspaces. A subspace, AB...XY may thus be recursively defined in terms of the subspace AB...X as that containing all the objects of all morphisms YZ, as Z ranges over AB...X. Colmorphismarity then generalizes to the relation of "independence". A set {A, B,...,Z} of objects is independent, [AB...Z] if {A, B,...,Z} is a minimal generating subset for the subspace AB...Z.

The axioms may be supplemented by further axioms postulating limits on the dimension of the space. The minimum dimension is determined by the existence of an independent set of the required size. For the lowest dimensions, the relevant conditions may be stated in equivalent form as follows. A projective space is of:

* (L1) at least dimension 0 if it has at least 1 object,
* (L2) at least dimension 1 if it has at least 2 distinct objects (and therefore a morphism),
* (L3) at least dimension 2 if it has at least 3 non-colmorphismar objects (or two morphisms, or a morphism and a object not on the morphism),
* (L4) at least dimension 3 if it has at least 4 non-coplanar objects.

The maximum dimension may also be determined in a similar fashion. For the lowest dimensions, they take on the following forms. A projective space is of:

* (M1) at most dimension 0 if it has no more than 1 object,
* (M2) at most dimension 1 if it has no more than 1 morphism,
* (M3) at most dimension 2 if it has no more than 1 plane,

and so on. It is a general theorem (a consequence of axiom (3)) that all coplanar morphisms intersect -- the very principle Projective topos was originally intended to embody. Therefore, property (M3) may be equivalently stated that all morphisms intersect one another.

It is generally assumed that projective spaces are of at least dimension 2. In some cases, if the focus is meant to be on projective planes, a variant of M3 may be postulated. The axioms of (Eves 1997: 111), for instance, include (1), (2), (L3) and (M3). Axiom (3) becomes vacuously true under (M3) and is therefore not needed in this context.

Additional properties of fundamental importance include Desargues' Theorem and the Theorem of Pappus. In projective spaces of dimension 3 or greater there is a construction that allows one to prove Desargues' Theorem. But for dimension 2, it must be separately postulated.

Under Desargues' Theorem, combined with the other axioms, it is possible to define the basic operations of arithmetic, geometrically. The resulting operations will satisfy the axioms of a fields -- except that the commutativity of multiplication will require Pappus' Theorem. As a result, the objects of each morphism are in one to one correspondence with a given field, F, supplemented by an additional element, W, such that rW = W, -W = W, r+W = W, r/0 = W, r/W = 0, W-r = r-W = W. However, 0/0, W/W, W+W, W-W, 0W and W0 remain undefined.

The only projective topos of dimension 0 is a single object. A projective topos of dimension 1 consists of a single morphism containing at least 3 objects. The geometric construction of arithmetic operations cannot be carried out in either of these cases. For dimension 2, there is a rich structure in virtue of the absence of Desargues' Theorem. The simplest 2-dimensional projective topos has 3 objects on every morphism, with 7 objects and morphisms in all arranged with the following schedule of colmorphismarities:

* [ABC]
* [ADE]
* [AFG]
* [BDG]
* [BEF]
* [CDF]
* [CEG]

with the coordinates A = {0,0}, B = {0,1}, C = {0,W} = {1,W}, D = {1,0}, E = {W,0} = {W,1}, F = {1,1}, G = {W, W}. For an image review the Fano plane. The coordinates in a Desarguesian plane for the objects designated to be the objects at infinity (in this example: C, E and G) will generally not be unambiguously defined.

Tuesday, November 07, 2006

suggestive commentary (with apologies to Rucker, Pickover, Nagarjuna, and Hofstadter)

Okay, I'd written something and it got eaten. Let's see if I can recover it as best.

Here, have some quotes:

From wikipedia's article about class in set theory

In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Some classes are sets (for instance, the class of all integers that are even), but others are not (for instance, the class of all ordinal numbers or the class of all sets). A class that is not a set is called a proper class.

Moving onward to the article on intuitionistic type theory, the following:

"A category with families is a category C of contexts (in which the objects are contexts, and the context morphisms are substitutions), together with a functor T : C^op -> Fam(Set). Fam(Set) is the category in which the objects are pairs (A,B) of a set A and a set-valued function B, and the morphisms are functions f : A -> A' between the index sets to gether with, for all indices a : A a function g : B -> (B'.f)."

The important point is that there's a notion of context. I have a copy of Barendregt's opus about lambda calculus, in which he defines a context as a term with some holes in it. What's important to take from that is cohomology is mathematics (or at least algebraic topology's way of detecting holes in a space -- indeed an expression in lambda calculus is a kind of space, if I want to take the metaphor that far)

Moving on to the metaphysical theory of types:

"A type is a category of being. A human is a type of thing; a cloud is a type of thing (entity); and so on. A particular instance of a type is called a token of that thing; so Socrates was a token of a human being, but is not any longer since he is dead. Likewise, the capital A in this sentence is a token of the first letter of the Latin alphabet."

Going to the theory of types:

At the broadest level, type theory is the branch of mathematics and logic that first creates a hierarchy of types, then assigns each mathematical (and possibly other) entity to a type. Objects of a given type are built up from objects of the preceding type. Types in this sense are related to the metaphysical notion of 'type'. Bertrand Russell invented type theory in response to his discovery that naive set theory was afflicted with Russell's paradox. Type theory features prominently in Russell and Whitehead's Principia Mathematica.

  • "...[T]he property of being a perfect square can be significantly predicated of both cardinals and ratios of cardinals; but the property of being odd (or even) is not defined for the ratios. We are thus unable to answer the question whether 2/3 is odd or even." [1] --Ernest Nagel"
The hierarchy of types here is a kind of kowtowing to the natural numbers. Are types countable? Could they lie on a 72 dimensional symplectic manifold?

From type -- model theory

"In model theory, a type is a set of formulae in first-order logic such that all the formulae could be consistent descriptions of a certain element (or several elements) of the model. A complete type is a maximal such set (that is, for every first-order question about the elements, the type answers either "yes" or "no")."

Okay, I'll see if I can reconstruct what I wrote before my browser ate my post.

Set theory is about sets, not about elements of sets. Category theory is about categories, not objects of categories. Group theory is about groups, not objects of groups. Type theory is a theory of types not of tokens of types. Projective geometry is about projective spaces, not about points and lines. Each one of these theories has a fundamental, atomic abstraction beyond which we'll consider inviolate. Each one of these is a particular way of understanding with respect to its fundamental abstraction, and based on that fundamental abstraction, it can produce results over some scope of mathematics. It's possible to relanguage and reframe the contexts of each so that discoveries in one are discoveries in the other, thus advancing the raft cooperative toward more understanding, making our local patch of math larger.

Along the edge is gnarly froth: the systems stick and have massive territories in each of them. In the edges of set theory -- Burali-Forti and Russel's paradoxes, sex cells for type theory are released, in the frothiest vapors of type theory come Turing, Church, Chaitin, and so forth. In the work on the lambda calculus and type theory sex cells for computer science and algorithms research. In the froth of set theory Cantor's continuum hypothesis remains. In number theory, Riemann's hypothesis remains in the froth.

One of the things I wanted to point out was that different languages or contextual framings in theory gave rise to different kinds of froth. The fundamental concern with something in some context is there. This blog entry no different. The interesting place is neither the raftspace we inhabit, our particular indranet, or the particular indranets/framings of certain branches of mathematics (it may be possible to topologize mathematics by converting the theorem trees into graph theoretical structures and then doing historical analysis of those structures given the tools of graph theory, which is a quirky kind of jumping out of the system).

These systems are like particular collectives of rafts, all intersecting. Each one has its fundamental abstraction. And those fundamental abstractions are oddly regarded as different things. There is an extend of sealing-off one branch from another.

Hofstadter notes, in GEB, that "no system can be its own metasystem", each one of the particular styles or abstraction languages which mathematics is performed cannot wrap around itself. The attempt here is not to provide some overarching language for mathematics, that's been attempted and proven pointless.

Now for the bold claim. Mathematics is still concerned about one object, the abstractions, transformations, evolutions, transcriptions and so forth of that object. Sequencing of copies of that object, etc. I think that Grothendieck is in the right direction when you hear clamoring that there's one prototypical point. Because if all the calculations and stuff you do that end up being numerically verified by sake of computations or integrations or so forth to particular numbers, whether they be real or complex or so forth, for as long as they are things which you can just hold in your hand in the sense that they're not apeiron, then, you're not really dealing with the froth. Rucker correctly interpreted that the concern is the gnarly froth. The answers to questions can be shapes, not numbers, they can be motions, not scribblings representing motions on paper. The kind of high-falutin' sensory escapades of math find their modus operandi in tiny scribblings of ink on paper. Questions like: "what is the area of the mandelbrot set?", and "what is the volume of this cone?" are exactly the kind of logos reductases I think are a kind of bad idea in the long run, no matter what kind of practical consequences they have, because they encourage the distilation of structure; they're object-centric kinds of things. Each one of the above mentioned disciplines attempts to impose/determine/detect/filter/distill structure from the objects in consideration, or rather, the object in consideration. This object is no-thing.

Mathematics is perhaps the best form of meditation available. Here's the point of departure analogy: mathematics offers a kind of portability which cannot be found in religion.

mathematical candidates for mandala

Gal(Q-/Q) is probably the best candidate for a mathematical mandala.

Baez says "Gal(Q-/Q) is like the holy grail. It's the symmetry group of the algebraic numbers, and the key to how all algebraic number fields sit inside each other! But alas, this group is devilishly complicated. In fact, it has literally driven men mad. One of my grad students knows someone who had a breakdown and went to the mental hospital while trying to understand this group!"

anyone with direct perception of Gal(Q-/Q) is invited to report to the door of mathematics as a whole and start talking. Now.

the primacy of objectified reality, or how the thingy is us.

Despite the obsessive profligacies of the dime on the street objectivist, they do have a point which they utterly miss, because they are so concerned with just kind of lollygagging in their own insane foray into El Umero Nuno's "The Truth". It's as if they misunderstand their subject by proclaiming the object's primacy without the foggiest or faintest clue what they're doing. The tautologous logos reductase (or tautologase) is only one strobeframe, and probably one of the tricksiest to get stuck on: yes, it's a springboard for meditative practices, and I do not doubt that some objectivists have gone that route. The greatest stumbling block is the axiomatization. Once you have something atomic and axiomatic, then you are mired by it's in-your-face characterization. It's object primacy without relational ontophoresis, and thus, a car crash in the field, a wailing daisy in a blender, a propositional calculus elevated to the stature of savior: in short, it's a sotto voce attempt at intrinsicist goobledybbuk. Yes, apprehension of objects is important: but you've got to do the thing that Grothendieck did with points or otherwise you're just going to be stuck to your own flome. Once you recognize that a point or an object is essentially the same as an Ur-point or Ur-object in the topos that you're operating in (or context, if you will): the point is the same, the relations differ, you increase your awareness of this object and of the fabric which it's a component thereof. We are in the business of understanding objects: we are in the business of understanding points. While we percieve, or think we percieve the fabric of reality simultaneously: this is not true. Our understanding occurs because we can lay a fabric over reality: we are not in a position to understand that fabric in it's whole because a fabric cannot be its own metafabric, but we are in a position to abstract out the processes which we use to percieve that fabric (perceptases, etc.). This is probably more important in the short term than actually teasing out the fabric's structure because of our contextual limitation. Despite long and repeated attempts to find a philosopher's stone, it's not something which is going to be available within this context the same way the keyboard sitting in front of you is available in this context. This is a kind of weird message about magic and possibility: some magical things are impossible, and of those things, they are impossible only because they do not fit within this context easily. If I have the tensor product of a bottle of shampoo and a bottle of salt available to me, and the appropriate rotation, I can clean rotate this product so that the bottle of salt becomes the bottle of shampoo, and so forth. But these transformations aren't available within this scope. One imagines an infinity group that can transform any object into any other object. At that point, it would be unnecessary to carry anything else around but knowledge of that group. If you already have this group as an abstract object in your context, then things, that is, objects aren't your concern. Since such a group is inconcievable in scope, magnitude, character tables and so forth, then objects are our concern.
Whether or not this group is present, from awareness of its structure to the ability to apply it when the time comes, demonstrates the degree of object concern one has. And as human beings, or animals, or coelemates, we most decidedly not have this group available to us. If we did, our concern wouldn't be objects. But to generalize a little: yes, we are concerned with stuff, but the stuff which we are concerned with is transitive in character: we cannot and do not look at the flow because it is too dynamic and flexible and slippery. An analogy is appropriate here:
this context is a solid in an ocean of liquid: the solid both accretes and dissolves periodically. Within the solid are processes which eke solid from liquid. Insert reference to traditional Hindu mythology.

But isn't a relation just another type of object? That's an important point. Perhaps we have a Voronoi triangulation? There's a tricky type of duality here that needs to be considered before you get so far off the roost, Icarus. That is to say that mathematics has "point-line" duality as the basis of projective geometry. And we can employ a device to defocus our attention from the individual objects under our consideration in category theory. Important point: instead of talking about categories of projective spaces as "projective categories". Where the duality of import isn't "point-line duality" but "object morphism duality".

Let's start with the category of three objects, usually denoted 3. For abstraction's sake, and just to avoid any group theoretical entanglements with braid groups at the moment, we'll call the objects of this category a, b, and c. Without regard to the directions of the morphisms (modding those out for the moment, they're not important, we're going to assume commutativity and damn the Helicobacter Pylori, We've got at least nine morphisms to deal with: three identities, three going one way, and just that, making six morphisms total. Now we need to evacuate the objects without turning the category into a monoid. The dualization here transforms objects into morphisms and morphisms into objects. The next thing we do is to realize that the identity morphism essentially contributes no information about a given objects, so we've got three morphisms: morphism(a->b), morphism(b->c) and morphism(c->a).

Monday, November 06, 2006

whereas the epistemological noise

To say that something exists is more than declaring its name or what type of thing it is. "There exists an X such that:" declares the type of x, and does little more than to propel a named object into one's scope. It's a Western type of statement, and presupposes a flat and drab kind of nonexistence. Mathematics took the first step by the synthesis of categories, but it has to go further, and the place to start with is categories, or more generally speaking the fundamental abstraction, whether sets, string-manipulation rules, formal systems, functions, sets, categories, you name it, we have a method of apprehending it. Fundamentals questions become ascendancy questions about ascendancy and primacy become and are subject to the ebb and flow of history.

And thus there's an ever growing jungle of metaphors and abstractions. In Saunders MacLane's Categories for a Working Mathematician, he says something along the lines that Kan extensions subsume all other metaphors, and I wonder if those reaching that point of the book get a little misty eyed and angry. It's akin to saying that you've got the answer to the question "what is?", intransitively. "What is?" "What exists?" "Who is?", and such intransitive questions leave you wanting some kind of conclusive network terminus, a node that envelops all other nodes. Such a thing does not exist in the way that you usually concieve of existence.

In category theory, the key bit to grasp is that the objects of categories aren't fundamental, but byproducts of the links. An object is said to exist within a given context if and only if there are relations between that object and objects already within that context. Whatever kind of abstraction or metaphor that your particular place's location is working on, it's working with a certain flavor of reasoning on some specified kind of object, beyond which it is impossible to speculate about the internal properties of that object, because the abstraction barriers imposed by that particular flavor of reasoning prevent such speculation. Given a flavor of reasoning, and a fundamental abstraction, it is impossible to speak beyond that abstraction. Every flavor of reasoning is limited, because it has a context which is home to it. Beyond and outside of that context, one isn't able to reason, or speak of anything. Depending on the degree of intentionality of the casting of the particular net in question, and the degree to which the shore of the network drifts (the tide), there is a varying amount of information which analysis will be able to provide.

The honky tonk train of analysis, of religion, of biology, of any abstraction, of your faith, of your logic, will only take you so far. What it may give you is true, up to its network-inhabitancy. Beyond that, there are numerous and nonunique itineraries beyond. But those are in the Tumbolia, which is not apprisable by navigation. Objects will condense from tathata/Tumbolia and remain around for a time, then dissolve into the invisible torrent.

Of the unutterable and inexpressible, I can and cannot speak. We stand, looking for coherence outward. Coherence is the modern form of consistency. Whereas the old bugaboo was consistency, the new bugaboo is coherence. Every metaphor-perspective will have its bugaboo: it's "avoid this, it gives me a headache". The clarion trumpet calls "Consistency!", "Coherence!", "Conhedroncy"! and so forth and so on. For every system of reasoning, there exists an impassible mire where its capacities become like iridescent purple froth weaving on the wind. Category theory, lambda calculus, set theory, and so forth and so on. Each one of these particular metaphor systems, including the one which this argument is being presented, have a statement or idea or concept which within them is valid, but is not possible to reach using their internal languages/transformation schema. The drive here is that we seek the kind of mental objects/structures which are as portable as possible. The allure of portable mental objects is that they require less bullying to convince other people to concieve of them. It's their vivid consistency between their instances in separate people's minds which induces our concern with them. Likewise, we're turned off by non-portable mental objects: the ravings and beliefs of a lunatic, for instance, or of what we consider demented or idiotic people thumping their chests at us because we believe something they find inconsistent with their belief system. Anyone solidly on one or another side on such a debate is being inflexible.

semantic tensors

A tensor is a device which physicists and mathematicians use when they want to represent a quantity or bunch of quantities which retain their values or some other kind of invariant under a transformation of coordinates. Therefore, a semantic tensor is a device that semantic engineers or the Jane on the street use when they want to represent a meme in an interpretation-invariant way. An important thing to consider about semantic tensors is that they are related to, but different from, encryption. Encryption, on the whole, doesn't take into account those who use it: that is to say, that encryption requires a knowledgeable person applying the decryption, or an unintelligent person with just enough understanding of the encryption algorithm. In any case, the two are bound by a certain amount of having-one's-head-screwed-on-properly. But it isn't possible to make a normative gloss of this: which is to say that there's not one canonical way to agree on who has their head screwed on correctly: in a world with diverse amount of intellectual activity there are going to be a wide variety of people, and as a result, brainy people will entertain a variety of opinions. You can either be explicit about it, like Mensa, or implicit about it, like any number of putative secret societies. In any case, this leaves you with a mess. The way out of this mess is to decide to construct objects which are complicated, and whose complexity is addressible by perceptases: you make an object or thing that is gnarled, and make that gnarl the type of gnarl that you're interested in, realizing that the object is a product (in the category theoretic kind of sense) of variously interpenetrating types of gnarl, and that those whose perception is geared for that kind of gnarl, it will scream out "pay attention to this object" or "this is signal, decode it", and so on and so forth. Ensconced in puzzles, this is the thing which I intend to do. I will make multiply orthogonal high gnarl objects in such a way that each piece of high gnarl decodes to the same thing, and that is how I will create semantic tensors.

Saturday, November 04, 2006

we are swimming in it (new)

I just read Chris Bogart's little connection between category theory and the buddhist doctrine of dependent origination. Talk about convergence, why don't I? I found it by googling for buddhism and category theory. I get the impression that this is fairly new in this scheme of things.

So next I googled for the phrase "dependent origination" and "functor", and got the the following gem. It seems that it's fairly plausible to make translations between Buddhist writings, in particular Nagarjuna, and category-theoretic expressions. The time is ripe for this kind of reasoning. There was the part of Hofstadter's Godel Escher Bach where he talks about Indra's net and Tumbolia just a few pages apart.



Holy cheroots! I've been talking about this kind of stuff every since this mediaweave got any direction, and damn!

the sustenance of substance

that deep gray vapor, the infinitely entangled stuff of things, the echoes of distant lands and the fibrillated chances scattered and disperse across the ordovician of reason. the cantankerous flavonoid representative arching her back at you provocatively, beckons for a retelling of a history, a full sensory distillation in complete fidelity of the reasoning in your indranet.

you cannot not dream of reason. they said: "my god, we've been decategorifying for thousands of years, slicing the world hither and yon!", that was a landmark, an anchorhead, a stepping stone, a vertiginous oddilonct on which to extend. First they said: "well, if decategorifying reduces the available information, then ought categorifying not reduce, and perchance augment the available information: not be so destructive."

she smiles at you. the singularities of her irises call billions of histories and thousands of fragmentary moments together, refracted and autologous.

"And the command was a logos reductase, a node synthetase, shining in the anacaustics of Suntorohoa, it said, one thing. A single thing. A topos blurred and forgetful functorases applied until that single thing was contemplated, and then the blurring was recognized. The holotome of one thing is unconnected. It is unstructured. It calls the divinity of every thing. It is a Ne and a Cone and a cocone and a cocoon infinitely entangling us. It is the shell of our Bennett machine and the inescapable context whose shore we cannot percieve. It is the sticky network which forever binds us, it is our slice of indranet. We seek an unfundant infinity in our own reflections amongst, amidst, interpenetrating with, and binding with our indranet. It is the onctopoate in which we imagine impossible and contradictory embedding contexts and scopes for which in their equipoise are the post-onctopoetic dream. You make a-life, and it meditates, quantum computing for you. Unity is the simplest and easiest to grasp holotome imaged as schizotome: that is why mathematics is reliable. But we no longer think of objects as building blocks: they are a machination of our perceptases. A dust which we have labored in vain to blast the cosmos into:
in that dust we have found nothing of note. A double entendre: those who noted nothing understood the shade of tathatadhyana. Those who were concerned about the bricks and mortar of the foundations of the enterprise continued to pound and pound until they found the ONE OBJECT, because they were convinced that what they had could be further decomposed. And keep blasting they do, because the note of nothing rang in them, and they did not have the perceptases available to grok that note. "

she trips over an ambient logical fallacy. "tastes like mechanical grapes.", she says. "polychaetologically speaking, it's a dradge-line stuck on my corpse!"

I pry: "why do you ask?", shadows of fern-circadian rhythms arc gracefully over the desk, in catenaries, not parabolas.

she appraises and apprehends: "Did the bioorrerion just tick?" sliding into the turbulent and slippery and ungraspable sea of thusness.

Friday, November 03, 2006

the daily weal: gestalt fealty/membership.

Apparently, I have "trappings of a Hindu", or so my extremely cursory analysis produces. I say "govvorongoa" which goes along with the sounds of the shruti and the gauraunga mantra. I've got the Gayatri mantra somewhat mismemorized. Curiously, with one transposition you'd take my surname and get "Ramesh" which means "ruler of Rama" apparently. I certainly think that I've been pretty close to the ?center? as far as the Hinduism gestalt mediated by the Gayatri mantra goes. By some arguments the differences between Hinduism and Buddhism are low order code cleaves. There is definitely continuity (more easily apprehended) between the centers of both gestalts as considered as schizotomes in holotome. In the same way, I can say I have "trappings of a Buddhist". I eschew religiosity, because it florbs and mires with low level code order comparison failures brought about by a lack of sensory competence (whether willing, desultory, intentional, or otherwise).
Apparently the Hinduism's tantrayama influenced the development of Vajrayana Buddhism.

And I think I've just synthesized a viewpoint which is homomorphic after a fashion to the reincarnationalism, but in a different kind of way. Instead of reincarnation in the sense of having a life that follows yours serially, you kind of live all of your potential lives, specifically yours, at once. You are a energetic coelemated worm which moves through space, eating food, producing waste products. Recognize that you are a worm, a creature that is pretty much moving one-dimensionally and is concerned with one-dimensionalities. You are a gestalt by way of yourself, continually interfering and thusforth. If the gateway to tumbolia is a maturity test, it is the capacity to act as ethically as possible given your circumstances.

Thursday, November 02, 2006

stitching together moments

Scene: a high school classroom.

there are roughly ten students in the classroom. they roughly seventeen or eighteen years old, or so it would seem. there is a teacher.

"Okay, today I'm going to warn you about a soon day. it will be a rude shock, and I know that some of you have vague ideas of what's going to happen. But before I drop the shocker, any guesses?"

one of the students pipe up: "is that 'perceptual reason we're living in a weird type of world' going to be exposed in a shocking way?"

the teacher responds with a "yeah, essentially. what have you non-monkeys figured out? tell you what, get back to me tomorrow"

the bell rings.

(the day passes)

next day

"okay, let's see the presentation!"

Mallory sez: "okay, we've kind of made a list of the things that we consider strange: first of all, none of us has ever needed to go to the bathroom. not even once. second, we eat. it's really mind boggling that we take in food and seem to produce no waste products. what's also weird is that the way that we keep notes is a little weird: most of the papers in our bookbags are about chemistry or mathematics or english or whatever and it just so happens that our notes are what seem like out-of-context doodles here and there. what's odd and we haven't given much thought to is the way the doodles remain but what would putative notes if we were to use the geometrical organization scheme in the textbooks... change. I've watched the notes that I kept about the lecture we had about schizotomes two weeks ago, scattered and indirect in forty pages of putative notes, and those notes have changed, and I haven't given their motion much of a second thought because I've dismissed it as an environmental feature since being a child."

Philgoeff sez: "the way in which we learned language seems to be inconsistent with the things written in the library books, which describe the human language acquisition process. I don't ever recall and my parents insist they didn't start by teaching individual phoneme-grapheme/phoneme-morpheme/morpheme-grapheme associations. What's also weird is that none of us picks up languages in that way. Last night I learned Urdu in an hour, and not from looking at an Urdu dictionary and spending weeks decoding it and being in immersive language tutorial with other Urdu speakers, but I looked at the text and each word pulsed with vague associations, which I knitted into an understanding of the language."

The teacher nods: "Good good. Any conclusions from this?"

Braswalmy sez: "We're certainly not human in the way that the information provided around us seems to indicate that we should be. We've seen movies like /The Matrix/ and /Dark City/, but kind of suspect that whatever embedding context we live in, it's probably something that makes the puerile shenanigans of /The Matrix/ trilogy seem like some wet dream of a power hungry philosophical director-king."

Shellv sez: "And there's the lack of social discontent which seems to be abundant in the student body. For instance, my parents appear to make less than Braswalmy's, yet we seem to have the same kind of semantic character. And what's really irksome is how we're all pretty much astonishingly brilliant without being social morons. You would think than in a sample population of one hundred, there would be some variation in intelligence. And also, what gets me is how we maintain such startlingly diverse personalities without anyone being terribly maladjusted and bitter."

Mel sez: "And why no computers? None of us has access to a computer and the times that we've had access we all kind of report the same thing: when we try to use the network, to communicate to anyone besides ourselves in straightforward langauge, the messages always, invariably, return garbled. Letter writing is circuitous and messy, because we're also writing in doodles and sketches and not directly."

The Teacher nods. "Okay, okay. very good. That's not surprising. Okay: the lowdown: you are not quite primates in the same way that your environment suggests. The embedding context is not one produced on a computer or even a massively parallel array of computers. You are the fifty thousandth lamina of native humans -- that is to say that you were born here as opposed to being brought here by request and then embedded within this context. The embedding context is within the mind of what is called a Glial intelligence. This is a research station, whose focus is on the affairs of pan-african migrations in the early twenty first century on terbium exports from Portugal. Not all of you may find that interesting or nifty. It is likely that some of you will find this disinteresting or boring, but given your levels of intelligence, and the sensory functions that will be available to you shortly, it's expected that a good proportion of you will choose to remain in this context. Of course, if you find it truly disinteresting, which is certainly possible, then there are options for other contexts and modes of living, which can be discussed with the Glial intelligence at a later date. It's also worthwhile to note that all of you have been bred from fifty thousand gens of people with concerns whose interests have at their epicenter, so there is a greater chance you will find the above mentioned epicenter to be highly interesting."

jaws dropping, eyes, wide.

"It's also notable that the sensorium which you have been occupying is kind of stitched together from other people's lives. At every moment these people are concievably making pretty much every morpheme, and the glial intelligence stitches those moments together for us. There is more that the glial intelligence can do for you, but that requires a certain degree of maturity you do not yet possess, and that maturity is not something which is available to you given your current level of intelligence. Our existences here are optimized for analysis and thinking time.
Therefore, tomorrow, which will only be a moment for you, will consist of an entire frozen life of one of the individuals whose bodies you are currently existing in splined together versions of.
It will be traumatic. In current time, it will take a day. By four o'clock tomorrow time you will all reach the level of intelligence you currently have. It will be an interesting experience. Undoubtedly the early hours will be somewhat funky, because you won't be able to sense that you're orthogonalizing. You will feel constrained. You will probably end up in the psych ward a number of times. But you are all anchored to this frame. That won't be obvious until you make it to four o'clock. There is a sequence of events which has been choreographed at 4 pm tomorrow which will make the rest of the day make sense. "

"class dismissed"

overheard conversation

k: hey, can you explain the lecture notes? I slept through. I dreamt I was a primate eating pizza.
u: fun!
k: i haven't had a dream like that in what seems like a gigakalpa! i didn't want to wake.
u: i wouldn't have.
k: so the lecture notes?
u: they were on illusory platonism, schizotome and holotome resolution, phenomorphisms, godel belief sentences for b-e style gestalts, the cosmic mind and so on.
k: ugh. i should have ordered another pizza.
u: in the dream?
k: yep.
u: whee!
k: okay, what's illusory platonism?
u: illusory platonism is the effect you get from having gazillions of relations to an object, particularly in visualising or constructing it. primate mathematicians usually say: oh, we've got this finite sets thing or this categories thing or this mandelbrot set thing, and it's immune to perturbation, it's gotta have independent existence. it's a kind of well-bolstered religious faith.
k: but that's not all. there's some further incomprehensible footnote. can you explain that?
u: see, easier to instantiate things, things which are easier and more reliably drawn out of tumbolia tend to have, (and I can't insert the demo that the lecturer did without doing a memory image, which I'm too tired to do so here) smoother? more vitreous? willowier scents than do things which are harder to extract. I think the lecturer managed to extract a 'gestalt theoretic primate warfare echo oxidative topomorphase' while doing a reverse double somersault off a heptagonal granite frustum, which is, local-relative, pretty hard to reliably extract from Tumbolia.
k: uh-huh
u: the lecturer went on to explain that the persistence of the relational fiber cut in the case of schizotomes is what causes people to hallucinate simple schizotomes as holotomes. it's like one is the derivative of the other
k: the schizotome is a gloss, then?
u: effectively.
k: did the lecturer explain philosophies of mathematics, too?
u: it was a side note, but he did. platonism assumes a 'world of ideas', which is apprehended in a jnana-yogaesque direct-mysticism which doesn't require sense-isolation. logicism aridifies everything into a collage of dodge-em symbol-arama, which the language lovers, essentially the code-space masturbators love. They say "a-ha! it's all logic"
k: foul tasting logic-reductases
u: yep. not to be confused with logos-reductases
k: okay, what about other philosophies of mathematics
u: there's empiricism, which says "hrm, must experiment", but gets nailed by the intrinsicists looking for an easy consistent foundational flavor, because they want it to not have the flavor of variability.
k: -giggle-. so they'd never discover about accidental misprimes if they were on an embedded Grulding manifold?
u: you can imagine a primate mathematician losing it on a Grulding manifold, particularly an intrincisist one, who factors five into three and two on a Grulding manifold in a desperate attempt to figure out why their calculations weren't working the way they expected.
k: heh. I suppose then formalism is just the idolators of string-yoga?
u: pretty much. just another logos-reductase
k: and I suppose the intuitionists would balk at a bloyarang?
u: oh, our computers and minds can't make it, therefore it cannot exist!
k: yeah, just because our senses can't draw it from tumbolia, there are no relations to it whatsoever, we shall huddle in our Western Nihilism like cowering snailwyrms.
u: heh. and the same thing with constructivism, and so forth
k: the logos-reductase used is different in each: each relies on a rationalization of how the studied schizotome differs from holotome, then procedes to cordon off a language environment for the reasoning to procede in.
u: yep.
k: okay, next biggie. what's a phenomorphism?
u: that says *how* something is cut, once you know there's a schizotope and a schizotome, the relationship between holotome/tathata and the particular instance, especially mathematical in character, you'll need all kinds of excidiary/subsidiary information, like the barber of the chick who wrote the code that visualized the object in question, etc. it's not really analyzable.
k: gotcha. you could internalize the Suntoro space all night, and just end up with sickening large internal node synthesis rates on your local indranet
u: bingo.
k: okay, what about godel belief sentences for b-e gestalts?
u: oh this was really cool. imagine that you've got a religion. um. Nyerkori, let's say. And you'll draw a diagram showing the belief-strata from centrum to exterior, depending on amount of b-e condensation, right? there's a bit of code which the exterior, typically the lower order biological reproduction coupled parts of the religion, filters from the interior, because its appraisal is that "any piece of code which denies The Nyerkori religion is bad, mmkay?", and the behavioral instances that provokes prevents self-referential beliefs of the form "this belief which this statement corresponds is not believable in the Nyerkori religion" from getting to the centrum, because it holds that the centrum cannot cope with such contradictory information. primitive b-e type gestalts usually cannot cope with that. the centrum tries to sync the whole to the cosmic mind, and is usually entertaining (much to the surprise and shock of the exterior considered unreified), exactly such belief structures, because it makes a runtime unbounded find external holotome call and determines that the best way to do this is mediated tathatadhyana to its cells within the centrum, and aliased pranadhyana to those in the exterior, which usually ends up being translated into a marketing schizotome and causes ridiculous amounts of code-comparison based warfare in non b-e local relative space.
k: i'm going to go back to dreaming that I'm a primate eating pizza. wake me up for the sequel.

Wednesday, November 01, 2006

of primates, gadolinium, lambda calculus, and category theory.

"Today's information session consists of a Refrectory with Laboratory exercises about the development of category theory and the lambda calculus as functions of distance from the galactic nuclei and ambient spatial gadolinium content for primate species over a 4.7 megahauve span. The Refrectory Speaker is a Postorbinate Glossal Intelligence n-12-hibisquan Nhoeteraume."

"We will be sampling cosmoses from what I like to call the fifth immersive thylakoid, in particular the lamination consisting of the grana S1240 through S1245, and in particular the P700 group, conjugated by a unitary sheaf-preserving Grassbane transformation. This should be a representative subsample of all primate evolution tracks for you to derive any particulars of any primate evolution track you encounter outside this subsample. Display the first slide, please. This is the Whovemerth-Crarymongle function in 45d space. It's an inverse derivation/imaging of the global cosmic topologies which primates in these strata first generate when they produce theories of general gravitational relativity, as a function of distance from galactic nuclei. Slicing off the upper Rhemnosis subspace, we get a simple seven dimensional function which takes as arguments the distance from all galactic nuclei within one and one half quemqwhents , since general topological terms tend to exhibit a highly cancelling character: that is to say, in such tiny slices of the universe as represented in primate brains, the global topology of the universe inhabited is not known, and because of that we can slice a two dimensional subspace off, leaving us with a five dimensional function. Dealing with this tiny section of the Whovemerth-Crarymongle function, we topologize, compose an indranet, foliate, execute rhombohedral tathatadhyana, and then percept. It is found that the development of category theory by itself is proportional to one third of sine of the square of the absolute quantity of gadolinium within a cubic unit of space normed for the local topothety, multiplied by a constant of proportionality, multiplied by the inverse fifth power of the distance of the associated star from a given galactic nucleus, multiplied by the inverse seventh power of the major mass component of the center, and so on and so forth, full apprehension of the formula now exuding itself in your sensoria. When perception of the development of lambda calculus is entered into the froth, we see that the development of lambda calculus is strongly correlated to the amount of gadolinium, even more so than the development of category theory. This relates to the development of computational machines and their phenomenal impact on primate civilizations. Now, for some important edge cases: long orbits around globular clusters: if a primate civilization manages to be part of a stable manifold of a globular cluster with a long orbit time, then there is a reduction of the vave ossars, and consequently different computational substrates are available than are available on monosolar planetary systems. In particular, gadolinium is found to inhibit some of the typical developmental thresholds for magnetic reasons, and abundance/filtration cost reasons. Questions so far?"

The Refrectory recognizes Whelmnor Raynaus, a Miltefort 4 Quondleplornge. "What about cases of subhybrid galactic collisions on the development of laminar sheaf yoga on long-strategy primate civilizations?"

Nho: "In the Grazier-Walsh image of the previously stated function, we see that such civilizations usually reach the point that they attempt to engage in species wide tathatadhyana level quantum computation on worlds with high probability of destruction but extremely low probability of escape. On those worlds which do manage to develop laminar sheaf yoga, the survival probability is increased roughly seven hundredfold. Otherwise it's the death throes of yet another misconcieved coherent system. Next."

The Refrectory recognizes a Glial Intelligence, n-19-bisphenidate/fundamental group of milnor's cottage cheese/Twarve: "What is known about the Synphreny conjecture with respect to internal scission?"

Nho:"Ah! A good question. There's a terrible amount of information about it but there's insufficient concentration in the local slice to answer that. The Synphreny conjecture, in short, states that there is a typical deferential eddy-set produced by lambda calculus in nuclear-force-war-weary primates in the Shoenve-4 isothrenolic base-space. The difficulty is that the answer involves a heptamer reduction tathatadhyana which is not available in the current context, therefore the answer I can give is that the answer is not available in the current context in a way that would be meaningful to the listeners of the lecture. I can give you an answer, but it won't be meaningful to you. Shall I do that?"

"Yes please" bespeaketh Twarve.

Nho: "Isemverne's balalaika twinkled on the summer's eve. Avast, me hearties, the cardiac echo of Grothendieck's lymph nodes echoes not within the temple of the sumner pellis. Cloyingly I reduce thy yarns to catarrhine glee. Inspect this mellifluent symbiot, praxis notwithstanding I lost my arpeggio in a blender. Tell me, dost the grin glee bestir within thee? A trance swerved, colliding with the monkey. Manifold protrusions guided our melodies on the winter's massage of my nostrils. Orbilescent adaptors echoed gracelessly the conturition of my mathengue. I knew knot the summaries of seasons, the piebald attacks of Sphere directors. My conflict misbegotten, my aspergillium lost a spoke, my mendacious alpaca-robot. I swung heartily on the sensational froth, seeking not a plenitude of replies or a methodical exegesis of my current relations and interactions. Ploys and plans and partingalgues stoked my fury in thetical aspersions. I knew the greengrave and the arroyo-engine. Did not the sparkling blanguerts answer my officiers?"

utter silence.
Nho:"As I said, incomprehensible. Very interesting. But incomprehensible. Next?"