nonsensica mathematica

It is notoriously easy to compose completely meaningless -- and meaningless in the sense that any world in which a given interpretation that the given thing under consideration makes sense would be one with sufficiently other rich pathological disabilities and instabilities that one would not be able to live long there without wondering why the first little autocatalytic thing didn't say "life, why bother, what's the point, we're all going to be attacked by random peroxide radicals this day or that" and then expired in the prebiotic melange -- arrangements of words to suit -- as in a hypothetical set of argumentum ad "this is meaningful to me" -- fallacies of significance, that is, things which seem at first to be important because they contains words or nouns or other fractured processes which are dear to our heart, but on further reflection, are not about anything at all. While I stridently enjoy this about-nothing character, I also find it particularly noisome, when it is used in lieu of a point. Beyond the point that the sand castles have no foundation and float in the air because someone is believing really hard that they float, I must scratch my head and say "um, okay." It is good to know that some people think this about art. But I have not spent enormous amounts of time considering art qua art. Most of my bugaboos concern irreplicable reasoning and such foibles a la Sokal. To wit: I'm concerned with generating random, meaningless argumentation for the hell of it, and because I think that it's useful to create sophisticated forgeries of good reasoning. The dihydrogen monoxide ballyhoo, the randomly generated paper made by some clever MIT folk, and so on: these sorts of snags wonderfully and delightfully illustrate where the argument to authority -- or at least to authoritative language -- crops up.

I find myself less and less convinced by long pages of incomprehensible mathematics -- I want to see pictures or movies which demonstrate the points which the text is making, rather than congealed masses of symbols -- those I don't trust, for example:

"In Theorem 3.4.1, we proved that a k-ary snark tree is only loosely inequivalent to a forked n-ary warthyff tree with 67 or less intrusions. Recently, Karmblen Nuychteff showed that a forked n-ary warthyff tree must have a prime number of intrusions if it loosely inequivalent to a k-ary snark tree if and only if the thneed signature of the forked n-ary warthyff tree is an odd multiple of seven or twelve less than a Mersenne prime. The proof is as follows: by Van Snordglington's lemma, there exists an exhaustive homomorphism from the space of forked n-ary warthyff trees that commutes with the extrusion codomain of the space of k-ary snark trees. Since Kan extensions are the most important concept, we take Kan extensions of both the exhaustive homomorphism, and the forestry service functor H7(p)(x), giving us a profinite module on the category of snark trees which resolves to the adjoint functor of the warthyff image tree. Then we take the combinatoric reduction of the forestry service functor and enumerate the correspondences between trees. By Van Kampen's theorem, the spaces are homogeneous and therefore we can fold-remove the forked trees, and that counts all of the nodes and branch-points up to the exhaustive homomorphism. Then we take cases: either the trees are Wilkes-Barre or the space isn't Hausdorff, and since there density extrusion matrix is singular only when the thneed signature is a multiple of seven or twelve less than a Mersenne prime, we're halfway there. Except that even multiples of seven don't work because of the unrenormalzed kappa expansion in the density extrusion matrix"

See? Completely and utterly meaningless. A better method might work by stitching five mathematics papers together, and changing them so their nouns and verbs are consistent with the nouns and verbs of the first. Oh well.

heliochromesis astringes Alcuinex

"The sliced holotome is not unique: it is ultimately and especially idiosyncratic and furthermore random in character: by its relations it is fractionally distilled from tathata: it is cracked, splintered, sundered, ensconced: but adjourning focus on the synthesis relations and moving to the effervescenzium, the following is found: to be conscious of something is quite explicitly to not be that thing. The Classical Errors number three: first, confusing consciousness with participation: if you are conscious, then you are seeing things tangentially: all that resolves to zero is an equivalence class across which ..."

the perils of persephone (obregard for annagran)

"The notion that other sentients of the same species have the same experience of the world as you is one of the tricksiest little bugaboos to properly digest: whereas peering on fragmentary detritus synthesized by the cantankerous outside-looking in schools of analysis creates contrails of prescriptive reality declarations: oh, our orthogonal and sloppy data collection reveals that your brain works some way only means that the way we have ensconced our theories and models of the functioning of your brain (and by proxy, your mind in our current abstractioneering campaign) are peer-accepted: whether these theories correspond to your sensorium is another question which is beyond the scope of our reasoning, but we don't admit this to ourselves: rather we will entangle ourselves with the notion that our theories are in fact the way the world works, and that our sensory experience is subsidiary to them. Regrettably this belief is not only exorbitantly wrong, but leads to map versus territory confusions as well as other mistaken congealments and faith based approximations and extensions to the horizon or to the local whatever high dimensional topological atlas and to even further vortices of confusion. You do not have any idea what other people are experiencing, or how they construct their world from their senses, only that the slice of their senses and your senses that are fibered over shared regions/spaces/etc. are homomorphic to another, or at least for you, what you see corresponds to your specific sense slicing of the world, and that other people's sense slicings of the world are different, and so forth and so on"

glaivesgreathes

the determination of the whole and organically encompassing experience determined by and for a generic experiencer in a particular foliation of a point of view synthesized in the long migration from here to there and thus thusness recoils in particulars expressing the isolated monad both parochially and sequestered sequelae of the current adumbrated excession to withdraw the major rhombent of experience in the field pleat of experienced reality sloshingly indexical to the mirror referent of the current world illocuently turrowed gannet sleids apprehended in the current plane pearl of expression to wonder whereas the knarves of elsewhence wail inconsegruities corroherent stalk parcels brick charged in jangling kalliphany atwixt and beheld the sussurancies of the arroven moment to retrieve the arrayed arroyos of the singular experience of the individual as yet unframed by language and other syllogistic in extensio res in transtensio media for formal prodromes and unhinged by slarky allotropy readying the phraseological synthesizer for endrumbant prospharetic calembricam oh Ohonckoa's clarion fanfaronade of the chordantly specified All Concept in the ears of polythesis plural from the unwhence to the transbegotten plural amharic charged copper sieves slarrayed in the theuropause upon the wholefroth of the Minor Cosmic Russell Moment of the irerratic
holoclines outgassing preponderant exiguum by trochlear misapproximation upon and beneath
the flow fold of the gate groves sharply and expungently exterrogatorily conefoaming the plurfect anhinged hold cloves of yesterwhy. Ohonckoa extensoris in abrupt tarhammeringly
birefringent asclepions to so and purely state with all channels in all substrates contingent upon the harmonious and unencumbered conceptual anchor to sarrow and make manifest the furrowed and the context reference by gainly and message consistency in all manner of interpretations so demanded and criticised by preference and clarity to ensconce and signify the shoreheres on the express and dubitously prime understanding for here and thusly made current and inhortative expressaria to wit one moment of all moments to clarify and elucidate most surfulgent and ferry without request by capstan plural and magnanimity grand cosmic in approbation and candor most transparent and without undue misrepresentation by sake of expression for the node index of the local gate cloak thereby exactly and harmoniously presenting the weave of now.

swim prep

"did you ask them what they believe? did they tell you? i think that they didn't say a thing. or else you misinterpreted. it's not the kind of answer that you can write on a piece of paper or a million pieces of paper or countably infinite pieces of paper. i quite expect you were out of your league. oh, don't get huffy with me. i keep telling you that you need to go there, or here, or elsewhence and pour tea down your kettle. no, it's not something you can buy in the store or demonstrate via religious faith, science, or epistemology, nor is it something that is a thing in the abstract or the concrete, nor is it nothing or nothingness. you keep complaining (quite loudly in fact) that all these cultural wars are nonproductive, and then you gingerly plunge into the fray loudly and drunkenly claiming your own epistemological ascendancy. see, i'm moving. at least, i'm staying still. i won't be here tomorrow. and neither will you, but i won't be elsewhere, and mayhaps you won't. no, it isn't magic, why would you think it was that? oh, argument from mystery! still harking back to logic as your own fundamental system of abstraction. you know i get more mileage out of arguing with objectivists! ok, ok, point taken, but it's not as if all of my marbles have crawled on the floor: there's a sequence of that's downright impossible: exclamations of incomprehensibility and multiply enfolded awareness, folded back onto itself, a transcendental blue purple vapor plume interpenetrating itself. dear gouraud shading no i'm not hawking this. you couldn't make a war from this stuff if you tried (at least not directly, and that's a separate question).

see, i can give you flotsam and jetsam, but it comes from and is gone to thusness: little farandolae, terraced chance-shelves, chromic engines, jumping color jacks, braided attention swirls, transelliptic orreries, and you'd be like "is that IT? is that the ULTIMATE THING?", and you wouldn't make the leap to jumping on the reference jungle and just losing yourself in the network, and that's precisely what you need to do: that's the essence of direct mystical experience, you stop trying to hold on, and just for once the boundaries between you and the universe, specifically, your ego and the universe evaporate. it's this chaotic froth weaving in and out all the time and you're just so immensely close, but everyone keeps saying that again and again: world doomsayers usually predict that tomorrow the world will end, and yes, tomorrow, the world does end. the world never ended, the world always ended, but that isn't the nub of the question, or the thing to hang your worries on.

if i were to use a change of contexts operation, you'd see that your opinions are meaningful: there are reasons that you hold them, either because they are authentic imagings of the world around you, or because in them you hold your stresses in order to survive -- even if they aren't authentic imagings of the world. but just staying in that mud puddle of yours is not productive. there's a, oh dear where do i start?

it's like this. you're a little civilization humming along, maybe you spend the first several thousand years acting and living like primates, and forming your own opinions of yourself as the highest form of life (as if!), and producing systems of self-appraisal which are hideously prescriptive and moreover aren't exhaustive catalogues of the type and kind of your creature that could exist because you spend so much time trying to bend the types of people you think there are to the types that you have imagined should be, and otherwise blasting the hell out of each other, and then finally, and invisibly, the world changes. this isn't the type of transformation that you can walk out in the street and see, or read about on the newspaper or the global information processing network, or even be aware of for a while until and only until it kind of gels that it has happened, and is happening, and will continue to happen. this is not a mystical or quasireligious or scientific or experiential transformation thus history will have a hard time coping with it. but more to the point once it has happened it takes much much deeper perspective to determine that it happened.

let's examine the impulse to find extraterrestrial life: why does this happen? because everything under the sun is dull, more to the point, people don't take the effort to avoid alienating other people, then fetishize the alien because it is the proverbial other. here 'extraterrestrial life' means 'sentient life'. but the whole (physical) space exploration program has a massive social cost, and it will mean bringing the problems of human beings to more places. your science fiction stories reflect this. the apposition of the desire of the different against the dull doldrums of the same when splayed against the primate territorial acquisition, defense, and resource control methodologies leaves a sorry taste in one's mouth. waxing poetic: you are part of larger structures. astronomers are the galaxy's way of understanding it's large term structure. it would be a sorry state of affairs if it were so easy for young species to continue their idiocies on grander and larger scales. what is farther away is so much more imminently close in the mind than it is in physical space. the anthroposopher Straywasp Gwongrive hypothesized that one reason for the general relativistic properties of space was to act as an in built speed limit of the territorial activities of primitive species. and make an anthropic argument you don't: you already know exactly what happens in universes where the speed of light is easy to cheat, just look at your science fiction. with great capacity, comes great responsibility, some philosopher said. the scope of your species' capacity to deal with energy corresponds to your species' capacity to deal with itself, and correspondingly to the length and reach of each member. there are shortcuts, but these do not involve phallic objects launched at ridiculous speeds to other worlds for the purposes of colonization. " -- Zanhai Arripongelet of the Codewarper Valve of the Fifth Thylakoid of the Greater Migrating Shenzoro Clade of Aunhex in the Yolk of Diparwanzion.

more mutterings

"... Carasweldgnahe antrempofzag asonclesse anyndrearhin dozmois u'lamparo theumas mahacryptais havouresis ferruly arnhenga dilleprouze azimun drandgepedran uzqe niorzem bal anhannehuve derzoran nogo dizunculi boen atauverre eryjoehvs bel drunnen tartomid vol'va
salmitropan astransgentian cin hollihoedreon amhaug jruzqwaysche dolpidro noehzhoi dunnet esprungum balispliris ehai nahengene eurhimde[4] jengeuear adhorinhe dzombe greungeon anhingas yelhirne ashokare belbillispin dysploris auhembe droyario aninculire ehaia jhengerai apilontophor dzeunxes tirincore ahaia avelvenirion dodenculi abhai dazdenzarpore doxynde jeunhe vovarin kellenga[2] harhimporaron esshai cingive mynzarevei daldroparin omhomen billabole hilule nahambe[3] zengheia hinzifloen dasmenin goronculi mahatryptahaia ahonifare jaspanellongis deldispafar ahare geverrengea aspingoa hohengafar gyngoehe xerynxe[1] dilhongiflora anhome uzqen a'lorzano juntheungil holoploraga?"

[1] A xerynx is a species of vyrhex, specifically Vyrhenxes xerynxior, otherwise known as the Sparse Iridescent Thistled Vyrhex, Henrietta Jaljuxe's Spectacular Vyrhex, or, to the native Arhaxen of the Greater Zervuli peninsula as The Quasi-Animal Which Emerges From The Chance-Plumes When The Night Is In The Third Purple Frond And The Catnip Frog Hibernates, Oh Ohonckoa, the Lemon-Bifurcated Astringent Tea Parity Bribe Engine Which ululates on the Transrational conductor in the Wavy Pentagonal-Odor Gravesieve, amongst other and various denotations and epithets. Despite the outcompetition of most species of vyrhex by parabiological Grothendieck-Teichmuller groups imported from Gondwanaland, the xerynx is actually doing better than it would be had the spurious groups not been imported.

The diet of the xerynx is rich in potassium, vitamin B17, deuterated ethanol, tritiated citric acid, fourteen dimensional wildebeest event sheaves, defunct French philosophical concepts, deoxygenated mafipulated phlogiston, sulfagne, xenon, GABA, water, antimatter carbon, Zervuli wharvethrushers (both the smectic, symplectic, and quasinematic varieties), dhyana phosphatases, iridium boride, hydrogen, and ennui chlorates.

[2] A kelleng is worth (k+48)/cos(12-k+1) wonzlos sterling, or 1.28179381011e-49 octavian pence.

[3] A nahambe is a unit of measurement equal to (e^sqrt(bet_9)) metaparasangs/joule, roughly 1.1180019491 yoctokalpas, or an eighth of a gregorium.


[4] The act of eurhimde is to violetly stave the rindwreathes harking betwixt the vexing hopefroth undergnarlingly approximating the ghostgrease beneath the grainflows while shudderingly overfoaming the bosonic bass chord in turbid and intentionally precise misapprehension against a (3,4,-0) comment cloak without birefringing the blue wreathpleat. It is not used metaphorically above. The bound ossar is "eryjoehvs" and the resolvent is "jruzqwaysche" (which is used metonymically and paraleptically but not analogically). The base point referent is "avelvenirion dodenculi abhai dazdenzarpore doxynde" and the migration curl coordinate system establishment tensor is given by "geverrengea aspingoa". It is worth noting that the Carsten-Heyzsler comma does not include the deity orgasm, and that the trees were most definitely very satisfied, but the wrengos disclaimed their lack of satiation most vociferously.

theraktys if then statement roguesponges

What did Marty Pringlestoinge say?

"Yarsfooth anna porengo! Bloongsblons-upon-Wtfologists? Hallongo O'Sfirika plyspfragh anthome annapanga d'liscinge etiaflasce jerrenx u'lorzano dolouk nelonzosse. Dranspruns allarza inszchians: arrambhei qe nhomeo addiaprenzis alossoflone o'neischzomeir dedroxe anyschgeo adamproen bischnonkeum drynzemfarehejn a'zazeue bifrioem. Alikzillim mhonhome zelzerris arrhymbe
aisonohero i'ue avazise anunculi oranhazilis i'anhe barzunculeo ganzet asperythem jeuzis enhem avrella na'aisue. Lynjem balaborano neischgjer pangue emheng ouvreisp i'brehve. Alhoenge hevroun dolpinkaro anheunge ascifthousis urrin ghen. Dxefjthe colam uezue drindroparo heuravve."

in Belgium...

Despite being a class IIIb/4 democracy and a bastion of freedom, various and sundry things are illegal in Belgium: toast, tensors, geometric algebra, isothiocyanates, swiss cheese, british cheese, wombats, nine dimensional bondage cages, semantics, cows, lemons, the LXN hallucinogen, rubik's cubes, fractals, sieves, antagonists, syllogisms, cartesian closed categories, lambda calculus, lisp, microsoft word, apple computers, salt, cantilevers, astrolabes, astrology, astronomy, orreries, planetaria, the common cold, multiples of sqrt(pi), amanuensis, horticulture, grapefruit, LASIK, caffeinated watermelons, japanese manufactured GPS recievers, homological algebra, nethack, self referential laws, centigrade, Look Around You, Dr. Seuss, protons, bikinis, high gain antenna rigs, debate, abridged dictionaries, stubble, plastic writing utensils, and onwards.

The following items have recently been legalized in Belgium: Arthur Dent, gravitons, Calabi-Yau manifolds, agar gel for petri dishes, John Horton Conway's game of life, algorithm optimization, bralessness, Bill Murray movies, A Wrinkle in Time, anti-censorship laws, formal power series, Foxtrot, crossword puzzles, ambiguity, pleonasm, video games, galagos, breeches, ceramics, nightmares, gerbils, cryptic commentary, dvds, paper, liquid crystals, problem sets, plot development, vitamin b-12, yttrium, having fun while being attacked by psychotic penguins, limes, indian food, and walnuts.

For more breaking developments on the absurd country of Belgium, read Tvongden press's Notices on Belgium Legality.

Stifled lemon strife

Rondeaux antebellum merengue seminal lassitude droll lemons snub benthic menhirs squarely in the maw ah lessees and dratted weasel demigods same arterioles sandcastles alas salsa massages meretricious managers muddle and mottle thinly yessir rambunctiousnesslessly misappropriating the green alien crystal vapor wedges for her own person to abuse. Natural laminar flows impede the stalk of the Toblerone God. Anthers and stamens and the Sex Pistils oh my! Mary mapped all the drumlins in Alaska at the command of the Mad Dromedary. Quotidienne Laura says to me that in the least timorous declaration of love she recieved from a smitten tho entirely unattractive bachelor, the suitor splayed his arms in the skull of a goose. Talay oh Mandalay!

copper plated dirigible machines

plesiosaurs simmer upon the nigh night drenched crater strewn scar-pitted landscape: awaiting thy ducks? It's a masterful miseloquencies: glue the guide's nostrils to the ackle-bird. Amung! Amung! Cry not the Insult Periscope for mayhaps only a massage let to gone and awaiting a dross reply. Oh pleugh! Oh plague carrying vesicles of ambiguity!

thundering pack animals storm Toronto

"...but the high-faluting esoteric philosophical stuff is just the springboard of false messiahs and preachers. You just can't get coherent information from our current perspective, and what's even more damning is the prevarications of these so-called mystical dilettantes who cream themselves on the tales of the impossible. I can't talk about anyone else's experiences: I'd much prefer to be small and fervently dull but all of this assorted material just turns up on my doorstop and much as I'd like to just remain in my little hovel I have to deal with it: and I've already got a full plate of issues of my own to cope with or to pretend to cope with. Now, for a moment, just setting aside my own psychological disorientation for one damn moment: there all these johnny and jane-come-latelies just proudly claiming some kind of metaphysical superiority when A. it's not clear how the culture wars are going to progress, B. nobody really wants to get stuck in the crossfire, C. anyone not making major philosophical or quasi-philosophical mazes for themselves and just doing interesting things is stuck explaining their mental objects to fashion-oriented brain dead rockstars, and D. the most interesting things are lost in a primate authority hierarchy maze of damning exotic states of mind..."

"...but every doomsday cult since the beginning of timekeeping has announced some kind of day in which everything changes in cataclysmic and apocalyptic magnitudes: an end of the world, an end of time and so on and so forth. None of their predictions came true, and their principal product was many disillusioned believers. So sell me on the singularity or sell me on science: I'm going to cautiously say that 'things, they aren't going to change', until I see it..."

"Regrettably it's a thorny puzzle. Nobody can extract any kind of information from the completely unstructured without filtering their recovery through their perspective. Then they'll try to sell you something. If you buy it, then you'll sell it to someone else. This is how religions get started. Ranting, raving people usually have this one important point which completely eludes the comprehension of rational people, and by the point that said ranting and raving people have been suitably classified and the ordinary world protected from exposure to them, the effects of that one thing which has caught their attention become scraggles on pieces of paper, and those who could actually elicit some kind of proper and holistic interpretation of their experience are usually in wide separation from those who have such experiences. It seems to me that you can't have your cake and eat it too. I don't know: I may herald from a too-depressing viewpoint"

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.

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.

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.
l

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).

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.