TX2.

Posted by: Nick Land

Tic Talk

The (Barkerian) Tic Xenotation provides a numerical semiotic adapted to the Naturals with special affinity to Euclid's Fundamental Theorem of Arithmetic. The TX constructs numbers in terms of their basic arithmetical features as primes or composites in a notation without modulus (base), place-value or numerals.

The exact circumstances among which D.C.Barker formulated the TX remain deeply obscure (for a number of reasons best explored elsewhere). For our immediate purposes it suffices to remark that the broad research context within which TX emerged was a highly abstract SETI-oriented investigation into minimally-coded intelligent signal, without presupposition as to origin (e.g. 'xenobiological organisms') or theme (e.g. 'cosmo-chemistry').

The investigation, situated in the jungles of Borneo, was entitled 'Project Scar' and received a high-level security classification. In keeping with this research topic, Barker proposed TX as a maximally abstracted or ultimately decoded numerical semiotic, stripped of all nonconstructive (or symbolic) conventions (and initially named 'Goedelian hypercode'.)

While the raw numeracy of TX is most accurately conceived as sub-qabbalistic, due to its indifference to modulus notation (the primary motor of qabbalistic occulturation), its very independence from convention makes it a valuable tool when investigating the basic features of numerical (arithmetical or qabbalistic) codes.

Among the notation-related features most prominently exposed to rigorous scrutiny by TX is ordinality.

AOsys

Within the Anglobal Oecumenon, the most pragmatically prevalent ordinal functions are alphabetical, utilizing the ordering convention of the Neoroman letters to arrange, sort, search and archive on the basis of Alphabetical or Alphanumerical Order, organizing dictionaries, encyclopaedias, lists and indexes 'lexicogrpahically.' The word 'alphabet' itself performs a (Greek) ordinal operation.

'Lexicography' - dictionary-type order - is used here (as in various fields, such as compilations of number series) to designate a mode of ordering (an ordinal-numeric function) rather than a definite topic ('words'). Although a relatively neglected numerical operation, lexicographic ordering plays a crucial role in concrete (popular-Oecumenic) ordinal practices. It is characterized by:

1) Popularity. Facility at lexicographic sequencing is considered a basic social competence, inherent - or even prior - to literacy, whilst pedagogically separate from the acquisition of numerical ('maths') skills. At the pedagogical level, Oecumenic societies tend to distribute ordinal/cardinal competences in accordance with the distinction between literacy/numeracy, thus establishing the basic division between linguistic/mathematical abilities from a primal nomofission (ordinal/cardinal differentiation). Literate citizens of the Oecumenon - those able to use a dictionary - are ordinally competent, through lexicographic conventions.
2) Pure ordinalism. Restricted entirely to sequencing problems, cardinal values remain entirely alien to lexicographic practices, to such an extent that rigorous ordinal-numeric operations are typically divorced entirely from numerical associations.The ordinal function of numerals (1st, 2nd, 3rd ...), in contrast, remains relatively impure - at least psychologically - since in this case a persistent cardinal temptation confuses sequencing function with the spectre of quantity. For this reason the alphanumerical subsumption of the numerals into lexicographic practices can be considered 'clarifying' in respect to ordinal operations.
3) Fractionality. Simulating lexicography within arithmetic requires the employment of modular (e.g. decimal) fractional values. Arithmetical listing by cardinality will be isomorphic with ordinal-lexicographic sequencing for all numbers of the format '0.n'.
4) Sequential diplocoding. Lexicographic systems require twin ordering conventions. They draw upon an alphabetical code and an ordinal place value convention (principally, left or right ordering, equivalent to the behavioural scheme for the movement of a reading-head). The alphabet instantiates the ordering scheme, but does not (internally) describe it - 'reading' the alphabet to extract the ordinal code ('abcd...' or 'zyxw...') itself presupposes an extrinsic sequencing convention (Alpha-Omega, from first to last).
5) Infinite potentiality. Any lexicographic system allowing interminable strings has a code potential (cardinally) equivalent to Aleph-0, with an infinitity of virtual Dedekind cuts (entry insertions) between any two terms, however close, and virtual isomorphy between any segment of the list/archive and the whole. It thus attests to a 'literate' infinity isomorphic with that of mathematics, drawing upon a common but culturally obscured digital source.

Intercoding Arithmetic

An intermediate semiotic attuned to purely demonstrative engagement with Euclid's Fundamental Theorem of Arithmetic (FTA) can be generated by transforming the standard Oecumenic decimal notation (*) by:
1) Employing the full Alphanumeric series 0-Z (0-35) for notational convenience, and
2) Raising all signs to their first hyprime power, from 0 = Prime-0 = 1 to Z = Prime-35 = 149.

The purpose of these transformations is to eliminate polydigit (place-value) numbering and expose the radical disorder implicit in the FTA. All integral numbers in the FTA intercode consist either of single figures or plexed-compounds of the form (...), with numerical clusters synthesized through multiplication rather than modular-positional construction.

Consider a number picked entirely at random, *86, disassembled by factorization in accordance with the FTA down to the listed components *2 and *43, the *1st and *14th primes, hence: 1E. The expression of this number is no longer under any positional constraint, '1E' or 'E1' are equally valid on numerical grounds and strictly equivalent. Shuffling a string of intercode figures (FTA components) of whatever length makes no difference whatsoever to the number designated, with the ordering of the series being subject only to an extrinsic convention (of minimal - even vanishing - importance from a (cardinal) arithmetical perspective, where it is relevant only 'psychologically', for convenience in assimilation and comparison).

Once the merely inertial and peudo-numerical order inherited from uninterrogated tradition is subtracted from FTA-intercode strings, dissociating all components from quantitative ordering, they are freed for lexicographic re-ordering as decoded series - an ordering which will deviate from the series of quantities, liberating an Autonomous Ordinality whilst de-cardinalizing the number line.

Consider *172, or 11E. Oecumenic-lexicographic procedures ensure this number precedes 1E (*86), as will all its successive binary multiples. Evidently, such procedures ensure that the infinite series of binary powers must be completed before arriving at 2 (*3). 'Natural' counting no longer has any prospect of reaching a nonbinary power, just as alphabetical-lexicographic 'counting' would proceed 'a, aa, aaa, aaaa ...' without ever arriving at 'b'. Reversing the problem and it is equally evident the lexicographic-ordinal line is never counted.

The Kantian assimilation of arithmetic to temporality models elementary time-synthesis as n+1, +1, +1 ... an intuition rendered questionable by the rigorous lexicographic disorganization of the number (listing) line. Once ordinally purified, the number line becomes uncountable by any supposed finite (temporalizing) subject, even from moment n to moment n+1. Instead, the line is synthesized by sorting (lexicographic sequencing) of prefabricated strings, whose quantities are determined on a different axis to their linear-positional codings. A prolongation of the time-arithmetic association would thus require a remodelling of time as nonprogressive synthesis without consistent scale or continuous-quantitative trend, no longer intelligible as passage or development. Such ordinal-lexicographic time maps a 'templexity' that is uncountable, fractured/fractional, erratic and heterogeneous, sequential but nonsuccesive.
Of course, all of this needs re-approaching on a far more rigorous basis, with a consistent focus on the topic of templexity - suffice it to say for 'now' that Kantian intuitions of number, time and their intermapping are themselves structured by notationally-problematizable constructions, since time-mapping has a hypothetical rather than essential relation to arithmetical common sense (with its undisturbed assumption of straightforward ordinal-cardinal interconvertability).

Note-1. Elevating this intermediate semiotic to a functional numeracy, with a semiotic power commensurate with the set of Naturals (including primes above Prime-Z), requires a final step:
3) Adopting Tic Xenotative plexion, where '(n)' = Prime-n.
Thus 0 = 1, (0) = Prime-1 = 2, ((0)) = Prime-2 = 3, etc.
The inefficiency of this semiotic relative to TX is demonstrated by its redundancy, most dramatically:
V = (B) = ((5)) = (((3))) = ((((2)))) = (((((1))))) = ((((((0))))))

Note-2. TX shares the intrinsic disorder of FTA-intercode. *86 = :(:(::)) or
(:(::)): or :((::):) ...

Out Of Order

TX/FTA-intercode numerical construction is indifferent to semiotic sequencing, position or grammar. A number expressed in either system could be distributed randomly within a space of n-dimensions, requiring only a cohesion convention (semiotic particles 'belong together' irrespective of order). Apprehended in their fully decoded potentiality as efficient number-signs, such formulae are clusters, not strings.

The TX case is still more extreme than that typical of FTA-intercode, however, since here even the spectral residue of sequential coding is erased. Given two complex TX-formulated numbers, correct order (quantitative comparison) requires - perhaps highly elaborate - calculation, eliminating entirely the practical usage of disordered TX clusters for ordinal operations.

For anything but small numbers, Euclidean cluster-stringing conventions (by ascending cardinalities) become procedurally complex, perhaps inoperable, for TX numerical formulas. This is evident even from small numbers, such as *149, TFA-intercode Z or (34), TX (((:))(::)). As the 35th prime, with 35 the product of *5 and *7, the sequencing of hyprime sub-factors (factors of the prime-ordinate, i.e. *35) is no longer facilitated by lexicographic codings drawn from the numeral sequence. That '5' precedes '7' is evident from the numeral code, but the ordering of ((:)) and (::) cannot similarly rely upon intrinsic lexicographic guidance. In the TX case, it is only by constructing the numbers and sequencing them arithmetically that the 'notational' question of their order can be resolved. In other words, the sequencing of the sign has ceased to be a notational or preliminary problem, becoming instead inextricable from the arithmetical construction of the number. This results inevitably from the elimination of notational redundancy in TX, with concomitant erasure of procedural 'intuition.'

Because TX number clusters are intrinsically disordered, a consistent and functional TX semiotic requires re-ordinalization through autonomous (extrinsic) lexicographic procedures, inevitably constructing a cardinally erratic 'number-line' or list/search sequencing protocol. The semiotic economy of TX makes this procedural problem easy to define. As an approximate AOsys analogue, lexicographic TX requires a variant of sequential diplocoding:
1) Cluster stringing. Sequencing the components of composite TX-formula numbers.
2) Number listing. Meta-sequencing of properly sequenced TX strings.

It might seem sensible to assume the Oecumenic left-to-right reading procedure, since the arbitrariness of this rule makes it unexceptionable, but the diplocoding option matrix necessitates a substantial question as to the consistency/inconsistency of this decision as between (1) and (2) above. Even allowing for this complication, the option matrix for a mechanical lexicographic TX ordering protocol remains highly constrained, consisting merely of twin decisions as to the sequencing of the tick [:], open plex [(] and close plex [)] signs.

Irrespective of the Cluster stringing decision, tick-precedence sequencing of the number list results in a the AOsys analogue previously mentioned (a, aa, aaa ...) 'counting' through the infinite series of binary powers before reaching any nonbinary number. The list is initiated by TX *2 = ':'.
Plex-precedence produces a far more anomalous list-line, one that is non-originating because it 'begins' with a series of arbitrarily large hyperplexed primes, notationally initialized by unending open-plex signs [((((((((((((...], since '...((' precedes '...(:'. Listing practices following a plex-precedence protocol necessarily begin in the middle.

[My assumption is that semiotic consistency (across clusters/lists) is to be preferred, with the sheer weirdness of plex-precedence sequencing making a strong case for its adoption. The 'alphabet' (ordinal code) would thus be described by TX *3 = (:).]

In his own brief comments on the cluster sequencing problem in the Project Scar report, Barker restricted himself to the observation that Euclidean (cardinally consistent) ordering was no more than a "provisional and arbitrary convention" which would quickly break down "given nondemonstrative numberical values [anything but very small Naturals]" that the problem should be considered "merely technical and extrinsic" and "probably best decided on communication-engineering grounds."

Given Barker's Project Scar research orientation, focused on "nonlinear recursively-embedded planar semionomic dot-groupings of cryptogeologic origin" - anomalous cryptoliths - it is not surprising that he came to the notational ordering problem late and distractedly. Just days after completing the "Appendix on Notation" Barker came entirely unstrung.

Stricken by revolting tropical diseases, increasingly obsessed with an interwoven tangle of cosmopolitical conspiracies of various scales, and multiplicitously agitated by teeming microparasites of dubious reality, Barker's plummet into noncommunicating delirium is charted by the digressions into doggerel annotating his Project Scar research report:

A chittering tide
Devouring my hide
Starting from the Outside

This is the slide ...

And Yet

In the same twitchy, spintered handwriting Barker remarks:

The xenotation continues to disorder itself as it condenses, tearing up the number line, devastating time and sleep. Perhaps it is a weapon from outer space. I say that seriously, even if it is a sickening kind of joke. There is no sleep, everything is broken, everything connects without joining, swarming, pulsing, dots, specks, dust particles dancing inside my eyes, continuously ripping ... thought has become a disease ... I even heard a voice (how ridiculous) saying: 'You must isolate the xenotation before it disintegrates the time-line.' It's just the fever of course, but the tic systems are all shuffled together now, shuffled together with this filthy disease and its cavernous speckled dreams and even Jolo admits that the markings are spreading over my skin, bites or rashes or maybe even colonies ... so the line has rotted through, disintegrated ... there's no line, that's the message, and yet ... And Yet. ... counting is ineluctable and unsurpassable ... You have to check it, re-check it continuously, but it's true. How could the hyprime indices be decided without a countable ordinality? They have to come from somewhere, from a matrix, a culture, even if the clusters seem to rip everything apart they MUST HAVE BEEN COUNTED at some stage, before dissimulating themselves and scattering again ... And yet we can only make sense of these dots and ripples by counting primes on a line that remains successive and integrated, developing reliably, communicable, they have a past, a true lineage, even if it's difficult to think, even if they tear it apart and make of it something shattered and insane, something diseased ... but really I don't blame them, NASA of course knew nothing, but even they knew nothing, they just arrived, why should they remember? Memory is impossible for them. In any case, it's just a disease, I understand that now. There's no malice ... not even real cruelty ...

Note. While there is no reason to believe Barker had exposure to, or interest in, the Anglossic Qabbala, the emphatic reiteration of 'And Yet' suggests he had feverishly identified it as a synonym for counting, perhaps even for temporality. (AND YET = 123).