Xenotation (#1)

From Euclid’s Fundamental Theorem of Arithmetic (FTA), or unique prime factorization theorem, we know that any natural number greater than one that is not itself prime can be uniquely identified as a product of primes. The decomposition of a number into (one or more) primes is its canonical representation or standard form.

Through the FTA, arithmetic attains the cultural absolute. Number is comprehended beyond all traditional contingency, as it exists for any competent intelligence whatsoever, human, alien, technological, or yet unimagined. We encounter the basic semantics of the Outside (comprehending all possible codes).

Insofar as numerical notation is constructed in a way that is extraneous to the FTA, we remain Greek. Our number signs fall lamentably short of our arithmetical insight, stammering deep patterns in a rough, ill-formed tongue. Stubbornly and inflexibly, we translate Number into terms that we know deform it, as if its true language was of no interest to us.

Yet, given only the FTA, the code of the Outside — or Xenotation — is readily accessible. Nothing is required except compliance with abstract reality.

A single operation suffices to count. In words, it matters little what we call it — implexion, envelopment, wrapping, or bracketing describe it with increasing vulgarity. For convenience, parenthesis — ‘( )’ — provides a sign. The semiotic (or purely formal) equation ‘( ) = 0’ offers additional economy. Xenotation needs nothing more.

One is redundant to the FTA. It begins with two, the first prime. This introduces our sole notational principle, and operation.

Every number has an ordinality and a cardinality (an index and a magnitude). Crudely represented, through a mixture of barbarous signs, we can see these twin aspects as they are relevant here:
First (Prime =) 2
Second (P =) 3
Third (P =) 5
Fourth (P =) 7

By wrapping an ordinate (or index), itself a number, the Xenotation marks a magnitude. So ‘(first)’ or ‘(1)’ = 2. One, we know, is superfluous, and thus economized: (1) = ( ) = 0. Remembering that ‘0’ is henceforth the sign for the initial implexion, and not the familiar (though cryptic) numeral, we can now depart from all notational tradition. [The further usage of decimal numerals, in hard brackets, will be strictly explanatory, and dispensable.]

An implexion signifies the number designated by the enclosed index. Once this rule is understood, Xenotation unfolds automatically.

0 [= 2]
(0) [= 3, the second prime]
((0)) [= 5, the third prime]
(((0))) [= 11, the fifth prime]

Compound numbers are signified in accordance with the FTA:

00 [= 2 x 2 = 4]
000 [= 2 x 2 x 2 = 8]
(0)0 [= 3 x 2 = 6]
((0))(0) [= 5 x 3 = 15]

For primes with compound indices, the procedure is unchanged:

(00) [= 7, the fourth (2 x 2) prime]
((0)0) [= 13, the sixth (3 x 2) prime]
((0)(0)) [= 23, the ninth (3 x 3) prime]

So the xenotated Naturals [from 2-31] proceed:

0, (0), 00, ((0)), (0)0, (00), 000, (0)(0), ((0))0, (((0))), (0)00, ((0)0), (00)0, ((0))(0), 0000, ((00)), (0)(0)0, (000), ((0))00, (00)(0), (((0)))0, ((0)(0)), (0)000, ((0))((0)), ((0)0)0, (0)(0)(0), (00)00, (((0))0), ((0))(0)0, ((((0)))) …

[That’s probably more than enough for now]

June 4, 2013admin 23 Comments »
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23 Responses to this entry

  • X Says:

    I’m glad to see some attention still being given to this work. I have been struggling with this distinction for about 2 years now, but, with respect to the initial formulation (quasi-pharmacologically speaking) of xenotation (2004?, accompanied by ‘tick marks’): was cardinality stripped out, leaving ordinality as autonomous and thereby positional without magnitude? If so, does xenotation, as formulated here, ‘follow suit’? If not, could the (rather Kantian) synthesis of magnitude and position be blocked or cut? and what effects would this have for its ‘numerical’ legibility? Nominal numbering (credit cards, etc)? If the semiotic sequence is non-linear (in terms of how it ‘reads’), how is this consistent on the numerical level? Your previous post on 0 was helpful (especially via ‘superbinary representation of the set of natural numbers’), thanks. Needless to say, I’m not at all mathematically adept, so excuse my need for clarification, which I’m worried is rather redundant only due to my epistemic incompetence…

    [Reply]

    admin Reply:

    The linguistic overlay doesn’t really matter — it’s just the framework for guiding apprehension of the notation. It certainly shouldn’t be considered significant philosophically.

    Your substantial questions are all appreciated (and well-directed), but they demand at least a few days of mulling time at this end. Part 2 will address them as comprehensively as possible.

    [Reply]

    Posted on June 4th, 2013 at 7:58 pm Reply | Quote
  • Contemplationist Says:

    If you have any interest in and knowledge of programming,
    you’ll enjoy this talk (from England) titled ‘Programming with Nothing.’

    Its a programmatic representation of your post (almost)
    http://rubymanor.org/3/videos/programming_with_nothing/

    [Reply]

    Posted on June 4th, 2013 at 8:18 pm Reply | Quote
  • X Says:

    @admin Thank you. I did have a hunch that the ‘linguistic overlay’ was merely a model or something like this for ‘us to grasp’. How far do you see its implementation? Which is to say, with what methodological frameworks could you ‘envision’ xenotation aiding (hopefully to get us out of the way as much as possible)? Perhaps I’ll wait for part II to go further with this line of inquiry.

    I agree that its philosophical significance is unimportant. However, I was thinking—and again, perhaps part II will stop me here—that if the synthetic and seemingly catalytic process of cardinal/ordinal fusion is blocked (by what?), and instead paused in favor of ordinality, could one be walking towards a fairly non-Kantian mathematical structure here? Unintuitive, transcendental (albeit without residuals, i.e. counts, since 0 is driver), and I assume (eventually, if assimilated properly) no longer amenable to so-called rational structures of organization? Not sure how philosophical this would be, if at all… I always could be ‘wrong’ here with respect to the philosophical side of things (ugh)

    [Reply]

    admin Reply:

    My only ‘answer’ to most of this right now is to multiply your question marks.

    Evidently, any culture that numbered exclusively in Xenotation would be incomprehensible to us — we need the scaffolding of more familiar numbering methods to climb out to it, and if the traditional structure is then dismantled behind us a lot of things become very difficult, whilst other things loom strangely into focus.

    I’ve put it up here because the basic operation — i.e. transcoding numbers into Xenotation clusters — is really very simple, and yet odd enough to spark up some previously unused brain circuits. It ‘wants’ to be propagating itself memetically in a much larger neural ocean, so what the hell. I’ve no idea what it will do …

    [Reply]

    Posted on June 4th, 2013 at 11:00 pm Reply | Quote
  • spandrell Says:

    Looks like the Burmese alphabet to me.

    [Reply]

    Posted on June 5th, 2013 at 1:14 am Reply | Quote
  • raptros_ Says:

    i don’t see a way to do any sort of computation (e.g. incrementing a number) without stepping outside of this representation, though.

    [Reply]

    admin Reply:

    If you’re thinking computation as “incrementing a number” then Xenotation is a senseless jungle of pain. It’s the infinitely expandable diagram for radically alien numerical intuitions, and if you want to explore it you have to start with what comes naturally:
    (a) Compounding — familiar processes of multiplication (easily reversed for division, but only by deduction of factors (without remainder)). This is insanely convenient.
    (b) Implexion, or numerical indexing (from an ordinate to its designated prime) — the truly alien capability, tracking the strange geography of the prime sequence (insanely obscure).
    I’m hoping to suck someone with mathematical expertise into a more rigorously formalized description of how this works. Calculating in Xenotation is terra incognita.

    [Reply]

    raptros_ Reply:

    yes, if you allow the empty string to represent the multiplicative identity, then appending and sorting by wrap count has the right properties.

    converting to this notation is obviously going to be only as tractable as prime factorization is in general. i think that also explains why addition will not be at all easy in this system.

    in short, you can wander far out into the primes easily, but it’ll take you a very long time to figure out where you are.

    [Reply]

    admin Reply:

    “you can wander far out into the primes easily, but it’ll take you a very long time to figure out where you are.” — that’s a great formulation. It raises the question, could a (fantastically alien) system of arithmetical intuitions take the prime landscape as ‘home’, such that regular, additive adjacency was conceived as an exotically distant peculiarity?

    raptros_ Reply:

    ugh i started throwing together some scala so I could play around with this notation and now I’ve stayed up way too late trying to find a good way to implement a prime numbers stream.

    Posted on June 5th, 2013 at 1:52 am Reply | Quote
  • raptros_ Says:

    so yes I have put together a xenotation implementation which you can all play with if you want.

    the sky’s the limit! or at least, the time and memory constraints of your physical machine.

    [Reply]

    Alan Liddell Reply:

    @Raptros_
    Scala? Is that what the cool kids are using these days?

    @admin
    I’m working on an analysis of this. I’m not a number theorist, but this sort of thing is always interesting to me.

    [Reply]

    admin Reply:

    [rubs hands with glee]

    [Reply]

    raptros_ Reply:

    scala is good stuff. it’s a good thing if it’s the cool thing.

    [Reply]

    Alan Liddell Reply:

    Oh, don’t get me wrong, I’m not criticizing your use of Scala. You just happen to be the first person I’ve come across who uses it.

    Apropos of the discussion on computation, addition and subtraction would be miserable for sure, but multiplication and division would be easier, especially if (on paper) you lined up the terms or if, supposing you were to represent it as some object in the object-oriented-programming sense by an array of exponents of each particular prime. After all, you just add and subtract exponents.

    Speaking of exponents, admin, it would be ever so much more tractable if instead of just concatenating, we had exponents. I had something like this in mind: 4 = 2×2 => 0^^ and 8 = 2x2x2 => 0^^^.

    Of course, we could go another, more perverse, route and fix a positional rule such that a smaller number coming before a larger number implies that that larger is the exponent of the smaller. Something like 8 = 2^3 => 0(0). So when you have 7^6 (or something), you’d have to write (00) back to back 6 times.

    Come to think of it, there are lots of places for that to fail. Compound exponents are also right out. Better scrap it.

    Now, it’s telling (or is it?) that in order to do any real work, we have to go back to thinking in our mundane number system (that seems to be the point, actually). These are just some preliminary comments, I’ll try and formulate some more interesting thoughts later.

    admin Reply:

    @ Raptros_
    Damn! My proxy won’t let me into that link. Hopefully somebody will feel motivated to say what is going on there.

    [I’m especially interested because it should be impossible to code into Xeno without an ordinated matrix of the prime series (designating each prime first, second, third etc.), and the predominant part of modern cryptography relies on the fact that this matrix is irreducible to a formula — so I’m wondering how your program is working]

    [Reply]

    raptros_ Reply:

    yes, that is all correct. that is what my comment about time and memory constraints was all about.

    my code defines a lazy sequence of primes, using some decent techniques for filtering based on what’s already been computed. could be improved, of course, but frankly I don’t think anyone’s going to need to be able to evaluate expressions so far out that they’d notice the slowdown. (if anyone wants to take a shot at improving it, just send me a merge request on github if you come up with anything.)

    the rest of the operations are defined on that sequence of primes.

    [Reply]

    raptros_ Reply:

    oh i just realized – if you want I can export a zip of the current repo state and send that by email or something.

    [Reply]

    Posted on June 5th, 2013 at 5:23 pm Reply | Quote
  • admin Says:

    @ Alan Liddell
    Thanks for those remarks — already offering some thoughts to chew over. On ideas related to fiddling with the notation, the ‘next step’ element to be considered is lexicographic order (as in a dictionary-type sorting algorithm). Ultimately, the single greatest difference of xenotation — counter-intuitive (for us) — is that its lexicographic and quantitative ordering diverge, disintegrating familiar conceptions of the number line.
    That said:
    (a) Experiment is always good
    (b) Your idea for a semantic usage of xeno cluster syntax (ordering) might be ultimately impractical (in these specifics), but its a sign that you’re heading deep down the rabbit hole …

    @ Raptros_
    What would be required to operationalize your thingummy? Assume I know nothing at all about what you’re up to at a technical level (and you’d be right). It might waste less of your time if I just listen in for a while at first …

    [Reply]

    raptros_ Reply:

    what you’d need to do to get it running is basically install sbt to let you build the source and then get a scala interactive session so you can play around with the definitions the code provides. can’t really think of a useful interface beyond that.

    [Reply]

    admin Reply:

    I’m going sluggish on this until I’ve done some serious sleep recovery

    [Reply]

    Posted on June 6th, 2013 at 10:54 pm Reply | Quote
  • Tentative Joiner Says:

    Reposting the link for future reference: TX clock.

    [Reply]

    Posted on May 5th, 2016 at 9:08 am Reply | Quote

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