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integral math

Last post 03-27-2007, 2:26 AM by ralphweidner. 20 replies.
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  •  08-24-2006, 11:45 AM 5216 in reply to 5144

    Re: integral math?

    So, I find it helpful to distinguish between the use of math and the process of math. The use of math is entirely about semantics. If the numbers don't relate to anything, they can't be used. For the process of math, semantics helps humans understand what they're doing, but the meat of it is the syntactic relationships. Then when an actual system gives a consistent semantic analog for the syntactic relations of your axioms, it will also be analogous to your results, which is when math becomes useful.

    So the question is, can we isolate the process of Integral Math? It would quite likely need a vision-logic syntax, which I'm having a hard time imagining. But more than that, will it be possible to syntactically characterize perspectives in such a way that the results of the syntactic relationships yield information that was not already available?

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  •  08-25-2006, 12:13 PM 5324 in reply to 5216

    Re: integral math?


    hey yotam!

    one of the great things about integral, i think, is that we can pursue what we're both very interested in here, apart from what everyone else may be interested in--if something comes up that does interest them, then, of course, they can always jump in. and, hopefully, we will come up with a thing or two that will interest others.

    i like your distinction between 'use' and 'process'. the terms i'm used to are 'pure' and 'applied', but as you well know, that is an arbitrary distinction: pure math gets applied, and applied math, as a syntax, develops its own purity.

    i'm definitely intrigued by ken's references to semiotics in IS (i just wish i could get hold of the two excerpt, E and F?, that were never made public!)--in particular, his remark that a holon is a sign. i don't quite know what that means. i thought a sign was an artifact. and maybe it still is, but it can point to holons as well as artifacts. of course, the notion of a holon, looking at the map instead of the territory, is itself an artifact!?

    anyway, it's important to keep this in mind, because when we're talking about artifacts instead of holons, then we want to say its quadrivia, not its quadrants: artifacts are not sentient beings, and don't take perspectives, but sentient beings can take a perspective of an artifact, as we frequently do.

    i wonder how 'process' and 'use' relate to the three strands of knowledge. once a mathematical syntax has been created, then we can 'use' it with respect to an appropriate referent. the syntax define the injunction (the 1st strand). the corresponding semantic would be how we've come to interpret the results of that injunction (the 2nd strand). if we're not satisfied with what the result tells us about the referent (3rd strand), then it may be time to develop the syntax, or possibly even the semantic, further, in order to get a better result.

    'process' has to do with developing syntax, which is the heart of the mathematical profession. like you've suggested, it can go beyond, and become relatively independent of, 'use', and referents as well. but something surprising sometimes results. these new, supposedly 'use'less syntaxes can turn out to have referents not clearly seen before--that only come clearly into view by means of these new syntaxes. so mathematics can actually promote evolution. unfortunately (i agree), that's been mostly flatland evolution thus far, with referents almost entirely in the right-hand quadrants or quadrivia.

    an exception comes immediately to my mind: the socratic dialectic plato fostered in his academy. of course this was philosophy rather than mathematics, but mathematics, specifically, geometry, served as the model. the socrates of plato's dialogues would ask some willing interlocuter he came across in the athenian forum, basically, what his axioms, his postulates, were for some left quadrant referent such as 'truth', 'justice', 'beauty', 'the good', and so on. with as much help as he could get from the interlocuter, he would then, using form-op, began building the corresponding syntax, until, typically, the implied semantic was obviously not doing justice to the original referent. at this point, he would go back to the postulates, telling the interlocuter that they needed to improve on them, so that they could build a better syntax, and so on.

    socrates was, in effect, using mature vision-logic, it seems to me--not just early, pluralistic vision-logic--because each iteration of syntax was set up to be better, by the addition of hindsight, than the previous one.

    the evolutionary history of mathematics has stumbled along a similar path, without the special insight of a plato. for one thing, it hasn't been very integral about syntax and referent.
    mathematicians have tended to focus on syntax and ignore (exclude) any possible referent. potential users of mathematics have tended to ignore syntax in their devotion to their particular referent. the two eventually get together, but a more integral approach, i feel, could be much more productive.

    as an example, mathematicians developed new, non-euclidean geometries by replacing the parallel postulate by a couple of alternatives, without any particular referents in mind, it seems to me. it was only later, when einstein came along with a referent, relativity of space and time, that a use for their exotic syntaxes was found--i hope i've got the history right here.

    couldn't the whole development of geometry be organized into a single syntax by going beyond form-op to vision-logic as the cognitive level for that syntax? there would be all the different geometries, as sub-syntaxes, which could be related according to whether one developed out of another.

    and how would integral math fit in? i don't know. i'm still having difficulty understanding the referents and the semantic.

    ralph

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  •  01-26-2007, 8:40 AM 18676 in reply to 1395

    • gerardy is not online. Last active: 11-27-2007, 10:46 PM gerardy
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    Re: integral math

    Hi,
    Hmm.. these posts were quite a while ago, so I don't know if anyone will notice, but that's what I get for showing up late to the party.

    Anyway, I thought this was an interesting discussion, both here in the forums, and in Integral Spirituality. In the book, I was struck by the way the he actually uses the notation, and in particular, that you need to multiply the two addresses together to make a pespective, which is what he says is the meaningful bit. This reminded me quite a bit of the Dirac "Bra-Ket" notation in quantum physics. (Here's a Wiki article on this if you're not familiar http://en.wikipedia.org/wiki/Bra-ket_notation).

    Indeed, there seem to be some parallels between the Integral Math (IM) as described in Integral Spirituality, and the Dirac math of quantum physics. In both cases you have the interaction (inner product, injunction) of two abstracted forms (states, wave equations, Kosmic addresses) which brings forth the acutal 'real' data (quantum observables, AQAL perspectives).

    I don't know how far the analogy would stretch, or if a true isomorphism could be made, but it did strike me as at least quite interesting. Dirac notation is terribly useful in quantum physics as it saves a lot of calculation and simplifies things considably. It would be interesting if this IM could really be fleshed out into a real mathematics and whether one could discuss the interactions in an Integral Post-Metaphysics from a purely abstracted level and still say anything meanful or useful.

    Anywho, just an observation...

    (Post #1, Whoo Hoo!)
    'Tis an ill wind that blows no minds -Principia Discordia
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  •  01-27-2007, 8:30 PM 18759 in reply to 18676

    Re: integral math


    hi gerardy,

    you wouldn't happen to be a physicist ?!

    i'm not, but i have enough of a background that i've somehow been able to read up to ch. 24 of roger penrose's 'the road to reality' [sic: physical reality], which just happens to have something to do with dirac and quantum mechanics. i was about ready to set the book aside, but having seen your post, i guess i'd better at least continue through ch. 24.

    you raise an interesting question, one, unfortunately, that remains an enigma to me, even with your suggestion. have you looked at g. spencer-brown's 'the laws of form'?

    going back to basics, with each occasion the four quadrants tetra-arise. the view through these quadrants gives us four different perspectives associated with each occasion. but each is a perspective of something, and that something has four quadrivia (or quadrants, if it's a sentient being). so how do we express this in IM?

    any ideas? i guess i'd better review the relevant pages in IS.

    ralph



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  •  03-21-2007, 10:45 AM 20920 in reply to 18759

    • gerardy is not online. Last active: 11-27-2007, 10:46 PM gerardy
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    Re: integral math

    Hi Ralph,
    Yep, my cover is blown already... Really I'm more of an astronomer or "astrophysicist" as the vogue term is now, but yes, I have a degree in physics and work in a physics department. My specialty isn't really quantum theory, but I've sat through a few courses and have a fair understanding of the basics.

    I haven't read either of the books you mention but both are respected authors.

    As for how to deal with quadrants/quadrivia in this Integral Math (as distinct from the IM described in the AQAL Journal... I haven't had a chance yet to digest that article but it looks interesting. As does the Integral Science bits, though I think I may find some of that harder to accept... still chewing on it.)... where was I? Ah yes, quadrants, etc.

    My first, relatively blind, take on this would be to represent quandrants, states, and types as relatively orthogonal dimensions which the overall "bra" or "ket" 'vector' (actually a big ugly tensor) would represent. Similarly developmental lines would also be dimensions included, the 'values' of which would be some representation of the 'level' of each line. All of this would be folded under the cover of the overlying many-dimensional object. The interaction of these "bra" and "ket" things would thus form some sort of linear algebra, the details of which would of course be terribly messy in practice. I'm just curious if the abstraction could produce some interesting results that don't depend much on the details of the messy perspectives themselves... The answer of course may be "no, not really".

    -Chris
    'Tis an ill wind that blows no minds -Principia Discordia
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  •  03-27-2007, 2:26 AM 21111 in reply to 20920

    Re: integral math


    hi chris,

    my guess is no, but it's still interesting to see why. the reason i guess so is that ken is not using the term 'dimension' to indicate physical dimensions except in the special case of the RQs. as he says, we cannot weigh a thought or a feeling or measure them in any other physical way. the quadrants are said to be equivalent dimensions of any occasion, but equivalent in the sense that they co-arise: no one takes precedence; no quadrant can be reduced to another. they're not one, and they're not two, that is, they are also not dual to each other: no chance for a bra/ket interaction.

    ralph

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