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integral math

Last post 03-27-2007, 2:26 AM by ralphweidner. 20 replies.
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  •  07-13-2006, 1:21 AM 1377

    integral math

    believe it or not, my first post here (#580 in 'let's get it started')did me alot of good. i'm not quite sure about the rest of you, though.

    i've been stewing over integral math since it appeared in excerpt C over three years ago now. it's so unlike ken to be obscure, but that's the way the integral math appeared to me, and i'm supposedly a math phd. when it reappeared in his inaugural address for ISC a year ago, i began to ask around, but no one could enlighten me.

    more recently, i heard ken using it in his dialogue with mark edwards, i saw some of you using it in some of the posts here for i-i commons, and it appears in slightly revised form in ch.1 of integral spirituality. and it still doesn't make sense to me.

    so it's time to get serious about this. any of you with some insight into this are more than welcome to help out here. i'm going to lay out what i understand in hopes that one of us will spot what's missing, or where i've gone astray. who knows? maybe we will catch ken in some horrible mistake?!

    that a math phd is clueless should clue us that this may not be so much about math as, perhaps, wilber-phase-5? so let's begin there.

    well, first phase-4, just to give us a running start. holons: challenged in every moment of their existence, and it becomes more complex, if not more challenging, once they've developed into modern holons. then, it's tetra-enact, or be erased from the Kosmos.

    so, now, how do we get from four quadrants to eight primordial perspectives? from enaction to taking perspectives? in taking a perspective of something we enact that something, don't we?
    and vice versa, in enacting something, don't we, in fact, take a perspective of it?

    what somethings, what entities can we take a perspective of? of a holon? of an aspect of a holon, say its interior, collective aspect? or just its collective aspect? of a single zone?

    not much of a start, but i'll keep working on it.

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  •  07-13-2006, 3:48 PM 1395 in reply to 1377

    Re: integral math

    I'll throw my two cents in here, if I may. (Though I think I may run out of cash soon, at this rate)

    First, thought, a gripe I've been carrying around for a year or so. Isaac Newton (and Leibniz) invented Integral Calculus. Ken Wilber invented something else!!!

    Now, I realize that he's stopped calling it Integral Calculus, but I haven't had a chance to whine about that to anyone who would care, and I can probably stop worrying about it now. Sometimes you need to just freak out and then let go.

    [deep breath]

    Okay. As for the actual substance of Integral Math - my understanding goes something like this (I'm gonna be really pedantic, not out of disrespect but because spelling everything out helps me be sure I know what I'm talking about) (I'm also mostly basing this on the use of Integral Math in Integral Spirituality. It's been a while since I've read the Vol2 excerpts on the subject, but I'll try to do that soon):

    I think it begins with the nature of second-tier thought. At that level, one begins to compare perspectives and integrate them with an awareness of whence the perspective is being taken. In order to formalize this process (the essence of math being formalizing thought processes), we need a way of describing perspectives, and perspectives of perspectives. Right? We need language to describe, not only what a person sees, but how they are looking, and then we need to describe those descriptions.

    The first piece to distinguish is which quadrant of the holon you're looking at (we mostly seem concerned with looking at holons). So you get the last term of an integral math string (1p, 3p, 1p pl, or 3p pl). This amounts to interior (1p) or exterior (3p), and singular (which goes unmarked) or plural (pl).

    Then Wilber basically says that for object of study (any object of which you're taking a perspective) you can either consider it "from the inside" or "from the outside." This is where it gets messy, though. Because inside and outside seem to mean different things depending on what we're looking at. So in the UL, inside means being the conciousness you're studying, and outside means being anything else. LL inside means being one of the people in the community, outside means observing the community without participation. And honestly, I don't completely understand inside and outside of the UR and LR. I believe that inside of UR means studying what information is going into the object, so to speak.

    And then once you have such a perspective, you can begin to study that perspective. And you can do this, again, either from the inside or the outside. This is the part I understand a little better. Any time I'm looking at someone else looking at something else, I can either study how they are looking or study what they see by looking. This is the outside or the inside of the perspective. And thus we build outward, kind of recursively. It's much like binary, so far. You take a given perspective and either stop, 3p it, or 1p it.

    But here's the mess. This, so far, is all notation. It isn't math. In some of the excerpts, Wilber starts using an = to denote resonance between perspectives, and with that you can start to get statements, as opposed to just nouns, but this is still very rough.

    In order for it to become the sort of math that you and I love, Ralph, it would need at least an axiom or two. Presumably we have the operation of "taking," but there could well be others. I hear rumors that Wilber does things on his own secret legal pads like integrating a bunch of perspectives to get another one, but I don't know of anything like that publicly available.

    It also seems to me that some way of denoting altitude would be helpful. A 3p x 1p x 1p of an action or person will differ significantly according to who is taking the perspective, and from what altitude. We could throw in some axioms about transcluding, and others about what happens when a perspective from altitude a takes as its object a perspective from altitude b when a > b, a < b, etc. But again, Wilber hasn't given us anything like this. What he has given us is a shorthand for noting to what extent, and in what way, the subject is associating with the object, and to what extent and in what way the subject is distancing itself from the object. And that is all.

    This is useful, surely. I enjoyed whipping out some integral math notation in a conversation here, but its main application so far is just to distinguish how different perspectives differ. The day when we can deduce SES on the back of a napkin are far, far away.

    If you've read the stuff about Kosmic Address and Integral Post-Metaphysics, we can have a lot more fun. Ideally one would abstract from any given level/line system, because as we've discussed they're a bit arbitrary in their divisions (Ken often mentions that you could use 30 levels, but he'll just stick with 3 or 5 or however many he needs), but you could start with a few basic axioms about relations between objects of different kosmic addresses and the same physical address, and do some very interesting work, I think.

    I'm very tired as I write all this, so I may need to edit it later and trim off a little speculative fat.


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  •  07-14-2006, 9:16 AM 1415 in reply to 1395

    Re: integral math

    On the other hand, this is really cool.
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  •  07-28-2006, 7:13 PM 2319 in reply to 1377

    Re: integral math

    Is it possible that there is Flatland because numbers are flat? The set of all numbers can be depicted on a two-dimension graph which would transcend and include in higher succession: natural, whole, integers, real and complex.

    There is no consciousness in mathematics. For example, if we are at a pizza shop and you ask to pass you the oregano, pepper and garlic how I pass it to you counts in our consciousness but not in math. That is, in math, 1 +2 = 3 which is indistinguishable from 2+1=3.  In the real world I pass you first the oregano, and then I pass you both the pepper and garlic together (1+2=3), and you have all three. I didn't pass you first the pepper and garlic together first, then the oregano (2+1=3) but flatland algebra says they are exactly the same where to us there was an obvious difference.

    To bring consciousness to flatland there has to be another type of number that transcends and includes complex numbers, which would inflate the two dimensional x-y plane of numbers into a three dimensional x-y-z cube that would take into account our experience at the pizza shop, namely that 1+2 is not equal to 2+1.

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  •  07-29-2006, 11:58 AM 2360 in reply to 2319

    Re: integral math

    hi large constant!

    i hope you aren't saying you're flat!?

    numbers didn't used to be flat. think of the pythagoreans! maybe numbers have been flattened by flatland.

    yes! mathematics is not a sentient being, and has no consciousness. but it can be a wonderful artifact. it can tell us, for example, that no matter how the spices are passed to us, 1+2 or 2+1, we're going to get all 3, and that's nice to know, isn't it?

    but anyway, you've come to the right thread, an integral math of perspectives, and not numbers. welcome!


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  •  07-29-2006, 11:49 PM 2402 in reply to 2360

    Re: integral math

    I'd like to continue the point that numbers are only as flat as we make them. Consider:

    You really only measured the number of spices that Ralph had at a specific time. If you were instead to offer a graph of spices over time, it would demonstrate the difference between 2+1 and 1+2 quite nicely.

    Alternatively, you made a choice to lump all spices into a single category. You could instead simply count how many containers of oregano Ralph has, and get a different sort of information. If you graph garlic, pepper, and oregano posession as their own dimensions on a 4d graph, you'd have all the information you could want and more.

    Alternatively, you could ignore the containers, and create a number measuring the quantity of garlic near Ralph, weighted by proximity in some way (amount of garlic at distance d divided by d, or some such) and notice that it spiked just before the pepper-near-Ralph number spiked.

    Any of these highlights some information and ignores some information, but so does any sentence in English. My last suggestion would give much more information about the quantity of garlic you passed to Ralph than your description did, because any  description, mathematical or not, makes decisions about what information to include and what to ignore.

    Unfortunately, just as I finished writing all this and was about to bring the point home about how non-flat numbers can be, I realized that numbers are rather useless when it comes to interiors. Anything on the right side of the holon map can be talked about numerically as richly as you like, but it's pretty hard to put numbers on interiors, except by extention from their exterior correlates.

    So now I'm not so sure anymore.

    On the other hand, Integral Math might be rather useless to describe exteriors. So they just might be a perfect match.

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  •  08-07-2006, 2:34 PM 3660 in reply to 1395

    Re: integral math

    yotam, nothing necessarily the matter with speculating. who knows? it might help me.

    i'm still completely lost. i've looked over the math posts from peter, balder and you, and you might as well be three blind men, as well as i can tell. but you at least have provided me some ideas to work with.

    i don't think the problem for any of us is the math so much as what the math refers to. i know i'm not clear on this, and i suspect none of you are either. so, as alice was advised, let's begin at the beginning: what is the kosmos made up of? holons, as in SES? i'm fine with that, but i vaguely remember the excerpts saying it's perspectives--that before you can have anything else, you first have to have a perspective. i'm not fine with that, that is, i don't get it. to take a perspective, don't you first have to have an injunction, an exemplar, a paradigm that enacts a world that you can then apprehend in taking a perspective? and who is this you who is enacting a world and then apprehending it? isn't this a holon?

    ok. a holon only arises from the event of enacting a world and taking a perspective, but it arises out of the holon that was before, the holon that was enacting and taking a perspective!?
    so it seems to me that the (manifest) kosmos is made up of events, or occasions, which are each comprised of an enactment and an apprehension by a holon, resulting in a new holon. holon, enactment and apprehension are necessary components of any event. taking a perspective (apprehension) implies that all the other elements must be there as well, but is itself only one element in the whole shebang.

    on the other hand, if we think of a holon as having some sense of self, then, even though it is transcended in an event, it can feel that the event was just something it experienced, along with enactment and taking a perspective, and it continues as basically itself. am i getting into the same trouble here that those who equate the four Qs with the persons i,we,it,its do?

    then there's the question of altitude, which you bring up. again, i feel it's simplest to see this as just one more aspect of an event. if we focus on the perspective, it will have a definite altitude, of course. and that's better than focusing on the holon, in this regard, since it will have a range of altitudes available to it (hopefully, i'm now engaging my higher self altitude!). however, again, it's the holon at the center of things, it seems to me, in deciding how to tetra-enact, for example, and what perspective to take.

    it could be that i'm simply reflecting here my own particular altitude, and someone at a different altitude will see this entirely differently, and we're both right, within limits. maybe ken is leaving this ambiguous so that we can each feel free to give our own interpretation to this!?

    so, what about the different possible altitudes? i personally feel thay go all the way down: less and less depth, but definitely some interiority, however shallow. if super-strings are real, then they can't be all exteriority, even if that's all we would be able to detect at this moment in history. how could a kosmos of only exteriority have possibly evolved into one with interiority? (if you're at an amber altitude, you're not allowed to answer that question!)

    anyway, it seems to me that a person who hasn't yet reached an altitude of orange is, by definition, incapable of the four distinct perspectives associated with the Qs. he will have essentially only a single quality of perspective whether he's introspecting or observing some 'it'. we can still talk about his 3-p and 1-p, but we need to keep in mind that they are, by definition (i.e. the map, not the territory), fused.

    my guess is that there are similar limitations on the 8 indigenous perspectives. we are only able to differentiate these in our own taking of perspectives, once we have reached a green altitude. and, coincidentally, we are only able to recognize that we can do this, once we have reached a teal or turquoise altitude, and have read ken wilber!?

    i better stop at this point to see what you or any others might say about this.

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  •  08-07-2006, 3:29 PM 3667 in reply to 3660

    Re: integral math

    Wow. Okay.

    I don't really follow a lot of what you're saying in the middle there. I guess that's sorta how it goes. You seem to be stuck basically in the chicken/egg issue of perspective and perspective taker. Is that about right?

    Let me see if I can express my understanding of how the universe could be made of perspectives in a way that isn't gibberish. The moment it made the most sense to me, I was thinking about it like this:

    There are (at least) two things we simply cannot apprehend. One is awareness itself - we cannot take our own awareness as an object of awareness except indirectly and skewedly. The other is what Kant called noumena, the actual reality of any object. No matter what, my understanding of the keyboard/spoon/apple/friend imperfect, even if I know it pretty well. It's skewed by the lenses of my perception and understanding.

    But notice that all of the action takes place between those two. It's when awareness meets object-as-it-really-is that we get experience. The whole world we live in, the world we experience, is stuck between those two. I think of them like the bounds on an open interval. The interval itself is where we live, and we can only hint at its endpoints.

    So here's where it gets weird - the endpoints don't exist. They aren't real. Let's call "real" what we can experience, what we can know. I summed it up, in a blazing moment of understanding, as "the world is the world percieved." What I really meant was "the world is the world perspected." REALITY, in the ontology I'm proposing here, the ontology I think Ken is proposing, is the world of experienced-object. And such things are real in as much as they are perspected. The reality of something IS, sorta, its perspectedness. This is its perspective of itself as well as others' perspectives of it. If a tree falls in a forest, the tree knows it fell. But it's in being known that the tree is real. So at the end of the day, the basic building block is the perspective. The act of taking a perspective of something (what I've been calling perspecting it) is participating in its co-creation.

    Does that make any sense?

    It has trouble. I'm still wrestling with it. It's one of those things where I know I've got the core, and I'm still working on some details. I thought this stuff up a few months before reading IS, and when I hit Integral Post-Metaphysics I literally cheered, because I knew that he was talking about what I had thought of. (I was also pissed at him for publishing first, but that happens :-) ).

    Just remember that inherent in being is being-perspected. And there is no truth of the being, other than truth-in-perspectives, so at the end of the day, the perspective is paramount. And not one perspective, but all.

    But that's all a step beyond Integral Math. I think the math that Ken uses in IS is just to help us differentiate the ways of looking at an object, and the ways of looking at a perspective. And, again, until it has operations and relations and axioms, it isn't really math, it's shorthand.

    I've never been sure about whether the quadrants can be seen from all levels. I know that it takes a certain level of cognition to recongize the irreducibility of system to parts (lower quadrants to upper) and vice versa, but I'm less certain about interiors vs. exteriors. Gene proposed explaining quadrants to a 5 year old with "you can see your hands and feet and tummy but you can't see your feelings, that's your inside and your outside.  everyone can see you and your family when you go places with them, but only you know what it's like to be a part of your family.  that's like an inside an outside of your family," and I don't know if that would work. At the very least, the quadrants must appear differently from every level on every line. I think that recognizing the mutal irreducibility but integrability of the 8 primordial perpectives takes some minimum altitude, though I don't know what it would be.

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  •  08-11-2006, 10:46 PM 4150 in reply to 3667

    Re: integral math


    if we want to avoid that detestable 'myth of the given', especially 'the view from nowhere', don't we need to factor in the perspective we bring to this discussion?

    if i understand you, in that first contraction from the oneness of nonduality, we get that first duality of awareness and noumena, i.e. Emptiness and Form. i have not consciously achieved nonduality nor the causal duality of Emptiness Witnessing Form, but i know of them both, somewhat, by description (as 3p). So what you say makes sense for me and, i guess you're saying, for you as well. our perspectives have never been from Emptiness, nor of Form. rather they have been from the manifest form, the holon we were, with whatever creative emptiness entered into that occasion of taking a perspective, and of some manifest, partial form, some noumena. our experience of taking this perspective is phenomenological, and not noumenological. what we see is according to who we are and not just what noumena there is to be seen. both of these forms, who we are and what is to be seen, contribute to the perspective. in the limiting case of pure Witness, they both become Form. in that case the Form we are supports and enables the Emptiness of Witnessing the Form that is. becoming conscious of nonduality is then simply recognizing that who we are is what we see?

    so i would say that noumena are the objects of the perspectives we take, but because of who we are, as subjects, is only form, and not Form as in pure Witnessing, what we experience phenomenologically is related to but not the same as the actual noumena. i think we're pretty much agreed up to this point. but even though i haven't gone there myself, i'm willing to trust ken, and others, that it's possible to go beyond where we've been all the way to what i understand, although he never describes it this way, as consciously experiencing nonduality. so, for me, what is (ontology) is not limited to what i've been able to see, but comprises Form and Emptiness, in a word, Nonduality.

    perhaps more in tune with your view, i just saw a replay of lily tomlin quipping 'reality is nothing but a collective hunch.' a good poke at 'the myth of the given'.
    yotam, i went back and looked at my previous post, and i have to agree with you: it's hard to follow. i'll try to do better.

    about altitude: a stanford professor who's name escapes me believes that the first humans to emerge in evolution did so with a language from which all subsequent languages have developed. if i remember correctly, pronouns were already in place. so there was at least an unconscious recognition of the quadrants from the very beginnings of our species.

    i think what gene said would work, but the 5 year old might interpret outside and inside as visible and invisible. she would say her heart is inside. i imagine she would also feel that any other 5 year old who did not know what it's like to be a part of his family as she did hers must be dumb, assuming she hadn't yet gotten beyond narcissism. but i liked very much what he said, especially about altitude.

    the differentiation of the quadrants is a remarkable accomplishment. it makes sense to me that it would require at least form-op thinking. to distinguish between what i feel and what the facts are can sometimes be a real challenge. in those moments, i guess, i feel the facts must be reducible to what i'm feeling. boomeritis triumphs again!

    i just read ch.7 and it's great! see you there,


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  •  08-15-2006, 10:29 AM 4358 in reply to 4150

    Re: integral math

    Hi Ralph,

    I never looked at the post-metaphysics stuff from the perspective (even hypothesized) of nonduality. I like that. I was just working from a Kantian kind of bit, acknowledging that our conceptions of reality are filtered through our limited perceptions. Let my try to phrase it in your language, though.

    The real essence of the idea, which I think you're getting, is that because there is never, ever, ever, form without emptiness, form is always form-in-emptiness. Noumena is always noumena phenomenized. And so built in to any object is the perspective(s) taken of it, and there is nothing of the object that is not perspected. Hence, deep down inside, the world is made of perspectives.

    You've won me over about the 5-year-old. It does take a certain age to get that other people don't know what it's like to be in your family. And it even takes a certain maturity to accept that those things adults talk about which you don't understand aren't stupid. So, yeah, I guess you need a certain altitude to see the quadrants.

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  •  08-16-2006, 10:34 AM 4438 in reply to 4358

    integral math?

    Attachment: blonde_equation.jpg
    integral math?
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  •  08-18-2006, 1:27 PM 4604 in reply to 4438

    Re: integral math?

    it looks like blondie math to me--straight from the big apple.

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  •  08-23-2006, 12:00 PM 5088 in reply to 4604

    Re: integral math?

    time, again, to get more serious:

    before we can effectively say what integral math is, we need to come to some agreement about what math is. i would think, yotam, that this would be easy for you and me, but maybe not.

    what is math? (the title of a great book by richard courant) if i understand ken correctly, it's a zone 2 methodology, pure math, that is, requiring at least a form-op altitude of consciousness. as practiced heretofore, it's been, in fact, a form-op methodology, hasn't it?

    for example, in euclidean geometry, we deal with interior objects such as points, lines, triangles, etc. these are UR signifiers pointing to UL signifieds that have LL meanings, at least for some of us, governed by LR syntax, formulated, for example, in the elements of euclid, and that syntax is form-op based: it is logical, as is any other, conventional, mathematical syntax. referents for its interior objects can easily be found in the exterior, orange world of the UR. for example, i'm sitting at a table whose edges suggest the notion of a (straight) line, its corners, that of a point.

    referents for other mathematical theories can be much more abstruse, but they can always be found, as well as i know, in the UR. actually, this can get a little tricky. the french mathematician rene' thom's controversial work on morphogenesis came out in the early 70's when i was a graduate student. he was attempting to, at least, partially model biological development, which we know is non-deterministic, in fact, creative to some extent.

    still, we know that an acorn can only become an oak tree, even if we can't predict exactly what oak tree it will become, in terms of shape, size, colors, etc. i think the way ken puts it is that the acorn, the subholon, sets the possibilities of what the future oak tree, the holon, can be. in these terms i think we can say thom was attempting to model to some extent the limits of those possibilities.

    even though i didn't know much biology at the time, i felt thom's morhogenetic syntax seriously failed to do justice to the referent, that is, biological development. so, even though it was a fascinating theory, it didn't make sense to me, in the absence of a viable referent, to pursue it.

    biological development, at least in lower forms of life, has been around for hundreds of millions of years, and, consequently, may have settled into fixed enough habits that it could be modeled in a partially deterministic (topological) way like thom attempted to do. the englishman d'arcy thompson investigated the mathematics of biological form, and, along with the biologist waddington, actually inspired thom's efforts, as i remember.

    the point, though, is that we're still dealing with referents in the UR. when we come to integral math, the referents are perspectives which, along with the requisite sentient beings, are prior to any particular quadrant. they define, in fact, the relevant quadrants and aspect of quadrants, inside or outside. as well as i can tell, ken is providing a classification scheme for the basic perspectives, which humanity has thus far systematically practiced. the four quadrants was from a modern point of view. the eight indigenous, primordial perspectives is from a more postmodern (developing out of and elaborating the modern) point of view. the modern point of view required form-op, or the capacity to take three perspectives (a la loevinger). the postmodern point of view requires early vision logic, or the capacity to take four perspectives. the integral or mature, vision-logic, point of view requires five in order to be able to take a perspective on a point of view, the postmodern one, that requires four perspectives in itself. the integral point of view could be called a meta-perspective, but ken says, i think, that he's gone beyond this, so maybe we better not use the term meta-perspective (even though he's called integral theory a meta-theory, and integral methodological pluralism is clearly a meta-methodology), because then there would be meta-meta and meta-meta-meta perspectives!

    at any rate, the point is that the referents for integral math aren't located in something as obvious as the UR. moreover, they can require more than a form-op structure of consciousness, so a form-op syntax may not even be capable of successfully representing them.

    what do you think?

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  •  08-23-2006, 1:13 PM 5093 in reply to 5088

    Re: integral math?

    First reaction only:

    My temptation is to say that perspectives take place in the UL, not everywhere. Because while the act of taking a perspective is certainly a four quadrant affair, the perspective itself, or the understanding that results from it, strikes me as primarily UL.

    I do agree that math is a form-op thing, especially, as they say, "formal math." And I agree that the objects being described are primarily Rq objects (though it is both UR and LR). But I'd like to emphasize that once the syntax and axioms are created, the math becomes effectively divorced from the objects originally being described.

    I think that in order to call something math, it needs those properties. It has to be a formal system, and it has to be governed by syntax, rather than semantics. The reason is that syntax is universal, in that it's self-contained. Semantics are too influenced by the cultural context.

    It seems possible that a formal language could be developed to describe vision-logic, instead of just form-op cognition. I'm not sure.

    It also seems possible that a formal language could be developed to model interiors rather than exteriors. That's what I hope Integral Math will be.


    Separate thing: Ken has mentioned that his Integral Math is based on the work of G Spencer Brown. Do you know anything about him?

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  •  08-23-2006, 10:02 PM 5144 in reply to 5093

    Re: integral math?

    my immediate reaction:

    taking a perspective is something only an individual holon can do, isn't it? and isn't it necessarily a tetra-enaction that effects all quadrants? even if it's just a perspective of the inside of the exterior of a social holon, say.

    you're right: referents for conventional math can come from the LR as well as the UR, and the math itself is divorced from the referents and is all syntax, but it yields meaning, a semantics, which that syntax points to. it is a formal language, isn't it? conventionally, one in which form-op is used to form the syntax. what if vision-logic, which includes form-op, is used instead? instead of the eternity (timelessness) of form-op, there would be, what?, evolution in time? what would that mean?

    imo, integral math includes exteriors as well as interiors. it's aq.

    g spencer brown? if i ever heard of him, i've forgotten. i looked him up, and the relevant work of his appears to be 'laws of form'. predictably, the portland library system had a copy up until two years ago, when it wasn't returned. it's a little expensive to be buying in hopes that it would enlighten us about integral math, but don't let me stop you.

    anyway, yotam, i enjoy discussing questions we don't know the answers to with you. i hope your internship continues.


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