INTRODUCTION Page 1 Page 2 Page 3 PART II Page 1 Page 2 Page 3 Page 4 PART III Page 1 Page 2 PART IV Page 1 Page 2 APPENDIX A APPENDIX B NOTES Notes 1-15 Notes 16-35 Notes 36-44 Notes 45-56

Appendix B. An Integral Mathematics of Primordial Perspectives

     Let me start this overview by repeating the paragraph in the text where integral mathematics is introduced:

     If we call this first event horizon a "first-person experience of first-person realities," we could represent it as (1p x 1p), where "1p" means "first person." That is the beginning of an integral mathematics based not on variables but on perspectives. For those interested, I will pursue this integral math in Appendix B and a series of ongoing endnotes. Honest, you don't have to follow this; it is simply a notional system useful for reminding us to honor all indigenous perspectives; but if mathematical notation is not your cup of tea, feel free to skip these indented sections and pick up the narrative--all you need to know is represented in figures 2 and 3.

     Briefly: "1p" is first person, "2p" is second person, and "3p" is third person--meaning actual but nonspecific persons--and "1-p" with a hyphen means a "first-person" perspective, whether that perspective is being taken by a first, second, or third person relative to the person making the assertion (and likewise "2-p" and "3-p"). Thus, for example, if I make an assertion, I would say that I am the first person (1p) speaking to you (2p). But your second person obviously is also an "I" or a first person from your vantage point; therefore, if I am making an assertion about you, and in order to honor your perspective, I would have to say that your second person has a first person: 2p(1p)--which means, the first person of the second person I am talking with. Likewise, "he" is not merely a third person (3p), but a third person who is also aware and prehensive--3p(1p)--which means: the first person of the third person I am talking about.

     Thus, if I am aware of you, it is not merely 1p x 2p, but rather, my first person is aware of you, which has its own first person: 1p(1p) x 2p(1p), which means, my first person is speaking with your first person.

     Of course, I can approach your consciousness as a subject in dialogue or an object to be studied--that is, I can be aware of your person in either a first-person (1-p) mode or a third-person mode (3-p), among others--thus, 1p x 1-p x 2p, for approaching you in a first-person mode, and 1p x 3-p x 2p, for approaching you in a third-person mode. Expanding each of those terms in the way just mentioned--where you are not just 2p but 2p(1p)--would give us: 1p(1p) x 1p(1-p) x 2p(1p), which means: I have a subjective view of you, or, spelled out: my first person knows, in a first-person mode, your first person; and 1p(1p) x 1p(3-p) x 2p(1p), which means, my first person knows, in a third-person mode, your first person, which would simply mean, I am seeing you in an objective fashion, I am taking up (or trying to take up) a third-person stance towards your first person. On the other hand, if I were a scientist trying to study you but only in a third-person mode, that would be, 1p(1p) x 1p(3-p) x 2p(3p), which means, my first person has a third-person view of your third person (or those aspects of you that are only objective and can be studied as an object, such as your mass, energy, biochemistry, etc.).

     Now, my first-person view, understanding, or interpretation of you, 1p(1p) x 1p(1-p) x 2p(1p) (which means, my first person has a first-person view of your first person), may or may not match your view of yourself, 2p(1p) x 2p(1-p) x 2p(1p) (which means, your first person has your first-person view of your first person). If those two perceptions do match, then we have

     1p(1p) x 1p(1-p) x 2p(1p) = 2p(1p) x 2p(1-p) x 2p(1p)

     which means, within the elements indicated, my first-person understanding of your first person equals your first-person understanding of your first person. This is called mutual understanding.

     The real world is not built of variables over domains whose operations can equal each other in a third-person mode, but rather of perspectives of sentient beings whose mutual reflections can resonate with each other. An integral mathematics of indigenous perspectives is meant to be a notational system for the real world, which is an Indra's Net of harmonic resonances among sentient beings prehending each other endlessly, and not a grid or lattice of third-person rocks clunking around in geometric space.

     Needless to say, this notational system can become quite complex quite quickly. It appears to be an entirely new form of mathematics that, of course, could take years to develop in its full dimensions. I am probably about 3% into this new landscape, but a few very arresting items have already surfaced (I've run around one hundred pages of equations so far, just to see what might be involved). Every now and then in the coming months (and years) I will post or publish a few excursions into the mathematics of perspectives, for those interested in such. Of course, the abstract portion of mathematics is notoriously a young male's game (the average age of the discoverer of break-through mathematical insights is 23: abstractions backed by raging testosterone seems to be the ticket here). But perhaps I can shed enough light on the initial stages of a sentient mathematics to get others started; and in giving a middle-aged version, it might be in a form diluted enough to be vaguely interesting to others who are also at something of a less-than-mathematically-zenith pitch.

     To begin with, and with reference to the earlier equations, we need to add singular and plural forms of each of those terms (e.g., first person plural is 1p*pl ["we" or "us"] and hyphenated first-person plural is 1-p*pl ["our"]). For example, "I think we agree that George is a fine person," one version of which is: 1p(1p) x 1p*pl(1-p*pl) x 3p(1p), which means, my first person has a perception of our (first-person plural) perception of George's first person. Naturally, you have your understanding of how we see George, 2p(1p) x 1p*pl(1-p*pl) x 3p(1p), which means, your second person has a perception of our view of the third person's first person. If you agree with my view of how we see George, then:

     1p(1p) x 1p*pl(1-p*pl) x 3p(1p) = 2p(1p) x 1p*pl(1-p*pl) x 3p(1p)

     The whole point of this type of mathematics is that, in the real world, holons (sentient beings) are connected to each other not merely by exterior topographical surfaces that can be represented as quantitative variables and abstract operations in mathematical equations, but by interior resonances, felt meanings, and shared perspectives, which can be represented by overlapping circles of qualitative event horizons suggested in the terms of an integral calculus of indigenous perspectives. Integral math is simply a tracing of what happens when sentient beings touch sentient beings: there is a first-, second-, or third-person perspective on first, second, and third persons indefinitely: a 123 of a 123 of a 123 of a 123..., which is why the Kosmos is constructed of perspectives, not perceptions, not events, not processes, not webs, not systems, for all of those are perspectives before they are anything else.

     Moreover, it appears that when those multi-person terms in the integral mathematics are all merely third-person terms, then in many cases the integral calculus collapses into typical abstract spaces captured in the various mathematics of surface representations (from Boolean algebra to differential calculus to imaginary numbers).

     What is so interesting about this is that the pronouns in language itself--which capture the reality of first-person, second-person, and third-person perspectives (e.g., I, we, him, her, she, they)--actually embed a universal integral mathematics in their own structure. A pronoun is not an actual person but a relative perspective that all actual persons can adopt. In the real world, I cannot be communicating to somebody if there are no second person anywhere; conversely, any time I take up a stance, a perception, a feeling, a view, an awareness, it is always already situated in relation to other actual sentient beings. These relationships are captured in pronouns, which, as the word itself suggests, are pro-nouns or even pre-nouns: something prior to nouns that all nouns must follow. The pronouns refer to positions/perspectives that sentient beings inhabit by virtue of existing only in a world of other sentient beings. The pronouns do not refer to actual people--they do not refer to John, Paul, George, or Ringo--but to the perspectives that all proper nouns (John, Paul, George, or Ringo ) have available to them, universally.

     A Nepalese has a first-person stance just as much as a New Yorker does. If a wolf is signaling another wolf about a prey they are hunting, that wolf is necessarily in a first-person stance to the other or second-person wolf about the prey (or third person). If a bacterium is signaling another bacteria using chemical messengers, that already is a first-, second-, and third-person situation.

     Following Peirce, I have in the past defined a sign as "any aspect of reality that stands for another, to another." What that actually means is that first-, second-, and third-person realities are built into any communication system whatsoever, all the way up, all the way down. In other words, there are no realities that are not always already perspectives.

      Because these perspectives, as they are captured in the pronouns of language, are abstractions to some degree (namely, "I" does not necessarily mean Ken Wilber, but any subject anywhere; and "he" does not necessarily mean Sue, but any third person anywhere), then language already embeds a universal mathematics. The relations among pronouns are relations among sentient beings wherever they arise. They are universal in that sense. These relations are therefore abstract, in the sense that they are not tied to any specific noun, and so they embed a universal or context-transcending aspect; but they are abstractions that only make sense or have content when inhabited by actual sentient beings. Call these abstract relations "demi-abstractions." The point is that natural languages have secreted within them a series of demi-abstractions that are a mathematics of the real world.

     These relations (or demi-abstractions) are the primary domain of an integral mathematics. This is a naturalistic mathematics that, in keeping with a post-metaphysical sweep, lends itself to real Kosmos representations, not dissociated ontological structures.

     But, as early suggested, if 1p and 2p are all collapsed to 3p, then the typical flatland, abstract spaces dealt with by ordinary mathematics come into view. In other words, the whole of typical abstract mathematics seems to be a limit case of an integral mathematics when the positions of the integral math are gutted of sentience and represented in their third-person dimensionality only. As usual, I am not saying those third-person dimensions are not there or are not real; they are merely one slice of a bigger Kosmos.

     One of the most interesting items about this Kosmic mathematics, or a mathematics of primordial perspectives, is that, in order to make an assertion, one must state the relation of the assertion to the sentient being making the assertion. That is to say, it is not merely that a first person sees a third person (that is still monological; that still embeds a prejudice that you can have perception or feeling on their own; but there is no such thing as feeling, perception, awareness, or interpretation--those are always already within perspectives). Integral math forces us to be honest about this. We have to say, not that a first person sees a third person, 1p x 3p, but that my first person sees a third person, 1p(1p) x 3p; if that third person is sentient, then I am seeing a third person who is also a first person in his or her own right: 1p(1p) x 3p(1p), and that third-person first person is a third person only in relation to me, who is making the assertion.

     In other words, I must always situate assertions in relation to the speaker of the assertions. This is what makes integral mathematics so novel, and also what prevents demi-abstractions from spinning out of the real world and into merely abstract or third-person third person realms, 3-p x 3p(3p)--not that those aren't there, only that they are third-person surfaces of the rest of the world.

     As one begins "running the equations" of this integral math, even if you trace out elaborate relations of perspectives to, say, the 7 ^th person degree (which is as far as I have taken it, and probably can take it), nonetheless ALL of the equations remain situated with reference to the first person making the equations or claims. If I am attempting to trace what that fourth person said, or that fifth person, or that sixth person, I must still assert them in the form, for example: 1p(1p) x 5p(1p), or my first person perceives the fifth person (who has a first person in his or her own right). Never will the integral math allow me to make an abstraction, only a demi-abstraction, and thus all universals are always already situated among sentient beings, who never have perceptions or feeling or consciousness, but only perspectives within which perceptions and feelings and consciousness arise.

     In the above equations, notice that we had three basic terms--for example, my subjective opinion of you: 1p(1p) x 1p(1-p) x 2p(1p), which means, my first person has a first-person perception of your first person. In each case where the integral math is taken to three terms like that, it turns out--in retrospect, or after the fact--that the first term defines a phenomenological space; the second term, a mode; and the third term, a dimension. That is, the first term is a space in which phenomena arise; the second, the mode in which they arise; and the third, the dimension that is arising.

     Thus, the notation that we just gave-- 1p(1p) x 1p(1-p) x 2p(1p)--can also be read: my I-space [1p(1p)] perceives you as a thou or second person [2p(1p)] when I adopt a first-person stance [1p(1-p)], that is, when I adopt the stance of somebody speaking to you as a thou, as a second person who bears a first person or "I."

     Or: in my I-space, your first person appears in its second-person dimension when I see you from a first-person perspective. Or: there exists an I-space such that your I-space appears as a second person when viewed from my first-person perspective.

     In other words, in assertions such as 1p(1p) x 1p(1-p) x 2p(1p), the first term describes the space in which phenomena are arising; the second term describes the perspective that is perceiving/enacting the phenomena; and the third term is the aspect, dimension, or perspective of the phenomena that is arising or being registered in that space. The first term asserts existence, the second, the mode of existence, and the third, the dimension of existence.

     Perhaps this begins to suggest that, because the Kosmos is built of perspectives (not perceptions, not feelings, not consciousness, not matter--those are all perspectives), an integral calculus can reconstruct this construction of a Kosmos out of perspectives. (This also suggests why an integral or root mathematics is prior to typical mathematics of third-person abstracted abstractions.)

     We start with a 1p, 2p, and 3p occasion, arising together. That is, a universe comes into being when a collection of sentient beings arises. G. Spencer Brown (whose Laws of Forms we will return to momentarily), famously said that a universe comes into being when an inside is marked from an outside--but that assumption merely embeds the monological prejudice and collapse. Conscious universes do not come into being that way; that is merely an abstraction away from what is always already the case with sentient manifestation. Not only is there is no inside without outside, there is no singular without plural; universes come into being when sentient beings come into being and perceive/touch each other. Thus, inside and outside is always already first and second persons; and singular and plural is always already we's and its. The absolute minimum you need to get a universe going is the four quadrants.

     Thus, a universe comes into being, not when an inside is marked from an outside, or a before is marked from an after, but a group of sentient holons arise. Even quarks have prehension, which means, the first quark is not a first particle but a first person. And whatever that quark registers is not a second particle but a second person. There is no way around this. The universe is built of perspectives.

      So we start with 1p, 2p, and 3p occasions, arising together, each of which registers the others in its own experiential or proto-experiential fashion--but none of them can register their existence in any way other than as a perspective. That is, there is never a subject that sees an object. There is no pure perception in which one entity sees another entity, for that is already a first-person perspective on a second or third person. In other words, there is no real space that is not always already a space-arising-as-a-perspective; therefore we cannot say that occasions (or holons or beings) come into existence and then see each other, because the "seeing each other" and the "existence" cannot be asserted apart from one another. To say that the quadrants arise simultaneously is to say that ontological dimensions and epistemological perspectives are one and the same thing, which is why we often call them dimension-perspectives.

     This does not mean "to be is to be perceived," for that implies there is being per se that can be perceived; nor is this to say that perception creates being, for that implies that perception itself exists apart from something perceived. This is rather to say that being and knowing are the same event within the set of perspectives arising as the event. The idea that being and knowing (or existing and prehending) are somehow different things arises only because we shift from one perspective-occasion to the other without realizing what we are doing. There is simply no perception that is not also a perspective, and therefore no appearance of being that exists other than as a phenomenal perspective. (If you are starting to get the sense that the phenomenal or manifest world is an infinite hall of mirrors, that is indeed the suggestion. Samsara is built of perspectives, not perceptions.)

     Since space is often taken to be ontological and time epistemological, then in third-person terms this amounts to saying that space and time are not separate but are rather a spacetime continuum. Fleshing that out with AQAL metatheory, we say that the exteriors of spacetime appear topographically as chains of mass-energy interlinked in various networks and systems, while the interiors appear as feelings and awareness interlinked in various cascades of intimacy. But they all arise together as perspective-occasions of the self-reflexive Kosmos (an assertion which is itself a third-person claim arising in this first-person space, but hopefully an assertion that is to some degree arising in a space of mutual understanding, such that my understanding of this and your understanding of this resonate with similar signification).

     We were saying that G. Spencer Brown, in his Laws of Form, stated that a universe comes into being when an inside is marked from an outside. Brown built his calculus based upon that distinction or that mark ("the value of the mark is the value of the mark; the value of the crossing is not the value of the crossing"). But, as we were also saying, an integral calculus of indigenous perspectives suggests that Brown's formal calculus hides a modernist prejudice, namely, that a singular inside can demark from a singular outside (i.e., a single boundary can be drawn which marks an inside from an outside), whereas not only is there no inside without outside, there is no singular without plural--that is, if a single anything arises, it arises in the plural; as even evolutionists are coming to realize, "There is no first instance"--which means, when a new something arrives on the scene, what actually arrives is a population of the new something. For example, when the first, say, elephant emerged, clearly there could not be merely one of them; at the least, both a male elephant and female elephant had to arise simultaneously: a population first showed up, not a single entity. (This is, of course, a massive mystery, which we summarize as emergent Eros.)

     The simpler point is merely that if we ever get to the point where there is subjectivity-inside and objectivity-outside, there is also and simultaneously intersubjectivity and interobjectivity. Or, as we are used to saying, the four quadrants arise simultaneously. Brown's Laws of Form, like Whitehead's prehension, privileges the monological subject, which, at best, can monologically dialogue, not dialogically dialogue.

     (Once you get locked into monological spaces of a subject prehending an object, you cannot have simultaneous co-presence or simultaneous prehension, but must build your universe with epicycles of subjects prehending objects which prehend each other, which never actually allows subjects to know each other as subjects, but only as objects of subjects. An integral calculus exposes Whitehead's prehension to be an abstraction, not a demi-abstraction, which is why true intersubjectivity escapes Whitehead; or, as Griffin put it, Whitehead's view is "partial dialogical," not "complete [or integral] dialogical.")

     An integral calculus starts instead with the simultaneous appearance of inside and outside in singular and plural (or the four quadrants, or simply a 123 world). That is, we start with a 1p, 2p, and 3p occasion, arising together, each of which registers the others in its own experiential or proto-experiential fashion. That gives us a 123 of a 123 (i.e., a first, second, or third person resonating/reflecting another first, second, or third person--with each necessarily quadratically registering the other--which is to say, sentient beings operating within the four quadrants of indigenous perspectives); and as these reflect upon and build upon each other--as evolution becomes more and more complex and differentiated-integrated--these native perspectives continue reflecting their reflections to greater degrees of consciousness, care, and compassion. By the time we get to a 123 of a 123 of a 123, we find spaces, modes, and dimensions (as briefly outlined above). When we get to a 123 of a 123 of a 123 of a 123, a complex Kosmos not only of the primordial perspectives but of highly elaborated paradigms and practices within those perspectives have emerged and are being engaged by the sentient beings at those waves.

     We capture several of those with the 8 major methodologies represented in figure 3, but again, those are simply representative examples.

     In other words, starting with the mere assumptions that: (1) a universe arises in singular and plural with insides and outsides, (2) singular insides are prehensive (i.e., panpsychism, or all individual holons are sentient beings), and (3) all sentient beings are situated relative to each other (i.e., all prehensions are always already perspectives), then, starting at that point--which might, for example, be a Big Bang or Big Bloom--we can (re)construct the essential features of a Kosmos as an AQAL matrix of indigenous perspectives; and a Kosmos that, in its upper self-reflexive modes, delivers the 8 major methodologies that human beings are already using to illumine the Kosmos that allows them to do so.

     This leads me to believe that the integral calculus is useful in elucidating the transcendental conditions necessary for sentience, conditions sedimented in the demi-abstractions embedded in natural languages. In other words, the fact that a matrix of indigenous perspectives eventually delivers the major methodologies already in existence, suggests they are indeed some of the most fundamental, perhaps the most fundamental, ingredients of such a universe.

     There are many ways to symbolize all this, and many different dimensions to which it can be iterated. We have been talking about first, second, and third persons (the minimum requirement). A "fourth person" means an actual fourth person (in addition to the first three actual persons), and it also means a "fourth-person perspective," which, although that can be defined in several different ways, means a person who can hold the other three perspectives in mind. Thus, when we say that there are first, second, and third persons (or simply 1, 2, and 3 persons), that itself is a fourth-person perspective. If we say that a 1, 2, or 3 person sees a 1, 2, or 3 person, that is a fifth-person perspective (i.e., a 123 of a 123). If we say a 123 person can have a 123 perspective of a 123 person, that is a sixth-person perspective (123 x 123 x123). And if we say that a 123 can have a 123 of a 123 seen from its 123, that is a seventh-person perspective.

     The integral math as I have developed it to date is a 7 ^th-person perspective of the many ways sentient beings touch--it is Indra's Net viewed to seven dimensions, if you will. Of course, Indra's Net is known in its reality or its Suchness only via a transmental or supramental One Taste, not a mental-perspectival conceptualization. Nonetheless, Indra's Net does manifest in the conventional domain--in fact, the entire manifest realm is said to be Indra's Net of multiple-interconnected dimensions--and, as such, various philosopher-sages have given mental-perspectival descriptions of it (from Plotinus to Aurobindo). Even Gebser's "integral-aperspectival" is actually fourth-person perspectival. However, to my knowledge, Indra's Net has never been described beyond a fourth-person perspective (not even in the Avatamsaka Sutra, considered the definitive statement of Indra's Net).

     If we attempt to do so--that is, if we attempt to articulate the structure of the manifest world--and we move from the fourth-person version (there exists a 123) to the fifth-person perspective (there is a 123 of a 123), that perspectival operation (denoted by "x"--as in: a 123 x 123) generates an explicit phenomenological space (an I-space, we-space, or it-space). Moving from fifth-person to sixth-person (a 123 of a 123 of a 123), we generate a mode or perspective (a first-person, second-person, or third-person perspective of 123 on 123, i.e., a 123p x 123-p x 123p). Iterating primordial perspectives once more, the seventh-person perspective brings forth and elucidates a specific dimension of that which is being perceived or felt (i.e., a 123p has a 123-p of a 123p x 123/p, which means, for example, that a second person has a first-person view of a third person seen in that person's first-person dimensions-- all of them still situated with reference to the first person making the assertion. I will come back to the fourth term in that equation--the symbol "123/p"--in a moment).

     Thus is a Kosmos built out of perspectives, with all other "things," "events," and "occurrences" in the Kosmos being generated out of iterations of primordial perspectives, perspectives that arrive on the scene simultaneously with whatever it is that arrives on the scene. We can't easily specify exactly what it is that arrived on the scene first--that arrived, say, within nanoseconds of a big bang--but we can say they arrived together as permutations and combinations of how they registered each other, pushed each other, bumped into each other, felt each other. If we build a Kosmos out of those possible perspectives, and not merely out of those possible particles, systems, or dynamic processes, then we build a universe of sentient beings, not a universe of insentient particles, processes, and networks--notions which are themselves nothing but third-person perspectives on the Kosmos taken by certain sentient beings.

     An integral calculus, then, is a calculus that honors sentient beings in their AQAL totality, or certainly attempts to. Most mathematical equations--both pure mathematics and chemical, physical, biological, systems, chaos mathematics--simply trace the exterior or topographical surfaces of possible holons across possible spaces, showing, eventually, how they fit with each other in some sort of third-person dimensional space. I am not saying those spaces aren't there, but simply that they are the integral calculus stripped of first persons and second persons--stripped of sentience--at which point it collapses into the flatland representational systems found in conventional mathematics, a mathematics that often accurately represents holons in the manifest world-- but if and only if those holons are viewed only in their third-person dimensions by a first person from that first person's third-person stance--which is why the integral calculus of indigenous perspectives collapses into conventional mathematical forms when the interior spaces are erased from the Kosmos (e.g., collapses into Brown's Laws of Form, and Boolean algebra, and differential calculus, and conventional "integral" calculus, which is merely a sum-total-of-surface-volumes-traversed calculus over the range specified).

     Equations in conventional mathematics represent ways that possible surfaces fit together in possible topographical spaces; integral mathematics represents those, plus the ways that possible interiors fit together in intentional spaces, spaces of sentience bringing forth event horizons within whose zones conventional mathematics itself can manifest in the first place. An equation in interior space is a measure of harmonic resonance or empathy between two holons, a registration of how they fit together in spheres of consciousness and not merely circles of geometry. Of course, integral mathematics is itself composed only and merely of third-person tokens, signs, and symbols; but those signs represent first and second and third persons (and fourth and fifth and sixth and seventh), which are not variables but perspectives, and which in the real world appear not as amounts and angles but as sentient beings with shared horizons (whose exteriors are amounts and angles).

     Equations in the real world of sentient beings are thus equations of mutual resonance. Even a mathematician, who writes (x = 3y), and shows it to another mathematician, who agrees that in that case, x does indeed equal 3y, is actually asserting the following: my first person has a first-person perception of a third-person abstraction [(x = 3y)], and I believe that this third-person abstraction is, or would be, true for all other persons who looked at it. Therefore, I am asserting that this abstraction is not merely true for me (or my first person), but is true for all other first persons; which means, if I take a third-person view of my third-person abstractions, I still believe that you will agree with me if you look at them in a third-person way yourself--and not only you, but all others who look at this dispassionately or objectively or rationally, will agree with me. I am actually claiming, then, that my first-person perception of my third-person abstractions is really a third-person (plural) perception [which is represented as (3-p*pl)] of this third-person abstraction: 1p(1p) x 1p(3-p*pl) x 1p(3p), which means, my first person has a third-person plural view of my third-person algebraic assertion. If you look at my algebra and attempt to take up a third-person (plural) view of it, then: 2p(1p) x 2p(3-p*pl) x 1p(3p), which means, your first person has your third-person (plural) view of my first person's assertion (which is the algebra, the third person we are considering, where "3p" in this case means "the assertion x = 3y," which is the third-person "it" we are discussing).

     The heart of the matter is that you might indeed agree with me that the algebraic equation is correct. If so, we have:

     1p(1p) x 1p(3-p*pl) x 1p(3p) = 2p(1p) x 2p(3-p*pl) x 1p(3p)

     That is what an equation in the real world looks like, even among mathematicians. Equations in the real world equate interiors (as well as exteriors), and thus they are built not just of exteriors that can be "equal" but of interiors that can be "equal," which is to say, can equal each other in mutual understanding or mutual resonance. That is what the equal sign means in the real world. The Kosmos vibrates with those equations of souls touching each other. That is what the Kosmos is made of.

     Notice in the above equation that--as usual--all of those terms are situated with reference to the first person who is making the assertion (in this case, me). In the right hand of that equation, even your first-person perceptions must be stated in reference to me who is making the claim. Likewise, were you to write a series of claims, I would always be the second-person first person to you. Thus, if I write: 2p(1p) x 1p(3p) [your first person sees my third person; that is, you are perceiving or touching the objective dimensions of my being-in-the-world], you would write that same statement as: 1p(1p) x 2p(3p) [my first person sees your third person]. Now the only way that those two perspectives can be entered in the same equation is if you and I can find a first-person plural space [(1p*pl)] in which we can agree that those are equivalent transforms. If we do so, each of us would still each have our individual understanding of this "we," even though we believe they overlap, so that if I then write an equation of our mutual understanding, one of its (many) forms would be:

      1p(1p) x 1p(1-p*pl) x [(1p*pl){ 1p(3p)}] = 2p(1p) x 2p(1-p*pl) x [(1p*pl){ 2p(3p)}]

     Which means, my first person has a first-person view of how we (first-person plural) see my third-person algebra; and you have a first-person view of how we see my algebra which is, in your space, a third person artifact produced by me, who is a second person to your first person.

     Although I would need to write out both sides of that equation in at least a four-term fashion to show the details of what is involved, the simpler point is that this equation is asserting the existence of a we-space in which the two sides of the equation are equivalent--it is asserting, that is, similar signification between the intersections of a first person and second person who enter a first-person plural space.

     If the Kosmos is built of perspectives, then the interactions in the Kosmos are built of similar signification or mutual resonance--which is why all holons have a Lower-Left quadrant of shared interiors, and not merely a Lower-Right quadrant of shared exteriors.

     The equations of integral mathematics all revolve, ultimately, around how holons actually relate--that is, not only with similar exteriors that can be added, subtracted, multiplied, divided, derivated, and so on, but with interiors that resonant with each other, or do not resonant with each other, or stand in a relation of understanding to each other, or stand in a relation of power to each other, or enfold the other with integration (in compound individuality), or subsume the other without integration, and so on.

     As one runs the integral calculus, various operations and functions emerge, including prehension (within perspective), mutual resonance, interpretation, telepathy, integration, differentiation, enfoldment, power over, transcendence, inclusion, and--most interestingly--the major validity claims (i.e., different equations begin to represent different types of validity claims or assertions of adequacy). I'm sure there are dozens, maybe hundreds more; maybe infinite.

     But, again, I am making a series of much simpler points in this introduction, so let me finish this brief intro by going back to the equation representing two mathematicians agreeing on the nature of an algebraic formula:

     1p(1p) x 1p(3-p*pl) x 1p(3p) = 2p(1p) x 2p(3-p*pl) x 1p(3p)

     In that equation of mutual understanding (i.e., in that real-world equation of two sentient beings agreeing on the nature of a third-person abstraction), the "3p" represents the third person, which in this case is an insentient third-person artifact, i.e., the algebraic formula (x = 3y). That equation says, my first person has an objective (third-person plural) view of my third-person artifact (the algebra) which is equal to (or mutually resonates with) your first person's perception, in an objective mode, of my artifact. (Don't confuse "3p," or the object-dimension that is being perceived/enacted--in this case, the artifact--with "3-p" or "3-p*pl," hyphenated, which is the mode in which the object is being perceived--in this case, a third-person plural mode.) So let's put that third person or artifact into the above equation (i.e., wherever we see "3p," we will substitute [x = 3y]):

     1p(1p) x 1p(3-p*pl) x 1p([x = 3y]) = 2p(1p) x 2p(3-p*pl) x 1p([x = 3y])

     Now, if we deny that all such assertions (in this case, the assertion that "x = 3y") are always already a perspective--that is, if we deny that there are any first or second persons involved in third-person assertions--then "1p" and "2p" and "3p" all become merely a number 1. That is, there are no first or second persons, and therefore no third persons, either; only insentient things and events and processes and abstract markers; and those abstract markers are not even "third persons" anymore, because there are no first or second persons to talk about them. So if we substitute the number "1" for 1p, 2p, and 3p in that equation, then that particular equation of mutual understanding between two souls becomes instead:

     1(1) x 1(1) x 1([x = 3y]) = 1(1) x 1(1) x 1([x = 3y])

     which obviously reduces to:

     [x = 3y] = [x = 3y]

     In other words, integral mathematics, stripped of sentience, collapses into the monological spaces of ordinary mathematics, where it merely asserts identity of abstract (third-person) markers. Those markers are real enough, but they only represent a narrow slice of the Kosmos, a slice generated from real-world sentient beings through a series of abstractions, collapses, and reductions, so that only a few of the dimensions of being-in-the-world are represented, and are represented in a way that deceptively appears that they are not perspectives of sentient beings but simply a view of "the way things are," or what Nagel so aptly called "the view from nowhere." This allows such collapsed cognitions to imagine a Kosmos built of abstract relations and insentient beings (which is itself a perspective of their sentience).

     There have been many attempts to arrive at a type of fundamental mathematics of the Kosmos that attempts to include items such as consciousness, interiority, mind, subtle energy, spirit, and so on. Many of these take the basics of conventional physics--such as the quantum vacuum potential, or fundamental matter waves, or string theory--and essentially equate those basics with consciousness or spirit. David Bohm, Arthur Young, Buckminster Fuller, Walter Russell, Milo Wolff, Ervin Laszlo, Wing Pon, William Tiller, among many others, have added to our understanding of how this might occur. But all of those approaches embed various degrees of the monological prejudice, and thus end up simply (and unfortunately) equating spirit with an implicate third-person holism (e.g., Bohm), or attempt to derive first-person consciousness from third-person operators (e.g., Fuller), or see consciousness emerging as a result of complex third-person systems interactions (e.g., Laszlo). Even the approaches that see consciousness or mind as fundamental (e.g., Russell), imbed the prejudice of perception or consciousness (which does not exist, as we have seen)--Whitehead being another example.

     In other words, all of them are pre-quadratic attempts to derive the essentials of the Kosmos from a starting point that prejudicially has already collapsed the essentials out of existence and thus must attempt to recover those essentials with epicycles of further abstractions. Again, I am not saying that aspects of their work are not true; I am saying that they have abstracted their conclusions out of the matrix of indigenous perspectives and then presented them as "the way things are," oblivious to the perspectives in which their "views from nowhere" actually arrive.

     This is certainly the case with "metaphysics" in general, whether we find it in Plotinus, Shankara, Asanga, Padmasambhava, Gurdjieff, Hegel, Rudolph Steiner, Carl Jung, William James, or the greatest of recent metaphysicians, Aurobindo. To the modernist and postmodernist critiques of metaphysics, we add the integral critique: their metaphysical systems are interpretations of their own spiritual experiences; the authenticity of the spiritual experiences is not in any way questioned, but the adequacy of their interpretations is: they have unconsciously abstracted, from the matrix of indigenous perspectives, a third-person overview that arrives on the scene secretly privileging the view from nowhere, even (or especially) when it emphasizes the importance of experience, spiritual awareness, feelings, or consciousness: all of those are, in fact, hidden low-order abstractions, and, as such, are the very heart of the metaphysical approach that all post-metaphysical integralism must struggle beyond. If "direct experience" and "consciousness" are already low-order abstractions mistaken for realities (and hence are metaphysical ghosts), the notions of "levels of being," "levels of knowing," "ontological planes," and so on are even worse: they are abstractions of abstractions of abstractions, even though the experiences that those interpretative frameworks are trying to represent are authentic enough. (In this regard, Aurobindo is the most offensive metaphysician in that he is the most accomplished; one can only stand in awe of his metaphysical system.)

     Again, I am not questioning their realization or enlightenment or spiritual experiences; I am questioning the framework that they used to interpret and conceptualize their experiences. Those metaphysical interpretive frameworks are simply not adequate to a postmodern integralism that has grown out of metaphysics but can no longer be contained by it (i.e., integralism transcends-and-includes metaphysics, such that integralism is external to metaphysics, or no longer constrained by its nexus-agency).

     Likewise, take an integral calculus of 7 dimension-perspectives; collapse it to 4 dimensions; set the domain on the interiors (first-person perspectives) to a specific stream in those interiors (such as the values stream) and then set that domain (or first-person space) to cover the specific range beige to orange; set the mode to register only 3-p perspectives; and those operators will generate a phenomenological space of scientific materialism. Set the domain range to green, set the mode to recognize only interiors, and you will generate an event horizon or phenomenological space of postmodern pluralism. And so on....

     Likewise with the great pre-quadratic metaphysical systems: their essentials can be derived from an AQAL matrix without the inadequacies of metaphysical interpretations, and thus their incredibly important insights can be taken into the modern and postmodern world without embarrassment.

     All of these operations are simply reminders, I believe, that the Kosmos is built of perspectives, whose fundamental operations include mutual interior resonance along with mutual crashing into each other externally; and that, therefore, any abstract notational systems can remind themselves of this by acknowledging an integral calculus of indigenous perspectives. When it comes to an integral mathematics itself, its starting point is the relations between the universal demi-abstractions embedded in pronoun-perspectives of natural languages, deposited there, we presume, by an evolution attuned to these real dimensions in the real world.

     One last point. We said we would return to the symbol "123/p." In some ways, this is the most interesting operator in integral mathematics. It means "stop." As one runs the integral equations, it soon becomes obvious that there is no fundamental perspective, no absolute Archimedean point, from which one can know anything. There is simply an ongoing cascade of perspectives on perspectives, all the way up, all the way down. The universe might be composed of holons--which I believe is the case--but "holon" is already a third-person symbol in a first-person prehension--i.e., it is already a perspective. Just as there is no such thing as consciousness, mind, feelings, awareness, things, events, or processes, there are no holons--for all of those are always already perspectives. And the only way we "know" any of those is that we arbitrarily and abruptly dig our heals into the cascading flow of infinite perspectives and we say, for example, "I see the tree!"

     Once we arbitrarily slam our foot down and perceive something, or feel something, or notice something, we have temporarily frozen the stream at that instant, and around that frozen singularity an AQAL matrix jumps into existence. Once I register another entity, a first and second person have jumped out of the stream; once we communicate about anything, third persons are everywhere--and all of that happens at the point, and only at the point, that I stutter the stream and temporarily stop the flow.

     The "stop" symbol (/p) in integral mathematics means: this is the occasion (the first, second, or third person event) where I arbitrarily stopped the stream and began my process of knowing in the midst of other sentient beings. The stop symbol means: "freeze frame." Freeze the flow at that frame, and let me start the knowing, feeling, perceiving of that event.

     Thus, with "I see the tree," we have, in simplified form: 1p(1p) x 1p(3-p) x 3p(3/p), which means, I have arbitrarily focused my attention on that tree over there, so I have stopped the cascade at the objective surfaces of that tree [3p(3/p)] and I have begun the knowing process there, so that now I will assert that my first person [1p(1p)] has an objective view [1p(3-p)] of that object over there [3p(3/p)], and THERE IT STOPS (which also means, and there it starts: the knowing process starts only when I dig my feet in and stop the flow). Without the 123/p moment (or the stopping moment), then perspectives cascade endlessly. In the manifest world, it is literally perspectives all the way up, all the way down, and without the arbitrary stopping moment or freeze frame, nothing gets registered. But initiate a stop, and the AQAL matrix jumps into existence around that point.

     This "jumping into existence," arbitrarily initiated, does not, however, have an altogether arbitrary form. As a sentient being, when I stop the flow and initiate (enact) a world, it is a world of other sentient beings; and therefore the form of the matrix of perspectives that can arise is constrained by all the other sentient beings who are also stopping streams and enacting worlds. All of our enactions have to inter-mesh, since they are co-creating each other. Hence we find the form of the AQAL matrix of primordial perspectives, which can be transcendentally deduced from the structure of our own everyday interactions, such as those embedded in natural languages (whose demi-abstractions can also drive an integral mathematics). The AQAL matrix is one view of the form of mutual enaction when sentient beings co-create each other in freeze frames of their own becoming: the AQAL matrix is the form of Spirit's lila.

     Well, so much for an overview, which I hope has at least suggested a few possibilities here. There are a hundred ways to take an integral mathematics, whose farthest reaches are surely beyond my capacities. But every now and then I will, as indicated, post or publish a few more preliminary stabs in this direction. If nothing else, I hope that this kind of notational system will act as another type of IOS or Integral Operating System--namely, a series of merely third-person symbols that nonetheless constantly remind us that there are actually first- and second- and third-person sentient beings in the real world. The Kosmos contains sentient beings, not particles or processes or webs or systems, and therefore the Kosmos is built of perspectives, not feelings or perceptions or consciousness, all of which are always already perspectives in Indra's endless Net......