Space-time continuums are simply connected frames of reference (connected to an ontological theory of reference wherein the first distinction is pure subjectivity, self-reference), notions of contents (programming objects, objects of consciousness) are connected by the logical inference (by George!) of the all-comprehending frame (polycontexturality) and hence constitute logical domains, which is precisely what Gotthard Gunther meant by introducing the term “contexture” in his thesis of “polycontexturality” as abstract life, pure subjectivity. So continuums are contextures (logical domains) because all connection is governed by the “laws of frame,” the same “Laws of Form” calculi (in fact, George Spencer-Brown calls the frame the “unmarked cross,” which is found to be in the unmarked state, if we objectify our subjectivity (and hence reify or un-reify depending on one’s reality thesis).). Having connected continuums and contextures in that context (as it is in my intuition), we can move on to more peculiar attributes of space and time.
Time, in the most fundamental sense, is not only the fourth dimension with which we are familiar
The Epochal Theory of Time (James and Whitehead) may be a thesis working on an intuition that temporal consensus (in Whitehead’s case, of subjective forms) is specific to one’s “cosmic epoch.” Having stated that time is not only the familiar fourth dimension of our experience, we can ask “what else?” I suggest, first of all, that time is the highest dimension of experience, since it is where consciousness runs along rather than through. In a two-dimensional space, as events pass, so pass planes of construction. In a one-dimensional space, events discard previous states of lines, throughout the motion of points and segments. George Spencer-Brown’s less-exclusive definition of time is “a one-way blindness” in reference to a less-exclusive “sight,” which is more fundamental than our familiar vision. In this sense of sight, I call formless subjectivity “sight without light,” an infinitesimal point of reference, self-indicating, but abstracted from the forms it traces in every act of drawing distinctions.
We Take as Given
by George Spencer-Brown
We take as given
The idea of a distinction
And that one cannot
Make an indication
Without drawing a distinction.
We take therefore
The form of distinction
For the form.
The form we take to
The idea] consists in seeing the universe as a language, a script. But it is a language in unending movement and change: each sentence breeds another sentence, each says something which is always different and yet says the same thing....The metaphor which consists in seeing the universe as a book is very ancient and appears also in the last canto of Dante's Paradise...
In that abyss I saw how love held bound
Into one volume all the leaves whose flight
Is scattered through the universe around;
How substance, accident and mode unite
Fused, so to speak, together, in such wise
That this I tell of is one simple light.
The pluralities of the world--leaves blown here and there--come to rest together in the sacred book; substance and accident in the end are joined. Everything is a reflection of that unity, not excluding the words of the poet who names it. In the next tercet, the union of substance and accident is presented as a knot, and this knot is the universal form enclosing all forms. This knot is the hieroglyph of divine love.
[Paz, Children, pp. 71, 75. This famous phrase we have referred to above: as Dante says in Canto XXXIII, 91: La forma universal di questo nodo...("The universal form of this knot..." or less precisely, "The form that knits the whole world....").]
THE FIFTH CROSSING
James Keys, poet, polymath, and alter ego of the mathematician G. Spencer Brown, in a profound footnote, rehearses the process of this--or any other--general program of Creation. He counts with technical precision the steps from the Void; but to follow his count it is essential to distinguish between cardinal and ordinal numbers--and this awareness has become very muddled by popular misconstructions and by the inattention of educators. In one part of his extensive comments, Keys outlines a rectification of the conventional archetypal sequence while he associates the formal, mathematical states with certain historical and cultural symbolic representations of them, as in with Buddha-states of the Tibetan cosmogony, or the Persons of the Trinity in Christian tradition.
The story of creation can of course be protracted indefinitely. To cut a long story short, it turns out that there are five orders (or "levels") of eternity, four of which are existent (although not of course materially existent, this comes later) and one which is non-existent.The non-existent order is of course the inmost, the one the Greeks called the Empyrean. In the mathematics of the eternal structure the five orders are plainly distinguishable, and it is a fact of some interest that the early Greek explorers, who were not so well equipped mathematically as we are today, nevertheless confirmed, from observation alone, that the number of eternal regions or "heavens" stands at five.
At the next level, travelling outwards from within, an extraordinary thing happens. As we come into the sixth level (i.e. the fifth order [Order number Five], recollecting that the first level is of order zero) by crossing the fifth "veil"--mathematically speaking a "veil" is crossed when we devise an "outer" structure that embodies the "rules" of the structure next within--when we cross this fifth veil, a strange thing happens. We find that we cannot in fact cross it (i.e. it is mathematically impossible to do so) without creating time.
The time we create first, like the first space [given the cardinal number One], is much more primitive and less differentiated than what we know in physical existence. The time we set our watches by is actually the third time. The first time is much less sophisticated. Just as the regions of the first space have no size, so the intervals of the first time have no duration. This doesn't mean, as it might suggest in physical time, that the intervals are very short, so short that they vanish. It means simply that they are neither short nor long, because duration is not yet a quality that has been introduced into the system. For the same reason, all the heavenly states, although plainly distinguishable from one another, are in reality neither large nor small, neither close together nor far apart.
Everything reflects in everything else, and the peculiar and fundamental property of the fifth order of being reflects itself all over the universe, both at the physical and metaphysical levels. An interesting reflexion of it in mathematics is the fact that equations up to and including the fourth degree can be solved with algebraic formulae. Beyond this a runaway condition takes over making it impossible to produce a formula to solve equations of the fifth or higher degrees. A similar "runaway" condition applies, as we shall see in a moment, when we cross the fifth "veil" outwards into the first time.
It requires only a moment's consideration to see that what we call time is in fact a one-way blindness, the blind side being called "the future." Once we proceed into any time, no matter how primitive, we come out of heaven, i.e. out of eternity, out of the region where there is no blindness and where, therefore, in any part of it, we can still see the whole. And as we proceed further and further out into each successive and less primitive time and space, our blindness at each crossing is recompounded. It is thus easy to come out, hard to find one's way back in.
[James Keys, Only Two Can Play This Game, Julian Press, New York (1972), footnote No. 1, pp. 123 ff.]
Although the world of AI (artificial intelligence) and the theoretical branch of computer design in general have been slow to grasp it, this grand iconic image offers a potentially rewarding tool and perhaps a clue for solving some of the complexities of parallel programming. In new models, simultaneous (parallel) processing transcends lineal tree logic, yet in designs for new-genereation supercomputers the requirements of physical proximity are increasingly difficult to tolerate as constraints on the speed of information processing. The key lies in our understanding the architecture of heaven, or eternity. The necessary arrangement of the heavenly or eternal realms (with a paradigmatic five-steps-from-the-void) can indeed be seen, but not while retaining our conventional attachments to habitual vision of the sort we find so useful in the everyday world. Given the special meanings of formal language, we might say of this empyrean exercise:
...to experience the world clearly, we must abandon existence to truth, truth to indication, indication to form, and form to void.
If we distinguish anything at all, then "all this"--including in the end the physical universe--is how it must eventually appear. In short, what I prove is that all universes, whatever their contents, are constructed according to the same formal principles.
[G. Spencer Brown, Laws of Form, p. 101. Keys, Only Two Can Play This Game, p. 110.]
These principles can be illustrated by the formal steps that must be taken ("all-at-once") in the orders of creation. This structure
corresponds to the void, the form, the axioms which see the form...Then you get the arithmetic, which is seeing what becomes of the axioms. And then you be it to do it, and in being it and doing it you find that, being and doing, you see the generalities of it, and that is the algebra. And while you are seeing you notice you have got equations...and suddenly you decide: "Aha! Supposing what it equals goes back into what it comes from?" Now you have generated time and matter all at once. There can be no matter without time. Time and matter come simultaneously. But this is the first matter in which the orders are counted, and it's called the "crystalline heaven," but it is not, really, a heaven.
In the construction of matter, all that happens is that we create the temporal and the material together by imagining that the outside feeds back into the inside. We then have a succession of marked and unmarked states generated by an oscillator function...Once you are in time, everything is a vibration.
[Keys, AUM Conference Transcript, pp. 96, 104, 106, 108.]
In the context of some brief reviews, James Keys drew parallels between these formal states or relationships and various literary, religious and artistic expressions, including Dante, the Gospel accord-ing to Thomas, and the author of The Divine Names, Dionysius the Areopagite, the Early Christian mystic to whom (mistakenly) St. Denis, the first Gothic church in the Ile-de-France was dedicated in 1144.
The secret sayings of Jesus of Nazareth, many of them so much deeper and stronger than what we find in the canonical gospels as to make it a different order of book. For example, it says much more clearly (gives an exact recipe, in fact) what you actually have to do to enter eternity. [In The Divine Names] the parallel accounts of the emergence of time, i.e. the statements of what we have to do to construct an element that doesn't exist in any of the five orders of eternity. We attempt to recount, in other words, what are the essential magic spells for creating a temporal existence, just as books such as the Gospel of Thomas aim to give the essential magic whereby these spells may be reversed.
[Keys, Only Two Can Play This Game, pp. 104, 108.]
In this order of complexity, this space we enter following the fifth crossing from the void, we discover--we are for the first time able to imagine--those entities commonly called numbers. They exist in what has been called the crystalline heaven, which is Order number Five (counting from the void as "zero"); that order is:
with the first time...what is called the astral plane in magic. It is the last of the material existences. Its structure is transparent and crystalline. In the middle ages it was projected out and called the crystalline heaven, although it is not, technically, an eternal region. It is where the eternal regions are first plotted and counted, for there are no numbers in eternity itself. You cannot count without time. When we proceed from here into the heavens themselves, we lose all numbers in a blinding flash as we return through the fifth veil into the outer heaven. From here on, if we are to survey what we see mathematically, we have to use Boolean elements, which are not numerical.
[Numbers] nevertheless, do exist. But not in the physical universe...Common arithmetic for university purposes, which for a less vulgar name is called the Theory of Numbers, one of the most beautiful sciences in all of mathematics, is the science of the individuality of numbers. A number theorist knows each number in its individuality. He knows about the relationships it forms, and so on, as an individual, as a constant. An algebraist is not interested in the individuality of numbers; he is interested in the generality of numbers. He is more interested in the sociology of numbers...he is not interested in individuals at all.
[Keys, Only Two, p. 134 f.; AUM Conference transcript, pp. 43, 45.]
Previously we sought to provide a link to certain basic information about number with our reference to Warren Sturgis McCulloch's essay "What Is a Number, that a Man May Know It, and a Man, that He May Know a Number?" Here, we justify our methodological use of number by telling where we may find a number and how to count it, literally, digitally. In one of the easiest ways to demonstrate this count:
Hold the palm of one hand in front of your face.
With the index finger of the other hand, count off the states or orders of eternity, beginning with your thumb.
Call the thumb, "Order Zero" (though it is the FIRST counted!)
Count the gap (or "valley") between the thumb and the adjacent index finger stands for the first crossing.
Call the index finger "Order One," which stands for the Form.
Then count the next interdigital gap as the second crossing.
Call middle finger "Order Two," the Axioms.
Then count the next gap as the third crossing.Call the ring finger "Order Three," the Arithmetic.Then count the next gap as the fourth crossing.
Call the little finger "Order Four," the Primary Algebra.
THEN count the fifth crossing, which, you see, is different from all the others, and not a gap or a valley at all because you can go past the wrist, all the way around the palm of your hand and return to your thumb. In the next state after the fifth crossing, "Order Five," the Algebra may contain equations of the second degree.