by Paul Pietsch
In hologramic theory, reasoning, thinking, associating or any equivalent of correlating the ring is matching the newly transformed transform with the back-transform. (The technical term for such matching is autocorrelation.)
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Let's shift our focus back to the nature of the phase code, which in our ring system is the preservation of ring information by periodic patterns. The ring memory is not limited to dots and stripes. When we react rings with too great a distance between their centers (if their center rings do not overlap), we do not produce stripes. Instead, something interesting happens. Notice that if we get the centers far enough apart, rings form in the readout:
;
Built into the higher frequency rings is a memory of rings closer to the center. Let me explain this.
First of all, as I've pointed out, the phase code isn't literally dots or stripes or dots[7] but a certain periodicity; i. e., a logic! Our rings are much like ripples on a pond; they expand from the central ring just as any wave front advances from the origin. Recall from Huygen's principle that each point in a wave contributes to the advancing front. The waves at the periphery contain a memory of their entire ancestry. When we superimpose sets of rings in the manner of the last figure, we back-transform those hidden, unsuspected "ancestral " memories from transform to perceptual space. The last picture demonstrates that no necessary relationship exists between the nature of a phase code and precisely how that phase code came into being. The "calculation" represented by the picture shows why hologramic theory fits the prescriptions of neither empirical nor rational schools of thought. In the experiments where we superimposed rings on rings the system had to "learn" the code; the two sets of rings had to "experience" each other within a certain boundary in order to transfer their phase variation into transform space. But the very same code also grew spontaneously out of the "innate" advance of the wave front. These, reader, are the reasons why I would not define memory on the basis of either learning or instinct. As trees are irreducibly wood, memory is phase codes: whether the code is "learned" or "instinctive" has no existential bearing on its mathematical--and therefore--necessary features.
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Consider something else our stripes, dots and rings reveal about the phase code. We can't assign memory to a specific structural attribute of the system. In hologramic theory, memory is without fixed size, absolute proportions or particular architecture. Memory is stored as abstract periodicity in transform space. This abstract property is the theoretical basis for the predictions my shufflebrain experiments vindicated, and for why shuffling a salamander's brain doesn't scramble its stored mind. My instruments cannot reach into the ideal transform space where mind is stored. For hologramic mind will not reduce directly to the constituents of the brain.
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chapter nine
The Holological Continuum
NORBERT WIENER, the mathematician who founded cybernetics just after World War II, once observed about computers, "the energy spent per individual operation is almost vanishingly small," and he went on to warn contemporary science that "Information is information, not matter or energy. No materialism which does not admit this can survive at the present day."[1] Reduction of the message to the medium, he realized, would eventually force scientists to do something they wouldn't (and don't) want to do: repeal the powerful and indispensable laws of thermodynamics. Wiener of course knew that information can't exist on nothing. But to stamp your foot and insist that information and mass-energy are one in the very same thing would quickly put the natural sciences in an untenable philosophical position. Wiener knew in his bones that abstract relationships--ratios-- within mass-energy are what create, encode and store information, not matter or energy as measured by the pound or erg.
With respect to the hologram, we have already arrived at Wiener's conclusion ourselves: functional relationships within the media encode phase information. Thus quite different physical entities, widely varying absolute energies or basically dissimilar chemical reactions can construct, store and decode the same hologram--as information! Initially, and by force of implication, we extended this principle of information to hologramic mind. But at the end of the last chapter, we came to this same conclusion directly, from hologramic theory, itself. A mind, the theory asserts, is not specific molecules, particular cells, certain physiological mechanisms or whatever may serve as the mind's media. Mind is phase information--relationships displayed in time and in perceptual space but stored as a function of time and a function of perceptual space in transform space. We have a subtle but pivotal distinction to make, then: molecules, cells, mechanisms, and the like are necessary to create, maintain or display those phase relationships. But the relationships are not reducible to a molecule, cell or mechanism any more than the message on a printed page is reducible to ink and paper. And when we investigate mind in molecules, on cells or via mechanisms, we have to be very careful about the words we chose to describe our results. Conjugates and equivalents of the verb "to do" are what we want when we employ test tubes, microscopes or electrodes. But when we ask the mind is, then we must turn to theory.
Does hologramic theory demand that different constituents and mechanisms of brain house memory? As Bertrand Russell maintained, theory is general. The moment we begin to talk about specific parts, we shift to the particular, to issues the experimentalist (e. g., the brain shuffler) must pursue. What hologramic theory does, though, is account for,say, how more than one class of things or events can serve as a medium for the very same memory. If a protein encodes the same spectrum of phase variations as occur in, for instance, a feedback loop around the hippocampus, then the same memory exists in or on both the protein and the feedback loop.
Again: the test tube, the microscope and the electrode work only in perceptual space. Of course, this restriction doesn't minimize their value, nor does it undermine the importance of experience. But to get inside the hologramic mind, to unravel its logic, to discover its plan, to figure out how the mind actually works, we must use abstract tools. Only with reason, with an assist from imagination, can we cross the boundary between the real and the ideal.
Yet we can use our imaginations to resynthesize in our own reality the relationships our reason uncovers in realms beyond. We were doing this with transforms, convolution theorem and Fourier series. The holographic engineer also does it with reconstruction beams. Pribram did it in the living, remembering brains of monkeys. I did it with shufflebrain, albeit unwittingly at first. If we use the theory with art as well as science, and exercise a little humility en passant, not only can we extend our comprehension beyond experience, but we can avoid imprisonment within the ideal and exile from our own reality. Hologramic theory does not dispense with the brain. Activated, mind perceives, thinks and drives behavior in perceptual space.
In its present form, though, hologramic theory will not serve our needs. Fourier transform space is too cramped for us to appreciate, for instance, the similarities and differences between our own mental cosmos and the minds of other creatures. Fourier transform space is too linear to explain the nonlinear relationship between the time we measure by the clock and the intervals that elapse in dreams, for example. Nor in Fourier and kindred transform spaces can we readily envisage the smooth, continuous movement of information from sensations to perceptions to memories to behavior to whatever. How can the information be the same while the events retain their obvious differences? Described only in the language of Fourier theorem, mind seems more like the inner workings of a compact disc player than the subjective universe of a living organism. If the reader has already felt that something must be wrong with our picture, it's because he or she knows very well that we mortals aren't squared-off, smoothed-down, case-hardened, linearly perfected gadgets.
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Thus far, the constraints on our understanding have two main causes. First, our theory is now anchored to the axioms and postulat
es of Euclid's geometry. Second, we haven't yet given sufficient theoretical identity to the particular, having been too preoccupied with escaping from the particular to give the attention it really deserves. In this chapter, I will first reassemble hologramic theory free of assumptions in previous chapters. Towards the end, I will draw the realm of the particular into our discussion. Ready? Take a deep breath!
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June 10, 1854 is an noteworthy date in the history of thought. It was also a momentous day in the brief life of mathematician Georg Friedrich Bernhard Riemann whose discoveries were crucial to Albert Einstein.[2] It was when Riemann stood before the distinguished professors of Göttingen's celebrated university to deliver his formal trial lecture as a probationary member of the faculty. Entitled, "On the Hypotheses Which Lie at the Foundations of Geometry," the lecture attacked a dogma that had ruled rational belief, "From Euclid to Legrendre," asserted the twenty-eight year old Privat-dozent. [3]
"It is well known," he continued, "that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It gives only nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relation [logic] of these presuppositions [the postulates of geometry] is left in the dark; one sees neither whether and in how far their connection [cause-effect] is necessary, nor a priori whether it is possible."
In the detached bon ton of scholars then and now, Riemann in effect was telling his august audience (including no less that his mentor, Karl Friedrich Gauss) that mathematicians and philosophers had flat-footedly assumed that space is just there. Like the gods in The Iliad who had an external view of the mortal realm, mathematicians and philosophers had inspected space in toto, had immediately brought rectilinear order to the nullity, with the flat planes of length, width and height, and thereby had known at a glance how every journey on a line, across a surface or into a volume must start, progress and stop. Geometric magnitudes--distance, area, volume--plopped inexorably out like an egg from laying hen's cloaca.
Ever since Newton and Leibnitz had invented calculus, infinitely small regions of curves had been open to mathematical exploration; but points on a straight line had eluded mathematician and philosopher, alike. Riemann doubted that the same fundamental elements of geometry--points--could obey fundamentally different mathematical laws in curves versus lines. He believed that the point in flat figures had become an enigma because geometry had been constructed from the top down, instead of from the bottom up: "Accordingly, I have proposed to myself at first the problem of constructing a multiply extended [many dimensional] magnitude [space] out of the general notions of quantity." He would begin with infinitely small relationships and then reason out the primitive, elemental rules attending them, instead of assuming those rules in advance. Then, not taking for granted that he knew the course before his journey into the unknown had begun, he would, by parts, follow the trail he had picked up, and he'd let the facts dictate the way.
To the best of my knowledge, the first formal principle of quantity remains to be found, if one really exists at all. Even Riemann's genius had to be ignited by intuition. Intuitively speaking, the basic notions of quantity imply measuring something with something else. As in our optical transform experiments in the last chapter, measuring to Riemann, "consists in superposition of the magnitudes to be compared." As in the case with the reference and object waves in the hologram, superposition of the two magnitudes or quantities occurs "only when the one is part of the other."[4] Riemann was speaking about continuity in the most exactingly analytic sense of the word.
Where can we find continuity? More important, how can we guarantee its existence in the relationship of, say, X and Y? To satisfy Riemann's requirements, we would have to show that at least one of the elements involved necessarily affects the other. Thus a frog on a lily pad won't do, especially if the animal is just sitting there enjoying the morning sunshine. We would have to find out if any change in either the animal or the plant forces a concomitant variation in the other. Thus the first requirement for establishing continuity is to get away from static situations and focus on dynamic--variable--relationships. Do they change together?
Suppose, though, that one unit of change in X procures a one-unit change in Y; that Y=X. If we graph the latter, the plot will look like a straight line. In a linear relationship, the ratio of Y to X, of course, remains constant no matter how large or small the values become. This constancy made mathematicians before Riemann shy away from points on the straight line. For an infinite number of points exist between any two points on a line; even as the values of X and Y approach zero we never close the infinite interval between two points on a straight line.
The curve is quite another story. What is a curve? My handworn 1964 edition of Encyclopedia Britannica characterizes it as "the envelop of its tangents." Remember that on a circle, the very embodiment of curvature, we can draw a tangent to a single point on the circumference. The same thing holds for tangents to a curve; and we do not draw a tangent to a straight line. Like the sine or cosine, the tangent is a function of an angle.[5] We might think of a tangent as a functional indicator of a specific direction. The points on a straight line all have the same direction; therefore, a tangent to a straight line would yield no information about changes in direction (because the directions are all the same). Neighboring points on a curve, by contrast, have different relative directions. Each point on a curve takes its own specific tangent. And the tangent to the curve will tell us something about how the directions of one point vary relative to neighboring points.[6]
Imagine that we draw a tangent to a point on the X-Y curve. The bend in the curve at that point will determine the slope of the tangent. If we could actually get down and take a look at our X-Y point, we'd find that its direction coincides with our tangent's slope. Of course, we can't reach the point. But we can continuously shrink X and Y closer and closer to our point. As we get nearer and nearer to the point, the discrepancy between the curve and the slope of the tangent becomes smaller and smaller. Eventually, we arrive at a vanishingly tiny difference between curve and tangent. We approach what Isaac Newton called--and mathematicians still call--a "limit" in the change of Y relative to X. The limit--the point-sized tangent--is much like what we obtain when we convert the value of pi from 22/7 to 3.14159...and on and on in decimal places until we have an insignificant but always persisting amount left over. The limit is very close to our point. The continuous nature of the change in Y to X permits us to approach the limit.
Finding limits is the subject of differential calculus. The principal operation, aptly called differentiation, is a search for limit-approaching ratios known as derivatives. The derivative is a guarantee of continuity between Y and X at a point. The existence of the derivative, in other words, satisfies Riemann's criterion of continuity: Y is part of X. The derivative is strictly a property of curves. For the derivative is a manifestation of changing change in the relationship of a point to its immediate neighbors. Derivatives, minuscule but measurable ratios around points, were the basis from which Riemann developed the fundamental rules of his new geometry.
Derivatives are abstractions. And, with one valuable exception, we can gain no impression of their character by representing them in perceptual space. The exception, though, will permit us to "picture" how Riemann discovered measurable relationships among points.
The exceptional derivative signals itself in mathematical discourse by an italicized lower case e. [7] The numerical value of e is 2.718218...(to infinity). It goes on and on forever, like pi. The curve made up of e is smooth in contour and sigmoid (S) in shape, and it relates Y to X, as follows: Y = eX. That is, Y equals 2.718218....if X is 1; the square of e if X is 2; the cube of e if X is 3....and so forth to infinity. In the latter expressions, e to some power of X is a function of X, meaning that e has a variational relationship to Y. But what makes e so very special to us is that its deriva
tive equals the function. In other words, when we look at a sigmoid curve, we see what we would see if we could actually plot Y = eX at a single point. Thus what we represent in perceptual space as a S-curve has validity for what we can't actually see at the point. [7a]
Imagine now that we have undertaken the task of exploring geometric figures. Assume that, like Riemann, we don't know the rules in advance and that our only metering device is e. To assist our imagery, envisage e 's as a string of pearls. Assume that we can bend and flex the string, increase or decrease the number of our e 's but can neither break nor stretch the string.
Okay, now suppose we come upon a flat surface and find two points, A and B. What would be the shortest distance between them? Remember, we must base our answers only on what we can measure. Gauged by our string of pearly e 's, the shortest path is the least number of e 's between A and B.
Suppose the distance from A to B is 12 pearls.
Now imagine that we put our string of pearls around some body's neck. Clearly, a path of 12 pearls remains a path of 12 pearls even though the new surface (neck) has a different shape from the first. Or with e 's as a gauge, we can relate a flat surface to a curved surface merely by finding their equivalents--the number of e 's. In our imaginary universe, round and square thus become variants of a common theme.
Let's return to the flat surface. This time, imagine that we run the hypotenuse of a large right triangle from A to B. When we lay the pearls between A and B, we find that the string fits very loosely on the hypotenuse. Even though we can always count pearls, we cannot gauge the hypotenuse very accurately because of the poor fit.