by Paul Pietsch
Suppose though that we reduce the number of e 's in our string and proportionally shrink the triangle; we continue to make the string and the triangle smaller and smaller but always remembering to take measurements by counting e 's. As we proceed, the slack between the string and the hypotenuse becomes smaller and smaller and eventually becomes so small as to be insignificant. Despite all the proscriptions of tradition, the hypotenuse--which looks like a straight line--has begun to approach a limit, relative to our pearls. We don't want to assert that the hypotenuse and our pearly e 's are "the same thing." If we do, we create the intolerable contradiction that the hypotenuse and the S-curve look alike, which we can see is false. But at infinity, the one is part of the other. The measurable part in question is little old e. Thus points in the hypotenuse have a measurable feature in common with points in our string of e 's. And that measurable feature is curvature.
Riemann also arrived at such an inference, although in a much more general, inclusive and rigorous way. "About any point," he discovered, "the metric [measurable] relationships are exactly the same as about any other point." A straight line, a flat surface or a rectilinear space consists of the same fundamental elements as a curve, circle or sphere; and magnitudes--round or square (pie or cornbread)--"are completely determined by measurements of curvature." Riemann showed that "flatness in smallest parts" represents "a particular case of those [geometric figures] whose curvature is everywhere constant." Flatness turned out to be zero curvature. In other words, a geometric universe constructed from elementary measurable relationships among points becomes an infinite continuum of curvatures, positive and negative, as well as zero.
Riemann demonstrated that "the propositions of geometry are not derivable from general concepts of quantity, but...only from experience." Out the window went the absolute prohibition against dealing with infinitely small regions of straight lines.[8] Out went the notion that parallel lines never! never! cross. Out went the universal dogma of even our own day that the shortest distance between two points is absolutely always the straight line.
The prohibition against points on lines was an artifact of trying to approach points from a flattened-down, squared-off, cut-straight universe; and the same had been true about parallel lines. And where did we ever come up with the idea that the shortest distance has to be the straight path? The argument in favor of straight lines has had its epistemological justification in the peroration of my little sister's ontology: "Because the teacher said!"
But how can all this Riemann stuff really be? Why don't our bridges all fall down?
The irony is that Riemann's system is not anti-Euclidean. As I pointed out earlier, Euclid's geometry belongs to the realm of experience. It is the geometry we invent and use for distances neither very great nor very small; for uncomplicated planes; for simple spaces; for universes of non-complex dimensions; for standard forms of logic (like Aristotle's). With Riemann's system as a guide, Euclid's propositions even acquire a valid a priori foundation. [9] Their limitations no longer dangerously hidden from us, Euclid's rules can serve our needs without tripping us up. Infinitesimal regions on a straight line are beyond the Euclidean approach, and thus we must use a different method for handling them; parallel lines don't intersect on a Euclidean plane; we can safely lay railroad track in the conventional manner. As far as the shortest distances thing is concerned, we may keep the protractor, the ruler and the T-square for geometric operations within the ken of experience. For in a Riemannian universe, the shortest distances between points is the path of least curvature. In the Euclidean realm of experience, that least curvature approaches zero curvature, and it coincides with what looks like a straight line.
I once worked on a crew with a graduate student in philosophy we all called Al-the-Carpenter. (His thesis was "God is Love.") Older and wiser and more generous than the rest of us, Al let scarcely a day pass without an enlightening observation or an unforgettable anecdote. "I met a little boy this morning," he once told us as we gathered around the time clock, "while I was making picnic tables outside the recreation hall. 'Is the world really round?' the little boy asked me. 'To the best of my knowledge,' I answered. 'Then tell me why you cut all the legs the same length. Won't your tables wobble?'"
"Yes," his tables would wobble, Al had replied. But the wobble would be infinitesimally tiny. The smile he usually wore broadened as he lectured like a latter-day Aristotle. "If you want to hear philosophical questions, pay attention to the children."
***
A theory constructed to fit the planes of experience will show its wobble when we invoke it on a very large, exceedingly small or extremely complex body of information.
Until now, we have viewed hologramic mind along Euclidean axes. But an understanding of hologramic mind requires the freedom only Riemann's simple but powerful ideas can give it.
***
Riemann's name and influence pervade mathematics (e.g., Riemann surfaces, Riemann manifolds, Riemann integrals, Riemannian geomery) for reasons that are evident in his qualifying lecture. The seeds of his subsequent work are present there. The elementary relationships at points later became the means by which scientists learned to conceptualize invariance. Toward the completion of the lecture, Riemann even anticipated general relativity. (Einstein's 4-dimensional space-time continuum is Riemannian.) Where did Riemann's insight come from? What intuitive spark caused his genius to push against the outermost fringes of human intellectual capability? I'm not sure I have the answer and only raise the question because I think my personal suppositions (i.e., hunch) will help us in our quest of hologramic mind.
***
Riemann, I believe, had a vivid concept of what I'll call "active" zero, the 0 between +1 and -1: the set we cross when we overdraw from the checking account--not the zero an erudite philosophy professor of mine, T. V. Smith used to call "what ain't." Judging from his own writings, Riemann seemed to have a crystal-clear intuitive idea of zero space--and even negative space![10] Let's try to develop a similar concept, ourselves. But so that what we say will have theoretical validity, let me make a few preliminary remarks.
There's a principle in logic known as Gödel's incompleteness theorem. The latter theorem tells us that we cannot prove every last proposition in a formal system. It's a sort of uncertainty principle of the abstract. We'll obey this tricky-- but powerful-- principle by always leaving at least one entire dimensional set beyond our reach. In fact, let's throw in two uncountable dimensions: negative and positive infinity which, by definition, are sets beyond our reach whether we admit our limitations or not. (Plus and minus infinity are by no means new ideas, incidentally.)
Consider a pretzel and a doughnut (or a bagel, if you're avoiding sweets). To keep the discussion simple, imagine them on a plane. Our doughnut has two apparent surfaces, the outside and the lining of the hole (Riemann called it a doubly connected surface). Our pretzel has four surfaces ("quadruply" connected) : one for each of the three holes, plus the exterior. Let's assume (with Riemann) that all things with the same number of surfaces belong to the same topological species, and let's not fret about whether the doughnut is round, oval or mashed down on one side. Also, let's consider surface to be a manifestation of dimension.
How can we convert a pretzel to a doughnut and vice versa? With the pretzel we could make the conversion with two cuts between two apparent surfaces. To go the other way, we (or the baker) can employ the term join.--two joins to regenerate a pretzel from a doughnut.
We've said that at either end of our universe there's a dimension (surface) we can't reach, positive and negative infinity. Thus, as we move up the scale (by joining), there's always one dimension more in the continuum than we can count. If we actually observe two surfaces on the doughnut, we know our overall system (or ideal universe) has at the very minimum three positive dimensions: the two we can count and the one we can't.[10a] What happens when we move in the other direction--when we apply (or add) one cut to our doughnut? We create a pancake with a si
ngle (singly connected) surface. But since we have the one countable surface, we know that still another surface must exist in the negative direction. And in the latter dimension is active zero.
Wait! We can't do that! you may insist. But look at it like this. We admitted that we can't count to infinity, right? If we don't put those uncountable dimensions on either end of our universe, then we end up in a genuine bind: either we can't count at all or we can reach unreachability. We can see for ourselves that we can cut or join--add and subtract (count)--to make pretzels and doughnuts out of each other. But to assert that we can reach unreachability may be okay for a preacher, but it's preposterous for a scientist.
Suppose we add a cut to a pancake--eat it up, for instance. Where are we? We now have another goofy choice: between active zero (zero surface) and "what ain't." We can't define "what ain't"; or, if we do define "what ain't" by definition, it won't be "what ain't" anymore. If we don't define it, it disappears from the argument, leaving us with good grammar and the zero-surfaced figure. And we can define the zero-surfaced dimension from our counting system: the dimension sandwiched between the one-surfaced pancake and the minus-one dimension.[11]
If we can conceive of a zero-dimensional surface, we can certainly appreciate a zero-curvature without making it "what ain't." And the zero-curvature, as part of the continuum of curvatures, is the curvature of the Euclidean world of experience.
***
Before we put hologramic mind into a Riemannian context, I would like to emphasize three important principles.
First, Riemann's success followed from in his basic approach. He didn't begin with an already-assembled coordinate system; he didn't erect the superstructure before he attempted to describe what the coordinates were like. In a system formulated from Riemann's approach, the elements define the coordinates, not the other way around as is usually the case, even in our own times.
The second principle is related to the first but is a direct outgrowth of Riemann's discovery that measurable relationships in the vicinity of one point are the same around all points. What does this mean in terms of different coordinate systems? If we can find those relationships in different coordinates, as we did in our imaginary experiments with pearly e 's, then the corresponding regions can be regarded as transformations of each other.
The third principle, an extension of the second, underlies our experiment with pearly e 's. An entire coordinate system becomes a transformation of any other coordinate system via their respective paths of least curvature. Thus the entire abstract universe is a single continuum of curvatures. Curvature is an abstraction. But we can always mimic curvature in our thoughts with our imaginary pearly pearly e 's.
Now let's make a preliminary first fitting of hologram mind to the world of Riemann.
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In terms of our search, a periodic event in perceptual space is a transformation, as a least curvature, to any other coordinates within the mental continuum. The same thing would be true of a series of periodic events. Since phase variation must be part of those events, memory (phase codes) becomes transformable to any coordinates in the mental continuum. A specific phase spectrum--a particular memory--becomes a definite path of least curvature in transformations from sensations to perceptions to stored memories to covert or overt behaviors, thoughts, feelings or whatever else exists in the mental continuum. If we call on our e 's for imagery, the least pearly path will not change merely because the coordinates change. Behavior, then, is an informational transform of, for instance, perception.
Transformation within the Riemann-style mental continuum is the means by which hologramic mind stores itself and manifests its existence in different ways. But we need some device for carrying out the transformation in question. For the latter purpose, I must introduce the reader to other abstract entities, quite implicit in Riemann's work but not explicitly worked out until some time after his death. These entities are known as tensors.
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The mathematician Leon Brillouin credits the crystal physicist W. Voigt with the discovery of tensors in the world of mass-energy.[12] It's no news that a crystal's anatomy will deform in response to stress or strain. But what stunned Voigt was that certain relative values within the distorted crystal remained invariant before and after the deformation. Tensors represent those invariant relative values. Like Riemann's invariant curvature relationships, tensors survive transformation anywhere, any place, any time. Just in time for Einstein, mathematicians worked out and proved the theorems for tensors. In the process they found tensors to be the most splendid abstract entities yet discovered for investigating ideal as well as physical changes. Tensors provided mathematicians with a whole new concept of the coordinate. And tensors furnished Einstein with a language in which to phrase relativity, as well as the means to deal with invariance in an ever-varying universe. As to their power and generality, Brillouin tells us: "An equation can have meaning only if the two members [the terms on either side of the equals sign] are of the same tensor character."[13] Alleged equations without tensor characteristics turn out to be empirical formulas and lack the necessity Benjamin Peirce talked about.
Tensors depict change, changing changes, changes in changing changes and even variations of higher orders. Conceptually, tensor resemble relative phase. Tensors relationships transform in the same way that relative phase does. This transformational feature affords us an impressionistic look at their meaning.
Do you recall the dot transform experiments from the last chapter:
Notice the diagonal arrangement of both the dots and the rings. If we rotate the rings so as to bring them into a horizontal orientation (tilt your head), the dots also take on a horizontal arrangement.
The rings and dots are transforms of each other. The fundamental direction of change remains constant in the transform as well as its back-transform. And during rotation, the basic orientation of the rings and dots--relative to each other--remains invariant. Absolute values differ enormously. But as we can see for ourselves, the relative values transform in the same way. This is the cardinal characteristic of the tensor: it preserves an abstract ratio independent of the coordinate system. If we stop to think about it, we realize that if tensors carry their own meaning wherever they go, they should be able to define the coordinate itself. And so they do.
Ordinary mathematical operations begin with a definition of the coordinate system. Let's ask, as Riemann did, what's the basis for such definitions? With what omniscience do we survey the totality of the real and ideal--from outside, no less!--and decide a priori just what a universe must be like? The user of tensors begins with a humble attitude. The user of tensors begins, as did Riemann, ignorant of the universe--and aware of his or her ignorance. The user of tensors is obliged to calculate the coordinate system only after arriving there and is not at all free to proclaim the coordinate system in advance. Tensors can work in the Cartesian systems of ordinary graphs; they can work in Euclid's world. But it is almost as though they were created to mediate travel in Riemann's abstract universes.
Change exists in two senses: co-variation, where changes proceed in the same direction (as when a beagle chases a jackrabbit); contra -variation, as exemplified in the two ends of a stretching rubber band. And (with tensors at least) the changes can be complex mixtures of covariance and contravariances (e. g., if the beagle gains on the jackrabbit as the quarry flags from fatigue or the rubber band offers progressively more resistance as tension increases). Tensors can attain higher rank--by simultaneously representing several variations. Packaged into one entity there can be an incredible amount of information about how things are changing. Ordinary mathematics become cumbersome beyond comprehension and eventually fail in the face of what tensors do quite naturally.
If you've seen equations for complex wave, you might be led to expect tensors to occupy several pages. But the mathematician has invented very simple expressions for tensors: subscripts denote covariance and superscript indicate contravariance.[13a] I
f, having applied the rules and performed the calculations, R in one place doesn't equal R in another, then the transformations aren't those of a tensor and from, Brillouin's dictum, represent local fluctuations peculiar to the coordinate system; they may have empirical but not analytical meaning. If the mathematician even bothers with such parochial factors at all, he'll call them "local constants."
Many features of holograms cannot be explained by ordinary transformations. In acoustical holograms, for instance, the sound waves in the air around the microphone don't linearly transform to all changes in the receiver or on the television screen. Tensors do. The complete construction of any hologram can be regarded as tensor transformations, the reference wave doing to the object wave what transformational rules do to make, say, Rij equal Rnm . Decoding, too, is much easier to explain with tensors than with conventional mathematics. With tensors, we can drop the double and triple talk (recall the transform of the transform), especially if we place the tensors within Riemann's continuum. The back-transform of phase codes from transform space to perceptual space becomes, simply, the shift (or parallel displacement) of the same relative values from a spatial to a temporal coordinate of the same continuum.
***
We imagined mind as a version of Riemann's theoretical world, as a continuous universe of phase codes. Now we add the concept of the tensor to the picture: Tensors represent phase relationships that will transform messages, independent of any coordinates within the universe. Indeed, phase relationships, as tensors, will define the coordinates--the perception, the memory, the whatever.[13b]
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I admit that a universe constructed from Riemann's guidelines is an exceedingly abstract entity. Diagrams, because of their Euclidean features, can undermine the very abstractions they seek to depict. But, as we did with perceptual and transform space, let's let our imaginations operate in a Euclidean world and, with reason, cautiously proceeding step by step, let's try to think our way to a higher level of intuitive understanding. Ready!