ZOOK: I hate to be rude to you young man, but I am not sure you are either culturally or emotionally prepared to hear what I might actually say on that subject. To begin with, I would not personally use 'metaphor' in this context. Also, while I have little use for dogma, per se, I cannot lend my name or intellect to the furtherance of antiscience, nor to sensationalism.
WRITER: Oh sir! That's not at all what I'm here for, at all, I assure you.
ZOOK: But...
WRITER: These are very exciting times in science, especially in biology and medicine, what with genes and DNA and computers and MRI and everything. Yet something seems missing, some seminal idea or crucial point of view or lost central chord or something. It's like a grand banquet with an undefined but indispensable course absent from the spread on the table. Like maybe smorgasbord without sour- creamed herring. Or a hot dog without a good tangy mustard. Your bones tell you something's just not there.
ZOOK: Can you be a little more explicit, please?
WRITER: I'm not really sure I can, sir. Forgive me about that. It's so hard to know about these things before they come into existence. But take something like AIDS, for instance. With everything the experts know about cells and molecular biology and with all the technology at their disposal today, why not more progress?
ZOOK: The immune systems is very complicated, and...
WRITER: I know the standard answers, sir. I'm not trying to zing anybody. And I don't mean to be hypercritical. Yet every time I hear the experts talk, I can't help but think, "Where's the beef," if you'll pardon the cliché.
ZOOK: The what?
WRITER: The beef! Don't you remember the TV commercial a few year back with the little old lady -- a bun the size of a catcher's mitt and the hamburger about as big as a half dollar. Microscopic amount of meat amid the mountainous quantity of fluff? Remember how Fritz Mondale clobbered poor Gary Hart with that?
ZOOK: I'm sorry, but I do not.
WRITER: Well... Look at Louis Pasteur. Didn't even know what a virus was when he tackled rabies. Maybe we need some wholly new ways of thinking about the living process? I don't mean with Ouija boards or anything like that. But maybe the very successes of the sciences of our times has forced blinders on the experts or freaked them into some kind of infinite logic loop: round and round and never even knowing it. Maybe it's like the flatworld Einstein had to get out of the way before he could see the space- time continuum.
ZOOK: I do not know how I can--
WRITER: This hologramic mind thing looked awful promising a few years ago.
ZOOK: The times are not right for it.
WRITER: That's what I'm kind of talking about, sir. Not that I'd be able to come up with those ideas, myself. But maybe out there in readerland some eleven- year old kid will get wind of the brain hologram and, Eureka!! will come up with the beef I'm talking about. Or think I'm talking about. Then again...Ah scheist, maybe I'm just being romantic a- hole. And...
ZOOK: You seem like a very nice young person, someone who could easily have been one of my own students. I will probably regret this but...Here, please have a seat.
Let me preface our discussion with a few words of Walt Whitman: "I give you fair warning before you attempt me further/ I am not what you supposed but far different."
WRITER: Gee! You really surprise me, Professor.
ZOOK: Occasionally, even an Asa Zook encounters an irresistible universe of meaning packed into a few simple lines of verse: In a manner of speaking, the verbal counterpart of a powerful equation.
WRITER: Speaking of poetry, you took exception to my use of 'metaphor' earlier. What word would you use for the hologram vis- a- vis the stored psyche?
ZOOK: There is nothing figurative about my usage of hologram. Obviously, the physical media differ, optical versus acoustical versus neural holograms. But the principle -- the logical sine qua non -- is the same wherever and whenever we find a hologram.
WRITER: Do you mean in the sense that, say, the number 6 can literally be applied to apples or oranges or wombat whiskers?
ZOOK: Ha! Ha! I mean the fundamental principle of the hologram as an informational rather than a physical entity.
WRITER: Informational and physical? Aren't they really the same?
ZOOK: Do you believe in the Second Law of Thermodynamics?
WRITER: Of course.
ZOOK: Obviously, information requires some medium for storage. But information may be used again and again without being used up. This is not true of matter.
WRITER: You can read a copy of the Times over and over. But you can only start a fire with it once. Is that what you mean?
ZOOK: Norbert Wiener first pointed to the critical distinction just after World War Two. Unfortunately, a generation of empirical scientists virtually ignored him. Perhaps, that is why you are here today.
WRITER: Anyway, the informational principle of the hologram is equipotentiality -- every single piece of the hologram plate can reproduce the entire original scene?
ZOOK: No! Definitely not!
WRITER: Wait! I always thought the equipotential business was the point of departure for advocates of the neural hologram.
ZOOK: Historically, yes. The survival of memory in a badly damaged brain seems analogous to the reconstruction of images from a small fragment of a hologram plate. But not all physical holograms exhibit this property, nor does all living memory. At the same time, the generic principle of the hologram permits equipotentiality and accounts for it when we happen to find it. But there is a vast difference between permitting and mandating. Equipotentiality is an ancillary consequence, not a primary antecedent of what the hologram must transcendentally be.
WRITER: Which is?
ZOOK: Phase! Harmonic phase. If you really want to explore the hologram's implications you'll have to comprehend phase. And if you want to understand how the hologram can exist at all -- how it can learn and remember -- you will have to know something about the fixed point.. But more about fixed point only after we look at phase! Are you still interested?
WRITER: Yessir.
ZOOK: What I shall say applies directly to waves but can easily be extended to any and all cyclical phenomena, periodic events or harmonic patterns.
WRITER: I'm aware of that, sir.
ZOOK: As you probably know, we can completely and precisely define any wave if we know its amplitude and its phase. How familiar are you with these terms?
WRITER: Amplitude is how high a wave rises and depends on how intense or bright it is, if the waves are light waves. It's the stuff or umph of a wave. I can understand amplitude, right off. Phase, I recently read, is how the stuff changes from place to place, how it's distributed.
But right from the getgo, I must admit, phase gives me a lot of trouble. I checked the definition just this morning before coming over here. Here's what I've written in my notes: "Phase is the fraction of a periodic interval a point on a wave passes after last having passed a reference mark..." Ugh! It's not just the language. But the definitions in the dictionaries and encyclopedias are given for simple harmonic motion, like a sweep hand on a watch or regular undulations -- sine and cosine waves. But what about complicated waves like, say, the profile of a face?
ZOOK: Let me remind you of Fourier theorem. Do you recall it?
WRITER: Any compound wave can be thought of as a series of simple, regular waves?
ZOOK: Close enough! A compound wave can be completely defined by its spectra of phases and amplitudes. Using your terms, the amplitudes are a function of the stuff -- the mass- energy of the waves. The phase spectrum determines how the stuff is distributed in space or time or the abstract equivalents thereof. Phase prescribes where the ups and the downs occur.
WRITER: I can see where phase is a function of time in moving waves. But what about, say, Claudette Colbert's hairdo? Standing waves?
ZOOK: Phase is about change, instantaneous change. In a traveling wave phase is a function of the time -- the instant -- a point on a wave arrives at a particular location. In standing waves, phase is a function of the transition, in space, from point to point to point; that is to say, the wave's slope.
WRITER: How steep the slope is? A function of how fast a sled would zip down its incline?
ZOOK: The change, or the algebraic sum in a spectrum of changes.
WRITER: Amplitude will determine, say, whether we're dealing with the shape of a statue in New York harbor or a figurine the size of organ grinder's monkey? But the phase spectrum will determine whether the profile belongs to Lady Liberty or King Kong, right?
ZOOK: Ha! Ha!
WRITER: I hate to say this, sir. And I hope you won't throw me out the door. But I still have trouble with phase.
ZOOK: I do too.
WRITER: What?
ZOOK: Intuitively, or in the absolute terms I suspect you are reaching for, I have no more idea of what phase is than I do a single geometric point or of an instant in time. The problem is -- and I suspect this is what makes even the optical hologram seem strange to empirically oriented scholars -- if we can actually see or touch, it is not phase, per se.
WRITER: Oxymoron: if we can grab it, it isn't what it is!
ZOOK: Phase is knowable as a relative entity. It is measured in relative terms, often as the phase difference between one wave and another, and is typically expressed as a function of an angle.
WRITER: The phase angle?
ZOOK: Very good! You can see examples of phase angles on the dial of your wristwatch.
WRITER: I'm sorry, but I'm wearing a digital watch.
ZOOK: That is too bad! But I think you are beginning to come to grips intellectually with phase.
WRITER: I'm just spouting words, sir, just words I'm afraid.
ZOOK: Perhaps you are still trying to reduce phase to concrete terms, which is not possible. How big is an angle? Can you heft a degree or stuff a radian into your pocket? Fill a basket with sines and cosines? Of course not. Degrees and radians and functions of angles are relative quantities. Relative attributes are said to be arbitrary.
WRITER: Arbitrary, not in the sense of capricious but as assigned by us, at our discretion? Right?
ZOOK: Phase is without absolute size. Of course, we need some amplitude for phase to occur at all in the physical world. But rather than phase as a creature of magic, think it as having to with relationships in mass- energy, not mass-energy in and of itself.
WRITER: Like a 90- degree angle that can exist between two 6- ton concrete slabs as well as between the gnat and its proverbial eyebrow? Incidentally, is it the arbitrary nature of phase that allows for equipotentiality among certain holograms?
ZOOK: In theory, a hologram can be almost as small as a geometric point, and thus repeated at every single location in the hologram plate. But a single hologram may also spread out over almost the entire universe.
WRITER: Almost? How much almost?
ZOOK: In theory, a single geometric point's worth.
WRITER: In practice?
ZOOK: Approaching the magnitude of a point
WRITER: Sukosi as all heck, then eh!
ZOOK: Sukosi?
WRITER: Small amount or wee.
ZOOK: The relative character of phase would permit the very same relative phase spectrum to be encoded on the head of a pin and simultaneously spread over an expanse equal to the surface of the sun.
WRITER: I'm puzzled, though. Isn't the hologram itself a contradiction? I mean, the hologram is phase captured, right? Aren't we back to our oxymoron?
ZOOK: There are systems that would permit us to regard the hologram as existing in an abstract transform space where phase becomes a function of a spatial rather than temporal coordinates. We would then speak in terms of the transform of phase, but not phase, per se. In reconstruction we would back- transform phase to the space of experience. One really needs equations to do justice to this approach.
A more intuitive approach, though, is to regard the hologram not as phase, per se, or even a transform thereof, but simply as coded information -- a memory -- from which the original phase spectrum can be regenerated. Which theoretical route do you prefer, the analytical or the intuitive?
WRITER: I'll take intuition any day, sir.
ZOOK: Are you familiar with the basic steps of optical holography?
WRITER: Let's see. There's an object beam -- light waves from the scene --
and a reference beam -- light from the same source as the object beam but
deflected with mirrors to miss the scene. The two sets of waves must be
coherent -- in step. Anyway, the object and reference beams collide and
interfere and create the hologram -- interference or diffraction patterns -- on
a photographic film. Sort of...May I use your black board, sir?
Sort of like this:
The scene's image can be reconstructed if the reference beam -- or a coherent
facsimile thereof -- is shined back through the hologram plate.
ZOOK: Understand that there can be configurations for the constructions of the hologram other than what you represent. An ordinary interference pattern is actually a primitive hologram, one in which the encoded message has a value of zero. Also there are other ways of producing even optical holograms.
WRITER: Can you explain one thing to me sir? What is coherency, really? I mean, I say "in step," but I really don't think I know what I'm talking about.
ZOOK: I will discuss coherency from the standpoint of the topologist's fixed point theory. But later! To avoid prolixity, let us immediately return to phase. Are you familiar with acoustical holography?
WRITER: I've read about it, yes, where sound provides the waves; the hologram is recorded with a microphone, displayed on a TV screen and photographed and the negative can be used for reconstruction.
ZOOK: An acoustical holographer named Alexander Metherell demonstrated that phase is the essence of the hologram. He invented what he called the "phase- only" hologram. He recorded the phase variations of the object wave but normalized the object wave's amplitude to a single quantity.
WRITER: That last part confuses me, sir.
ZOOK: Metherell had to use some amplitude -- stuff in your vernacular -- or he could not have operated in the physical world. Wave- borne messages are carried as variations -- modulations -- in the waves. But a single quantity will carry zero message. Metherell sent a single amplitude into his recording circuit but recorded the phase spectrum. His experiment was very well executed, I might add. He took as an object an enlarged letter, R. As a control, he made a conventional hologram of the R; that is, he recorded its amplitude and phase. Then he made a phase- only hologram of the same R. Finally, he compared the quality of the reconstructed images and found that of the phase- only acoustical holograms to be equal to or better than their conventional counterpart.
WRITER: What about optical holograms? Can they be phase- only?
ZOOK: Light is mass- energy at maximum velocity, under which circumstances phase and amplitude are inextricable. Phase- only holograms would not be possible with light, per se. Appreciate, however, that the inextricability in question -- or the absence thereof -- is an attribute of the medium, not a universal feature of all wave phenomena.
WRITER: Gee, as I stop and think about it, without the acoustical hologram the indispensability of phase might have been missed entirely. Am I all wet on that score?
ZOOK: All wet? I am unfamiliar with your idiom. But it seems probable that without the phase- only hologram, the philosophical implications of the hologram in general might not be evident.
WRITER: Wow! Do you think the neural hologram is phase- only?
ZOOK: Given the minuscule energy levels of biological systems, my opinion is that the neural hologram is phase- only, in Metherell's sense of the term. The important thing, though, is not my opinion but this. If we find a hologram, in whatever medium or circumstance, it is a memory of phase. Other attributes -- whether an amplitude spectrum is simultaneously encoded; what the nature of the code's distribution happens to be; what special features are imposed by quirks of the medium -- these are not ontological necessities but are accidental features.
WRITER: Accidental, in the cause- effect, not the slip-on-the-banana-peel sense of the word, right?
ZOOK: Ha! Ha! Can I offer you a coffee or a tea?
WRITER: This is great! Szechwan?
ZOOK: Lipton.
WRITER: Now early on, sir you said something about tackling coherency with the fixed point.
ZOOK: Yes! We can profitably use various theorems and corollaries of fixed point theory to simplify and explain much about the hologram that would otherwise remain beyond rational inquiry. The proof of the most famous fixed- point theorem was published in 1912 by an angry, rebellious and, at the time, young Dutchman, L. E. J. Brouwer.
WRITER: If fixed point tells us how holograms can learn and remember, I'm all ears.
ZOOK: Fixed- point theorems guarantee , with a few permissive constraints that, in a true continuum -- such as a wave -- at least one point must retain the same position before and after any continuous transformation. The theorems do not specify just which point shall be fixed, only that one or more must be -- if, indeed, we are dealing with a truly continuous system.
WRITER: You mean if I stir my tea, one point won't change positions?
ZOOK: As long as you do not splash the liquid from the glass, one point must occupy the same locus after the stirring as it did before it.
WRITER: That doesn't sound possible! And guaranteed?
ZOOK: Guaranteed! Several proofs of fixed- point theorems exist today, some very general and quite astonishingly interesting, but rather arcane, and more abstract than we actually need for our discussion. Let me employ a relatively simply, unsophisticated one to illustrate why a continuous system must have a fixed point.
Assume for the sake of simplicity that our system is finite and bounded. Assume also that as long as we remain within the boundaries, we can arbitrarily make any transformations we please; that is, we can move between and among any points and along any paths we choose, providing we do not violate the system's boundaries. Should something block any given path, we shall have violated a hidden boundary.
Now assume that our system contains only 3 points: A, B and C. Let us elect to move points along only the shortest paths. Now let us move point A onto point B to produce the new point AB, remembering to proceed along the shortest path. Now we get ready to move AB on C. But just before we arrive at C, C shifts. Then what?
WRITER: If C shifts, we can't move along the shortest path from AB.
ZOOK: Either we cannot make any transformation we please -- which would violate a boundary -- or we cannot shift C.
WRITER: What if we force AB onto C anyway?
ZOOK: Then we prove that our system is not a true continuum; in such circumstances AB and C may be juxtapositional, or contiguous, but not continuous.
WRITER: Can you give me a concrete example of this juxtaposition business?
ZOOK: A deck of cards. Any situation where shuffling can scramble the order.
WRITER: A stack of children's blocks?
ZOOK: Yes.
WRITER: They eventually topple! What you're saying, in effect, is that if two -- or more -- sets of things enjoy a continuous relationship they must share a fixed point?
ZOOK: The fixed point property -- fpp * of some texts -- is a necessary and sufficient condition of continuity.
WRITER: You said waves are continuous. Does that mean a wave must have a fixed point?
ZOOK: Always. And the fixed point remains invariant upon diffraction of the wave by objects.
WRITER: Do you mean like when the object wave reflected off the scene one point in it wasn't changed?
ZOOK: Yes.
WRITER: I'm beginning to sniff something real good here.
ZOOK: Does the name Riemann mean anything to you?
WRITER: The fellow who invented the geometry Albert Einstein used on the space-time continuum?
ZOOK: Riemann had a penchant for raising the most fundamental of questions, questions of the sort scholars are often loath to pose about their fields.
WRITER: Loath, for fear of the answers they might get?
ZOOK: Riemann once asked how we can measure a quantity at all. He postulated that we can do so through the superposition of one thing on another.
WRITER: Flopping one thing on top of another?
ZOOK: Not quite. Not simply bringing things in apposition but uniting them such that one becomes the other at at least one point. "When one is part of the other," in Riemann's own words. Fixed- point theorems vindicate Riemann's postulation.
WRITER: The King of Clubs does not define the Queen of Hearts merely by lying on top of her.
ZOOK: Not unless both enjoy the same fixed- point property, fpp .
WRITER: Is there any sexier term than fpp?
ZOOK: Pardon me?
WRITER: Never mind, sir. A bad habit of mine. I'll stick with fpp.
ZOOK: Now reconsider your diagram of holography. The object and reference waves O and R, respectively, come from the same source. They originated as part of the same continuum. Therefore, before striking the object, O equals R. Consequently, O and R enjoy the same fpp. Beyond the scene, one region of O that will remain unaffected by the diffractions exerted by the objects is the fixed point, which exists in R.
WRITER: Wait! If I follow you correctly, our fpp is tantamount to coherency.
ZOOK: Yes.
WRITER: And when we carry out the reconstruction process, when we shine the reference waves back through the hologram, and regenerate the scene's image, we...we reunite the fixed points in R and O!
ZOOK: In a manner of speaking, yes..
WRITER: So that's what remembering comes down to -- to reuniting fixed points so as to redefine the original phase spectrum in the realm of experience. Gee, that reminds me of a Japanese greeting I've always liked and which translates, "I am you." Boy! It's like saying holologic is some kind of cosmological true love.
ZOOK: I do not understand?
WRITER: Never mind, sir. I was only kidding. Sort of.