NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page -             -      -Author. Article nameMen Who Made a New...
NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page -   -   -Author. Article nameIntroduction                   ...
NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page -   -   -Author. Article name      self-consciousness.∗     ...
NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page -   -   -Author. Article namemicrotubules.                  ...
NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page -           -   -Author. Article name    colors. More freque...
NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page -   -   -Author. Article nameyet these familiar, characteris...
NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page -   -   -Author. Article namefoundations of quantum theory, ...
NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page -      -   -Author. Article nameI do not know, of course — I...
NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page -   -   -Author. Article nameNonetheless, we are left with p...
NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page -       -   -Author. Article namephysical theory. Here is an...
NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page -   -   -Author. Article name         Curiously, colors and ...
NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page -   -   -Author. Article name         Let us now pay special...
NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page -   -   -Author. Article name         In brief, then, it see...
NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page -   -   -Author. Article namevector in Hilbert space as well...
NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page -   -   -Author. Article name         Now, this is quite a c...
NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page -   -   -Author. Article name         Returning to the place...
NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page -           -   -Author. Article nameReferences             ...
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On the Unification of Mind & Matter

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I discuss color vision as an exemplar of more general ideas concerning the mind/body problem.

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On the Unification of Mind & Matter

  1. 1. NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page - - -Author. Article nameMen Who Made a New Science Invited Article On the Unification of Mind and Matter 1 Brian J. Flanagan Abstract The “hard problem” is considered vis-à-vis Gödel’s work on formal systems, tensor network theory, the vector character of sensory qualities, the symmetries and phase relations of those qualities and Heisenberg’s matrix formulation of quantum theory. An identity is considered vis-à-vis the secondary properties of perception and (1) the hidden variables of quantum theory; (2) the internal spaces of gauge theory; and (3) the additional dimensions of M-theory. Key Words: quantum field theory, mind/body problem, color, vision, Gödel, EPR, hidden variables, M-theory, mind-brain identity theory, tensor network theory, secondary qualities, Einstein, Bohr, Heisenberg, Pellionisz, Llinas, Churchland NeuroQuantology 2007; 4: - The assembling of all these elements has been effected by century by century, in past ages down to our own time… Al-Kindi (ca. 800–870 CE)Corresponding author:1Brian J. FlanaganAddress: Sentient Technologies, 434 S Johnson Street #1, Iowa City, IA 52240 USAe-mail: sentek1@yahoo.comISSN 1303 5150 www.neuroquantology.com
  2. 2. NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page - - -Author. Article nameIntroduction which reposes directly on a “minimal brainA little over 20 years ago, when I first event,” viz., a photonic state vector. Longbegan to air my ideas regarding the before that idea came to light, I made anquantum substrate of mind and brain, the amusing blunder, however.topic was hardly on the map. For a long In my youthful explorations ofwhile, I took comfort in the famous dictum physics it became pretty clear that theof Freeman Dyson, who advises us that, color of a thing didn’t seem to depend on“For any speculation which does not at its motion in space-time — except in thefirst glance look crazy, there is no hope.” important case of Doppler shifts. Other Today, thanks to Sultan Tarlaci, it things being equal, however, the color of aseems clear that the subject has attained a thing does not change as our world spinscertain maturity and I should like to thank on its axis or flies along through thehim for this fine honor. It seems to me that interstellar regions.no reward can equal or surpass the Today I see, with the benefit ofappreciation of one’s peers. I should also hindsight, that these constants in thelike to thank my teachers for their patient appearances of things are instances ofguidance and instruction. symmetries or invariants — and so part of My parents bought me a toy robot Klein’s rich legacy. But back then I feltwhen I was a little tyke. A terror in stymied and so turned to neuroscience anddiapers, I took it apart right away — such to the logic and philosophy of science.contrivances being beneath me — to see I soon found the seminal work bywhat made the thing go. I got as far as the McCulloch and made a lengthy study ofmotor, where I discovered coiled wire and mathematical logic, which led to Russell &magnets. I then read up on Faraday and Whitehead, Tarski, Turing and Gödel.Maxwell until I encountered the latter’s Trying to get a handle on Gödel’s workequations and was thoroughly mystified. quickly led me to the famous essay by Years later, I went to see the Nagel & Newman, where I encounteredpremier of 2001: A Space Odyssey. I was the following passage:quite taken with HAL, the sentientcomputer. It was about this time that I How did Gödel prove hisbegan to wonder how one might engineer conclusions? Up to a point, thesuch a device. At 16, reading Bertrand structure of his demonstration isRussell, I was arrested by the problem of modeled, as he himself noted, on thehow the brain can perceive color, which reasoning involved in one of theseemed to be related to light — a physical logical antinomies known as thething — but which had long been held to “Richard Paradox,” first propoundedreside in the mind — a mental thing. This by the French mathematician, Julesproved a fortunate choice of problems; Richard, in 1905 […]years later, I came across a remark from The reasoning in the RichardWilliam James: Paradox is evidently fallacious. Its construction nevertheless suggests The ultimate of ultimate problems, of that it might be possible to “map” course, in the study of the relations of (or “mirror”) metamathematical thought and brain, is to understand why statements about a sufficiently and how such disparate things are comprehensive formal system into connected at all […] We must find the the system itself. If this were minimal mental fact whose being reposes directly on a brain-fact; and we possible, then metamathematical must similarly find the minimal brain statements about a system would be event which will have a mental represented by statements within the counterpart at all. system. Thereby one could achieve the desirable end of getting the Eventually it came together for me that formal system to speak about itselfa color vector is a “minimal mental fact” — a most valuable form ofISSN 1303 5150 www.neuroquantology.com
  3. 3. NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page - - -Author. Article name self-consciousness.∗ different, I had another inspiration. Years before, in the course of many happyThat last bit fired my thinking. The brain, readings of the Churchlands’ works, Ibeing a physical thing, is presumably a encountered the tensor network theory of(quantum!) mechanical thing, and can Pellionisz and Llinas. I’d known for sometherefore be modeled by a formal system. time that Heisenberg’s approach toMoreover, our brains describe their states quantum mechanics employed matrices— they talk about themselves. They talk operating on vectors. Riding the bus thatabout objects of perception, which are day, it hit me all at once that the matricesrepresented in us as configurations of of tensor network theory and the matricescolors and sounds — which properties of QM are the same matrices and thatappear to be elemental in the sense that we our perceptual fields just are quantumcannot define them in respect of simpler fields. Most of my subsequent efforts wereentities. Years later, it hit me that, rather devoted to seeing whether this inspirationthan try to reduce these elemental could stand up to the light of day. In allproperties to simpler entities, that maybe this, I was guided by the ethos ofwe ought to take nature at her word and simplicity admirably framed by Einstein:ask what sort of geometry might result if “When the solution is simple, God iswe were to alter the axiom of physical answering.” I believe that the answer to thescience which has it that colors and sounds mind/body duality is quite simple —exist only in the mind. What if we were to though sweeping in its ramifications —regard colors and sounds as the elements and in what follows I have tried to be easyof a formal theory (T), where the on the reader, mindful of theinterpretation of T was an EPR-complete interdisciplinary nature of the work and ofquantum theory? the real difficulties encountered in the I was an undergraduate when I contributing fields.first began to explore these notions and Where do we go from here? I amspoke about them to anyone who would profoundly gratified to see that Dr.listen. No one understood what I was on Tarlaci’s pioneering efforts have beenabout and I suppose they suspected I didn’t, followed by other success stories, aseither. Fortunately for me, Douglas witness the publication in the ProceedingsHofstadter’s wonderful book on Gödel, of the Royal Society of Henry Stapp’s ideasEscher, Bach came out about that time. A and by the forthcoming symposium onsimilar episode occurred a few years later; Quantum Interaction, hosted by StanfordI’d published a paper on the quantum University and the AAAI.substrate of the mind and brain in 1985 and Although the “quantum mind”presented another in 1987. The former was movement continues to find its detractors,greeted with the customary silence, but the their comments tend to evince only alatter work prompted quite a lot of interest. passing acquaintance with quantum theoryAfter my little talk, I had lunch with a and/or neuroscience and strike me as thebright young PhD student from California, kind of rearguard actions which Kuhnwho told me, “Yeah, you could tell — at discusses in his landmark work on Thefirst, everyone was wondering what planet Structure of Scientific Revolutions. I thinkyou stepped off of; then they were all there are good reasons for thinking so; onereally interested.” reads, e.g., that Nature Neuroscience has Then, in 1989, Penrose published just now published an article telling us thathis first big book on these topics and the the “noise” in the brain — often thought toball really began to roll, with scholarly preclude quantum effects — is actually justconferences organized and fiery rhetoric what one might expect to find if the braindetonated. A little later on, while riding a is engaged in Bayesian computations. Andbus and working on something completely then, it seems as though scarcely a week goes by, lately, but one reads about the∗ My emphasis remarkable electronic properties ofISSN 1303 5150 www.neuroquantology.com
  4. 4. NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page - - -Author. Article namemicrotubules. chemical binding is electromagnetic in On the whole, I see good reasons for origin, and so are all phenomena of nerveoptimism. impulses.” If we should suppose, as wouldSimplicity seem altogether plausible, that consciousQuantum field theory (QFT) enjoys an processes just are phenomena of nerveunfortunate reputation for being difficult. impulses, then it seems to follow as aWithout diminishing the very real ready consequence that those consciouschallenges presented by that body of processes are electromagnetic in origin.thought, it helps to bear in mind that a field Can we put flesh on the bones of thisis basically quite a simple thing, as Gerard conjecture?‘t Hooft reminds us: “a field is simply a Dyson wrote that a field is “is anquantity defined at every point throughout undefined something which existssome region of space and time.” Freeman throughout a volume of space.” MaxwellDyson helps us further: tells us that color is undefined, in the sense that colors are so very simple, they are This is the characteristic mathe- incapable of analysis: matical property of a classical field: it is an undefined something which When a beam of light falls on the exists throughout a volume of space human eye, certain sensations are and which is described by sets of produced, from which the possessor numbers, each set denoting the field of that organ judges of the color and strength and direction at a single luminance of the light. Now, though point in the space. everyone experiences these sensations and though they are the Dyson also tells us, with the foundation of all the phenomena ofsimplicity of genius, that “There is nothing sight, yet, on account of theirelse except these fields: the whole of the absolute simplicity, they arematerial universe is built of them.” incapable of analysis, and can never What has QFT to do with become in themselves objects ofconsciousness, though? In the helpful case thought. If we attempt to discoverof visual perception we often refer to the them, we must do so by artificialvisual field, where colors would seem to be means and our reasonings on themakin to Dyson’s “undefined something must be guided by some theory.which exists throughout a volume ofspace.” Russell and Whitehead back up In this wise it is intriguing to Maxwell: “Thus ‘this is red,’ ‘this is earlierreflect upon a terse observation from than that,’ are atomic propositions.”Wittgenstein: So far as colors existing through- out a volume of space goes, we are on safe A speck in the visual field, though ground, thanks to Riemann, who, at the it need not be red must have some beginning of his famous habilitation color; it is, so to speak, surrounded by lecture on the foundations of geometry, color-space. Notes must have some points out that colors and the positions of pitch, objects of the sense of touch objects both define manifolds: some degree of hardness, and so on. So few and far between are the On a mind/brain identity theory, occasions for forming notions whosesuch as we find in Bohm, Chalmers, Feigl, specialization make up a continuousLockwood and Russell, it is tempting to manifold, that the only simplesuppose that our perceptual fields just are notions whose specialization form aelectromagnetic fields. Abdus Salam helps multiply extended manifold are theus along here, reminding us that “all positions of perceived objects andISSN 1303 5150 www.neuroquantology.com
  5. 5. NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page - - -Author. Article name colors. More frequent occasions for whose eye it is the sensation of the creation and development of yellow. these notions occur first in the higher mathematics. How did we get here? The Definite portions of a manifold, spectrum of colors is arrayed, in a distinguished by a mark or a predictable, reliable manner, with the boundary, are called Quanta [...]. spectrum of wavelengths found in visible light. And even quite intelligent peopleHere is Hermann Weyl to help us out a bit will hasten to tell you that color just ismore on manifolds: “The characteristic of wavelength — not knowing that colors arean n-dimensional manifold is that each of vectors, whereas wavelengths are scalars,the elements composing it (in our having only magnitude.† Weyl reminds usexamples, single points, conditions of a of the essential facts:gas, colors, tones) may be specified by thegiving of n quantities, the ‘co-ordinates,’ To monochromatic light corres-which are continuous functions within the ponds in the acoustic domain themanifold.” simple tone. Out of different kinds Dyson wrote that that fields are of monochromatic light composite“described by sets of numbers, each set light may be mixed, just as tonesdenoting the field strength and direction at combine to a composite sound. Thisa single point in the space.” That sounds takes place by superposing simplelike a vector and it just so happens that oscillations of different frequencycolors behave like vectors, too, as with definite intensities.Grassmann, Maxwell, Schrödinger, Weyland Feynman all tell us — and as thetechnology behind our color TVs andcomputer monitors makes plain.∗ Appealing to Feigl’s formulationof mind/brain identity theory, we might saythat configurations of colors representother objects in the natural world, as whenwe describe an apple as a red, roundishthing. Already we encounter a difficulty,however, for we want to know where the Figure 1. Wave superpositionredness comes in. Schrödingers equationis often said to contain, in principle, all thatcan be known about a physical system.Schrödinger disagreed, however: If you ask a physicist what is his idea of yellow light, he will tell you that it is transversal electromagnetic waves of wavelength in the neighborhood of 590 millimicrons. Figure 2. Vector superposition; If you ask him: But where does notice that the normalized yellow vector is a precise metamer for yellow comes in? he will say: In my yellow at 580 nm. picture not at all, but these kinds of vibrations, when they hit the retina of a healthy eye, give the person We all know these things — and∗ † The seminal works by Grassmann, Maxwell and Sounds also come to us in spectra, with remarkableSchrödinger can be found in the superb collection relations between tone and number, already known toedited by MacAdam. Pythagoras — though still awaiting a good explanation.ISSN 1303 5150 www.neuroquantology.com
  6. 6. NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page - - -Author. Article nameyet these familiar, characteristic (eigen)color vectors and tones (predictably related The qualities of size, figure (orto “simple oscillations of different shape), number and motion are forfrequency with definite intensities”) these Galileo the only real properties ofcolors and tones are not yet properly objects. All other qualities revealedincorporated into the body of physical in sense perception — colors, tastes,theory. So, again: How did we come to this odors, sounds, and so on — existpass? only in the sensitive body, and do not qualify anything in the objects themselves. They are the effects ofHistory the primary qualities of things on theThe answer can be found in the classic senses. Without the living animalworks of Burtt, Duhem, and Lovejoy, but sensing such things, these second-Weyl puts the matter succinctly: ary qualities (to use the term introduced by Locke) would not The idea of the merely subjective, exist. immanent nature of sense qualities, Much of modern philosophy has as we have seen, always occurred in devolved from this fateful history woven together with the distinction. While it was scientific doctrine about the real undoubtedly helpful to the physical generation of visual and other sense sciences to make the mind into a perceptions […] Locke’s standpoint sort of dustbin into which one could in distinguishing primary and sweep the troublesome sensory secondary qualities corresponds to qualities, this stratagem created the physics of Galileo, Newton, and difficulties for later attempts to Huyghens; for here all occurrences arrive at some scientific in the world are constructed as understanding of the mind. In intuitively conceived motions of particular, the strategy cannot be particles in intuitive space. Hence an reapplied when one goes on to absolute Euclidean space is needed explain sensation and perception. If as a standing medium into which the physics cannot explain secondary orbits of motion are traced. One can qualities, then it seems that any hardly go amiss by maintaining that science that can explain secondary the philosophical doctrine was qualities must appeal to explanatory abstracted from or developed in principles distinct from those of close connection with the rise of this physics. Thus are born various physics. dualisms. Galileo and Newton carried theAusten Clark elaborates on the essential day, along with their philosophicalsituation wherein we find ourselves: admirers, viz., Locke, Hobbes and Descartes — over the objections of Leibniz The world as described by natural and also Hume — for the simple reason science has no obvious place for that their new science worked, and colors, tastes, or smells. Problems marvelously so. And one might well ask, if with sensory qualities have been the new physics was really so flawed, why philosophically and scientifically does it continue to work so well? (That is, troublesome since ancient times, and until we try to frame a science of in modern form at least since perception?) Galileo in 1623 identified some sensory qualities as characterizing Completeness nothing real in the objects them- Thoughtful persons will see that we have selves […] encountered the former question at theISSN 1303 5150 www.neuroquantology.com
  7. 7. NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page - - -Author. Article namefoundations of quantum theory, in the The proposal that I put forth overEinstein-Bohr debates, in EPR and in the 20 years ago is that the “secondaryliterature that followed. qualities” of color and sound and so forth, Now, a notoriously intractable long ago banished to the misty realm of theissue in contemporary thought on the mind, just are EPRs missing elements ofmind/body problem concerns what reality — variables only hidden in plainChalmers has famously dubbed the “hard sight, their true nature obscured by 300problem,” viz., the question of why some years of dogma. As scientists, we take inbrain states have perceptual states attached this doctrine with our mothers’ milk; itto them. Others, whom I suspect betray a forms part of what “everybody knows,”guilty conscience, have attempted to wave and so constitutes a brainwashing moreaway the issue, dismissing it as mere talk thorough than anything Heisenberg andregarding “the redness of red.” Often Bohr ever devised — and this despite themissing from these discussions is any fact (picked up at the time by Hume andreference to the scientific literature Leibniz) that this doctrine lies at the veryconcerning color. This is a bit odd, given crux of the scientific worldview.that such illustrious persons as Newton, I dont want to be too hard on BohrYoung, Helmholtz, Grassmann, Maxwell, and Heisenberg, whose place in theSchrodinger, Weyl, Einstein and Feynman pantheon of physics is, of course, beyondhave devoted serious thought to the nature question and whose thoughts on theof color. questions at hand I will now exploit. As noted above, Schrödingers As David Bohm informs us in thewave function is often said to contain, in introduction to his congenial book onprinciple, all that can be known about a Quantum Theory, “Bohr suggests thatphysical system. Einstein disagreed: thought involves such small amounts of energy that quantum-theoretical limita- one arrives at very implausible tions play an essential role in determining theoretical conceptions, if one its character.” Bohr also wrote to the effect attempts to maintain the thesis that that nothing is so fundamental to the the statistical quantum theory is in worldview of quantum theory as the principle capable of producing a demolition of the notion that physics has to complete description of an do with what is out there — i.e., the individual physical system. Newtonian ideal of a world where a detached observer presents us with no The seminal work of EPR raised problems.the possibility that QM is incomplete, in Henry Stapp has put Heisenberg’sthe sense that not all “elements of reality” ideas about QM and the observer problemare incorporated into the body of quantum to good use, arguing that quantum theorytheory. Those missing elements have come has essentially to do with our knowledge ofto be known as “hidden variables,” and I physical states and systems. He has made aam happy to see that these variables are bold and intriguing conjecture — drawingonce again on the front burner of on William James’s ideas about thetheoretical physics, thanks to Hartle, selective nature of attention — that it is ourHolland, t Hooft, Peres and Smolin, conscious attention that is behind theamong others. I should like to pause here Quantum Zeno Effect (QZE). For my part,to pay tribute to the path finding work of I tend to agree with J.S. Bell, who thoughtDavid Bohm and J. S. Bell, who, nearly that the observer ought not to bealone, kept this question alive for 60 years, “ontologically privileged.” I expect that thewhen the foundations of quantum theory QZE is due to nonlocal effects — but then,languished in a moribund state, thanks to it may well not be a case of either/or: Ifthe “brainwashing” which, as Gell-Mann consciousness does consist of quantumtells us, Bohr and Heisenberg induced in a field processes, those processes might thengeneration of physicists. have nonlocal effects, producing the QZE.ISSN 1303 5150 www.neuroquantology.com
  8. 8. NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page - - -Author. Article nameI do not know, of course — I don’t think in our own experience, butanyone does at this juncture — but it something very different. Theseems a fascinating possibility. observer, when he seems to himself I will return to Heisenberg later on, to be observing a stone, is really, ifbut would like to point out that five other physics is to be believed, observingprominent physicists also devoted serious the effects of the stone upon himself.thought to mind and matter, namely Thus science seems to be at warEinstein, Pauli, Schrodinger, Weyl and with itself: when it means to be mostWigner. Aside from Pauli, whose thoughts objective, it finds itself plunged intoon these issues have only recently come to subjectivity against its will.light, I have remarked in another placeupon the contributions of these men, and Einstein replies:so I only touch upon them here. Wigner,e.g., wrote that physics would have to be Apart from their masterful formula-transformed in order to account for tion these lines say something whichconsciousness. Pauli, in a recently had never previously occurred to me.translated series of letters to Carl Jung, For, superficially considered, thelooked forward to a unitary description of mode of thought of Berkeley andmind and matter, informed by quantum Hume seems to stand in contrast totheory. the mode of thought in the natural sciences. However, Russells justLogic cited remark uncovers a connection:Getting back: Einstein, in his “Remarks on If Berkeley relies upon the fact thatBertrand Russell’s Theory of Knowledge,” we do not directly grasp the “things”quotes Russell in this connection, going to of the external world through ourthe heart of the matter: senses, but that only events causally connected with the presence of “things” reach our sense-organs, then this is a consideration which gets its persuasive character from our confidence in the physical mode of thought. Figure 3. Russell We all start from naive realism, i.e., the doctrine that things are what they seem. We think that grass is green, that stones are hard, and that Figure 4. Einstein snow is cold. But physics assures us that the greenness of grass, the Now, our usual “confidence in the hardness of stones, and the coldness physical mode of thought” stems from its of snow, are not the greenness, conceptual beauty, logical clarity, hardness, and coldness that we know predictive power and practical utility.ISSN 1303 5150 www.neuroquantology.com
  9. 9. NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page - - -Author. Article nameNonetheless, we are left with public, Symmetryreproducible and reliable facts of The move sketched in just now will strikegreenness, hardness, and coldness. some as a little outlandish; nonetheless, we As noted earlier, these properties already have in hand the kind of theoryseem to be elemental on account of their suggested above; the primary qualities ofextreme simplicity. Let us take a page from extension in space and duration in time areGödel, then, and frame a formal theory T, already part of the foundations of physicsof a very general nature, constrained only and are encoded in the metric of generalby the laws of simple logic. relativity (GR). Take for the elements of T both Einstein showed us how to derivethe primary qualities and secondary gravitation from the metric tensor — andqualities of sensory perception. It then here the primary quality of mass alsofollows as a consequence of simple logic enters the theory. Minkowski, who wasthat, no matter how complex this Einstein’s teacher, said that “relativity”essentially mechanical thing we call T may was a misnomer, for it was really a theorybe — it may have all the complexity of a of invariants, given that relativity flowshuman brain — T will be unable to define from the invariance of the speed of lightits elements, for the simple reason that its for all observers. Today we call these kindselements would not then be elements and of invariants by the more suggestive namewe would have a simple contradiction. of symmetries — and these symmetries are In some such fashion, perhaps, are of truly fundamental importance. Nobelwe unable to define the elements of our laureate Steven Weinberg has stated theperceptions, those self-same elemental case with admirable precision: “Today, it issensory qualities. increasingly clear that the symmetry group Can we find sensible places at the of nature is the deepest thing that wefoundations of physical theory where understand about nature.”additional variables might be introduced As will be seen, group theory alsowithout damaging the structure above — holds sway over the realm of colors andor, better yet, where additional variables sounds.might put that edifice on a more secure Weyl did for electromagnetismfooting? I have already suggested that (EM) what Einstein did for gravity. Kaluzahidden variables hold out that promise — demonstrated how, by adding an additionalfor colors and sounds are surely simple spatial dimension, both gravity and Weyl’sthings, as Maxwell, Wittgenstein, Russell EM could flow from the same metric.and Whitehead have argued. Being so Weyl got off to a bit of a false start, assimple, they might justly be regarded as Einstein pointed out, thinking that lengthelemental. Given that these properties are (or “gauge”) was the quantity in question.evidently part of reality, they might then be A little later on, it was pointed out that thethought of as EPR’s “elements of reality,” phase of the wave function was thewhich have, again, yet to be explicitly quantity in question, but “gauge” stuck.incorporated into the body of quantum How might we coordinate thetheory. I say “explicitly incorporated” secondary properties with space-time? Ibecause we already commonly ascribe have already suggested that these sensorythese properties, in everyday speech, to properties are the “hidden” variables (HVs)various forms of EM energy — as when of quantum theory. This whole realm ofwe speak of warm sunshine, red light and discourse continues to be fraught withso forth. Indeed, these properties are controversy, however — the quite recenttypically the most salient aspects of light. work by Hartle, Holland, Smolin, Peres, ‘tFeynman, in his little book on QED, writes Hooft et al. notwithstanding. What otherabout blue photons and we all know what options do we have?he means — until we start to think about it. Let us see whether we can tighten up just what we need to do so far coordinating the secondary qualities withISSN 1303 5150 www.neuroquantology.com
  10. 10. NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page - - -Author. Article namephysical theory. Here is an excerpt from wave function.Lockwood’s wonderful book, Mind, Brain Ryder’s brilliantly lucid introduction to& Quantum; Lockwood and I arrived at QFT provides us with a handy table thatmany of the same conclusions and by summarizes this correspondence.similar routes — and wholly independentlyof one another (private communication). Gauge theory General relativity Take some range of phenomenal Gauge Co-ordinate transformations transformations qualities. Assume that these qualities can be arranged according to some Group of all abstract n-dimensional space, in a Gauge group co-ordinate way that is faithful to their perceived transformations similarities and degrees of similarity [...] Then my Russellian proposal is Gauge potential, Aµ Connection coefficient, that there exists, within the brain, Γκµν some physical system, the states of which can be arranged in some Field strength, Gµν Curvature tensor, Rκλµν n-dimensional state space [...] And the two states are to be equated with Bianchi identity: Bianchi identity: each other: the phenomenal qualities are identical with the states of the Σ Dρ Gµν = 0 Σ Dρ Rκλµν = 0 corresponding physical system. ρµν ρµν cyclic cyclicThis is precisely right, I think; surprisingly,the mathematics of fiber bundle theorywould seem to offer us just what we need The formulae above are a bit daunting forin the way of state spaces, as the following some; let us attend Cao, who makes theexcerpt from Atiyah suggests (where R4 is essential points in an easy-going manner:the familiar four-dimensional space-time): The internal space defined at each We shall now recall the data of a space-time point is called a fiber, and classical theory as understood by the union of this internal space with physicists and then reinterpret them space-time is called fiber-bundle in geometrical form. space. Then we find that the local Geometrically or mechanically we gauge symmetries remove the can interpret this data as follows. flatness of the fiber-bundle space Imagine a structured particle, that is since we assume that the internal a particle which has a location at a space directions of a physical system point x of R4 and an internal at different space-times points are structure, or set of states, labeled by different. elements g of G. So the local gauge symmetry also requires the introduction of gauge Is the parallel between the state potentials, which are responsible forspaces of Lockwood and those of Atiyah the gauge interactions, to connectmore than suggestive? Today, all of internal directions at different space-particle theory is governed by gauge theory, time points. We also find that the rolewhich attaches an “internal space” to each the gauge potentials play inelementary particle — a space governed by fiber-bundle space in gauge theory isa mathematics precisely analogous to that exactly same as the role the affinefound in relativity. The symmetries of this connection plays in curved space-internal space mirror those found in time in general relativity.space-time and these internal symmetrieshelp guide the evolution of the particle’sISSN 1303 5150 www.neuroquantology.com
  11. 11. NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page - - -Author. Article name Curiously, colors and sounds also seems to have escaped Schrödinger,exhibit these fundamental symmetries. judging from the handful of scientificThus, other things being equal, colors and papers he wrote about color.sounds are symmetric or invariant under If we then assign red, green andsuch physically important operations as blue (or some other triple) to the principaltranslations, rotations and reflections. The axes of a unit sphere, we can then generatecelebrated theorem of Emmy Noether all the other colors by simple vectorrelates these kinds of symmetries to the addition. The inverse of x is, again, just x,conservation of matter, charge and angular but pointing in the opposite direction. Ofmomentum. Ramond makes this point course, colors come in differentexplicit in his classic introduction to field magnitudes, and so we need to weight thetheory: vectors. It is a most beautiful and awe- Projection inspiring fact that all the Let us revisit Weyl in a moment, where he fundamental laws of Classical points out that composite colors and tones Physics can be understood in terms are made “by superposing simple of one mathematical construct called oscillations of different frequency with the Action. It yields the classical definite intensities.” First, here is Dirac on equations of motion, and analysis of the subject of superposition in quantum its invariances leads to quantities theory: conserved in the course of the classical motion. In addition, as When a state is formed by the Dirac and Feynman have shown, the superposition of two other states, it Action acquires its full importance will have properties that are in some in Quantum Physics. vague way intermediate between those of the original states and that Here is Weinberg again to let us in approach more or less closely toon the joke: “Furthermore, and now this is those of either of them according tothe point, this is the punch line, the the greater or less ‘weight’ attachedsymmetries determine the action. This to this state in the superpositionaction, this form of the dynamics, is the process. The new state is completelyonly one consistent with these defined by the two original statessymmetries.” when their relative weights in the In brief, what I would like to superposition process are known,suggest here, above all else, is this: In the together with a certain phaseobservable symmetries of color and sound, difference, the exact meaning ofwe have a route leading from our direct weights and phases being providedexperience of these traditionally “mental” in the general case by theproperties straight into the heart of mathematical theory.contemporary physical theory. Colors and sounds also exhibit A little reflection should serve towell known relations to the phases of the persuade one that Dirac’s remarks applywaves with which they are associated. equally will to those color states which areMoreover, color vectors and sound vectors predictably concomitant with photon states.behave like the elements of a group under In manufacturing color TV screens,addition: If one makes the physically millions of colors can be produced byintuitive choice of assigning the inverse of superposing the light from red, green andx (for any color or sound) to –x, where –x blue pixels in fixed amounts or “weights.”is just x, but 180° out of phase, and where We also know that any light of a giventhe zero elements are just darkness and color will be brightened or darkenedsilence — or, no light and no sound. I am according to whether the waves bearingrather pleased to have figured this out, as it that color are in phase or out of phase.ISSN 1303 5150 www.neuroquantology.com
  12. 12. NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page - - -Author. Article name Let us now pay special attention to it is only the one fundamentalDirac’s comment concerning how, when relation: The point C lies on theone quantum state arises from the segment AB. In projective geometrysuperposition of two other states, it will no notions occur except such as areexhibit properties “that are in some vague defined on this basis. On the otherway intermediate between those of the side, there is given a second systemoriginal states.” What might we make of Σ2 of objects — the manifold ofthis “vague” assertion? colors — within which certain Weyl clarifies matters for us again, relations R, R’, [...] prevail whichpointing out that the “betweenness” of shall be associated with those of thethese intermediate states can be readily first domain by equal names,modeled by an axiom from projective although of course they havegeometry: “Thus the colors with their entirely different intuitive content.various qualities and intensities fulfill the Besides the continuous connection,axioms of vector geometry if addition is it is here the funda- mental relation:interpreted as mixing; consequently, C arises by a mixture from A and B;projective geometry applies to the color let us therefore express it somewhatqualities.” strangely by the same words we This is an important point and so used in projective geometry: ‘Thelet us linger a moment or two over Weyl’s color C lies on the segment joiningreasoning. the colors A and B.’ If now the elements of the second system Σ2 are made to correspond to the elements of the first system Σ1 in such a way, that to elements in Σ1 for which the relation R, or R’ […] holds, there always correspond elements in Σ2 for which the homonymous relation is satisfied, then the two domains of objects are isomorphically represented on one another. In this sense the projective plane and the color continuum are isomorphic with one another. Every theorem which is correct in the one system Σ1 is transferred unchanged Figure 5. Weyl to the other Σ2. A science can never Mathematics has introduced the determine its subject matter except name isomorphic representation for up to an isomorphic representation. the relation which according to The idea of isomorphism indicates Helmholtz exists between objects the self- understood, insurmountable and their signs. I should like to carry barrier of knowledge. It follows that out the precise explanation of this toward the “nature” of its objects notion between the points of the science maintains complete projective plane and the color indifference. Thus, e.g., what qualities On the one side, we have a distinguishes the colors from the points of the projective plane one manifold Σ1 of objects — the points can only know in immediate alive of a convex section of the projective intuition. plane — which are bound up with one another by certain fundamental relations R, R’, ...; here, besides the continuous connection of the points,ISSN 1303 5150 www.neuroquantology.com
  13. 13. NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page - - -Author. Article name In brief, then, it seems as though special cases of projective geometry.spectral colors respect the laws of No wonder that Cayley exclaimed,projective vector geometry. Following “Projective geometry is all geo-Maxwell’s reasoning and common usage, metry.”we can map color vectors to a sphere in amanner which recapitulates their group What I am trying to suggest is thatbehavior under superposition, where red, the projective geometry of perceptualgreen and blue (or some other triplet of space is the true geometry of the world and“orthogonal” colors) can be used to that the world only appears flat anddesignate the principal axes. Euclidean to us because we typically This is all rather suggestive, I experience it at speeds far below that ofthink, given the prominence of projective light and in fairly weak gravitational fields.spaces in quantum theory, Kaluza-Klein On this view, projective relativity is atheory, M-theory and the projective better fit to the much-rumored real worldrelativity of Einstein-Cartan. It is also — and this is, again, quite suggestive,interesting to note that, as with the vectors given the prominence of projectivein quantum theory, color vectors at any geometry in Kaluza-Klein theory and itsgiven point of space-time are jointly descendants in string/M-theory. Anexclusive and mutually exhaustive — one adequate exploration of these connectionsof the principal reasons why vectors are so would take us too far a field, however, anduseful in quantum theory, as Hughes tell us. so must wait for another day. For now, IThat is, a “point” (patch) in space can not will simply remind the reader that, on thebe red and green at the same time. And one hand, the additional dimensions ofthen, the dual language of forms would Kaluza-Klein theory and its descendantsseem to allow for colored patches — such await a real-world interpretation. While onas are directly observed — while also the other hand, the secondary properties ofreminding us that projective geometry perception respect important symmetriesprovided us with the first instance of those and phase relations and also fiber over (“sit“dualities” that are so prominent in over”) their respective spaces.string/M-theory. Then, too, it seems fairly clear that Matrixvisual space also has a projective character, I said I would return to Heisenberg, whoseonce one considers that parallel railroad matrix mechanics rivaled Schrödinger’stracks appear to us as converging in the wave mechanics until Dirac demonstrateddistance. Poincaré wrote that “Perceptual that the two formulations were fairlyspace is only an image of geometric space, equivalent. In QM, a physical state isan image altered in shape by a sort of assigned a vector on a unit sphere inperspective.” I would like to suggest that Hilbert space. Operators then rotate thePoincaré had it turned around somewhat, it vector, carrying it to a new state. And thisbeing more accurate to say that the usual is interesting, because colors can also beEuclidean space is a special case of mapped to a unit sphere, as previouslyprojective space, as Kline makes explicit: discussed, and in a manner faithful to their group behavior under superposition. [It] became possible to affirm that Heisenberg’s formulation of QM projective geometry is indeed would then seem to perform quite as well logically prior to Euclidean geometry in describing the behavior of color, a and that the latter can be built up as a traditionally mental object, where every special case. Both Klein and Cayley physical operation which changes the color showed that the basic non-Euclidean of a thing corresponds to a rotation of its geometries developed by Lobachev- color vector in color space. sky and Bolyai and the elliptic Thus, for example, in an operator non-Euclidean geometry created by corresponding to the Doppler Effect, a Riemann can also be derived as red-shift would rotate the photonic stateISSN 1303 5150 www.neuroquantology.com
  14. 14. NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page - - -Author. Article namevector in Hilbert space as well as its or symmetric — as the earth rotates on itsconcomitant color vector, by a fixed axis and the solar system goes flying alongnumber of degrees and in a certain through space-time. Colors, sounds, and sodirection — doing so predictably, reliably, forth are, then, symmetric underand quantifiably. translations and rotations (and, it would The picture which emerges seem, reflections) in space-time. Similarly,portrays color space as a subspace of an those operators which do change colorEPR-complete Hilbert space. vectors look like matrix-valued tensors, Now, since a red-shift (blue-shift) since these operators must preserve thecorresponds to a lowering (raising) of the relevant symmetries.energy of the photon, we have a ready If I am not much mistaken, then,correspondence with the Hamiltonian we seem to have here the germ of a general(energy) operator, which then ought to lead correspondence principle between whatto an EPR-complete Lagrangian, which might be called a primary and a secondarythen ought to embody what might be called mechanics, akin to Bohr’s familiarsecondary symmetries. Noether’s theorem correspondence relations between classicalis a helpful guide, here: Energy, being a and quantum mechanics. Again, notice thatconserved quantity, gives us a constant we already have in hand the kinds offrequency, by the fundamental relation: vectors and matrices in question; they are employed in the manufacture of color TVs E = hν and stereo systems. These working artifacts provide, moreover, a substantialThe frequency being constant, we fully sort of proof of the notions alluded toexpect the color to be invariant. Another herein.basic equation tells us that the energy It is quite intriguing to note, again,operator is the Hamiltonian, so a change in in analogy with the pixels on a TV screenenergy due to the Doppler Effect (or to a or computer monitor, that color spacegravitational field) rotates the photonic appears to fiber over (“sit over”) thestate vector in a predictable way, yielding a manifold of visual perception, where apredictable change of color. copy of color space is assigned to each point of visual space-time. The analogy Ψ Ψ EΨ = HΨ falters, for what we really need is a picture wherein, to paraphrase Wittgenstein, everyMoving to the Lagrangian picture, the pixel may be of any color. For thoseconservation laws lead us along a natural unfamiliar with Greene’s entertaining bookpath to action equations and variational on The Elegant Universe, here is anprinciples, as when we consider that, for illustration of the fiber-bundle concept.constant E, a photon is stationary in colorspace — i.e., its vector continues to pointin the same direction in color space. Noticealso that, with color vectors, matrices andtensor operators in hand, we can plausiblyavail ourselves of the tools of Riemanniangeometry. So it seems we have come fullcircle with respect to my youthful blunder;the joke is on me, since it now seems fairlyclear that the symmetries of the secondaryproperties lead us to the equations of Figure 6. Calab-Yau space.motion. We also immediately recover thefamiliar observable fact that, other thingsbeing equal, the secondary properties ofthings do not change — they are invariantISSN 1303 5150 www.neuroquantology.com
  15. 15. NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page - - -Author. Article name Now, this is quite a coincidence,because Greene is here exhibiting thosebeasties known as Calabi-Yau spaces —the additional spatial dimensions ofM-theory, usually thought to bevanishingly small because we do not “see”them — a notion inherited from the Figure 7. Color spacepioneering work of Kaluza and Klein andoften stated in passing — and quite in Conclusionkeeping with the dogma from physical This shift of Gestalt will prove a jolt fortheory (which we can trace directly to the many, running counter to a worldview thatpioneering work of Galileo and Newton) Galileo et al. inherited from Democritusthat the world we see is “obviously” and the atomists of classical antiquity. Thefour-dimensional (4D). But the world we shift is facilitated by the durable, reliable“see” is manifestly not 4D, for then the nature of the secondary qualities orworld would be quite literally invisible. properties — i.e., they are not going awayWhat we “see” are, in fact, a flux of and their behavior is open to inspection,colored surfaces — a fact which ought to test, prediction and falsification. To choosebe of interest to those who pursue two instructive examples, consider thatChern-Simons theory, which links gauge colors and sounds cannot reveal whethertheory to M-theory by way of a bridge one is at rest or in a uniform state ofsupplied by topology. motion. Likewise, the “red” of a red-shift Minkowski clarifies the essential alone cannot tell us whether the source issituation in his famous essay on relativity: receding or in a gravitational field. It seems, therefore, as though We will try to visualize the state of colors and sounds respect the symmetries things by the graphic method. Let x, embodied in both the special and general y, z be rectangular co-ordinates for theories of relativity. space, and let t denote time. The There are also a few simple objects of our perception invariably physical facts which seem to have eluded include places and times in combin- many of our critics. ation. Nobody has ever noticed a Given the evident symmetries and place except at a time, or a time phase relations of colors and sounds, except at a place. […] The together with the apparent fact that their multiplicity of all thinkable x, y, z, t respective spaces fiber over their systems of values we will christen associated sensory spaces, one wonders, the world. again, whether colors and sounds (along with the other secondary properties) mightTo paraphrase a bit, “the objects of our occupy the internal spaces of gauge theoryperception” invariably exhibit the truth of and/or the additional spatial dimensions ofWittgenstein’s observations concerning M-theory.how “a speck in the visual field, though it Notice that this move addressesneed not be red must have some color one of the most persistent complaints[…]” Once one accepts that colors and regarding M-theory, namely its failure tosounds and so forth are simple physical provide observable predictions. For on theentities which define manifolds, it is but a view suggested here, we really do “see”short step to “seeing” that color space the additional dimensions — they havefibers over visual space — and so with simply been obscured by centuries ofsounds and aural space, etc. On this view, a dogma.subspace of Calabi-Yau space looks likethis:ISSN 1303 5150 www.neuroquantology.com
  16. 16. NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page - - -Author. Article name Returning to the place where webegan: Apropos the seminal work ofAndras Pellionisz and Rudolpho Llinas,the Churchlands and others have arguedfor a so-called “tensor network theory” ofneural function. It seems as though theviews of these authors comport well withthose expressed above, especially inrespect of: 1) The neural implementation of matrix operations on input vectors; and the view that 2) the cerebellum’s job consists in “the systematic transformation of vectors in one neural hyperspace into vectors in another neural hyperspace”; and the notion that 3) “the tensor calculus emerges as the natural framework with which to address such matters,” together with 4) the characterization of phenomenal properties as vectors; and 5) the physics and mathematics of the secondary properties. We arrive at a picture whereinboth “mental” perception and its“physical” substrate are mediated by oneset of matrix-valued tensors which operateon sensory input vectors to yield both whatwe perceive and what we do by way ofbehavioral responses. (Or, as I like to say,neural form follows quantum function.)Given the fractal character of dendriticarborizations, we might expect to find thiskind of self-similarity across temporal andspatial scales.AcknowledgementsI should like to thank my teachers for their patientguidance. I have been fortunate in having many fineinstructors, among whom I must include RobertWoerner, Frank Kosier, George Nelson, HaroldBechtoldt and Isidore Gormezano. I am also gratefulto the late John S. Bell and to Michael Lockwood fortheir kind encouragement and to my friends andfamily for their support. I am especially indebted toAlfredo Pereira for many stimulating discussions.ISSN 1303 5150 www.neuroquantology.com
  17. 17. NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page - - -Author. Article nameReferences Lovejoy, Arthur O. The Revolt Against Dualism. Open Court: LaSalle, IL, 1960.Atiyah, MF Geometry of Yang-Mills Fields. Minkowski, H. “Space and Time,” The Principle of Accademia Nazionale Dei Lincei Scuola Relativity, ed., H. A. Lorentz, Methuen & Co.: Normale Superiore: Pisa, Italy, 1979. London, 1923.Bohm, D. and Hiley, BJ. The Undivided Universe, Poincaré, H. “Space and Geometry,” The Routledge: London, 1993. Foundations of Science. The Science Press:Borst, CV, ed. The Mind-Brain Identity Theory. St. New York, NY, 1913. Martins Press, New York, NY , 1970. Riemann, B. “On the Hypotheses Which Lie at theBurtt, EA. The Metaphysical Foundations of Bases of Geometry,” Nature, Vol. VIII, Modern Science. Garden City, NY: 183-4. Doubleday, 1954. Ryder, LH. Quantum Field Theory. Cambridge,Cao, T. “Gauge Theory,” Philosophical Foundations 1985. of Quantum Field Theory. eds., Brown, H. Salam, A. Unification of Fundamental Forces. and Harré, R. Oxford, 1988. Cambridge, 1990.Chalmers, DJ. “Puzzle of Conscious Experience,” Schrödinger, E. “Mind and Matter,” What Is Life? Scientific American, 12/95. Cambridge, 1996.Churchland, PM. A Neurocomputational Perspective. t Hooft, G. “Gauge Theories of the Forces between MIT, 1989. Elementary Particles,” Scientific American,Churchland, PS. Neurophilosophy: Toward a 6/80. Unified Understanding of the Mind-Brain. Weinberg, S. “Towards the final laws of physics,” MIT, 1986. Elementary Particles and the Laws of Physics,Clark, A. Sensory Qualities. Clarendon: Oxford, Cambridge, 1987. 1993. Weyl, H. Mind and Nature, University ofDirac, PAM. The Principles of Quantum Mechanics. Pennsylvania Press, 1934. Oxford, 1958. Wheeler, J. and Tegmark, M. “100 Years ofDuhem, P. The Aim and Structure of Physical Quantum Mysteries,” Scientific American, Theory Atheneum: New York, NY, 1974. 2/01.Dyson, FJ. “Field Theory,” Scientific American, Wittgenstein, L. Tractatus Logico-Philosophicus. 4/53. Humanities Press: Atlantic Highlands, NJ,Einstein, A, “On Bertrand Russell’s Theory of 1974. Knowledge,” Albert Einstein Philosopher-Scientist. ed., Schilpp, Open Court: LaSalle, IL, 1970.Einstein, A., Podolsky, B. and Rosen, N. “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” Physical Review, Vol. 47, 777 (1935).Feigl, H. "Mind-body, not a pseudo problem" The Mind-Brain Identity Theory. ed., Borst, CV, New York, NY: St. Martins Press, 1970.Flanagan, B. “Secondary Qualities and Hidden Variables,” Microphysical Reality & Quantum Formalism. Vol. 1 & 2. eds., Van Der Merwe, et al., Kluwer Academic: Boston, MA, 1987.Flanagan, B. “Are Perceptual Fields Quantum Fields?” Information & Cognition, Vol. III, No. 3, 2001.Flanagan, B. “Multi-scaling, Quantum Theory and the Foundations of Perception,” NeuroQuantology, Vol I, No. 4, 2003.Holland, P. The Quantum Theory of Motion. Cambridge, 1993.Hughes, RIG. The Structure and Interpretation of Quantum Mechanics. Harvard: Cambridge, MA, 1989.Kaluza, T. “On the Unity Problem of Physics,” Modern Kaluza-Klein Theories. Addison-Wesley: Menlo Park, 1987.Kline, M. “Projective Geometry,” Scientific American, 1/55.Lockwood, M. Mind, Brain and the Quantum. Basil Blackwell Ltd: Cambridge, MA, 1989.ISSN 1303 5150 www.neuroquantology.com

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