On the Unification of Mind & Matter

NeuroQuantology | XXXXXXXXXXXXXXXXX | Vol. 4 | Issue 4 | Page - - -
Author. Article name

Men 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. Flanagan
Address: Sentient Technologies, 434 S Johnson Street #1, Iowa City, IA 52240 USA
e-mail: sentek1@yahoo.com

ISSN 1303 5150 www.neuroquantology.com


Introduction which reposes directly on a “minimal brain
A little over 20 years ago, when I first event,” viz., a photonic state vector. Long
began to air my ideas regarding the before that idea came to light, I made an
quantum substrate of mind and brain, the amusing blunder, however.
topic was hardly on the map. For a long In my youthful explorations of
while, I took comfort in the famous dictum physics it became pretty clear that the
of 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 the
first 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 a
seems clear that the subject has attained a thing does not change as our world spins
certain maturity and I should like to thank on its axis or flies along through the
him for this fine honor. It seems to me that interstellar regions.
no reward can equal or surpass the Today I see, with the benefit of
appreciation of one’s peers. I should also hindsight, that these constants in the
like to thank my teachers for their patient appearances of things are instances of
guidance and instruction. symmetries or invariants — and so part of
My parents bought me a toy robot Klein’s rich legacy. But back then I felt
when I was a little tyke. A terror in stymied and so turned to neuroscience and
diapers, 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 by
what made the thing go. I got as far as the McCulloch and made a lengthy study of
motor, 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 work
equations and was thoroughly mystified. quickly led me to the famous essay by
Years later, I went to see the Nagel & Newman, where I encountered
premier of 2001: A Space Odyssey. I was the following passage:
quite taken with HAL, the sentient
computer. It was about this time that I How did Gödel prove his
began to wonder how one might engineer conclusions? Up to a point, the
such a device. At 16, reading Bertrand structure of his demonstration is
Russell, I was arrested by the problem of modeled, as he himself noted, on the
how the brain can perceive color, which reasoning involved in one of the
seemed to be related to light — a physical logical antinomies known as the
thing — but which had long been held to “Richard Paradox,” first propounded
reside in the mind — a mental thing. This by the French mathematician, Jules
proved a fortunate choice of problems; Richard, in 1905 […]
years later, I came across a remark from The reasoning in the Richard
William 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 itself
a color vector is a “minimal mental fact” — a most valuable form of



self-consciousness.∗ different, I had another inspiration. Years
before, in the course of many happy
That last bit fired my thinking. The brain, readings of the Churchlands’ works, I
being a physical thing, is presumably a encountered the tensor network theory of
(quantum!) mechanical thing, and can Pellionisz and Llinas. I’d known for some
therefore be modeled by a formal system. time that Heisenberg’s approach to
Moreover, our brains describe their states quantum mechanics employed matrices
— they talk about themselves. They talk operating on vectors. Riding the bus that
about objects of perception, which are day, it hit me all at once that the matrices
represented in us as configurations of of tensor network theory and the matrices
colors and sounds — which properties of QM are the same matrices and that
appear to be elemental in the sense that we our perceptual fields just are quantum
cannot define them in respect of simpler fields. Most of my subsequent efforts were
entities. Years later, it hit me that, rather devoted to seeing whether this inspiration
than try to reduce these elemental could stand up to the light of day. In all
properties to simpler entities, that maybe this, I was guided by the ethos of
we 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 is
we were to alter the axiom of physical answering.” I believe that the answer to the
science 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 easy
of a formal theory (T), where the on the reader, mindful of the
interpretation of T was an EPR-complete interdisciplinary nature of the work and of
quantum 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 am
spoke 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 been
about and I suppose they suspected I didn’t, followed by other success stories, as
either. Fortunately for me, Douglas witness the publication in the Proceedings
Hofstadter’s wonderful book on Gödel, of the Royal Society of Henry Stapp’s ideas
Escher, Bach came out about that time. A and by the forthcoming symposium on
similar episode occurred a few years later; Quantum Interaction, hosted by Stanford
I’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 a
latter work prompted quite a lot of interest. passing acquaintance with quantum theory
After my little talk, I had lunch with a and/or neuroscience and strike me as the
bright young PhD student from California, kind of rearguard actions which Kuhn
who told me, “Yeah, you could tell — at discusses in his landmark work on The
first, everyone was wondering what planet Structure of Scientific Revolutions. I think
you stepped off of; then they were all there are good reasons for thinking so; one
really interested.” reads, e.g., that Nature Neuroscience has
Then, in 1989, Penrose published just now published an article telling us that
his first big book on these topics and the the “noise” in the brain — often thought to
ball really began to roll, with scholarly preclude quantum effects — is actually just
conferences organized and fiery rhetoric what one might expect to find if the brain
detonated. A little later on, while riding a is engaged in Bayesian computations. And
bus 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 of



microtubules. chemical binding is electromagnetic in
On the whole, I see good reasons for origin, and so are all phenomena of nerve
optimism. impulses.”
If we should suppose, as would
Simplicity seem altogether plausible, that conscious
Quantum field theory (QFT) enjoys an processes just are phenomena of nerve
unfortunate reputation for being difficult. impulses, then it seems to follow as a
Without diminishing the very real ready consequence that those conscious
challenges 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 this
is basically quite a simple thing, as Gerard conjecture?
‘t Hooft reminds us: “a field is simply a Dyson wrote that a field is “is an
quantity defined at every point throughout undefined something which exists
some region of space and time.” Freeman throughout a volume of space.” Maxwell
Dyson 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 of
simplicity of genius, that “There is nothing sight, yet, on account of their
else except these fields: the whole of the absolute simplicity, they are
material universe is built of them.” incapable of analysis, and can never
What has QFT to do with become in themselves objects of
consciousness, though? In the helpful case thought. If we attempt to discover
of visual perception we often refer to the them, we must do so by artificial
visual field, where colors would seem to be means and our reasonings on them
akin to Dyson’s “undefined something must be guided by some theory.
which exists throughout a volume of
space.” Russell and Whitehead back up
In this wise it is intriguing to Maxwell: “Thus ‘this is red,’ ‘this is earlier
reflect 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 whose
such as we find in Bohm, Chalmers, Feigl, specialization make up a continuous
Lockwood and Russell, it is tempting to manifold, that the only simple
suppose that our perceptual fields just are notions whose specialization form a
electromagnetic fields. Abdus Salam helps multiply extended manifold are the
us along here, reminding us that “all positions of perceived objects and



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 people
Here is Hermann Weyl to help us out a bit will hasten to tell you that color just is
more on manifolds: “The characteristic of wavelength — not knowing that colors are
an n-dimensional manifold is that each of vectors, whereas wavelengths are scalars,
the elements composing it (in our having only magnitude.† Weyl reminds us
examples, single points, conditions of a of the essential facts:
gas, colors, tones) may be specified by the
giving of n quantities, the ‘co-ordinates,’ To monochromatic light corres-
which are continuous functions within the ponds in the acoustic domain the
manifold.” 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 tones
denoting the field strength and direction at combine to a composite sound. This
a single point in the space.” That sounds takes place by superposing simple
like a vector and it just so happens that oscillations of different frequency
colors behave like vectors, too, as with definite intensities.
Grassmann, Maxwell, Schrödinger, Weyl
and Feynman all tell us — and as the
technology behind our color TVs and
computer monitors makes plain.∗
Appealing to Feigl’s formulation
of mind/brain identity theory, we might say
that configurations of colors represent
other objects in the natural world, as when
we describe an apple as a red, roundish
thing. Already we encounter a difficulty,
however, for we want to know where the Figure 1. Wave superposition
redness comes in. Schrödinger's equation
is often said to contain, in principle, all that
can 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 remarkable
Schrödinger can be found in the superb collection relations between tone and number, already known to
edited by MacAdam. Pythagoras — though still awaiting a good explanation.



yet these familiar, characteristic (eigen)
color vectors and tones (predictably related The qualities of size, figure (or
to “simple oscillations of different shape), number and motion are for
frequency with definite intensities”) these Galileo the only real properties of
colors and tones are not yet properly objects. All other qualities revealed
incorporated into the body of physical in sense perception — colors, tastes,
theory. So, again: How did we come to this odors, sounds, and so on — exist
pass? only in the sensitive body, and do
not qualify anything in the objects
themselves. They are the effects of
History the primary qualities of things on the
The answer can be found in the classic senses. Without the living animal
works 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 the
Austen Clark elaborates on the essential day, along with their philosophical
situation 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 the



foundations of quantum theory, in the The proposal that I put forth over
Einstein-Bohr debates, in EPR and in the 20 years ago is that the “secondary
literature that followed. qualities” of color and sound and so forth,
Now, a notoriously intractable long ago banished to the misty realm of the
issue in contemporary thought on the mind, just are EPR's missing elements of
mind/body problem concerns what reality — variables only hidden in plain
Chalmers has famously dubbed the “hard sight, their true nature obscured by 300
problem,” viz., the question of why some years of dogma. As scientists, we take in
brain states have perceptual states attached this doctrine with our mothers’ milk; it
to them. Others, whom I suspect betray a forms part of what “everybody knows,”
guilty conscience, have attempted to wave and so constitutes a brainwashing more
away the issue, dismissing it as mere talk thorough than anything Heisenberg and
regarding “the redness of red.” Often Bohr ever devised — and this despite the
missing from these discussions is any fact (picked up at the time by Hume and
reference to the scientific literature Leibniz) that this doctrine lies at the very
concerning color. This is a bit odd, given crux of the scientific worldview.
that such illustrious persons as Newton, I don't want to be too hard on Bohr
Young, Helmholtz, Grassmann, Maxwell, and Heisenberg, whose place in the
Schrodinger, Weyl, Einstein and Feynman pantheon of physics is, of course, beyond
have devoted serious thought to the nature question and whose thoughts on the
of color. questions at hand I will now exploit.
As noted above, Schrödinger's As David Bohm informs us in the
wave function is often said to contain, in introduction to his congenial book on
principle, all that can be known about a Quantum Theory, “Bohr suggests that
physical 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’s
the sense that not all “elements of reality” ideas about QM and the observer problem
are incorporated into the body of quantum to good use, arguing that quantum theory
theory. Those missing elements have come has essentially to do with our knowledge of
to be known as “hidden variables,” and I physical states and systems. He has made a
am happy to see that these variables are bold and intriguing conjecture — drawing
once again on the front burner of on William James’s ideas about the
theoretical physics, thanks to Hartle, selective nature of attention — that it is our
Holland, 't Hooft, Peres and Smolin, conscious attention that is behind the
among 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 thought
David Bohm and J. S. Bell, who, nearly that the observer ought not to be
alone, kept this question alive for 60 years, “ontologically privileged.” I expect that the
when 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: If
the “brainwashing” which, as Gell-Mann consciousness does consist of quantum
tells us, Bohr and Heisenberg induced in a field processes, those processes might then
generation of physicists. have nonlocal effects, producing the QZE.



I do not know, of course — I don’t think in our own experience, but
anyone does at this juncture — but it something very different. The
seems a fascinating possibility. observer, when he seems to himself
I will return to Heisenberg later on, to be observing a stone, is really, if
but would like to point out that five other physics is to be believed, observing
prominent physicists also devoted serious the effects of the stone upon himself.
thought to mind and matter, namely Thus science seems to be at war
Einstein, Pauli, Schrodinger, Weyl and with itself: when it means to be most
Wigner. Aside from Pauli, whose thoughts objective, it finds itself plunged into
on these issues have only recently come to subjectivity against its will.
light, I have remarked in another place
upon 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 which
consciousness. Pauli, in a recently had never previously occurred to me.
translated series of letters to Carl Jung, For, superficially considered, the
looked forward to a unitary description of mode of thought of Berkeley and
mind and matter, informed by quantum Hume seems to stand in contrast to
theory. the mode of thought in the natural
sciences. However, Russell's just
Logic cited remark uncovers a connection:
Getting back: Einstein, in his “Remarks on If Berkeley relies upon the fact that
Bertrand Russell’s Theory of Knowledge,” we do not directly grasp the “things”
quotes Russell in this connection, going to of the external world through our
the 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.



Nonetheless, we are left with public, Symmetry
reproducible and reliable facts of The move sketched in just now will strike
greenness, hardness, and coldness. some as a little outlandish; nonetheless, we
As noted earlier, these properties already have in hand the kind of theory
seem to be elemental on account of their suggested above; the primary qualities of
extreme simplicity. Let us take a page from extension in space and duration in time are
Gödel, then, and frame a formal theory T, already part of the foundations of physics
of a very general nature, constrained only and are encoded in the metric of general
by the laws of simple logic. relativity (GR).
Take for the elements of T both Einstein showed us how to derive
the primary qualities and secondary gravitation from the metric tensor — and
qualities of sensory perception. It then here the primary quality of mass also
follows as a consequence of simple logic enters the theory. Minkowski, who was
that, 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 theory
be — it may have all the complexity of a of invariants, given that relativity flows
human brain — T will be unable to define from the invariance of the speed of light
its elements, for the simple reason that its for all observers. Today we call these kinds
elements would not then be elements and of invariants by the more suggestive name
we would have a simple contradiction. of symmetries — and these symmetries are
In some such fashion, perhaps, are of truly fundamental importance. Nobel
we unable to define the elements of our laureate Steven Weinberg has stated the
perceptions, those self-same elemental case with admirable precision: “Today, it is
sensory qualities. increasingly clear that the symmetry group
Can we find sensible places at the of nature is the deepest thing that we
foundations of physical theory where understand about nature.”
additional variables might be introduced As will be seen, group theory also
without damaging the structure above — holds sway over the realm of colors and
or, better yet, where additional variables sounds.
might put that edifice on a more secure Weyl did for electromagnetism
footing? I have already suggested that (EM) what Einstein did for gravity. Kaluza
hidden variables hold out that promise — demonstrated how, by adding an additional
for colors and sounds are surely simple spatial dimension, both gravity and Weyl’s
things, 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, as
simple, they might justly be regarded as Einstein pointed out, thinking that length
elemental. 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 the
thought of as EPR’s “elements of reality,” phase of the wave function was the
which have, again, yet to be explicitly quantity in question, but “gauge” stuck.
incorporated into the body of quantum How might we coordinate the
theory. I say “explicitly incorporated” secondary properties with space-time? I
because we already commonly ascribe have already suggested that these sensory
these properties, in everyday speech, to properties are the “hidden” variables (HVs)
various forms of EM energy — as when of quantum theory. This whole realm of
we speak of warm sunshine, red light and discourse continues to be fraught with
so forth. Indeed, these properties are controversy, however — the quite recent
typically the most salient aspects of light. work by Hartle, Holland, Smolin, Peres, ‘t
Feynman, in his little book on QED, writes Hooft et al. notwithstanding. What other
about 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 with



physical 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 that
many of the same conclusions and by summarizes this correspondence.
similar routes — and wholly independently
of 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 cyclic

This is precisely right, I think; surprisingly,
the mathematics of fiber bundle theory
would seem to offer us just what we need The formulae above are a bit daunting for
in the way of state spaces, as the following some; let us attend Cao, who makes the
excerpt 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 for
spaces of Lockwood and those of Atiyah the gauge interactions, to connect
more than suggestive? Today, all of internal directions at different space-
particle theory is governed by gauge theory, time points. We also find that the role
which attaches an “internal space” to each the gauge potentials play in
elementary particle — a space governed by fiber-bundle space in gauge theory is
a mathematics precisely analogous to that exactly same as the role the affine
found 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 symmetries
help guide the evolution of the particle’s



Curiously, colors and sounds also seems to have escaped Schrödinger,
exhibit these fundamental symmetries. judging from the handful of scientific
Thus, other things being equal, colors and papers he wrote about color.
sounds are symmetric or invariant under If we then assign red, green and
such physically important operations as blue (or some other triple) to the principal
translations, rotations and reflections. The axes of a unit sphere, we can then generate
celebrated theorem of Emmy Noether all the other colors by simple vector
relates 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. Of
momentum. Ramond makes this point course, colors come in different
explicit in his classic introduction to field magnitudes, and so we need to weight the
theory: 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 to
on the joke: “Furthermore, and now this is those of either of them according to
the point, this is the punch line, the the greater or less ‘weight’ attached
symmetries determine the action. This to this state in the superposition
action, this form of the dynamics, is the process. The new state is completely
only one consistent with these defined by the two original states
symmetries.” 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 phase
observable symmetries of color and sound, difference, the exact meaning of
we have a route leading from our direct weights and phases being provided
experience of these traditionally “mental” in the general case by the
properties straight into the heart of mathematical theory.
contemporary physical theory.
Colors and sounds also exhibit A little reflection should serve to
well known relations to the phases of the persuade one that Dirac’s remarks apply
waves with which they are associated. equally will to those color states which are
Moreover, 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 by
intuitive choice of assigning the inverse of superposing the light from red, green and
x (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 given
the zero elements are just darkness and color will be brightened or darkened
silence — or, no light and no sound. I am according to whether the waves bearing
rather pleased to have figured this out, as it that color are in phase or out of phase.



Let us now pay special attention to it is only the one fundamental
Dirac’s comment concerning how, when relation: 'The point C lies on the
one quantum state arises from the segment AB'. In projective geometry
superposition of two other states, it will no notions occur except such as are
exhibit properties “that are in some vague defined on this basis. On the other
way intermediate between those of the side, there is given a second system
original states.” What might we make of Σ2 of objects — the manifold of
this “vague” assertion? colors — within which certain
Weyl clarifies matters for us again, relations R, R’, [...] prevail which
pointing out that the “betweenness” of shall be associated with those of the
these intermediate states can be readily first domain by equal names,
modeled by an axiom from projective although of course they have
geometry: “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 fundamental 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 somewhat
qualities.” strangely by the same words we
This is an important point and so used in projective geometry: ‘The
let us linger a moment or two over Weyl’s color C lies on the segment joining
reasoning. 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,



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 a
manner which recapitulates their group What I am trying to suggest is that
behavior under superposition, where red, the projective geometry of perceptual
green 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 and
designate the principal axes. Euclidean to us because we typically
This is all rather suggestive, I experience it at speeds far below that of
think, given the prominence of projective light and in fairly weak gravitational fields.
spaces in quantum theory, Kaluza-Klein On this view, projective relativity is a
theory, M-theory and the projective better fit to the much-rumored real world
relativity of Einstein-Cartan. It is also — and this is, again, quite suggestive,
interesting to note that, as with the vectors given the prominence of projective
in quantum theory, color vectors at any geometry in Kaluza-Klein theory and its
given point of space-time are jointly descendants in string/M-theory. An
exclusive and mutually exhaustive — one adequate exploration of these connections
of the principal reasons why vectors are so would take us too far a field, however, and
useful in quantum theory, as Hughes tell us. so must wait for another day. For now, I
That is, a “point” (patch) in space can not will simply remind the reader that, on the
be red and green at the same time. And one hand, the additional dimensions of
then, the dual language of forms would Kaluza-Klein theory and its descendants
seem to allow for colored patches — such await a real-world interpretation. While on
as are directly observed — while also the other hand, the secondary properties of
reminding us that projective geometry perception respect important symmetries
provided 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 Matrix
visual space also has a projective character, I said I would return to Heisenberg, whose
once one considers that parallel railroad matrix mechanics rivaled Schrödinger’s
tracks appear to us as converging in the wave mechanics until Dirac demonstrated
distance. Poincaré wrote that “Perceptual that the two formulations were fairly
space is only an image of geometric space, equivalent. In QM, a physical state is
an image altered in shape by a sort of assigned a vector on a unit sphere in
perspective.” I would like to suggest that Hilbert space. Operators then rotate the
Poincaré had it turned around somewhat, it vector, carrying it to a new state. And this
being more accurate to say that the usual is interesting, because colors can also be
Euclidean space is a special case of mapped to a unit sphere, as previously
projective 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 state



vector in Hilbert space as well as its or symmetric — as the earth rotates on its
concomitant color vector, by a fixed axis and the solar system goes flying along
number of degrees and in a certain through space-time. Colors, sounds, and so
direction — doing so predictably, reliably, forth are, then, symmetric under
and 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 color
EPR-complete Hilbert space. vectors look like matrix-valued tensors,
Now, since a red-shift (blue-shift) since these operators must preserve the
corresponds 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 what
to an EPR-complete Lagrangian, which might be called a primary and a secondary
then ought to embody what might be called mechanics, akin to Bohr’s familiar
secondary symmetries. Noether’s theorem correspondence relations between classical
is a helpful guide, here: Energy, being a and quantum mechanics. Again, notice that
conserved quantity, gives us a constant we already have in hand the kinds of
frequency, 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 substantial
The frequency being constant, we fully sort of proof of the notions alluded to
expect 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 screen
energy due to the Doppler Effect (or to a or computer monitor, that color space
gravitational field) rotates the photonic appears to fiber over (“sit over”) the
state vector in a predictable way, yielding a manifold of visual perception, where a
predictable 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, every
Moving to the Lagrangian picture, the pixel may be of any color. For those
conservation laws lead us along a natural unfamiliar with Greene’s entertaining book
path to action equations and variational on The Elegant Universe, here is an
principles, as when we consider that, for illustration of the fiber-bundle concept.
constant E, a photon is stationary in color
space — i.e., its vector continues to point
in the same direction in color space. Notice
also that, with color vectors, matrices and
tensor operators in hand, we can plausibly
avail ourselves of the tools of Riemannian
geometry. So it seems we have come full
circle with respect to my youthful blunder;
the joke is on me, since it now seems fairly
clear that the symmetries of the secondary
properties lead us to the equations of Figure 6. Calab-Yau space.
motion.
We also immediately recover the
familiar observable fact that, other things
being equal, the secondary properties of
things do not change — they are invariant



Now, this is quite a coincidence,
because Greene is here exhibiting those
beasties known as Calabi-Yau spaces —
the additional spatial dimensions of
M-theory, usually thought to be
vanishingly small because we do not “see”
them — a notion inherited from the
Figure 7. Color space
pioneering work of Kaluza and Klein and
often stated in passing — and quite in
Conclusion
keeping with the dogma from physical This shift of Gestalt will prove a jolt for
theory (which we can trace directly to the
many, running counter to a worldview that
pioneering work of Galileo and Newton) Galileo et al. inherited from Democritus
that the world we see is “obviously” and the atomists of classical antiquity. The
four-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 or
world would be quite literally invisible. properties — i.e., they are not going away
What 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 choose
be of interest to those who pursue two instructive examples, consider that
Chern-Simons theory, which links gauge colors and sounds cannot reveal whether
theory to M-theory by way of a bridge one is at rest or in a uniform state of
supplied by topology. motion. Likewise, the “red” of a red-shift
Minkowski clarifies the essential alone cannot tell us whether the source is
situation 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) might
To paraphrase a bit, “the objects of our
occupy the internal spaces of gauge theory
perception” invariably exhibit the truth of and/or the additional spatial dimensions of
Wittgenstein’s observations concerning M-theory.
how “a speck in the visual field, though it
Notice that this move addresses
need 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 to
sounds and so forth are simple physical
provide observable predictions. For on the
entities 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 have
fibers over visual space — and so with simply been obscured by centuries of
sounds and aural space, etc. On this view, a dogma.
subspace of Calabi-Yau space looks like
this:



Returning to the place where we
began: Apropos the seminal work of
Andras Pellionisz and Rudolpho Llinas,
the Churchlands and others have argued
for a so-called “tensor network theory” of
neural function. It seems as though the
views of these authors comport well with
those expressed above, especially in
respect 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 wherein
both “mental” perception and its
“physical” substrate are mediated by one
set of matrix-valued tensors which operate
on sensory input vectors to yield both what
we perceive and what we do by way of
behavioral responses. (Or, as I like to say,
neural form follows quantum function.)
Given the fractal character of dendritic
arborizations, we might expect to find this
kind of self-similarity across temporal and
spatial scales.

Acknowledgements
I should like to thank my teachers for their patient
guidance. I have been fortunate in having many fine
instructors, among whom I must include Robert
Woerner, Frank Kosier, George Nelson, Harold
Bechtoldt and Isidore Gormezano. I am also grateful
to the late John S. Bell and to Michael Lockwood for
their kind encouragement and to my friends and
family for their support. I am especially indebted to
Alfredo Pereira for many stimulating discussions.



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