Here the impossible union of spheres Of existence is made actual. ~TS Eliot 
Bohm, Hiley, Weyl, Russell, Einstein, Bergmann, Helmholtz, Poincaré, Nagel, Newman  
Those robots are effectively blind. ~Peter Corke 


Bohm Hiley Weyl 
One
may
then ask what is
the relationship between the physical
and the mental processes? The
answer that we propose here is that there are not two processes.
Rather, it is being suggested that both are essentially the same.
Bohm
& Hiley
Einstein
It
has
often been said,
and certainly not without justification, that the man of science makes
a poor philosopher. Why then should it not be the right thing for the
physicist to let the philosopher do the philosophizing? Such indeed
might be the right thing at a time when the physicist believes he has
at his disposal a rigid system of fundamental concepts and fundamental
laws which are so well established that waves of doubt can not reach
them; but, it can not be right at a time when the very foundations of
physics itself have become problematic as they are now [...] In
contrast to psychology, physics treats only of sense experiences and of
the 'understanding' of their connection. But even the concept of the
'real external world' of everyday thinking rests exclusively on sense
impressions.
Einstein The
first fullfledged exposition of a mathematical theory of vision is
found in the Optica
of Euclid (fl. 300 B.C.). Indeed, Euclid's approach to vision was so
strictly mathematical as to exclude all but the most incidental
references to those aspects of the visual process not reducible to
geometry—the
ontology of visual radiation and the physiology and psychology of
vision. Lejeune comments that Euclid's Optica
systematically ignores every physical and psychological aspect of the
problem of vision. It restricts itself to that which can be expressed
geometrically. [...]
Its model is the treatise on pure geometry, and its method that of the Elements: a few postulates all fully necessary, from which follow deductively and with full mathematical rigor a series of theorems of a traditional form. Lindberg
The notion of duality is important in projective geometry, having profound consequences for its development. Felix Klein's Erlanger lectures had promoted the idea that the geometry of transformation groups was the proper setting for generalizations of this relationship. Bryant
Mathematics
has introduced the name isomorphic representation for the relation
which according to Helmholtz exists between objects and their signs. I
should like to carry out the precise explanation of this notion between
the points of the projective plane and the color qualities [...] the
projective plane and the color continuum are isomorphic with one
another. Every theorem which is correct in the one system Σ_{1}
is transferred unchanged to the other Σ_{2}.
A science can never determine its subject matter except up to an
isomorphic representation. The idea of isomorphism indicates the
selfunderstood, insurmountable barrier of knowledge. It
follows
that toward the "nature" of its objects science maintains complete
indifference. This for example what distinguishes the colors from the
points of the projective plane one can only know in immediate alive
intuition [...]
Weyl
Let
me try to explain my own view of the difference between geometry and
algebra. Geometry is, of course, about space; of that there is no
question. If I look out at the audience in this room I can see a lot,
in one single second or microsecond I can take in a vast amount
of information and that is, of course, not an accident. Our brains have
been constructedin such a way that they are extremelyc oncerned with
vision. Vision, I understand from friends who work in neurophysiology,
uses up something like 80 or 90 percent of the cortex of the brain.
There are about 17 different centers in the brain, each of which is
specialized in a different part of the process of vision: some parts
are concerned with vertical, some parts with horizontal, some parts
with color, perspective, finally some parts are concerned with meaning
and interpretation. Understanding, and making sense of, the world that
we see is a very important part of our evolution. Therefore spatial
intuition or spatial perception is an enormously powerful tool,
and that is why geometry is actually such a powerful part of
mathematics — not only for things that are obviously geometrical, but
even for things that are not. We try to put them into geometrical form
because that enables us to use our ntuition. Our intuition is our most
powerful tool. That is quite clear if you try to explain a piece of
mathematics to a student or a colleague. You have a long, difficult
argument and finally the student understands. What does the student
say? The student says, "I see!"

The processes on
the retina produce excitations which are conducted to
the brain in the optic nerves, maybe in the form of electric currents.
Even here we are still in the real sphere. But between the physical
processes which are released in the terminal organ of the nervous
conductors in the central brain and the image which thereupon appears
to the perceiving subject, there gapes a hiatus, an abyss which no
realistic conception of the world can span. It is the transition from
the world of being to the world of appearing image or of consciousness.
Weyl Russell
So
long
as we adhere to
the conventional notions of mind and matter, we are condemned to a view
of perception which is miraculous. We suppose that a physical process
starts from a visible object, travels to the eye, there changes into
another physical process, causes yet another physical process in the
optic nerve, and finally produces some effect in the brain,
simultaneously with which we see the object from which the process
started, the seeing being something "mental", totally different from
the physical processes which precede and accompany it. This view is so
queer that metaphysicians have invented all sorts of theories designed
to substitute something less incredible.
Russell
There are other kinds of unitary field theories, including some that today claim a great deal of interest. These utilize, in some way or other, an increase in the number of dimensions of spacetime. One famous example is Kaluza's proposal. He increased the number of dimensions to five, without changing the Riemannian character of the model. He was thus able to increase the number of components of the metric so as to accommodate the electromagnetic field as well. He set one extra component equal to a constant, because he had no use for it. To account for the observed fourdimensionality of spacetime, he assumed that no field depended on the fifth coordinate. Bergmann Perceptual space is only an image of
geometric space,
an image altered in shape by a sort of perspective. ~Poincaré
How
did
Gödel prove his conclusions? Up to a point, the structure of his
demonstration is modeled, as he himself noted, on the reasoning
involved in one of the logical antinomies known as the "Richard
Paradox," first propounded by the French mathematician, Jules
Richard,
in 1905 [...] The reasoning in the Richard Paradox is evidently
fallacious. Its construction nevertheless suggests that it might be
possible to "map" (or "mirror") metamathematical statements about a
sufficiently comprehensive formal system into the system itself. If
this were possible, then metamathematical statements about a system
would be represented by statements within the system. Thereby one could
achieve the desirable end of getting the formal system to speak about
itself — a most valuable form of selfconsciousness.
The idea of such mapping is a familiar one in mathematics. It is employed in coordinate geometry, which translates geometric statements into algebraic ones, so that geometric relations are mapped onto algebraic ones. The idea is manifestly used in the construction of ordinary maps, since the construction consists in projecting configurations on the surface of a sphere onto a plane. Nagel & Newman
Thus the colors with their various qualities and intensities fulfill
the axioms of vector geometry if addition is interpreted as mixing; consequently, projective geometry applies to the color qualities. ~Weyl The Langlands program has, along these lines, something called the geometric Langlands program, which replaces the number fields by Riemann surfaces. It is a very interesting theory which is much easier than the number field case, but not trivial and still quite big. It is developed by using the theory of vector bundles on Riemann surfaces. Atiyah


Einstein Russell Euclid Bergmann Klein Poincaré Gödel Nagel Newman Mach 

Optica Elements 

Erlangen  
projective isomorphism 
