action manifold field
vision
vision
Here the impossible union of spheres
Of existence is made actual. ~TS Eliot


light spectra symmetry




Bohm, Hiley, Weyl, Russell, Einstein, Bergmann, Helmholtz, Poincaré, Nagel, Newman








Those robots are effectively blind. ~Peter Corke
















Bohm

Hiley





Weyl









artifivial retina


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
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
 

sense impressions



The first full-fledged 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 geometrythe 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 

  
Riemann


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 Σ1 is transferred unchanged to the other Σ2. A science can never determine its subject matter except up to an isomorphic representation. The idea of isomorphism indicates the self-understood, 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


digital brain


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!"

Atiyah


Calabi-Yau

































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As above, so below


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
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
  


symmetry



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 space-time. 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 four-dimensionality of space-time, he assumed that no field depended on the fifth coordinate.

Bergmann


perspective on color


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 self-consciousness.

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
 

color_sphere


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













retina

brain




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Euclid





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nerve

mental

physical
Optica

Elements
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