Here the impossible union of spheres
Of existence is made actual. (TS Eliot)
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Here the impossible union of spheres Of existence is made actual. (TS Eliot) |
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| Bohm, Hiley, Weyl, Russell, Einstein, Bergmann, Helmholtz, Poincaré, Nagel, Newman | |||||||
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  |
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 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
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Einstein Russell |
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 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
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| Euclid |
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 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
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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
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Optica Elements |
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| Bergmann Klein |
 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 transfor- mation groups was the proper setting for generalizations of this relationship. Bryant
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Erlangen | ||||
Poincaré Gödel Nagel Newman Mach |
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 S1
is transferred unchanged to the other S2.
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 [...]
§
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
 A
color is a physical object a soon as we consider its dependence, for
instance, upon its luminous source, upon temperatures, and so forth. Mach
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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") meta-mathematical 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 |
projective isomorphism metamathematics |
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