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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 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 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
 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 metamath- ematical 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  |
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Einstein 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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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 transformation groups was the proper setting for generalizations of this relationship. Bryant |
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 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 [...]
§
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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projective isomorphism |
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