| Smart Remarks 1: Einstein, Bohr, Bohm, Feynman, Dirac, Atiyah & Weyl | ||||||
We are accustomed to regarding as real those sense perceptions which are common to different individuals, and which therefore are, in a measure, impersonal. The natural sciences, and in particular, the most fundamental of them, physics, deal with such sense perception.
  Bohr
Every great and deep difficulty bears
in itself its own solution. It forces us to change our thinking in order to find it. 
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I believe that the first step in the setting of a "real external world" is the formation of the concept of bodily objects and of bodily objects of various kinds. Out of the multitude of our sense experiences we take, mentally and arbitrarily, certain repeatedly occurring complexes of sense impression (partly in conjunction with sense impressions which are interpreted as signs for sense experiences of others), and we attribute to them a meaning—the meaning of the bodily object. Considered logically this concept is not identical with the totality of sense impressions referred to; but it is an arbitrary creation of the human (or animal) mind. On the other hand, the concept owes its meaning and its justification exclusively to the totality of the sense impressions which we associate with it.
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| Bohr Bohm |
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Bohr suggests that thought involves such small amounts of energy that quantum- theoretical limitations play an essential role in determining its character. Bohm  |
color quantum |
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wave | |||||
Feynman Weyl |
It is just like the mathematics of the addition of vectors, where (a, b, c) are the components of one vector, and (a', b', c' ) are those of another vector, and the new light Z is then the "sum" of the vectors. This subject has always appealed to physicists and mathematicians. In fact, Schrödinger wrote a wonderful paper on color vision in which he developed this theory of vector analysis as applied to the mixing of colors.
When a state is formed by the superposition of two other states, it will have properties that are in some vague way intermediate between those of the original states and that approach more or less closely to those of either of them according to the greater or less 'weight' attached to this state in the superposition process. The new state is completely defined by the two original states when their relative weights in the superposition process are known, together with a certain phase difference, the exact meaning of weights and phases being provided in the general case by the mathematical theory. When a state is formed by the superposition of two other states, it will have properties that are in some vague way intermediate between those of the original states and that approach more or less closely to those of either of them according to the greater or less 'weight' attached to this state in the superposition process. The new state is completely defined by the two original states when their relative weights in the superposition process are known, together with a certain phase difference, the exact meaning of weights and phases being provided in the general case by the mathematical theory.
Dirac 
Gauge theory first appeared in physics in the early attempt by H. Weyl to unify general relativity and electro-magnetism. Weyl had noticed the conformal invariance of Maxwell's equations and sought to exploit this fact by interpreting the Maxwell field as the distortion of relativistic length produced by moving around a closed path. Weyl's interpretation was disputed by Einstein and never generally accepted. However after the advent of quantum mechanics with its all-important complex wave-functions it became clear that phase rather than scale was the correct concept for Maxwell's equations, or in modern language that the gauge group was the circle rather than the multiplicative numbers.
Atiyah
Calabi-Yau space
Our discussion will focus on compactified string theory, which as we shall see, requires the compact portion of space-time to meet certrain stringent constraints. Although there are more generalsolutions, we shall study the case in which the extra “curled-up” dimensions fill out an n-dimensional manifold that has the following properties:
• it is compact,
• it is complex, • it is Kahler, • it has SU(d) holonomy,
where d = n/2. For much of these lectures, n will be 6 and hence d = 3. Manifolds which meet these conditions are known as Calabi-Yau manifolds, for reasons which will become clear shortly.
Greene
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To monochromatic light corresponds in the acoustic domain the simple tone. Out of different kinds of monochromatic light composite light may be mixed, just as tones combine to a composite sound. This takes place by superposing simple oscillations of different frequency with definite intensities. Weyl
Thus the colors with their various qualities and intensities fulfill the axioms of vector geometry if addition is inter- preted as mixing; consequently, projective geometry applies to the color qualities.
Weyl
Dirac
The characteristic of an n-dimensional manifold is that each of the elements composing it (in our examples, single points, conditions of a gas, colors, tones) may be specified by the giving of n quantities, the "co-ordinates," which are continuous functions within the manifold.
Weyl
Weyl
Epistemologically it is not without interest that in addition to ordinary space there exists quite another domain of intuitively given entities, namely the colors, which forms a continuum capable of geometric treatment.
§
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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light neural nets sound |
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Dirac Atiyah Calabi Yau Greene |
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manifold phase |
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| Calabi-Yau |
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| More smartness 2: Einstein, Noether, Russell, Helmholtz, Lockwood, Churchland, Atiyah, Weyl | ||||||
