Q&C

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Galileo to Husserl

Galileo
Galileo Galilei

In questions of science the authority
of a thousand is not worth the humble reasoning of a single individual.



'D' wing of R&D




In questions of science the authority of a thousand is not worth the humble reasoning of a single individual.

§

Hence I think that these tastes, odors, colors, etc., on the side of the object in which they seem to exist, are nothing else than mere names, but hold their residence solely in the sensitive body...

Galileo


The Copernican astronomy and the achievements of the two new sciences must break us of the natural assumption that sensed objects are the real or mathematical objects. They betray certain qualities, which, handled by mathematical rules, lead us to a knowledge of the true object, and these are the real or primary qualities, such as number, figure, magnitude, position and motion [...] qualities which also can be wholly expressed mathematically. The reality of the universe is geometrical; the only ultimate characteristics of nature are those in terms of which certain mathematical knowledge becomes possible. All other qualities, and these are often far more prominent to the senses, are secondary, subordinate effects of the primary.

Burtt


Galois

Evariste Galois


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Galois in 1831 was the first to really understand that the algebraic solution of an equation was related to the structure of a group, le groupe of permutations related to the equation. By 1832 Galois had discovered that special subgroups (now called normal subgroups) are fundamental. He calls the decomposition of a group into cosets of a subgroup a proper decomposition if the right and left coset decompositions coincide. Galois then shows that the non-abelian simple group of smallest order has order 60.

Gauss

Carl Friedrich Gauss



I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect [...] geometry should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics.


non-Euclidean

We cannot confuse what to us appears
unnatural with the absolutely impossible.




Godel
Kurt Gödel

Either mathematics is too big for the human mind or the human mind is more than a machine.




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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 [...]


Map


The basic fact which underlies all these mapping procedures is that an abstract structure of relations embodied in one domain of objects is exhibited to hold between "objects" in some other domain. In consequence, deductive relations between statements about the first domain can be established by exploring (often more conveniently and easily) the deductive relations between statements about their counterparts. For example, complicated geometrical relations between surfaces in space are usually more readily studied by way of the algebraic formulas for such surfaces.  

Newman & Nagel

Hameroff
Stuart Hameroff


'D' wing of R&D

Early quantum experiments led to the conclusion that quantum superpositions persisted until measured or observed by a conscious observer, that "consciousness collapsed the wave function," This became known as the "Copenhagen interpretation," after the Danish origin of Niels Bohr, its primary proponent. The Copenhagen interpretation placed consciousness outside physics!

To illustrate the apparent silliness of this idea, Erwin Schrödinger in 1935 formulated his famous thought experiment now known as Schrödinger's cat. Imagine a cat in a box. Outside the box a quantum superposition (e.g. a photon both passing through and not passing through a half-silvered mirror) is coupled to release of a poison inside the box. According to the Copenhagen interpretation the poison would be both released and not released, and the cat would be both dead and alive until the box was opened and the cat observed. Only at that instant would the cat be either dead or alive. Schrödinger intended his thought experiment to show how ludicrous was the Copenhagen interpretation, however to this day there is no accounting for reduction or collapse of a large scale, isolated quantum superposition.

Hameroff


Hamilton
William Rowan Hamilton

Who would not rather have the fame of Archimedes than that of his conqueror Marcellus?

On earth there is nothing great but man; in man there is nothing great but mind.

§

(Sir) William Rowan Hamiltonleast action principle. (1805-1865) was a child prodigy in languages and mathematics who submitted his first paper to the Royal Irish Academy when he was 17. He entered Trinity College where, at 22, he was elected a professor in astronomy and royal astronomer of Ireland while still an undergraduate. He invented quarternions, breaking with the tradition of commutative algebras and discovered conical refraction, but his great contribution to ray optics, based on the work of Fermat, was the least action principle.



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conics


The discovery was amazing. Mechanics is the study of moving bodies— cannonballs moving in a parabolic arc, pendulums swinging regularly from side to side, and planets moving in ellipses around the Sun. Optics is about the geometry of light rays, reflection and refraction, rainbows and prisms and telescope lenses. That they were connected was a surprise; that they were the same was unbelievable.

It was also true. And it led directly to the formal setting used today by mathematicians and mathematical physicists, not just in mechanics and optics but in quantum theory too: Hamiltonian systems. Their main feature is that they derive the equations of motion of a mechanical system from a single quantity, the total energy, now called the Hamiltonian of the system. The resulting equations involve not just the position of the parts but how fast they are moving — the momentum
of the system. Finally, the equations have the beautiful feature that they do not depend on the choice of coordinates.




rainbow



In the radical relation thus contemplated by Descartes, in his view of algebraical geometry, the related things are elements of position of a variable point which has for locus a curve or a surface; and the number of these related elements is either two or three. In the relation contemplated by me, in my view of algebraical optics, the related things are, in general, in number, eight: of which, six are elements of position of two variable points of space, considered as visually connected; the seventh is an index of color; and the eighth, which I call the characteristic function, — because I find that in the manner of its dependence on the seven foregoing are involved all the properties of the system, — is the action between the two variable points; the word action being used here, in the same sense as in that known law of vision which has been already mentioned. I have assigned, for the variation of this characteristic function, corresponding to any infinitesimal variations in the positions on which it depends, a fundamental formula; and I consider as reducible to the study of this one characteristic function, by the means of this one fundamental formula, all the problems of mathematical optics...


Hamilton


Hamiltonian approach: Consider space and time separately. We have a Hilbert spaceof states, and a self-adjoint 'Hamiltonian" operator H acting on ℋ. The evolution is given by the unitary operator eitH on ℋ.

Atiyah

H = E

red-shift

energy relation

If E is constant, v (frequency) will also be constant, giving us a constant 
i.e., invariant or symmetric  color vector. Changing E rotates the color vector.

In a closed system, how can we tell whether the changes in
the color vector are due to gravity or acceleration?

Relativity would seem to suggest that we cannot tell.




Heisenberg
Werner Heisenberg

The violent reaction on the recent development of modern physics can only be understood when one realises that here the foundations of physics have started moving; and that this motion has caused the feeling that the ground would be cut from science.


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matrix mechanics


Heisenberg looked first at the connection between the observable properties of the emitted light—its color (frequency) and the intensity—and the motion of the charged ball according to the classical mechanics of Newton. Then he considered the quantum properties of the observed light and reinterpreted the classical formulas for the motion in order to give the observed frequencies and intensities.

§

Heisenberg: "We cannot observe electron orbits inside the atom [...] Now, since a good theory must be based on directly observable magnitudes, I thought it more fitting to restrict myself to these, treating them, as it were, as representatives of the electron orbits."

"But you don't seriously believe," Einstein protested, "that none but observable magnitudes must go into a physical theory?"

"Isn't that precisely what you have done with relativity?" I asked in some surprise...

"Possibly I did use this kind of reasoning," Einstein admitted, "but it is nonsense all the same ... In reality the very opposite happens. It is the theory which decides what we can observe."         



Helmholtz
Hermann von Hemhholtz

Whoever in the pursuit of science, seeks after immediate practical utility may rest assured that he seeks in vain.


'D' wing of R&D


laser
Noether
How far does the laser move in color space?

Similar light produces, under like conditions, a like sensation of color. ~Helmholtz


If you take a physical state and do something to itlike rotating it, or like waiting for some time deltatyou get a different state. We say, "performing an operation on a state produces a new state." We can express the same idea by an equation:


|phi> = A|psi>.


An operation on a state produces another state. The operator A stands for some particular operation. When this operation is performed on any state, say |psi>, it produces some other state |phi>.
Feynman


What we see is the solution to a computational problem; our brains compute the most likely causes from the photon absorptions within our eyes.

§

The natural scientist no more than the philosopher can ignore epistemological questions when he is dealing with sense perception or when he is concerned with the fundamental principles of geometry, mechanics, or physics.

Helmholtz




Matrix mechanics


A speck in the visual field, though it need not be red must have some color; it is, so to speak, surrounded by color-space. Notes must have some pitch, objects of the sense of touch some degree of hardness, and so on.

Wittgenstein


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 

Hilbert
David Hilbert

Every mathematical discipline goes through three periods of development: the naive, the formal, and the critical.


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unit sphere


The key to the representation is the fact that Pythagoras’ theorem, or its analogue, holds in any vector space equipped with an inner product. Consider the space R3. For any vector v in R3,

v = vx + vy + vz


Here vx, vy, and vz are the projections of v onto an orthogonal triple of rays spanning R3— or, as we can call them, the axes of our coordinate system.

Pythagoras’ theorem tells us that

|vx|2 + |vy|2 + |vz|2 = |v|2,

and so, if v is normalized,

|vx|2 + |vy|2 + |vz|2 = 1

Let us now assume that we wish to represent three mutually exclusive events that together exhaust all possibilities, and that each event has a certain probability. If we use the axes of R3 to represent the events x, y, and z, we can construct a normalized vector v to represent any probability assignment to these events. We simply take vectors vx, vy, and vz along these axes such that:

|vx|2= p(x), |vy|2 = p(y) and |vz|2= p(z),

and then add them (vectorially) to yield v.

Since the events x, y, and z are mutually exclusive and jointly exhaustive, we know that p(x) + p(y) + p(z) = 1 and it follows ... that v is normalized. This almost trivial construction lies at the heart of the use of vector spaces in physical theory.

Hughes

color sphere


Then it turns out that the mixture of the two lights (it is one of the consequences of the laws that we have already mentioned) is obtained by taking the sum of the components of X and Y:

Z = X + Y = (a + a')A + (b + b')B + (c + c')C.

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.

Feynman







Hume
David Hume

From the succession of ideas and impressions we form the idea of time. It is not possible for time alone ever to make its appearance.


'D' wing of R&D
The fundamental principle of that philosophy is the opinion concerning colors, sounds, tastes, smells, heat and cold; which it asserts to be nothing but impressions in the mind, deriv'd from the operation of external objects, and without any resemblance to the qualities of the objects. […]

Thus there is a direct and total opposition betwixt our reason and senses […] When we reason from cause and effect, we conclude, that neither color, sound, taste, nor smell have a continued and independent existence. When we exclude these sensible qualities there remains nothing in the universe, which has such an existence.
Hume


Hume saw clearly that certain concepts, for example that of causality, cannot be deduced from our perceptions of experience by logical methods.

§

This line of thought had a great influence on my efforts, most specifically Mach and even more so Hume, whose Treatise of Human Nature I studied avidly and with admiration shortly before discovering the theory of relativity.

Einstein





Husserl
Edmund Husserl

Only one need absorbs me: I must win clarity else I cannot live; I cannot bear life unless I can believe that I will achieve it.


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Psychology, on the other hand, is science of psychic Nature and, therefore, of consciousness as Nature or as real event in the spatiotemporal world.

§

Pure phenomenology claims to be the science of pure phenomena. This concept of the phenomenon, which was developed under various names as early as the eighteenth century without being clarified, is what we shall have to deal with first of all.

§

If all consciousness is subject to essential laws in a manner similar to that in which spatial reality is subject to mathematical laws, then these essential laws will be of most fertile significance in investigating facts of the conscious life of human and brute animals.


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