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Aharonov
Yakir Aharonov




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color sphere

Hamiltonian


In classical mechanics, we recall that potentials cannot have such significance because the equation of motion involves only the field quantities themselves. For this reason, the potentials have been regarded as purely mathematical auxiliaries, while only the field quantities were thought to have a direct physical meaning.

In quantum mechanics, the essential difference is that the equations of motion of a particle are replaced by the SchrÖdinger equation for a wave. This SchrÖdinger equation is obtained from a canonical formalism, which cannot be expressed in terms of the fields alone, but which also requires the potentials. Indeed, the potentials play a role, in the SchrÖdinger equation, analogous to that of the index of refraction in optics.

Aharonov & Bohm



EM Potential


We now proceed to the derivation of the laws of geometrical optics from the laws of wave optics. A rigorous and general derivation, however, requires a considerable amount of advanced mathematics so that we shall not go into details here. We shall be satisfied with observing the essential points by taking the simple example given above.

The problem is again that of the refraction of monochromatic light [...]

   

holds, as is apparent from the figure. In other words, the refractive index n is inversely proportional to the wavelength of the light wave in the respective media; i.e.,

n = k''/l

The proportionality constant k'', however, may depend on the frequency or, in other words, on the color of the light.
Tomonaga



prism

Furthermore, and now this is the point, this is the punch
line, the symmetries determine the action. ~Weinberg



JS Bell
JS Bell

[...] conventional formulations of quantum theory, and of quantum field theory in particular, are unprofessionally vague and ambiguous. Professional theoretical physicists ought to be able to do better. Bohm has shown us a way.



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Theoretical physicists live in a classical world, looking out into a quantum-mechanical world. The latter we describe only subjectively, in terms of procedures and reults in our classical domain. This subjective description is effected by means of quantum- mechanical state functions state functions, which characterize the classical conditioning of  quantum- mechanical systems and permit predictions about subsequent events at the classical level. [...]

Now nobody knows just where the boundary between the classical and quantum domains is situated. Most feel that experimental switch settings and pointer readings are on this side. But some would think the boundary nearer, others would think it farther, and many would prefer not to think about it.

JS Bell


Quantum theory is the most successful scientific theory of all time. Many of the great names of physics are associated with quantum theory. Heisenberg and Schrödinger established the mathematical form of the theory, while Einstein and Bohr analysed many of its important features. However, it was John Bell who investigated quantum theory in the greatest depth and established what the theory can tell us about the fundamental nature of the physical world.

Moreover, by stimulating experimental tests of the deepest and most profound aspects of quantum theory, Bell's work led to the possibility of exploring seemingly philosophical questions, such as the nature of reality, directly through experiments.

PhysicsWeb


Non-locality in quantum theory is discussed in terms of the global form of the wave function, and as subtle set of necessary and sufficient conditions on the 2-matrix or reduced density matrix. In addition to manifesting itself through the well known Bell's inequalities non-locality also appears as a macroscopic coherence length in condensed and coherent systems. By examining the structure of the 2-matrix, a connection between these two forms of non-locality is made. It is suggested that subtle enfolded orders and non-local forms may have a wider implication and be relevant for a variety of living systems.

F. David Peat


It is shown that the matrix models which give non-perturbative definitions of string and M-theory may be interpreted as non-local hidden variables theories in which the quantum observables are the eigenvalues of the matrices while their entries are the non-local hidden variables. 

Lee Smolin



Niels Bohr
Niels Bohr

In our description of nature the purpose is not to disclose the real essence of phenomena but only to track down as far as possible relations between the multifold aspects of our experience.


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spectra

The mathematical machinery of quantum mechanics 
became that of spectral analysis... ~Steen


[It] was found possible to account for the atomic stability, as well as for the empirical laws govern- ing the spectra of the elements, by assuming that any reaction of the atom resulting in a change of its energy involved a complete transition between two so-called stationary quantum states and that, in particular, the spectra were emitted by a step-like process in which each transition is accompanied by the emission of a monochromatic light quantum of an energy just equal to that of an Einstein photon.

Bohr  


Bohr suggests that thought involves such small amounts of energy that quantum-
theoretical limitations play an essential role in determining its character.

Bohm




Niels Bohr brainwashed a whole generation of physicists into believing that the problem (of the interpretation of quantum theory) had been solved fifty years ago.

Gell-Mann









David Bohm
David Bohm


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


But in 1952 I saw the impossible done. It was in papers by David Bohm. Bohm showed explicitly how parameters could indeed be introduced, into nonrelativistic wave mechanics, with the help of which the indeterministic description could be transformed into a deterministic one. More importantly, in my opinion, the subjectivity of the orthodox version, the necessary reference to the ‘observer,’ could be eliminated. ...

But why then had Born not told me of this ‘pilot wave’? If only to point out what was wrong with it? Why did von Neumann not consider it? More extraordinarily, why did people go on producing ‘‘impossibility’’ proofs, after 1952, and as recently as 1978? ... Why is the pilot wave picture ignored in text books? Should it not be taught, not as the only way, but as an antidote to the prevailing complacency? To show us that vagueness, subjectivity, and indeterminism, are not forced on us by experimental facts, but by deliberate theoretical choice?
JS Bell










Max Born
Max Born


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Anyone dissatisfied with these ideas may feel free to assume that there are additional parameters not yet introduced into the theory which determine the individual event.

§

I am now convinced that theoretical physics is actually philosophy. [...] I believe there is no philosophical high-road in science, with epistemological signposts. No, we are in a jungle and find our way by trial and error, building our road behind us as we proceed.

Born


Quantum mechanics is very impressive. But an inner voice tells me that it is not yet the real thing. The theory yields a lot, but it hardly brings us any closer to the secret of the Old One. In any case I am convinced that He doesn't play dice.

Einstein, Letter to Born

Calabi
Eugenio Calabi


Cartan

Elie Cartan

Elie Cartan is one of the most influential of 20th-century geometers. At one point he had an intense correspondence with Einstein on general relativity.  His "Cartan geometry"refraction idea is an approach to the concept of parallel transport that predates the widely used Ehresmann approach (connections on principal bundles).  It simultaneously generalizes Riemannian geometry and Klein's Erlangen program, in which geometries are described by their symmetry groups:

EUCLIDEAN GEOMETRY  -------------->  KLEIN GEOMETRY

      |                                                         |
      |                                                         |
      |                                                         |
      |                                                         |
      v                                                        v

  RIEMANNIAN GEOMETRY  -----------> CARTAN GEOMETRY


This Week's Finds, by John Baez    
  







David Chalmers
David Chalmers




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The abstract notion of information, as put forward by Claude E. Shannon of MIT, is that of a set of separate states with a basic structure of similarities and differences between them. We can think of a 10-bit binary code as an information state, for example. Such information can be embodied in the physical world. This happens whenever they correspond to physical states (voltages, say); the differences between them can be transmitted along some pathway, such as a telephone line.

color vectors


We can also find information embodied in conscious experience. The pattern of color patches in a visual field, for example, can be seen as analogous to that of pixels covering a display screen. Intriguingly, it turns out that we find the same information states embodied in conscious experience and in underlying physical processes in the brain. The three-dimensional encoding of color spaces, for example, suggests that the information state in a color experience correspond directly to an information state in the brain. We might even regard the two states as distinct aspects of a single information state, which is simultaneously embodied in both physical processing and conscious experience.

Chalmers        




Patricia Churchland
Patricia Smith Churchland

A complex of coefficients of this type is comparable with a matrix such as occurs in linear algebra. (Heisenberg)

A trained-up network is one in which, for appropriate input vectors, the network gives the correct response, expressed in terms of an output vector. Training up a network involves adjusting the many weights so that this end is achieved. This might be done in a number of different ways. One might hand-set the weights, or the weights might be set by a back- propagation of error or by an unsupervised algorithm. Weight configurations too are characterizable in terms of vectors, and at any given time the complete set of synaptic values defines a weight state space, with points on each axis specifying the size of a particular weight. [...]


It is conceptually efficient to see the final resting region in weight space as embodying the total knowledge stored in the network. Notice that all incoming vectors go through the matrix of synaptic connections specified by that weight-space point. [...]

A matrix is an array of values, and the elements of an incoming vector can be operated on by some function to produce an output vector.



NN


Paul Churchland
Paul Churchland


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.
(Dirac)


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What we are looking at, then, is a multistage device for successively transforming an initial sensory activation vector into a sequence of subsequent activation vectors embodied in a sequence of downstream neuronal populations. Evidently, the basic mode of singular, ephemeral, here-and-now perceptual representation is not the propositional attitude at all; it is the vectorial attitude. And the basic mode of information processing is not the inference drawn from one propositional atttitude to another; it is the synapse- induced transformation of one vectorial attitude into another, and into a third, a forth, and so on, as the initial sensory information ascends the waiting information-processing hierarchy.


That highly trained processing hierarchy embodies the network's general background knowledge of the important categories into which Nature divides itself and many of the major relations between them. That is to say, the brain's basic mode of representing the world's enduring structure is not the general or universally quantified propositional attitude at al; it is the hard-earned configuration of weighted synaptic connections, those that transform the activation vectors at one neuronal population into the activation vectors of the next. It is these myriad connections that the learning process was originally aimed at configuring, and it is these connections that subsequently do the important computational work of the matured network.

§

[The] tensor calculus emerges as the natural framework with which to address such matters ...


color matrix







d'Alembert

d'Alembert

The laws of physics must be valid in all systems of coordinates. They must thus be expressible as tensor equations. Whenever they involve the derivative of a field quantity, it must be a covariant derivative. The field equations of physics must all be rewritten with the ordinary derivatives replaced by covariant derivatives. For example, the d’Alembert equation quablaV = 0 for a scalar V becomes, in covariant form

dAlembertian

Dirac

The D'Alembertian is the equivalent of the Laplacian in Minkowskian geometry. It is given by:

d'Alembertian

PlanetMath

blue sphere
wave equation
wave equation


Klein-Gordon equation
Klein-Gordon equation



Leonardo da Vinci
Leonardo da Vinci

Simplicity is the ultimate sophistication.





If the front of a building, or any piazza or field, which is illuminated by the sun, has a dwelling opposite to it, and if in front that does not face the sun you make a small round hole, all the illuminated objects will send their images through that little hole and will appear inside the dwelling on the opposite wall, which should be white. And there they will be, exactly and upside down ... If the bodies are of various colors and shapes, the rays forming the images will be of various colors and shapes, and of various colors and shapes will be the representations on the wall.


da Vinci



De Broglie

Louis De Broglie
The question which finally must be answered is knowing (Einstein has often emphasized this point) whether the present interpretation, which uses solely the psi-wave with its statistical character is a "complete" description of realityin which case it would be necessary to assume indeterminism and the impossibility of representing reality on the atomic level in a precise way in the framework of space and timeor if, on the contrary, this interpretation is incomplete and hides behind itself, as the older statistical theories of classical physics doa perfectly determinate reality, describable in the framework of space and time by variables which would be hidden from us, i.e., which would escape experimental determination by us.
De Broglie

prism

The aspects of things that are most important for us are hidden because of their simplicity and familiarity.
Wittgenstein


Bohmian mechanics, which is also called the de Broglie-Bohm theory, the pilot-wave model, and the causal interpretation of quantum mechanics, is a version of quantum theory discovered by Louis de Broglie in 1927 and rediscovered by David Bohm in 1952. It is the simplest example of what is often called a hidden variables interpretation of quantum mechanics. In Bohmian mechanics a system of particles is described in part by its wave function, evolving, as usual, according to Schrödinger's equation. However, the wave function provides only a partial description of the system.





Descartes
René Descartes

If you would be a real seeker after truth, it is necessary that at least once in your life you doubt, as far as possible, all things.

color coordinates

The long chains of simple and easy reasonings by means of which geometers are accustomed to reach the conclusions of their most difficult demonstra- tions, had led me to imagine that all things, to the knowledge of which man is competent, are mutually connected in the same way, and that there is nothing so far removed from us as to be beyond our reach, or so hidden that we cannot discover it, provided only we abstain from accepting the false for the true, and always preserve in our thoughts the order necessary for the deduction of one truth from another. And I had little difficulty in determining the objects with which it was necessary to commence, for I was already persuaded that it must be with the simplest and easiest to know, and, considering that of all those who have hitherto sought truth in the sciences, the mathematicians alone have been able to find any demonstrations, that is, any certain and evident reasons, I did not doubt but that such must have been the rule of their investigations.
Descartes          



PAM Dirac
Paul Dirac


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It seems clear that the present quantum mechanics is not in its final form [...] I think it very likely, or at any rate quite possible, that in the long run Einstein will turn out to be correct.


superposition


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    

Harmonic Analysis

Dirichlet

Dirichlet
As we saw earlier, the Kaluza-Klein states of the compactified 11-dimensional M-theory are equivalent to the nonperturbative 10-dimensional Dirichlet 0-brane states.



Dirichlet gave the first rigorous proof of Fourier’s theorem. Riemann (1868) says that this
paper was the first in which the fact that Fourier series are non-absolutely convergent was noticed and dealt with correctly.

Riemann states this in the historical introduction to his paper on Fourier analysis, which was his Habilitationsschrift (probationary essay) at Göttingen in 1854. It remained unpublished until 1868, when Dedekind discovered it after Riemann’s death. Riemann actually says more: He says that Dirichlet was the first person to discover the phenomenon of conditional convergence. This, however, cannot be true. Cauchy was evidently aware of the difference between absolutely and conditionally convergent series, and we already noted that Abel took careful account of this distinction in 1826. It is true, however, that Dirichlet begins his paper by pointing out that one of Cauchy’s attempted proofs of convergence for Fourier series fails on just this point. Riemann based his history in this paper on a long conversation with Dirichlet, so this probably represents Dirichlet’s memory of the essential difficulty in the proof. Dirichlet’s proof worked for functions that were “piecewise monotonic,” and guaranteed convergence at points of continuity of such functions. Dirichlet believed the proof could be extended to prove convergence for any continuous function.
Offner


Vibrating Drum

Vibrating strings and membranes produce characteristic (eigen) sound vectors.


Freeman Dyson
Freeman Dyson

Mind and intelligence are woven into the fabric of our universe in a way that altogether surpasses our comprehension.


Mind and intelligence are woven into the fabric of the universe in a way that altogether surpasses our comprehension.
§

This is the characteristic mathematical property of a classical field: it is an undefined something which exists throughout a volume of space and which is described by sets of numbers, each set denoting the field strength and direction at a single point in the space.

§

There is nothing else except these fields: the whole of the material universe is built of them.

§

The universe shows evidence of the operations of mind on three levels. The first level is elementary physical processes, as we see them when we study atoms in the laboratory. The second level is our direct human experience of our own consciousness. The third level is the universe as a whole. Atoms in the laboratory are weird stuff, behaving like active agents rather than inert substances. They make unpredictable choices between alternative possibilities according to the laws of quantum mechanics. It appears that mind, as manifested by the capacity to make choices, is to some extent inherent in every atom. The universe as a whole is also weird, with laws of nature that make it hospitable to the growth of mind.

Dyson    



Einstein
Albert Einstein

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.



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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.

§

The overcoming of naive realism has been relatively simple. In his introduction to his volume, An Inquiry Into Meaning and Truth, Russell has characterized this process in a marvellously pregnant fashion:

We all start from 'naive realism,' i.e., the doctrine that things are what they seem. We think that grass is green, that stones are hard, and that snow is cold. But physics assures us that the greenness of grass, the hardness of stones, and the coldness of snow, are not the greenness, hardness, and coldness that we know in our own experience, but something very different. The observer, when he seems to himself to be observing a stone, is really, if physics is to be believed, observing the effects of the stone upon himself. Thus science seems to be at war with itself: when it means to be most objective, it finds itself plunged into subjectivity against its will.

Apart from their masterful formulation these lines say something which had never previously occurred to me.
Einstein  

     


Euler

Leonhard Euler

Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena.



least action
least action



The calculus of variations seeks to find the path, curve, surface, etc., for which a given function has a stationary value (which, in physical problems, is usually a minimum or maximum). Mathematically, this involves finding  of integrals of the form:

calculus of variations
Wolfram MathWorld  

Furthermore, and now this is the point, this is the punch line, the symmetries determine the action.  This action, this form of the dynamics, is the only one consistent with these symmetries [...]  This, I think, is the first time that this has happened in a dynamical theory: that  the symmetries of the theory have completely determined the structure of the dynamics, i.e., have completely determined the quantity that produces the rate of change of the state vector with time.
Weinberg
symmetry



Feigl
Herbert Feigl

There's no question that Feigl recognized the difficulty of the problem. His problem of "sentience" is of course just the problem I'm concerned with. (David Chalmers)


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I can here only briefly indicate the lines along which I think the 'world knot'-to use Schopenhauer's striking designation for the mind-body puzzles may be disentangled. The indispensable step consists in a critical reflection upon the meanings of the terms 'mental' and 'physical', and along with this a thorough clarification of such traditional philosophical terms as 'private' and 'public', 'subjective' and 'objective', 'psychological space(s)' and 'physical space', 'intentionality', 'purposiveness', etc. The solution that appears most plausible to me, and that is consistent with a thoroughgoing naturalism, is an identity theory of the mental and the physical, as follows: Certain neurophysiological terms denote (refer to) the very same events that are also denoted (referred to) by certain phenomenal terms. ... I take these referents to be the immediately experienced qualities, or their configurations in the various phenomenal fields.

Feigl



There's no question that Feigl recognized the difficulty of the problem. His problem of "sentience" is of course just the problem I'm concerned with.




Fermat
Fermat

And perhaps, posterity will thank me for having shown it that the ancients did not know everything.

Among the more or less general laws, the discovery of which characterize the developmentof physical science during the last century, the principle of Least Action is at presentcertainly one which, by its form and comprehensiveness, may be said to have approachedmost closely to the ideal aim of theoretical inquiry. Its significance, properly understood,extends, not only to mechanical processes, but also to thermal and electrodynamicproblems. In all the branches of science to which it applies, it gives, not only an explanation of certain characteristics of phenomena at present encountered, but furnishes rules whereby their variations with time and space can be completely determined. It provides the answersto all questions relating to them, provided only that the necessary constants are known andthe underlying external conditions appropriately chosen.

§

It was a long time before it was clearly explained, and the principle of least action correctly understood. If the principle be said to have been discovered at this time, the honour should be given to Lagrange. This, however, would be an injustice to other men who had prepared the way for Lagrange to bring the work later to a satisfactory completion. Of these, the first was Leibniz; indeed, he was the chief, according to a letter dated 1707, the original of which has been lost. Then came Maupertuis and Euler. It was chiefly Moreau de Maupertuis (appointed president of the Prussian Academy of Sciences (1746-1759) by Frederick the Great) who not only recognized the existence and significance of the principle, but used his influence in the scientific world and elsewhere to procure its acceptance.

§

Maupertuis’s exposition of the principle of least action asserted no more than “that the action applied to bring about all the changes occurring in Nature is always a minimum.” Strictly, this formulation does not admit any conclusions to be drawn regarding the laws governing the changes, for as long as no statement of the conditions to be satisfied is made, no deductions can be made as to how the variations are balanced. Maupertuis had not the faculty of analytical criticism necessary to discern this want. The failure will be more easily understood when it is realized that Euler himself, a brilliant mathematician, did not succeed in producing a correct formulation of the principle, though he was assisted by many
colleagues and friends.

Maupertuis’s real service consisted in his search for a principle that would be, above all, a minimum principle. That was the real object of his investigation. To this end he made use of Fermat’s principle of quickest arrival, although its bearing upon the principle of least action was very indirect and, at all events, unknown to the physics of his time.

Planck, The Principle of Least Action (PDF)


Least action



Feynman
Richard Feynman

Nature has a great simplicity
and therefore a great beauty.



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If you take a physical state and do something to itlike rotating it, or like waiting for some time deltat you 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>.


matrix operator


I would like to again impress you with the vast range of phenomena that the theory of quantum electrodynamics describes: It's easier to say it backwards: the theory describes all the phenomena of the physical world except the gravitational effect [...] and radioactive phenomena, which involve nuclei shifting in their energy levels. So if we leave out gravity and radioactivity (more properly, nuclear physics) what have we got left? Gasoline burning in automobiles, foam and bubbles, the hardness of salt or copper, the stiffness of steel. In fact, biologists are trying to interpret as much as they can about life in terms of chemistry, and as I already explained, the theory behind chemistry is quantum electrodynamics.

Fourier
Fourier

There is a very simple technique for analyzing any curve, no matter how complicated it may be, into its constituent simple harmonic curves. It is based on a mathematical theorem known as Fourier's theorem [...]

The theorem tells us that every curve, no matter what its nature may be, or in what way it was originally obtained, can be exactly reproduced by superposing a sufficient number of simple harmonic curves—in brief, every curve can be built up by piling up waves. (Sir James Jeans)   


Fourier components

The mathematician Fourier proved that any continuous function could be produced as an infinite sum of sine and cosine waves. His result has far-reaching implications for the reproduction and synthesis of sound. A pure sine wave can be converted into sound by a loudspeaker and will be perceived to be a steady, pure tone of a single pitch. The sounds from orchestral instruments usually consists of a fundamental and a complement of harmonics, which can be considered to be a superposition of sine waves of a fundamental frequency f and integer multiples of that frequency.

The process of decomposing a musical instrument sound or any other periodic function into its constituent sine or cosine waves is called Fourier analysis. You can characterize the sound wave in terms of the amplitudes of the constituent sine waves which make it up. This set of numbers tells you the harmonic content of the sound and is sometimes referred to as the harmonic spectrum of the sound. The harmonic content is the most important determiner of the quality or timbre of a sustained musical note.



Galileo
Galileo Galilei

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


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...

§

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.



General Relativity

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

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 Nils 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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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

Hamiltonian

Hamiltonian operator

energy relation

color sphere


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 E, v
and 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.


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

laser

Noether
How far does the laser move in color space?

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

§

The natural scientist no more than the philosopher can ignore epistemological questions when he is dealing with sesnse 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 colour; it is, so to speak, surrounded by colour-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 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.

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





David 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.

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    

Doppler

By E = hv, Doppler changes in frequency —> changes in energy —> rotations of color vector


sphere


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.


Jacobi
Carl Jacobi

Hamilton-Jacobi equation

William James
William James

The ultimate of ultimate problems, of course, in the study of the relations of thought and brain, is to understand why and how such disparate things are connected at all […] We must find the minimal mental fact whose being reposes directly on a brain-fact; and we must similarly find the minimal brain event which will have a mental counterpart at all.


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When we're asked "What do 'red', 'blue', 'black', 'white' mean?" we can, of course, immediately point to things which have these colours,—but that's all we can do: our ability to explain their meaning goes no further.
   Wittgenstein


§

Thus "this is red," "this is earlier than that," are atomic propositions.

Russell & Whitehead


RGB



All other psychological phenomena are derived from the combinations of these ultimate psychological elements, as the totality of words may be derived from the totality of letters. Completion of this task would provide the basis for a Characteristica universalis of the sort that had been conceived by Leibniz, and before him, by Descartes.

Brentano

There is a branch of mathematics known as 'harmonic analysis' which deals with the converse problem of sorting out the resultant curve into its constituents. Superposing a number of curves is as simple as mixing chemicals in a test-tube; anyone can do it. But to take the final mixture and discover what ingredients have gone into its composition may require great skill.

Fortunately the problem is easier for the mathematician than for the analytical chemist. There is a very simple technique for analyzing any curve, no matter how complicated it may be, into its constituent simple harmonic curves. It is based on a mathematical theorem known as Fourier's theorem ...

The theorem tells us that every curve, no matter what its nature may be, or in what way it was originally obtained, can be exactly reproduced by superposing a sufficient number of simple harmonic curves—in brief, every curve can be built up by piling up waves.

Sir James Jeans




Fourier

In May 1926 Schrödinger published a proof that matrix and wave mechanics gave equivalent results: mathematically they were the same theory.



Kaluza
Theodor Kaluza



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Calabi-Yau
Calabi-Yau space



Alongside the metric tensor of the 4-dimesnional manifold (interpreted as a tensor potential for gravity) a general relativistic description of world-phenomena requires also an electromagnetic four-potential  Am.

The remaining dualism of gravity and and electricity, while not lessening the theory's enthralling beauty, nevertheless sets the challenge of overcoming it through a fully unified view.

A few years ago H. Weyl made a surprisingly bold thrust towards the solution of this problem, one of the great favorite ideas of the human spirit. Through a new radical revision of the geometric foundations, he obtains along with the tensor gmn a kind of fundamental metric vector which he then interprets as the
electromagnetic potential Am. The complete world metric then becomes the common source of all natural events.

§

It is true that our previous physical experience contains hardly any hint of the existence of an extra dimension...

Kaluza

red

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.

§

The aspects of things that are most important for us are hidden because of their simplicity and familiarity.
Wittgenstein


Well, obviously the extra dimensions have to be different somehow because otherwise we would notice them.
Green


 

Now it may be asked why these hidden variables should have so long remained undetected.

Bohm




Felix Klein
Felix Klein


Projective geometry has opened up for us with the greatest facility new territories in our science, and has rightly been called the royal road to our particular field of knowledge.

§

It became possible to affirm that projective geometry is indeed logically prior to Euclidean geometry and that the latter can be built up as a special case. Both Klein and Arthur Cayley showed that the basic non-Euclidean geometries developed by Lobachevsky and Bolyai and the elliptic non- Euclidean geometry created by Riemann can also be derived as special cases of projective geometry. No wonder that Cayley exclaimed, "Projective geometry is all geometry."
Kline

projection





Oscar Klein
Oscar Klein



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In a former paper, the writer has shown that the differential equation underlying the new quantum mechanics of Schrödinger can be derived from a wave equation of a five- dimensional space, in which h does not appear originally, but is introduced in connection with a periodicity in x0. Although incomplete, this result, together with the considerations given here, suggest that the origin of Planck's quantum may be sought just in this periodicity in the fifth dimension.
Klein


Klein's adaptation of Kaluza's work had a major difference from the original in that the extra or fifth dimension was curled up into a ball that was on the order of the Planck length, 10-33 cm. It is important to note, however, that the extra dimension, though curled up, was still Euclidean in nature.

 Ian T Durham


The highlight of that conference, at least with the hindsight of history, was the remarkable paper by Oscar Klein in which he proposed a unified model of electromagnetism and the nuclear force based on Kaluza-Klein ideas. This paper stands out in its originality and its brilliance from the other contributions to the conference and it foreshadowed the later developments of non-Abelian gauge theories that are the foundation of our present theory of particle physics.
David J Gross

Lagrange
 Joseph-Louis Lagrange

As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection.

action integral

If the total energy is conserved, then the work done on the particle must be converted to potential energy, conventionally denoted by V, which must be purely a function of the spatial coordinates x, y, z, or equivalently a function of the generalized configuration coordinates X, Y, and possibly the derivatives of these coordinates, but independent of the time t. (The independence of the Lagrangian with respect to the time coordinate for a process in which energy is conserved is an example of Noether's theorem, which asserts that any conserved quantity, such as energy, corresponds to a symmetry, i.e., the independence of a system with respect to a particular variable, such as time.)


Laplace
Pierre-Simon Laplace

Such is the advantage of a well constructed language that its simplified notation often becomes the source of profound theories.


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The Laplacian is extremely important in mechanics, electromagnetics, wave theory, and quantum mechanics, and appears in Laplace's equation

del ^2phi==0,

The wave equation is the important partial differential equation
del ^2psi==1/(v^2)(partial^2psi)/(partialt^2)
(1)

that describes propagation of waves with speed v. The form above gives the wave equation in three-dimensional space where del ^2 is the Laplacian, which can also be written

v^2del ^2psi==psi_(tt).
(2)

An even more compact form is given by

 square ^2psi==0,
(3)

where  square ^2 is the d'Alembertian, which subsumes the second time derivative and second space derivatives into a single operator.

A version of the Laplacian that operates on vector functions is known as the vector Laplacian, and a tensor Laplacian can be similarly defined. 

A function psi which satisfies Laplace's equation is said to be harmonic. A solution to Laplace's equation has the property that the average value over a spherical surface is equal to the value at the center of the sphere (Gauss's harmonic function theorem). Solutions have no local maxima or minima. Because Laplace's equation is linear, the superposition of any two solutions is also a solution.

Wolfram MathWorld


Leibniz
Gottfried Wilhelm von Leibniz

The art of discovering the causes of phenomena, or true hypothesis, is like the art of decyphering, in which an ingenious conjecture greatly shortens the road.

Mechanical causes


Besides, it must be confessed that Perception and its consequences are inexplicable by mechanical causes; that is to say, by figures and motions. If we imagine a machine so constructed as to produce thought, sensation, perception, we may conceive it magnified — to such an extent that one might enter it like a mill. This being supposed, we should find in it on inspection only pieces which impel each other, but nothing which can explain a perception. It is in the simple substance, therefore, — not in the compound, or in the machinery, — that we must look for that phenomenon [...]




John Locke
John Locke


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These I call original or primary qualities of the body, which I think we may observe to produce simple ideas in us, viz., solidity, extension, figure, motion or rest, and number.

Secondly, such qualities which in truth are nothing in the objects themselves, but powers to produce various sensations in us by their primary qualities, i.e. by the bulk, figure, texture, and motion of their insensible parts, as colour, sounds, tastes, etc., these I call secondary qualities.

§

Earthly minds, like mud walls, resist the strongest batteries; and though, perhaps, somethimes the force of a clear argument may make some impression, yet they nevertheless stand firm, keep out the enemy, truth, that would captivate or disturbe them.



Michael Lockwood



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state space



Take some range of phenomenal qualities. Assume that these qualities can be arranged according to some abstract n-dimensional space, in a way that is faithful to their perceived similarities and degrees of similarity — just as, according to Land, it is possible to arrange the phenomenal colors in his three-dimensional color solid. Then my Russellian proposal is that there exists, within the brain, some physical system, the states of which can be arranged in some n-dimensional state space [...] And the two states are to be equated with each other: the phenomenal qualities are identical with the states of the corresponding physical system.


Lockwood    


Calabi-Yau

We shall now recall the data of a classical theory as understood by physicists and then reinterpret them in geometrical form. Geometrically or mechanically we can interpret this data as follows. Imagine a structured particle, that is a particle which has a location at a point x of R4 and an internal structure, or set of states, labeled by elements g of G.

Atiyah       

elements of G


After all, our very definition of a particle or metastable nuclear state is based on its classification as the carrier of a definite representation of the Poincaré group [...]

Ne'eman

Ernst Mach
Ernst Mach

A color is a physical object as soon as we consider its dependence, for instance, upon its luminous source, upon temperatures, upon spaces, and so forth.

§

Without renouncing the support of physics, it is possible for the physiology of the senses, not only to pursue its own course of development, but also to afford to physical science itself powerful assistance.

James Clerk Maxwell
James Clerk Maxwell


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 absolute simplicity

When a beam of light falls on the human eye, certain sensations are produced, from which the possessor of that organ judges of the color and luminance of the light. Now, though everyone experiences these sensations and though they are the foundation of all the phenomena of sight, yet, on account of their absolute simplicity, they are incapable of analysis, and can never become in themselves objects of thought. If we attempt to discover them, we must do so by artificial means and our reasonings on them must be guided by some theory.

McCulloch
Warren McCulloch



In 1943 Warren McCulloch and Walter Pitts proposed a general theory of information processing based on networks of binary switching or decision elements, which are somewhat euphemistically called "neurons," although they are far simpler than their real biological counterparts [...]

McCulloch and Pitts showed that such networks can, in principle, carry out any imaginable computation, similar to a programmable, digital computer or its mathematical abstraction, the Turing machine.
Muller, Reinhardt


Isaac Newton
Isaac Newton

If I have ever made any valuable discoveries, it has been owing more to patient attention, than to any other talent.


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Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things.

§
... the science of colours becomes a speculation as truly mathematical as any other part of physics.
§

Are not gross bodies and light convertible into one another; and may not bodies receive much of their activity from the particles of light which enter into their composition? The changing of bodies into light, and light into bodies, is very conformable to the course of Nature, which seems delighted with transmutations.



Emmy Noether
Emmy Noether


In the judgment of the most competent living mathematicians, Fraulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present day younger generation of mathematicians.

Einstein  

Noether     


Wolfgang Pauli
Wolfgang Pauli


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For the invisible reality, of which we have small pieces of evidence in
both quantum physics and the psychology of the unconscious, a symbolic psychophysical unitary language must ultimately be adequate, and this is the far goal which I actually aspire. I am quite confident that the final
objective is the same, independent of whether one starts from the psyche (ideas) or from physis (matter). Therefore, I consider the old distinction between materialism and idealism as obsolete.

Pauli, letter to Jung
 

psyche & physis

Pellionisz
Andras Pellionisz


The incorrect perception that the quantum system has only microscopic manifestations considerably confused this subject. As we have seen in preceding sections, manifestation of ordered states is of quantum origin. When we recall that almost all of the macroscopic ordered states are the result of quantum field theory, it seems natural to assume that macroscopic ordered states in biological systems are also created by a similar mechanism.

Umezawa      

fractal net

Does fractal neural form follow quantum function?

Penrose
Roger Penrose


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Can it really be true that Einstein, in any significant sense, was a profoundly "wrong" as the followers of Bohr maintain? I do not believe so. I would, myself, side strongly with Einstein in his belief in a submiscroscopic reality, and with his conviction that present-day quantum mechanics is fundamentally incomplete.
Penrose

Poincare
Henri Poincaré

It is often said that we "project" into geometric space the objects of our external perception; that we "localize" them.

Has this a meaning, and if so what?

Does it mean that we represent to ourselves external objects in geometrical space?

Our representations are only the reproduction of our sensations; they can therefore be ranged only in the same frame as these, that is to say, in perceptual space.

It is as impossible for us to represent to ourselves external bodies in geometric space, as it is for a painter to paint on a plane canvass objects with their three dimensions.

Perceptual space is only an image of geometric space, an image altered in shape by a sort of perspective [...]


Karl Pribram
Karl Pribram


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The text of this volume claims that the mathematical formulations that have been developed for quantum mechanics and quantum field theory can go a long way toward describing neural processes due to the functional organization of the cerebral cortex.

Riemann
Bernhard Riemann

[So] few and far between are the occasions for forming notions whose specialisations make up a continuous manifold, that the only simple notions whose specialisations form a multiply extended manifold are the positions of perceived objects and colors. More frequent occasions for the creation and development of these notions occur first in the higher mathematic.

Definite portions of a manifold, distinguished by a mark or a boundary, are called Quanta [...]

    quanta

Bertrand Russell
Bertrand Russell

I think we ought always to entertain our opinions with some measure of doubt. I shouldn't wish people dogmatically to believe any philosophy, not even mine.



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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          

§

Take some range of phenomenal qualities. Assume that these qualities can be arranged according to some abstract n-dimensional space, in a way that is faithful to their perceived similarities and degrees of similarity — just as, according to Land, it is possible to arrange the phenomenal colors in his three-dimensional color solid. Then my Russellian proposal is that there exists, within the brain, some physical system, the states of which can be arranged in some n-dimensional state space ... And the two states are to be equated with each other: the phenomenal qualities are identical with the states of the corresponding physical system.

Lockwood        

color sphere

Salam
Abdus Salam


[All] chemical binding is electromagnetic in origin, and so are all phenomena of nerve impulses.

Salam    
    

Neural net

Erwin Schrodinger
Erwin Schrödinger


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If you ask a physicist what is his idea of yellow light, he will tell you that it is transversal electromagnetic waves of wavelength in the neighborhood of 590 millimicrons. If you ask him: But where does yellow come in? he will say: In my picture not at all, but these kinds of vibrations, when they hit the retina of a healthy eye, give the person whose eye it is the sensation of yellow.

laser

John Searle
John Searle

The question we wanted to ask is this: 'Can a digital computer, as defined, think?' That is to say, 'Is instantiating or implementing the right computer program with the right inputs and outputs, sufficient for, or constitutive of, thinking?' And to this question, unlike its predecessors, the answer is clearly 'no.' And it is 'no' for the reason that we have spelled out, namely, the computer is defined purely syntactically. But thinking is more than just a matter of manipulating meaningless symbols, it involves meaningful semantic contents. These semantic contents are just what we mean by 'meaning.'
Searle        

Henry Stapp
Henry Stapp


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All of these materialistic-type theories are now known to be false: they provide no adequate 
basis for understanding the structure of our experience.

Quantum theory is the newest contender. It accomodates all of the empirical evidence accepted
by the scientific community.

It differs from the materialistic theories in three essential respects.

First, it is not designed to be a description of a reality that exists independently
of human observers. Rather it is designed to be a computational procedure that allows
us to form expectations about our future experiences — as we describe them to ourselves
and to our colleagues in a suitable language — on the on the basis of knowledge gleaned
from previous similarly described experiences.

Second, it is an incomplete description, in the sense that there is a random element whose
origin and mode of acting is not specified.

Third, the ontological character of the "object" described by the theory is, on the basis
of how it behaves, more like "knowledge" than like the matter of the materialistic theories:
it makes sudden "jumps"that extend over all of space when an increment in knowledge occurs. Thus the theory is not about matter, as matter was conceived of in, say, the classical physical theory stemming from the works of Newton and Maxwell.

§


The fundamental logical structure of
quantum theory expresses the information that describes the actual state of any system S large or small, in terms of the plus one or zero eigenvalues of a set of projection operators in the Hilbert Space associated with that system S! These projection operators must be orthogonal (Pi P j = Pi   δij ) and sum to unity: (i.e., be a decomposition of unity.) For any such set there is basis in the Hilbert space such that each of the projectors in this set is represented by a matrix that is filled with zeros and ones, with all the ones located only on the diagonal. Different choices of basis allow different “mutually consistent” sets of P’s to be defined. But the key point is that only sets of P’s that are all mutually orthogonal give allowed “mutually consistent” sets of Yes/No bits of information. Two projection operators that cannot be expressed in terms of a set of ones lying on the diagonal in some one common basis are not mutually compatible: no actual state can be specified by giving, simultaneously, their eigenvalues.

Stapp   


projection

Projection

Gerard 't Hooft

Gerard 't Hooft

A theory that yields "maybe" as an answer should be recognized as an inaccurate theory.

§

A field is simply a quantity defined at every point throughout some region of space and time.

't Hooft

Turing
Alan Turing


I propose to consider the question, 'Can machines think?' This should begin with definitions of the meaning of the terms 'machine 'and 'think'.

Turing machine

I do not wish to give the impression that I think there is no mystery about consciousness. There is, for instance, something of a paradox connected with any attempt to localise it.

Turing          

John von Neumann
John von Neumann


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First, it is inherently entirely correct that the measurement  or the related process of the subjective perception is a new entity relative to the physical environment and is not reducible to the latter. Indeed, subjective perception leads us into the intellectual inner life of the individual, which is extra-observational by its very nature (since it is taken for granted by any conceivable observation or experiment). Nevertheless, it is a fundamental requirement of the scientific viewpointthe so-called principle of the psycho-physical parallelismthat it must be possible to describe the extra-physical process of the subjective perception as if it were in reality in the physical worldi.e., to assign to the parts equivalent physical processes in the objective environment, in ordinary space.


A new, essentially logical, theory is called for in order to understand high-complication automata and, in particular, the central nervous system. It may be, however, that in this process logic will have to undergo a pseudomorphosis to neurology to a much greater extent than the reverse.
Von Neumann        

Herman Weyl
Hermann Weyl


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.

RGB


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 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.

wave superposition


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.

.
color superposition



Whitehead
Alfred North Whitehead



The sense-object is the simplest permanence which we trace as self-identical in external events. It is some definite sense-datum, such as the color red of a definite shade. We see redness here and the same redness there, redness then and the same redness now. In other words, we perceive redness in the same relation to various definite events, and it is the same redness which we perceive. Tastes, colors, sounds, and every variety of sensation are objects of this sort.


Whitehead

Thus "this is red," "this is earlier than that," are atomic propositions.

Russell & Whitehead

redness of redredness of redredness of red

A = B & B = C 
> A = C
  

Wigner
Eugene Wigner


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Since matter clearly influences the content of our consciousness, it is natural to assume that the opposite influence also exists, thus demanding the modification of the presently accepted laws of nature which disregard this influence.
Wigner    
Hidden symmetry

Ludwig Wittgenstein
Ludwig Wittgenstein

The feeling of an unbridgeable gulf between consciousness and brain-process: how does it come about that this does not come into the considerations of our ordinary life? This idea of a difference in kind is accompanied by slight giddiness,—which occurs when we are performing a piece of logical sleight-of-hand. (The same giddiness attacks us when we think of certain theorems in set theory.) When does this feeling occur in the present case? It is when I, for example, turn my attention in a particular way on to my own consciousness, and, astonished, say to myself: THIS is supposed to be produced by a process in the brain!

§

The aspects of things that are most important for us are hidden because of their simplicity and familiarity.

§

Is there such a thing as a 'natural history of colors' and to what extent is it analogous to a natural history of plants? Isn't the latter temporal, the former non-temporal?

§

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.

§

When we're asked "What do 'red', 'blue', 'black', 'white' mean?" we can, of course, immediately point to things which have these colors,—but that's all we can do: our ability to explain their meaning goes no further.

Yau
Shing Tung Yau




Thomas Young
Thomas Young

Polymaths have always posed a problem in academia. How do they relate to specialization and interdisciplinarity, genius and dilettantism, inspiration and perspiration? Robert Hooke, Benjamin Franklin and Alexander von Humboldt were among those who were too academically wide-ranging for posterity to cope with, and their scientific reputations suffered as a consequence. Individual curiosity is the driving force of science, but when insatiable, can it hamper the intellectual? The life and work of the polymath Thomas Young (1773-1829) illuminates the issue perhaps more acutely than that of any other scientist. Today, views of Young span the spectrum from near-universal genius to dabbling dilettante.

ScienceWeek  


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RGB superposition

Color and Stereoscopic Vision

Color vision is based on the ability to discriminate between the various wavelengths that constitute the spectrum. The Young-Helmholtz theory, developed in 1802 by Thomas Young and H. L. F. Helmholtz, is based on the assumption that there are three fundamental color sensations—red, green, and blue—and that there are three different groups of cones in the retina, each group particularly sensitive to one of these three colors. Light from a red object, for example, stimulates the cones that are more sensitive to red than the other cones. Other colors (besides red, green, and blue) are seen when the cone cells are stimulated in different combinations. Only in recent years has conclusive evidence shown that the Young-Helmholtz theory is, indeed, accurate. The sensation of white is produced by the combination of the three primary colors, and black results from the absence of stimulation.


RGB sensitivities