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 Schrödinger equation, analogous to that of the index of refraction in optics. 

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

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

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

 

 

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" 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 (see "week213"), 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    
  

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        

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.

 

 

 

 

§

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

Algebra is generous: she often gives more than is asked for.

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.

Leonardo da Vinci, Codex Atlanticus, folio 372v.

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.

The long chains of simple and easy reasonings by means of which geometers are accustomed to reach the conclusions of their most difficult demonstrations, 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          

 

 

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.

§

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    

 

Laplace's equation in two dimensions is given by

 

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    

 

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.

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:

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.

Chalmers

 

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.

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.

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 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.

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.

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 [...]

 

 

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.

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

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 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."                    

 

 

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

 

How far does the laser move in color space?

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

 

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 (P_i P _j = P_i   δ_ij ) 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   

 

 

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      

 

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    

E = hv 

Doppler changes in frequency 

—> changes in energy 

—> rotations of color vector

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.

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.

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.

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

 

 

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

Green

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)

 

MathPages

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

 

The wave equation is the important partial differential equation

 

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

An even more compact form is given by

where  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  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

 

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 [...]

 

 

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 color, 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 disturb them.

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    

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  

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

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.

 

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.

 

 

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

§

[The] science of colors 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.

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  

     

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 finalobjective 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

 

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.

Neural form follows quantum function?.

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 [...]

 

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.

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

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

 

    

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        

 

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

Salam    
    

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.

 

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        

All of these materialistic-type theories are now known to be false: they provide no adequatebasis 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.

Stapp  

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

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  

        

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        

 

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.

 

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.

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.

.

 

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

 

  

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    

 

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 colours' 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 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.

§

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.

 

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.