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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. (with Bohm)

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



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.

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.

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

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.

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

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.

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.

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

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.

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        

Wolfram MathWorld  

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.


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.

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 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 Analysis and Synthesis

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.

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

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.

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.

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.

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

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

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

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)

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.)

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 . 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 [...]

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.

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.

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.

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.

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.

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.

§

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.

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

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.'

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 accomodate 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.

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.