Re: sources  
Yakir Aharonov
Search me!

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

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.

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
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. Nonlocality 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 2matrix or reduced density matrix. In addition to manifesting itself through the well known Bell's inequalities nonlocality also appears as a macroscopic coherence length in condensed and coherent systems. By examining the structure of the 2matrix, a connection between these two forms of nonlocality is made. It is suggested that subtle enfolded orders and nonlocal forms may have a wider implication and be relevant for a variety of living systems. It is shown that the matrix models which give nonperturbative definitions of string and Mtheory may be interpreted as nonlocal hidden variables theories in which the quantum observables are the eigenvalues of the matrices while their entries are the nonlocal hidden variables. 

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. 
The
mathematical machinery of quantum mechanics [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 socalled stationary quantum states and that, in particular, the spectra were emitted by a steplike 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. GellMann 

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 
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 highroad 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


Eugenio Calabi 

Elie Cartan 
Elie
Cartan is one of the most influential of 20thcentury 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


RIEMANNIAN GEOMETRY > CARTAN GEOMETRY
This Week's
Finds, by John
Baez


David Chalmers 
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 10bit 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. 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 threedimensional 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 Smith Churchland A complex of coefficients of this type is comparable with a matrix such as occurs in linear algebra. (Heisenberg) 
A trainedup 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 handset 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 weightspace 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. 

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) 
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, hereandnow 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 informationprocessing 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 hardearned 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 ... 

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 V = 0 for a scalar V becomes, in covariant form Dirac The D'Alembertian is the equivalent of the Laplacian in Minkowskian geometry. It is given by:  
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  
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 _{}wave
with its statistical character is a "complete" description of
reality—in 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 time—or if, on the
contrary, this interpretation is incomplete and hides behind itself, as
the older statistical theories of classical physics do—a 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 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 BroglieBohm theory, the pilotwave 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.  
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.
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


Paul Dirac 
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


Dirichlet 
As
we saw earlier, the KaluzaKlein states of the compactified
11dimensional Mtheory are equivalent to the nonperturbative
10dimensional Dirichlet 0brane 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 nonabsolutely 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. 

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


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


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 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:
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


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) 
I can here only briefly indicate the lines along which I think the 'world knot'to use Schopenhauer's striking designation for the mindbody 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 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 (17461759) 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.


Richard Feynman Nature has a great simplicity and therefore a great beauty. 
If you take a physical state and do something to it—like rotating it, or like waiting for some time t—
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: _{}> = A_{}>. 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 _{}>, it produces some other state _{}>. 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 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) 
The mathematician Fourier proved that any continuous function could be produced as an infinite sum of sine and cosine waves. His result has farreaching 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 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


Evariste Galois 
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 nonabelian simple group of smallest order has order 60. 

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.


Kurt Gödel Either mathematics is too big for the human mind or the human mind is more than a machine. 
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") metamathematical 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 selfconsciousness. 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. Newman
& Nagel


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 halfsilvered 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


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. (18051865) 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. 
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. 

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. 
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." 

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.
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 A speck in the visual field,
though it need not be red must have some
colour; it is, so to speak, surrounded by colourspace. 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 selfunderstood, 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


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

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


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. 
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.


Carl Jacobi 

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 brainfact; and we must similarly find the minimal brain event which will have a mental counterpart at all. 
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 testtube; 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.
In
May 1926 Schrödinger
published a proof that matrix and wave mechanics gave equivalent
results: mathematically they were the same theory.


Theodor Kaluza 
CalabiYau space Alongside
the metric tensor of the 4dimesnional manifold (interpreted as a
tensor potential for gravity) a general relativistic description of
worldphenomena requires also an electromagnetic fourpotential A_{m}.
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 g_{mn} a kind of fundamental metric vector which he then interprets as the electromagnetic potential A_{m}. 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 A speck in the visual field, though it need not be red must have some color; it is, so to speak, surrounded by colorspace. 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 
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 nonEuclidean 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


Oscar Klein 
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 x^{0}.
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 KaluzaKlein 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
nonAbelian gauge theories that are the foundation of our present theory of particle physics.
David J Gross


JosephLouis 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. 
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.)


PierreSimon Laplace Such is the advantage of a well constructed language that its simplified notation often becomes the source of profound theories. 
The Laplacian is extremely important in mechanics, electromagnetics, wave theory, and quantum mechanics, and appears in Laplace's equation
that describes propagation of waves with speed . The form above gives the wave equation in threedimensional 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. 

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. 
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 
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 
Take some range of phenomenal qualities. Assume that these qualities can be arranged according to some abstract ndimensional 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 threedimensional 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 ndimensional 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 

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 
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. 

Warren McCulloch 


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


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


Wolfgang Pauli 
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 

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
Does fractal neural form follow quantum function? 

Roger Penrose 
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 presentday
quantum mechanics is fundamentally incomplete.
Penrose 

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 
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.


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 [...]


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. 
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 ndimensional 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 threedimensional 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 ndimensional 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


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

Erwin Schrödinger 
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.


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 
All of these materialistictype theories are now known to be false: they provide no adequate 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 (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. Projection 

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  
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'. 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 
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 extraobservational 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 viewpoint—the socalled principle of the psychophysical parallelism—that
it must be possible to describe the extraphysical process of the
subjective perception as if it were in reality in the physical world—i.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 highcomplication 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


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. 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 ndimensional 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
"coordinates," 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.
. 

Alfred North Whitehead  The senseobject is the simplest permanence which we trace as selfidentical in external events. It is some definite sensedatum, 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
 
Eugene Wigner 
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


Ludwig Wittgenstein 
The feeling of an unbridgeable gulf between consciousness and brainprocess: 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 sleightofhand. (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 nontemporal? §
A speck in the visual field, though it need not be red must have some color; it is, so to speak, surrounded by colorspace. 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. 

Shing Tung Yau 

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 wideranging 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 (17731829) illuminates the issue perhaps more acutely than that of any other scientist. Today, views of Young span the spectrum from nearuniversal genius to dabbling dilettante. 

