Dendrite to Eye
In dualism, ‘mind’ is contrasted with ‘body,’ but at different times, different aspects of the mind have been the centre of attention. In the classical and mediaeval periods, it was the intellect that was thought to be most obviously resistant to a materialistic account: from Descartes on, the main stumbling block to materialist monism was supposed to be ‘consciousness,’ of which phenomenal consciousness or sensation came to be considered as the paradigm instance.
The principle of duality in projective geometry states that we can interchange point and line in a theorem about figures lying in one plane and obtain a meaningful statement. Moreover, the new or dual statement will itself be a theorem — that is, it can be proven. On the basis of what has been presented here we cannot see why this must always be the case for the dual statement. However, it is possible to show by one proof that every rephrasing of a theorem of projective geometry in accordance with the principle of duality must be a theorem. This principle is a remarkable characteristic of projective geometry. It reveals the symmetry in the roles that point and line play in the structure of that geometry.
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 two main approaches for the description of sound fields are methods derived from solutions of the wave equation and geometric methods based on analogies to ray optics. Their mathematical representations are reviewed and it is shown that representations by projective geometry and descriptions by Fourier acoustics lead to similar parametric representations of sound fields.
Annibale et al.
Very recently, in the last two years,Witten and his collaborators (mainly Kapustin and Gukov, two young Russians) have managed to deduce what is required for the geometric Langlands program from non-abelian dualities in physics. The kind of dualities they use are close to the dualities in Donaldson’s theory and to the dualities in the mirror symmetry, and are based (at least) on the electric-magnetic duality. The original dualities in physics are those between electricity and magnetism.Atiyah
was based upon three fundamental hypotheses:
A. Stationary states are determined by complex valued wave functions
(q), which remain finite everywhere in q-space.
B. The functions satisfy a differential equation
in which the operator H is obtained from the classical Hamiltonian H(p; q)
by replacing every momentum p by
C. The eigenvalues E are the energy values. To these three hypotheses,
Schrödinger added Bohr’s postulate:
This theory was presented in Schrödinger’s first and second communications
on “Quantisierung als Eigenwertproblem” in Annalen der Physik 79. The first
communication was received on 27 January, and the second on 23 February
On the other hand, matrix mechanics was invented by Heisenberg in June
1925, and presented in a fully developed form in Dirac’s first paper on
quantum mechanics(received 7 November 1925) and also in the famous
“three-men’s paper” of Born, Heisenberg and Jordan (received 16 November
1925). This theory was based upon four mechanical hypotheses and two
radiation hypotheses. The mechanical hypotheses are:
1. The behaviour of a mechanical system is determinedby the matrices
p and q (one matrix q forevery coordinate q, and one p for every
if p belongs to the same coordinate q, otherwise equal to 0.
3. H(p, q) = W = diagonal matrix, having diagonal elements En,
the energy values.
4. Equations of motion:
~BL van der Waerden [PDF]
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 1 is transferred unchanged to the other 2. 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...
All chemical binding is electromagnetic in origin, and so are all phenomena of nerve impulses.
Fundamental electromagnetic interactions occur between any two particles that have electric charge. These interactions involve the exchange or production of photons. Thus, photons are the carrier particles of electromagnetic interactions.
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.
It followed from the special theory of relativity that mass and energy are both but different manifestations of the same thing - a somewhat unfamiliar conception for the average mind. Furthermore, the equation E = mc², in which energy is put equal to mass, multiplied by the square of the velocity of light, showed that very small amounts of mass may be converted into a very large amount of energy and vice versa.
theory that yields "maybe" as an answer should be recognized as an
Can it really be true that Einstein, in any significant sense, was a profoundly "wrong" as the followers of Bohr maintain? I do not believe so. I would, myself, side strongly with Einstein in his belief in a submiscroscopic reality, and with his conviction that present-day quantum mechanics is fundamentally incomplete.
In attempting to judge the success of a physical theory, we may ask ourselves two questions: (1) “Is the theory correct?” and (2) “Is the description given by the theory complete?” It is only in the case in which positive answers may be given to both of these questions, that the concepts of the theory may be said to be satisfactory. The correctness of the theory is judged by the degree of agreement between the conclusions of the theory and human experience...Whatever the meaning assigned to the term complete, the following requirement for a complete theory seems to be a necessary one: every element of the physical reality must have a counterpart in the physical theory.
Thus "this is red," "this is earlier than that," are atomic propositions.
Russell & Whitehead
The world as described by natural science has no obvious place for colors, tastes, or smells.
Until recently my view about these things was pretty similar to Feynman’s with maybe a bit of Dirac’s; namely: Quantum mechanics is so confusing that I can’t even tell if there is a problem, but maybe it’s all ok because it works. There is probably not much profit in thinking about “interpretations” and even less in arguing about them.
But over the last two years I’ve come to see it differently. Now I feel that our current views of quantum mechanics are provisional; it’s the best we can do without a much deeper understanding of its connection with gravity, but it’s not final. The reason involves a very particular development, the so called ER = EPR principle. ER = EPR tells us that the immensely complicated network of entangled subsystems that comprises the universe is also an immensely complicated (and technically complex) network of Einstein-Rosen bridges. To me it seems obvious that if ER = EPR is true it is a very big deal, and it must affect the foundations and interpretation of quantum mechanics.
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.