The reciprocal relationship of epistemology and
A = A
The first full-fledged exposition of a mathematical theory of vision is found in the Optica of Euclid (fl. 300 B.C.). Indeed, Euclid's approach to vision was so strictly mathematical as to exclude all but the most incidental references to those aspects of the visual process not reducible to geometry — the ontology of visual radiation and the physiology and psychology of vision. Lejeune comments that Euclid's Optica systematically ignores every physical and psychological aspect of the problem of vision. It restricts itself to that which can be expressed geometrically. [...]
Its model is the treatise on pure geometry, and its method that of the Elements: a few postulates all fully necessary, from which follow deductively and with full mathematical rigor a series of theorems of a traditional form.
Thus "this is red," "this is earlier than that," are atomic propositions.
Russell & Whitehead
Among expressions (i.e., finite cocatenations of symbols) we distinguish terms and formulas. The simplest, so-called atomic terms are the variables and the individual constants; a compound term is obtained by combining n simpler terms by means of an operation symbol of rank n. Similarly, an atomic formula is obtained by combining n arbitrary terms by means of a predicate of rank n; compound formulas are built from simpler ones be means of sentential connectives and quantifier expressions.
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.
...every element of the physical reality must have a counterpart in the physical theory.
A color is a physical object a soon as we consider its dependence, for instance, upon its luminous source, upon temperatures, and so forth.
The ultimate of ultimate problems, of course, in the study of the relations of thought and brain, is to understand why and how such disparate things are connected at all … We must find the minimal mental fact whose being reposes directly on a brain-fact; and we must similarly find the minimal brain event which will have a mental counterpart at all.
The sense-object is the simplest permanence which we trace as self- identical in external events. It is some definite sense-datum, such as the color red of a definite shade. We see redness here and the same redness there, redness then and the same redness now. In other words, we perceive redness in the same relation to various definite events, and it is the same redness which we perceive. Tastes, colors, sounds, and every variety of sensation are objects of this sort.
The characteristic of an n-dimensional manifold is that each of the elements composing it (in our examples, single points, conditions of a gas, colors, tones) may be specified by the giving of n quantities, the "co- ordinates," which are continuous functions within the manifold.
This is the characteristic mathematical property of a classical field: it is an undefined something which exists throughout a volume of space and which is described by sets of numbers, each set denoting the field strength and direction at a single point in the space.
There is nothing else except these fields: the whole of the material universe is built of them.
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.
[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.
Increasingly, many of us have come to think that the missing element that has to be added to quantum mechanics is a principle, or several principles, of symmetry. A symmetry is a statement that there are various ways that you can change the way you look at nature, which actually change the direction the state vector is pointing, but which do not change the rules that govern how the state vector rotates with time. The set of all these changes in point of view is called the symmetry group of nature. It is increasingly clear that the symmetry group of nature is the deepest thing that we understand about nature today.
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.
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 — is transferred unchanged to the other. 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 [...]
The science of colors becomes a speculation as truly
mathematical as any other part of physics. ~Newton
There is in the absolute differential calculus a kind of law of reciprocity or duality in accordance with which we can deduce from every theorem or formula a reciprocal theorem or formula by interchanging the words covariant and contravariant, and lowering or raising the indices.
From the above brief review, we find there are three versions of geometrization of non- gravitational gauge interactions:
1. Fibre-bundle version, in which the gauge interactions are correlated with the geometrical structures of internal space. [...] the essence of the internal space is still a vexing problem: Is it a physical reality as real as space-time, or just a mathematical structure?
2. Kaluza-Klein version, in which extra space dimensions which compactify in low-energy experiments are introduced and the gauge symmetries by which the forms of gauge interactions are fixed are just the manifestation of the geometrical symmetries of the compactified space. [...] The assumption of the reality of the compactified space is substantial and is in principle testable [...]
3. Superstring version, in which the introduction of extra compactified space dimensions is due to different considerations from just reproducing the gauge symmetry.
While a proper understanding of M-theory still eludes us, much is now known about it. In particular the various geometric results that have emerged from string theory become related in interesting but mysterious ‘dualities’ whose real meaning has yet to be discovered.
The reciprocal relationship of epistemology and science is of noteworthy kind. They are dependent upon each other. Epistemology without contact with science becomes an empty scheme. Science without epistemology is – insofar as it is thinkable at all – primitive and muddled. However, no sooner has the epistemologist, who is seeking a clear system, fought his way through to such a system, than he is inclined to interpret the thought-content of science in the sense of his system and to reject whatever does not fit into his system. The scientist, however, cannot afford to carry his striving for epistemological systematic that far. He accepts gratefully the epistemological conceptual analysis; but the external conditions, which are set for him by the facts of experience, do not permit him to let himself be too much restricted in the construction of his conceptual world by the adherence to an epistemological system. He therefore must appear to the systematic epistemologist as a type of unscrupulous opportunist: he appears as realist insofar as he seeks to describe a world independent of the acts of perception; as idealist insofar as he looks upon the concepts and theories as the free inventions of the human spirit (not logically derivable from what is empirically given); as positivist insofar as he considers his concepts and theories justified only to the extent to which they furnish a logical representation of relations among sensory experiences. He may even appear as Platonist or Pythagorean insofar as he considers the viewpoint of logical simplicity as an indispensable and effective tool of his research.
Like the efforts of an Anton Chekhov character, Lobachevsky's effort to disseminate his non-Euclidean geometry was utterly futile. He knew he had invented something profoundly important, and he worked for years to perfect and demonstrate its mathematical significance. The tragedy of his life is that nobody listened.
Some call it the greatest revolution in mathematics since the time of the Greeks — the discovery that changed the definition of a straight line. In ordinary, flat, Euclidean geometry, a straight line is exactly that — its "straightness" defined by the singular unbending direction that it follows for its length. In non-Euclidean geometry, a straight line is defined simply by the fact that it joins two points within a given space. Straight lines may actually be curved. Janos Bolyai, who called himself Euclid's phoenix, defined absolute geometry as that form of geometry in which the theorems were true regardless of whether they were Euclidean or non-Euclidean.
Klein fully developed the idea of a general geometry as being about the invariant properties of a group of defined transformations. His work fundamentally advanced non-Euclidean geometry from a rigorous, if fanciful, subject into something that was on equal footing with Euclidean geometry. Klein also gave Gauss so much credit for non- Euclidean geometry that it would become his standard due.
Riemann gave a lecture to his fellow faculty members at Gottingen on June 10, 1854, shortly before Gauss died. Riemann's thesis was titled "On the Hypotheses Which Lie at the Basis of Geometry." Gauss attended the lecture, which set in motion a change in mathematics that continues to unfold today.
The most fundamental conceptual breakthrough Riemann made was to determine that bodies in a physical space are not simply occupants of this space but actors that bend and shape the space itself by their very presence. In doing so he anticipated the central concept and laid out the mathematical foundation of Einstein's general relativity theory by more than sixty years and even suggested that space could be measured by its physical masses. This insight was astounding considering that Riemann was a classical physicist working more than fifty years before the advent of relativity.
I start, further back, by asking some philosophical or metaphysical questions. If we end up with a coherent and consistent unified theory of the universe, involving extremely complicated mathematics, do we believe that this represents “reality’’? Do we believe that the laws of nature are laid down using the elaborate algebraic machinery that is now emerging in string theory? Or is it possible that nature’s laws are much deeper, simple yet subtle, and that the mathematical description we use is simply the best we can do with the tools we have? In other words, perhaps we have not yet found the right language or framework to see the ultimate simplicity of nature.
To get a better idea of what I am trying to say, let us consider GR as a description of gravity. To a mathematician this theory is beautifully simple but yet subtle. Moreover, it is highly nonlinear so that it is extremely complicated in its detailed implications. This is no doubt why it appeals to both Einstein and Penrose as a model theory. Is it not possible that something having the same inherent simplicity (and nonlinearity) can explain all of nature?
While everyone might agree that this would be an ideal philosophical ambition, there appears to be the insuperable obstacle presented by QM. To get round this will require some conceptual leap, and such leaps have in the past only come when one is prepared to sacrifice some accepted dogma, such as Einstein did with the separation of space and time.
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.
A = B
It seems useful to me to develop a little more precisely the "geometry" valid in the two-dimensional manifold of perceived colors. For one can do mathematics also in the domain of these colors. The fundamental operation which can be performed upon them is mixing: one lets colored lights combine with one another in space...
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.
This leaf (given to me in the present act of
perception) has this definite green color (given
to me in this very perception). ~Weyl
Our focus in the present paper is on the geometric Langlands program for complex Riemann surfaces. We aim to show how this program can be understood as a chapter in quantum field theory. No prior familiarity with the Langlands program is assumed; instead, we assume a familiarity with subjects such as supersymmetric gauge theories, electric-magnetic duality, sigma- models, mirror symmetry, branes, and topological field theory. The theme of the paper is to show that when these familiar physical ingredients are applied to just the right problem, the geometric Langlands program arises naturally. Seemingly esoteric notions such as Hecke eigensheaves, D-modules, and so on, appear spontaneously in the physics, with new insights about their properties.
A speck in the visual field, though it need not be red must have some color; it is, so to speak, surrounded by color-space. Notes must have some pitch, objects of the sense of touch some degree of hardness, and so on.
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 [...]
projective plane RP1 =
If we take an "upper hemisphere," this meets each line in a unique point which we may take as the representative of that point of the projective plane— except for opposite points on the equator which have to be "identified."
If antipodal points have 180 degrees of phase difference, adding them gives 'darkness' or 'no light' or 'zero.' We then have a natural inverse operation—all we need to give colors a group structure, the other requirements being obviously met.
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. ~Weyl
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.
A projection matrix P is an n x n square matrix that gives a vector space projection from Rn to a subspace W. The columns of P are the projections of the standard basis vectors, and W is the image of P. A square matrix is a projection matrix iff P2 = P.
Our basic ontology is that all systems, macroscopic
structures included, are quantum fields. ~Saunders
A field is simply a quantity defined
at every point throughout some region
of space and time. ~'t Hooft
The second principle of color mixing of lights is this: any color at all can be made from three different colors, in our case, red, green, and blue lights. By suitably mixing the three together we can make anything at all, as we demonstrated [...]
Further, these laws are very interesting mathematically. For those who are interested in the mathematics of the thing, it turns out as follows. Suppose that we take our three colors, which were red, green, and blue, but label them A, B, and C, and call them our primary colors. Then any color could be made by certain amounts of these three: say an amount a of color A, an amount b of color B, and an amount c of color C makes X:
X = aA + bB + cC.
Now suppose another color Y is made from the same three colors:
Y = a'A + b'B + c'C.
Then it turns out that the mixture of the two lights (it is one of the consequences of the laws that we have already mentioned) is obtained by taking the sum of the components of X and Y:
Z = X + Y = (a + a')A + (b + b')B + (c + c')C.
is just like the mathematics of the addition of vectors, where (a, b, c
) are the components of one vector, and (a', b', c' ) are those of
another vector, and the new light Z is then the "sum" of the vectors.
This subject has always appealed to physicists and mathematicians. In
fact, Schrödinger wrote a wonderful paper on color vision in which he
developed this theory of vector analysis as applied to the mixing of
What we learn from our whole discussion and what indeed has become a guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure endowed entity Σ, try to determine its group of automorphisms, the group of those element-wise transformations which leave all structural relations undisturbed.
You can expect to gain a deep insight into the constitution of Σ in this way. After that you may start to investigate symmetric configurations of elements, i.e., configurations which are invariant under a certain subgroup of the group of all automorphisms [...]
is a most beautiful and awe-inspiring fact that all the fundamental
laws of classical physics can be understood in terms of one
mathematical construct called the action.
It yields the classical
equations of motion, and analysis of its invariances leads to
quantities conserved in the course of the classical motion. In
addition, as Dirac and Feynman have shown, the action acquires its full
importance in quantum physics.
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 [...]
Pythagoras could be called the first known string theorist. Pythagoras, an excellent lyre player, figured out the first known string physics — the harmonic relationship. Pythagoras realized that vibrating Lyre strings of equal tensions but different lengths would produce harmonious notes (i.e. middle C and high C) if the ratio of the lengths of the two strings were a whole number.
Fourier's theorem is probably the most far-reaching
principle of mathematical physics. ~Feynman
monochromatic light corresponds in the acoustic domain the simple
tone. Out of different kinds of monochromatic light composite light may
be mixed, just as tones combine to a composite sound. This takes place
by superposing simple oscillations of different frequency with definite
We cannot confuse what to us appears unnatural
with the absolutely impossible. ~ Gauss
Riemann discovered that the physics of music was the key to unlocking the secrets of the primes. He discovered a mysterious harmonic structure that would explain how Gauss's prime number dice actually landed when Nature chose the primes.
What Riemann discovered was that Gauss's graph is like the fundamental note played by an instrument, but that there are special harmonic waves that, when added to this graph, gradually change it into the true graph or "sound" of the primes, just as the harmonics of the clarinet change the sine wave into the square wave.
Harmonic analysis in music is the study of chords, and of how they are used in combination to create musical effects. Harmonic analysis in mathematics takes on a somewhat different meaning. It too has roots in music, or at least in the mathematical analysis of sound. However, the term can also mean a kind of universal duality that runs throughout mathematics, as well as other sciences.
Different musical instruments make different kinds of sound, even when they play the same note. The analysis of this phenomenon can be very complicated, but to a first approximation, it is the shape of the instrument that determines the sound it creates. Shape is an obvious geometric property. Sound is an example of what is often called a spectral phenomenon. The term spectral has a precise meaning in mathematics, but we can also think of it in the everyday sense of the word-ghostlike, hard to pin down-the very opposite of geometric. We thus have an example of something geometric the shape of a musical instrument-which corresponds to something spectral-the sound produced by the instrument. Harmonic analysis, in the broadest sense, refers to a general principle in mathematics that links geometric objects with spectral objects. The two kinds of phenomena are sometimes related by explicit formulae, sometimes by parallel mathematical theories and sometimes by laws of physics. The principle seems to be a profound aspect of the inner structure of mathematics, of which sound is but one special case.
Light is another kind of spectral phenomenon. It comes in different colors, or wave lengths, which we can observe by passing sunlight through a prism. We speak of the spectrum of light, or more generally of the electromagnetic spectrum, to mean the range of possible wave lengths. This is the scientific origin of the term spectral. One can devise physical experiments with radiation in which the physical or geometric properties of the equipment govern the wavelengths of the radiation that is produced.
The most far-reaching examples from physics are perhaps the ones in quantum mechanics. Quantum mechanics tells us that there is a wave-particle duality for the basic constituents of matter.
The suggestion that there may be such "hidden variables" is as old as the probabilistic interpretation of the state vector. It was made by Born [...] a few months after he first proposed that interpretation: "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." But almost as old is the denial that such hidden variables can exist.