|Wittgenstein, Minkowski, Riemann, Maxwell, Russell, Whitehead, Kline, Feigl, Brentano, Poincaré, Gödel|| |
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.
In the language of physics, theories where forces are explained in terms of curvature are called `gauge theories'. Mathematically, the key concept in a gauge theory is that of a `connection' on a `bundle'. The idea here is to start with a manifold M describing spacetime. For each point x of spacetime, a bundle gives a set Ex of allowed internal states for a particle at this point.
So few and far between are the occasions for forming notions whose specializations make up a continuous manifold, that the only simple notions whose specializations form a multiply extended manifold are the positions of perceived objects and colors.
Definite portions of a manifold, distinguished by a mark or a boundary, are called Quanta [...]
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.
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 Σ1 is transferred unchanged to the other Σ2. 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 [...]
[It] became possible to affirm that projective geometry is indeed logically prior to Euclidean geometry and that the latter can be built up as a special case. Both Klein and Arthur Cayley showed that the basic non-Euclidean geometries developed by Lobachevsky and Bolyai and the elliptic non-Euclidean geometry created by Riemann can also be derived as special cases of projective geometry. No wonder that Cayley exclaimed, "Projective geometry is all geometry."
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.
"[The] projective plane and the color continuum are isomorphic with one another." (Weyl)
We begin with a piece of late-19th-century physics. The vacuum Maxwell equations for the electric and magnetic fields E and B,
have a symmetry under
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.
How did Gödel prove his conclusions? Up to a point, the structure of his demonstration is modeled, as he himself noted, on the reasoning involved in one of the logical antinomies known as the "Richard Paradox," first propounded by the French mathematician, Jules Richard, in 1905 [...] The reasoning in the Richard Paradox is evidently fallacious. Its construction nevertheless suggests that it might be possible to "map" (or "mirror") meta-mathematical statements about a sufficiently comprehensive formal system into the system itself. If this were possible, then metamathematical statements about a system would be represented by statements within the system. Thereby one could achieve the desirable end of getting the formal system to speak about itself—a most valuable form of self-consciousness.
The idea of such mapping is a familiar one in mathematics. It is employed in coordinate geometry, which translates geometric statements into algebraic ones, so that geometric relations are mapped onto algebraic ones. The idea is manifestly used in the construction of ordinary maps, since the construction consists in projecting configurations on the surface of a sphere onto a plane [...]
The basic fact which underlies all these mapping procedures is that an abstract structure of relations embodied in one domain of objects is exhibited to hold between "objects" in some other domain. In consequence, deductive relations between statements about the first domain can be established by exploring (often more conveniently and easily) the deductive relations between statements about their counterparts. For example, complicated geometrical relations between surfaces in space are usually more readily studied by way of the algebraic formulas for such surfaces.
Newman & Nagel
A group in mathematics is an object that measures symmetry in the same way that a number measures quantity. As the recent paper explains, groups can be used to cast light on neo-Riemannian theory, which is a topic in music theory that gives some insight into progressions of major and minor triad chords.
Neo-Riemannian theory is not named after the famous 19th- century German mathematician Bernhard Riemann, but rather after the German music theorist Hugo Riemann, who lived somewhat later. The three basic operations in the theory, known as P, R and L, transform triad chords into other triad chords as follows.
The P transformation exchanges a major triad with the parallel minor triad: for example, it exchanges a C major triad (C-E-G) with a C minor triad (C-E flat-G).
The R transformation exchanges a major triad with the relative minor triad: for example, it exchanges a C major triad (C-E-G) with an A minor triad (A-C-E).
The L transformation exchanges a triad for its leading-tone exchange: for example, it exchanges a C major triad (C-E-G) with an E minor triad (E-G-B).
The animation shows the notes of the scale on a torus.
a vector is a magnitude and a particular direction from some point, a
tensor gives a magnitude for every direction from that point. A tensor
field is something that assigns a tensor to every point in space.
Riemann calls a system in which one individual can be determined by n measurments an n fold extended aggregate, or an aggregate of n dimensions. Thus the space in which we live is a threefold, a surface is a twofold, and a line is a simple extended aggregate of points. Time also is an aggregate of one dimension. The system of colors is an aggrgate of three dimensions, inasmuch as each color, according to the investigations of Thomas Young and Clerk Maxwell, may be represented as a mixture of three primary colors in definite quantities.
It is easy to imagine a global color symmetry. The quark colors, like the isotopic-spin states of hadrons, might be indicated by the orientation of an arrow in some imaginary internal space.
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.
A geometrical-physical theory as such is incapable of being directly pictured, being merely a system of concepts. But these concepts serve the purpose of bringing a multiplicity of real or imaginary sensory experiences into connection in the mind. To "visualize" a theory therefore means to bring to mind that abundance of sensible experiences for which the theory supplies the schematic arrangement.
We will try to visualize the state of things by the graphic method. Let x, y, z be rectangular co-ordinates for space, and let t denote time. The objects of our perception invariably include places and times in combination. Nobody has ever noticed a place except at a time, or a time except at a place. [...] The multiplicity of all thinkable x, y, z, t systems of values we will christen the world.
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.
We said at an earlier place, that every difference in experience must be founded on a difference of the objective conditions; we can now add: in such a difference of the objective conditions as is invariant with regard to coordinate transformations, a difference that cannot be made to vanish by a mere change of the coordinate system used.
The feeling of an
unbridgeable gulf between conscious-ness and brain-process: 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 sleight-of-hand. [...] 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!
Thus "this is red," "this
is earlier than that," are
Some of the early work was motivated by the hope that the fifth dimension could provide the hidden variables that would eliminate indeterminacy from quantum mechanics. Despite the many generalizations amd changes in emphasis that have occurred, I will refer generically to theories in which gauge fields are unified with gravitation by means of extra, compact dimensions as Kaluza-Klein theories.
I can here only briefly indicate the lines along which I think the 'world knot' — to use Schopenhauer's striking designation for the mind-body puzzles may be disentangled. The indispensable step consists in a critical reflection upon the meanings of the terms 'mental' and 'physical', and along with this a thorough clarification of such traditional philosophical terms as 'private' and 'public', 'subjective' and 'objective', 'psychological space(s)' and 'physical space', 'intentionality', 'purposiveness', etc. The solution that appears most plausible to me, and that is consistent with a thoroughgoing naturalism, is an identity theory of the mental and the physical, as follows: Certain neuro- physiological 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.
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 [...]
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.
Sir James Jeans
Any physical system is completely described
by a normalized vector (the state vector or
wave function) in Hilbert space. ~Byron & Fuller