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| Wittgenstein, Minkowski, Riemann, Maxwell, Russell, Whitehead, Kline, Feigl, Brentano, Poincaré, Gödel | ||||||
Wittgenstein Minkowski Riemann Weyl Maxwell Russell Whitehead Kline Brentano Feigl Poincaré
Fourier Feynman Nagel Newman |
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 non-temporal?
§ 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.
Wittgenstein
[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 [...]
Riemann
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.
Maxwell
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 [...]
Weyl
pinhole camera [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.
Kline
"[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
Witten
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
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
conse- quence, 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 |
 |
Riemann calls a system in which one individual can be determined by n measurments an nfold 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. Helmholtz 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. 't Hooft 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. Minkowski 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. Weyl
Weyl
Wittgenstein
Russell
& Whitehead
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.
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. Witten
I can here only briefly indicate the lines along which I think the 'world knot' — to use Schopenhauer's striking designation for the mind-body puzzles may be disentangled. The indispensable step consists in a critical reflection upon the meanings of the terms 'mental' and 'physical', and along with this a thorough clarification of such traditional philosophical terms as 'private' and 'public', 'subjective' and 'objective', 'psychological space(s)' and 'physical space', 'intentionality', 'purposiveness', etc. The solution that appears most plausible to me, and that is consistent with a thoroughgoing naturalism, is an identity theory of the mental and the physical, as follows: Certain neurophysiological terms denote (refer to) the very same events that are also denoted (referred to) by certain phenomenal terms. [...] I take these referents to be the immediately experienced qualities, or their configurations in the various phenomenal fields. Feigl
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 [...] Poincaré
Fortunately the problem is easier for the mathema- tician 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
Fourier's theorem is probably the most far-reaching principle of mathematical physics. Feynman
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manifolds space-time projective
monism harmonic |
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