A work based only on a line concept is scarcely more than a illustration; it fails to achieve pictorial structure. Pictorial structure is based on a plane concept. The line originates in the meeting of two planes ... we can lose ourselves in a multitude of lines, if through them we lose our senses for the planes.
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"Terms" p. 71Hans Hofmann
Since one cannot create "real depth" by carving a hole in the picture, and since one should not attempt to create the illusion of depth by tonal gradation, depth as a plastic reality must be two dimensions in a formal sense as well in the sense of color. "Depth" is not created on a flat surface as an illusion, but as a plastic reality. The nature of the picture plane makes it possible to achieve depth without destroying the two-dimensional essence of the picture plane. ... A plane is a fragment in the architecture of space. When a number of planes are opposed one to another, a spatial effect results. A plane functions in the same manner as the walls of a building. ... Planes organized within a picture create the pictorial space of its composition. ... The old masters were plane-consciousness. This makes their pictures restful as well as vital...
Hans Hofmann
A line cannot control pictorial space absolutely. A line may flow freely in and out space, but cannot independently create the phenomenon of push and pull necessary to plastic creation. Push and pull are expanding and contracting forces which are activated by carriers in visual motion. Planes are the most important carriers, lines and points less so ... the picture plane reacts automatically in the opposite direction to the stimulus received; thus action continues as long as it receives stimulus in the creative process. Push answers with pull and pull with push. ... At the end of his life and the height of his capacity Cézanne understood color as a force of push and pull. In his pictures he created an enormous sense of volume, breathing, pulsating, expanding, contracting through his use of colors.
Hans Hofmann
The main objection to the theory of pure visualization is our thesis that the non-Euclidean axioms can be visualized just as rigorously if we adjust the concept of congruence. This thesis is based on the discovery that the normative function of visualization is not of visual but of logical origin and that the intuitive acceptance of certain axioms is based on conditions from which they follow logically, and which have previously been smuggled into the images. The axiom that the straight line is the shortest distance is highly intuitive only because we have adapted the concept of straightness to the system of Eucidean concepts. It is therefore necessary merely to change these conditions to gain a correspondingly intuitive and clear insight into different sets of axioms; this recognition strikes at the root of the intuitive priority of Euclidean geometry. Our solution of the problem is a denial of pure visualization, inasmuch as it denies to visualization a special extralogical compulsion and points out the purely logical and nonintuitive origin of the normative function. Since it asserts, however, the possibility of a visual representation of all geometries, it could be understood as an extension of pure visualization to all geometries. In that case the predicate "pure" is but an empty addition, since it denotes only the difference between experienced and imagined pictures, and we shall therefore discard the term "pure visualization." Instead we shall speak of the normative function of the thinking process, which can guide the pictorial elements of thinking into any logically permissible structure.
Hans Reichenbach
It is remarkable that this generalization of plane geometry to surface geometry is identical with that generalization of geometry which originated from the analysis of the axiom of parallels. ...the construction of non-Euclidean geometries could have been equally well based upon the elimination of other axioms. It was perhaps due to an intuitive feeling for theoretical fruitfulness that the criticism always centered around the axiom of parallels. For in this way the axiomatic basis was created for that extension of geometry in which the metric appears as an independent variable. Once the significance of the metric as the characteristic feature of the plane has been recognized from the viewpoint of Gauss' plane theory, it is easy to point out, conversely, its connection with the axiom of parallels. The property of the straight line as being the shortest connection between two points can be transferred to curved surfaces, and leads to the concept of straightest line; on the surface of the sphere the great circles play the role of the shortest line of connection... analogous to that of the straight line on the plane. Yet while the great circles as "straight lines" share the most important property with those of the plane, they are distinct from the latter with respect to the axiom of the parallels: all great circles of the sphere intersect and therefore there are no parallels among these "straight lines". ...If this idea is carried through, and all axioms are formulated on the understanding that by "straight lines" are meant the great circles of the sphere and by "plane" is meant the surface of the sphere, it turns out that this system of elements satisfies the system of axioms within two dimensions which is nearly identical in all of it statements with the axiomatic system of Euclidean geometry; the only exception is the formulation of the axiom of the parallels. The geometry of the spherical surface can be viewed as the realization of a two-dimensional non-Euclidean geometry: the denial of the axiom of the parallels singles out that generalization of geometry which occurs in the transition from the plane to the curve surface.
Hans Reichenbach
It's all for the planned redistribution of wealth which is also stated in this document, the redistribution of wealth which is based on a new concept called equity. And it says this: we must not lose sight of equity, or fairness based on need. Where have you heard that here, today? From each according to his ability, to each according to his need.
Michele Bachmann
Hofmann, Hans
Hofstadter, Douglas
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