Frases de Hans Reichenbach (41 citas, página 2)

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Hans Reichenbach: Frases en inglés

“If E1 is the cause of E2, then a small variation (a mark) in E1 is associated with a small variation in E2, whereas small variations in E2 are not associated with variations in E1. If we wish to express even more clearly that this concept does not contain the concept of temporal order, we can express it in the following form, where events that show a slight variation are designated E*: E1E2, E1*E2*, E1E2* and never the combination E1*E2.”

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Hans Reichenbach

The Philosophy of Space and Time (1928, tr. 1957)

“We define: any two events which are indeterminate as to their time order may be called simultaneous…. Simultaneity means the exclusion of causal connection…. Yet we must not commit the mistake of attempting to derive from it the conclusion that this definition coordinates to any given event at a given different place. This would be the case only for a special form of causal structure, a form that does not conform to physical reality.”

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Hans Reichenbach

The Philosophy of Space and Time (1928, tr. 1957)

“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.”

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Hans Reichenbach

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.
The Philosophy of Space and Time (1928, tr. 1957)

“We must… maintain that mathematical geometry is not a science of space insofar as we understand by space a visual structure that can be filled with objects - it is a pure theory of manifolds.”

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Hans Reichenbach

The Philosophy of Space and Time (1928, tr. 1957)

“…the mathematician uses an indirect definition of congruence, making use of the fact that the axiom of parallels together with an additional condition can replace the definition of congruence.”

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Hans Reichenbach

The Philosophy of Space and Time (1928, tr. 1957)

“The concept of congruence in Euclidean geometry is not exactly the same as that in non-Euclidean geometry. …"Congruent" means in Euclidean geometry the same as "determining parallelism," a meaning which it does not have in non-Euclidean geometry.”

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Hans Reichenbach

The Philosophy of Space and Time (1928, tr. 1957)

“Occasionally one speaks… of signals or signal chains. It should be noted that the word signal means the transmission of signs and hence concerns the very principle of causal order…”

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Hans Reichenbach

The Philosophy of Space and Time (1928, tr. 1957)

“…the famous assertion by Einstein that the length of a rod depends on its velocity and on the chosen definition of simultaneity. …is based on the fact that we do not measure the length of the rod, but its projection on a system at rest. How the length of the projection depends on the choice of simultaneity can be illustrated by reference to a photograph taken through a focal-plane shutter. Such a shutter… consists of a wide band with a horizontal slit, which slides down vertically. Different bands are photographed successively on the film. Moving objects are therefore strangely distorted; the wheels of a rapidly moving car for instance, appear to be slanted. The shape of the objects in the picture will evidently depend on the speed of the shutter. Similarly, the length of the moving segment depends on the definition of simultaneity. One definition of simultaneity differs from another because events that are simultaneous for one definition occur successively for another. What may be a simultaneity projection of a moving segment for one definition is a "focal-plane shutter photograph" for another.”

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Hans Reichenbach

The Philosophy of Space and Time (1928, tr. 1957)

“Occasionally one speaks… of signals or signal chains.”

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Hans Reichenbach

It should be noted that the word signal means the transmission of signs and hence concerns the very principle of causal order...
The Philosophy of Space and Time (1928, tr. 1957)

“The surfaces of three-dimensional space are distinguished from each other not only by their curvature but also by certain more general properties. A spherical surface, for instance, differs from a plane not only by its roundness but also by its finiteness. Finiteness is a holistic property.”

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Hans Reichenbach

The sphere as a whole has a character different from that of a plane. A spherical surface made from rubber, such as a balloon, can be twisted so that its geometry changes. ...but it cannot be distorted in such a way as that it will cover a plane. All surfaces obtained by distortion of the rubber sphere possess the same holistic properties; they are closed and finite. The plane as a whole has the property of being open; its straight lines are not closed. This feature is mathematically expressed as follows. Every surface can be mapped upon another one by the coordination of each point of one surface to a point of the other surface, as illustrated by the projection of a shadow picture by light rays. For surfaces with the same holistic properties it is possible to carry through this transformation uniquely and continuously in all points. Uniquely means: one and only one point of one surface corresponds to a given point of the other surface, and vice versa. Continuously means: neighborhood relations in infinitesimal domains are preserved; no tearing of the surface or shifting of relative positions of points occur at any place. For surfaces with different holistic properties, such a transformation can be carried through locally, but there is no single transformation for the whole surface.
The Philosophy of Space and Time (1928, tr. 1957)