„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.“

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

Última atualização 22 de Maio de 2020. História
Hans Reichenbach photo
Hans Reichenbach
professor académico alemão 1891 - 1953

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„We find in the history of ideas mutations which do not seem to correspond to any obvious need, and at first sight appear as mere playful whimsies — such as Apollonius' work on conic sections, or the non-Euclidean geometries, whose practical value became apparent only later.“

—  Arthur Koestler Hungarian-British author and journalist 1905 - 1983

as quoted by Michael Grossman in the The First Nonlinear System of Differential and Integral Calculus (1979).
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„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."“

—  Hans Reichenbach American philosopher 1891 - 1953

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

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„Development of Western Science is based on two great achievements, the invention of the formal logical system (in Euclidean geometry) by the Greek philosophers, and the discovery of the possibility to find out causal relationships by systematic experiment (Renaissance). In my opinion one has not to be astonished that the Chinese sages have not made these steps. The astonishing thing is that these discoveries were made at all.“

—  Albert Einstein German-born physicist and founder of the theory of relativity 1879 - 1955

Letter to J.S. Switzer (23 April 1953), quoted in The Scientific Revolution: a Hstoriographical Inquiry By H. Floris Cohen (1994), p. 234 http://books.google.com/books?id=wu8b2NAqnb0C&lpg=PP1&pg=PA234#v=onepage&q&f=false, and also partly quoted in The Ultimate Quotable Einstein edited by Alice Calaprice (2010), p. 405 http://books.google.com/books?id=G_iziBAPXtEC&lpg=PP1&pg=PA405#v=onepage&q&f=false
1950s

Hans Reichenbach photo

„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.“

—  Hans Reichenbach American philosopher 1891 - 1953

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)

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