Hans Reichenbach Quotes

If we wish to express our ideas in terms of the concepts synthetic and analytic, we would have to point out that these concepts are applicable only to sentences that can be either true of false, and not to definitions. The mathematical axioms are therefore neither synthetic nor analytic, but definitions. ...Hence the question of whether axioms are a priori becomes pointless since they are arbitrary.

If heat were the affecting force, direct indications of its presence could be found which would not make use of geometry as an indirect method. ...direct evidence for the presence of heat is based on the fact that it affects different materials in different ways. ...The forces... which we have introduced... have two properties: (a) They affect all materials in the same way. (b) There are no insulating [or isolating] walls. ...the definition of the insulating wall may be added here: it is a covering made of any kind of material which does not act upon the enclosed object with forces having property a. Let us call the forces which have the properties a and b universal forces; all other forces are called differential forces. Then it can be said that differential forces, but not universal forces, are directly demonstrable.

There is no pure visualization in the sense of a priori philosophies; every visualization is determined by previous sense perceptions, and any separation into perceptual space and space of visualization is not permissible, since the specifically visual elements of the imagination are derived from perceptual space. What led to the mistaken conception of pure visualization was rather an improper interpretation of the normative function... an essential element of all visual representations. Indeed, all arguments which have been introduced for the distinction of perceptual space and space of visualization are base on this normative component of the imagination.

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.

Carnap calls such concepts as point, straight line, etc., which are given by implicit definitions, improper concepts. Their peculiarity rests on the fact that they do not characterize a thing by its properties, but by its relation to other things. Consider for example the concept of the last car of a train. Whether or not a particular car falls under this description does not depend on its properties but on its position relative to other cars. We could therefore speak of relative concepts, but would have to extend the meaning of this term to apply not only to relations but also to the elements of the relations.

...the differential element of non-Euclidean spaces is Euclidean. This fact, however, is analogous to the relations between a straight line and a curve, and cannot lead to an epistemological priority of Euclidean geometry, in contrast to the views of certain authors.

Common to the two geometries is only the general property of one-to-one correspondence, and the rule that this correspondence determines straight lines as shortest lines as well as their relations of intersection.

Some philosophers have believed that a philosophical clarification of space also provided a solution of the problem of time. Kant presented space and time as analogous forms of visualization and treated them in a common chapter in his major epistemological work. Time therefore seems to be much less problematic since it has none of the difficulties resulting from multidimensionality. Time does not have the problem of mirror-image congruence, i.e., the problem of equal and similarly shaped figures that cannot be superimposed, a problem that has played some role in Kant's philosophy. Furthermore, time has no problem analogous to non-Euclidean geometry. In a one-dimensional schema it is impossible to distinguish between straightness and curvature. ...A line may have external curvature but never an internal one, since this possibility exists only for a two-dimensional or higher continuum. Thus time lacks, because of its one-dimensionality, all those problems which have led to philosophical analysis of the problems of space.

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.

...the principle of the limiting character of the velocity of light. This statement... is not an arbitrary assumption but a physical law based on experience. In making this statement, physics does not commit the fallacy of regarding absence of knowledge as evidence for knowledge to the contrary. It is not absence of knowledge of faster signals, but positive experience which has taught us that the velocity of light cannot be exceeded. For all physical processes the velocity of light has the property of an infinite velocity. In order to accelerate a body to the velocity of light, an infinite amount of energy would be required, and it is therefore physically impossible for any object to obtain this speed. This result was confirmed by measurements performed on electrons. The kinetic energy of a mass point grows more rapidly than the square of its velocity, and would become infinite for the speed of light.

We can... treat only the geometrical aspects of mathematics and shall be satisfied in having shown that there is no problem of the truth of geometrical axioms and that no special geometrical visualization exists in mathematics.

For the Lorentz transformation spatial measurements are also changed, because they are obtained relative to a moving system. In our example only time was transformed, while the distances between points at rest remained the same; the spatial coordinates, therefore, retain their identity.

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. 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 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.

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.