Hans Reichenbach Quotes

Visual forms are not perceived differently from colors or brightness. They are sense qualities, and the visual character of geometry consists in these sense qualities.

...the stereographic projection of the spherical surface. From the north pole P we draw radial lines to project every point of the surface of the sphere upon the horizontal plane [below, perpendicular to a line joining it to P and the sphere's center]. In general this transformation is unique and continuous , although the metrical relations are distorted; for the point P, however, it shows a singularity. Point P is mapped upon the infinite; i.e., no finitely located point of the plane corresponds to it. It can be shown that every transformation possesses a singularity in at least one point. The surface of the sphere is therefore called topologically different from the plane. Only a "sphere without a north pole" [point] would be topologically equivalent to a plane. ...such a sphere has a point-shaped hole without a boundary and is no longer a closed surface.

Euclidean geometry can be easily visualized; this is the argument adduced for the unique position of Euclidean geometry in mathematics. It has been argued that mathematics is not only a science of implications but that it has to establish preference for one particular axiomatic system. Whereas physics bases this choice on observation and experimentation, i.e., on applicability to reality, mathematics bases it on visualization, the analogue to perception in a theoretical science. Accordingly, mathematicians may work with the non-Euclidean geometries, but in contrast to Euclidean geometry, which is said to be "intuitively understood," these systems consist of nothing but "logical relations" or "artificial manifolds". They belong to the field of analytic geometry, the study of manifolds and equations between variables, but not to geometry in the real sense which has a visual significance.

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.

Clocks are inherently four-dimensional instruments, since the endpoints of their unit distances are events. Measuring rods, on the other hand, are three-dimensional measuring instruments; their end points are space points and they can be changed into four-dimensional measuring instruments only if events are produced at their end points according to a special rule.

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.

...the order of betweenness does not depend on mutual distances... betweenness is purely a relational order.

Although it is admitted that certain differences cannot be verified by experiment, we should not infer from this fact that they do not exist. ...we are accused of having confused subjective inability with objective indeterminacy.

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.

Light signals alone provide the metrical structure of the four-dimensional space-time continuum. The construction may be called light axioms.

...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.

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...

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.

Perceptual space is not a special space in addition to physical space, but physical space which we endow with a special subjective metric. ...apart from the definition of congruence in physics and that based on perception, there is no third one derived from pure visualization. Any such third definition is nothing but the definition of physical congruence to which our normative function has adjusted the subjective experience of congruence.

...the relation of betweenness on the torus is undetermined for curves that cannot be contracted to a point [e.g., circles around a doughnut hole], i.e., for three of such curves it is not uniquely determined which of them lies between the other two. ..This indeterminateness... has the consequence that such a curve [alone] does not divide the surface of the torus into two separate domains; between points to the "right" and to the "left" of the line.