People From Hamburg

"If the definition of simultaneity is given from a moving system, the spherical surface will result when Einstein's definition with є = 1/2 is used, since it is this definition which makes the velocity of light equal in all directions."

"Why is Einstein's theory better than Lorentz's theory? It would be a mistake to argue that Einstein's theory gives an explanation of Michelson's experiment, since it does not do so. Michelson's experiment is simply taken over as an axiom."

"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."

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

"Once a definition of congruence is given, the choice of geometry is no longer in our hands; rather, the geometry is now an empirical fact."

"This fact... proves that space measurements are reducible to time measurements. Time is therefore logically prior to space."

"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 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."

"...absolute time would exist in a causal structure for which the concept indeterminate as to time order lends to a unique simultaneity, i.e., for which there is no finite interval of time between the departure and return of a first-signal..."

"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."

"...introduce the auxiliary concept of first-signal...defined as the fastest message carrier between any two points in space. We now send a first-signal from P, calling the event of departure E1... The event of its arrival at P' is called E'. Simultaneously with the arrival of this signal, another first signal is sent from P'. The arrival of this signal at P is the event E2. ...the time interval between E1 and E2 is coordinated to the event E', [E1 is earlier than E' and E2 is later than E'] and every event of this time interval except for the endpoints is inderterminate as to the time order relative to E'."

"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 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."

"Whereas the conception of space and time as a four-dimensional manifold has been very fruitful for mathematical physicists, its effect in the field of epistemology has been only to confuse the issue. Calling time the fourth dimension gives it an air of mystery. One might think that time can now be conceived as a kind of space and try in vain to add visually a fourth dimension to the three dimensions of space. It is essential to guard against such a misunderstanding of mathematical concepts. If we add time to space as a fourth dimension it does not lose any of its peculiar character as time. ...Musical tones can be ordered according to volume and pitch and are thus brought into a two dimensional manifold. Similarly colors can be determined by the three basic colors red, green and blue... Such an ordering does not change either tones or colors; it is merely a mathematical expression of something that we have known and visualized for a long time. Our schematization of time as a fourth dimension therefore does not imply any changes in the conception of time. ...the space of visualization is only one of many possible forms that add content to the conceptual frame. We would therefore not call the representation of the tone manifold by a plane the visual representation of the two dimensional tone manifold."

"...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."

"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."

"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."

"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."

"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."

"If along the path of truth, success (which was often near-failure unnoticed) is subjected to the same scrutiny and desire for improvement as failure, we may find ourselves in closer proximity to trees."

"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."

"...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."

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

"Fortunately analysis is not the only way to resolve inner conflicts. Life itself still remains a very effective therapist... The therapy effected by life itself is not, however, within one's control. Neither hardships nor friendships nor religious experience can be arranged to meet the needs of the particular individual. Life as a therapist is ruthless; circumstances that are helpful to one neurotic may entirely crush another."

"[The neurotic] feels caught in a cellar with many doors, and whichever door he opens leads only into new darkness. And all the time he knows that others are walking outside in sunshine. I do not believe that one can understand any severe neurosis without recognizing the paralyzing hopelessness which it contains. … It may be difficult then to see that behind all the odd vanities, demands, hostilities, there is a human being who suffers, who feels forever excluded from all that makes life desirable, who knows that even if he gets what he wants he cannot enjoy it. When one recognizes the existence of all this hopelessness it should not be difficult to understand what appears to be an excessive aggressiveness or even meanness, unexplainable by the particular situation. A person so shut out from every possibility of happiness would have to be a veritable angel if he did not feel hatred toward a world he cannot belong to."

"Rationalization may be defined as self-deception by reasoning."

"Through the eclipse of large areas of the self," says Karen Horney in Our Inner Conflicts, "by repression and inhibition as well as by idealization and externalization, the individual loses sight of himself; he feels, if he does not actually become, like a shadow without weight and substance."

"It is amazing how obtuse otherwise intelligent patients can become when it is a matter of seeing the inevitability of cause and effect in psychic matters. I am thinking of rather self-evident connections such as these: if we want to achieve something, we must put in work; if we want to become independent, we must strive toward assuming responsibility for ourselves. Or: so long as we are arrogant, we will be vulnerable. Or: so long as we do not love ourselves, we cannot possibly believe that others love us, and must by necessity be suspicious toward any assertion of love. Patients presented with such sequences of cause and effect may start to argue, to become befogged or evasive."

"Taking again as an example the need to appear perfect, I would be interested primarily in understanding what this trend accomplishes for the individual (eliminating conflicts with others and making him feel superior to others), and also what consequences the trend has on his character and his life. The latter investigation would make it possible to understand, for example, how such a person anxiously conforms with expectations and standards to the extent of becoming a mere automaton, and yet subversively defies them; how this double play results in listlessness and inertia; how he is proud of his apparent independence, yet actually is entirely dependent on the expectations and opinions of others; how he is terrified lest anyone should discover the flimsiness of his moral strivings and the duplicity which has pervaded his life; how this in turn has made him seclusive and hypersensitive to criticism."