Educators from Germany

510 quotes found

"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."

- Hans Reichenbach

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"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."

- Hans Reichenbach

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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." 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."

- Hans Reichenbach

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"Yes it was 1949. How I came to that. That's like how one gets to know a human being. It so happens that I've always had a preference – as everyone has prejudices and preferences – for the square as a shape in preference to the circle as a shape. And I have known for a long time that a circle always fools me by not telling me whether it's standing still or not. And if a circle circulates you don't see it. The outer curve looks the same whether it moves or does not move. So the square is much more honest and tells me that it is sitting on one line of the four, usually a horizontal one, as a basis. And I have also come to the conclusion that the square is a human invention, which makes it sympathetic to me. Because you don't see it in nature. As we do not see squares in nature, I thought that it is man-made. But I have corrected myself. Because squares exist in salt crystals, our daily salt. We know this because we can see it in the microscope. On the other hand, we believe we see circles in nature. But rarely precise ones. Nature, it seems, is not a mathematician. Probably there are no straight lines either. Particularly not since Einstein says in his theory of relativity that there is no straight line, rod knows whether there are or not, I don't. I still like to believe that the square is a human invention. And that tickles me. So when I have a preference for it then I can only say excuse me."

- Josef Albers

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"Nobody knows the exact figure because records were not kept, but it seems certain that during a three hundred year period between three and five million women were tortured and killed by the “Holy Inquisition,“ an institution founded by the Roman Catholic Church to suppress heresy. This sure ranks together with the Holocaust as one of the darkest chapters in human history. It was enough for a woman to show a love for animals, walk alone in the fields or woods, or gather medicinal plants to be branded a witch, then tortured and burned at the stake. The sacred feminine was declared demonic, and an entire dimension largely disappeared from human experience. Other cultures and religions, such as Judaism, Islam, and even Buddhism, also suppressed the female dimension, although in a less violent way. Women's status was reduced to being child bearers and men's property. Males who denied the feminine even within themselves were now running the world, a world that was totally out of balance. The rest is history or rather a case history of insanity... The female form is less rigidly encapsulated than the male, has greater openness and sensitivity toward other lifeforms, and is more attuned to the natural world... If the balance between male and female energies had not been destroyed on our planet, the ego's growth would have been greatly curtailed. We would not have declared war on nature, and we would not be so completely alienated from our Being."

- Eckhart Tolle

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"The ancient pagans were not so very unlike the Christians; e. g., Istar, like the Virgin Mary, represented at the same time eternal virginity and motherhood, and the name of the temple on the Acropolis might truly be translated “Church of the Holy Virgin,” for Parthenon is derived from παρθένος, “virgin.” In prehistoric times there was more reverence for the female deity than for the male god. So Ares (or Mars) is the god of fight, of combativeness, while Athene is the teacher of the art of warfare, of generalship, of strategy in battle... The character of Aphrodite as a universal principle was never lost sight of. She was and remained the giver of life, joy, love, loveliness, grace, fertility, increase, exuberance, rejuvenescence, springtime, restoration of life, immortality, prosperity and the charm of existence,—and all this she was in one, all as a universal principle and in its cosmic significance.... Eros is said to have existed prior to Aphrodite, for when she rose out of the sea, Eros met her at the shore, while according to another version he was regarded as her son. The notion that Aphrodite is the cosmic principle of love has found expression in poetry and philosophy, but her mythical nature has never been definitely settled. Homer, who calls Aphrodite Cypris (Κύπις) speaks of her in the Iliad (V, 312) as the daughter of Zeus[10] and Dione, the goddess."

- Paul Carus

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"Mathematics because of its nature and structure is peculiarly fitted for high school instruction [Gymnasiallehrfach]. Especially the higher mathematics, even if presented only in its elements, combines within itself all those qualities which are demanded of a secondary subject. It engages, it fructifies, it quickens, compels attention, is as circumspect as inventive, induces courage and self-confidence as well as modesty and submission to truth. It yields the essence and kernel of all things, is brief in form and overflows with its wealth of content. It discloses the depth and breadth of the law and spiritual element behind the surface of phenomena; it impels from point to point and carries within itself the incentive toward progress; it stimulates the artistic perception, good taste in judgment and execution, as well as the scientific comprehension of things. Mathematics, therefore, above all other subjects, makes the student lust after knowledge, fills him, as it were, with a longing to fathom the cause of things and to employ his own powers independently; it collects his mental forces and concentrates them on a single point and thus awakens the spirit of individual inquiry, self-confidence and the joy of doing; it fascinates because of the view-points which it offers and creates certainty and assurance, owing to the universal validity of its methods. Thus, both what he receives and what he himself contributes toward the proper conception and solution of a problem, combine to mature the student and to make him skillful, to lead him away from the surface of things and to exercise him in the perception of their essence. A student thus prepared thirsts after knowledge and is ready for the university and its sciences. Thus it appears, that higher mathematics is the best guide to philosophy and to the philosophic conception of the world (considered as a self-contained whole) and of one’s own being."

- Christian Heinrich von Dillmann

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"Walter Gieseking was a victim -- artistically, at least -- of World War II. When the Germans started the war, Gieseking (1895-1956) was among the greatest pianists alive. When Germany was defeated six years later, Gieseking, though only 50 years old, was a shadow of his former self. Although he was later cleared by an Allied court, Gieseking -- whose world fame would have made him welcome anywhere -- willingly collaborated in the cultural endeavors of the Third Reich.What remained of him pianistically, however, made it seem as if he had been punished by a higher court. Although his reputation as a great pianist remained until his death in 1956, Gieseking's numerous postwar recordings -- many of which continue to be available on the EMI label -- have always called that reputation in doubt. Even though some of those recordings, particularly those of the music of Debussy and Ravel, are distinguished enough, none justifies Gieseking's huge reputation.One is grateful, therefore, that this year's Gieseking centennial has brought forth several of the pianist's prewar recordings, most recently the first two volumes (a third is expected in the next few months) of the pianist's concerto legacy (APR) and another disc that collects four of the Beethoven piano sonatas Gieseking recorded between 1931-39.These performances show us a pianist who was not merely a great virtuoso, but the man who liberated the pedal. Like the two pianists most influenced by his example -- Sviatoslav Richter and Arturo Benedetti Michelangeli -- Gieseking's imaginative use of the pedal, combined with his sophisticated ear, permitted him to cultivate a tonal palette without antecedent in its range and subtlety of color and dynamics. And while Gieseking may not have been a profoundly emotional interpreter, he had a profoundly musical mind that rarely failed to bring music to life."

- Walter Gieseking

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"God was English, though – since God was not always kind – this did not mean that everything was always going well. But ill fortune did not affect the national conviction of the superiority of the English, a visible hallmark of the century. It is found, for instance, in Richard Morison's writings in the 1530s, perhaps the first sign of this kind of thing; it is fully ripe in John Foxe and in similar writers of the Elizabethan era. God has singled out the English for his own, as the true elect nation. Morison, for instance pointed out that the English ate beef while the French lived on broth and vegetables, a plain proof of English superiority. And this was the view of a man who, I ought to emphasise, had lived many years abroad. We are not taking about ignorant men; we are talking about men who, having seen both sides, were (and I do not know that they were necessarily wrong) content to believe that the country they had been born into was especially blessed. That conviction is very marked among the Elizabethans and Jacobeans... The convictions I speak of are found widely diffused in popular consciousness, among the aristocracy, the gentry and the people at large, whether travellers or stay-at-homes. They might dislike one another, trouble one another, and be discontented with one another, but relative to the foreigner, relative to the poor and depressed subjects of supposedly despotic powers, they knew themselves specially favoured... The English thought England was good and elsewhere was inferior."

- Geoffrey Elton

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"In crediting Emmy Noether with her share in this transformation of mathematics, most biographers have followed Hermann Weyl's analysis... noting that it falls in three periods, of which the first, lasting until about 1919, was one of "relative dependence," whereas the other two were characterized by the algebraic work for which she is remembered. ...[D]ifficulties arise in drawing a sharp distinction between... "relatively dependent" and the rest, however. One can find examples of originality in her early work, and many instances of dependence in her later period... the exclusion of "dependent" work from consideration makes it impossible to study any process of conceptual change. ...The work that was most influential was done when she was in her forties; The "Noether school" of those who collaborated with her in attempting to make algebra the tool and foundation of all mathematics consists of individuals who knew her only in the last decades of her life. In short, her historic influence in effecting conceptual change is based on the events in the last decade of her life. Her stature as a creative mathematician is better understood if we examine her mathematical career in its entirety, however. Only then can we appreciate to what extent Emmy Noether's work fits Poincare's famous description of mathematical creativity..."

- Emmy Noether

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"Anarchism had belonged to the most active resistance fighters against , and their numbers had been decimated by the ruthless National Socialist persecution. Between 1919 and 1923 there had been approximately 1500,000 anarchists in . By the end of the Weimar Republic, about 50,000 activists remained. In 1945 their numbers were down to 15,000, and many of those were seriously ill as consequence of torture and persecution. Hence, anarchist groups in the immediate post-war period had no more than about 5,000 members. In its pre-1933 centres such as , , /, Berlin, and groups were formed who tried to rivive German anarchism organisationally and intellectually. In the Soviet zone of occupation, they soon clashed with the Soviet military authorities and the SED. Their leading figures such as Alfred Weiland or Willi Jelinek were kidnapped by the and imprisoned on trumped-up charges. Whether Jelinek's death in Bautzen prison had natural causes or was murder is still unclear today. By 1948-49 there subsequent waves of persecution had uprooted anarchism in the Soviet zone to such an extent that it was organisationally extinct. In the West, the 1950s saw maany attempts to unite the diverse groups into one German federation. The absence of the leading intellects on the movement's ability to regenerate its energies. Many, like , had been murdered by the Nazis. Others, like Rudolf Rocker or , were still in exile. Furthermore, Rocker was arguably more influential in Spain than in Germany, ant the same can be said in relation to Souchy and Latin America. Despite the many organisational and ideological fissures which characterised post-war German anarchism, an attempt to the movement finally succeeded at a conference in Neviges in August 1959. Yet the emerging Association of Free Socialist and Anarchists was not successful in reviving the fortunes of German anarchism. It failed to overcome the strong ideological differences between the diverse anarchist groups leading a rather shadowy existence in subsequent years. By the mid-1960s, anarchism was marginal political phenomenon in the Federal Republic, and the very words 'anarchism' and 'anarchic' had become bywords for disorganisation rather than signifying one of the few genuine alternatives which had existed in the history of the German left to reformist Social Democracy on the one hand and authoritarian Communism on the other."

- Stefan Berger

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"During the winter and spring of 1933, the Nazis made a strenuous effort to present themselves as in harmony with conservative German and Prussian traditions, or even as the natural result and outgrowth of these traditions. The nazis made the conservative Prussian past serviceable to their need for political legitimation to an extent hitherto unprecedented. Long before the Second World War, Prussian values became National Socialist values, judged to epitomize the German character, and help up as models to emulate: austerity, thrift, tenacity in the pursuit of one's goals, a preparedness for personal sacrifice, and a willingness to lay down one's life in the service of a higher cause that would win out in the end, even in the face of overwhelming odds. Above all, there was the concept of duty; it was imperative to "fulfill" one's duty of the ' and the . Among other things, made the Prussian past and the values it imputed to it palpable in the form of Grand historic films that enjoyed mass audiences. It was already during the period of the seizure of power that conservatives lost the Deutungshoheit, that is, the prerogative to interpret the great traditions and historical figures of the past, to the Nazis. From 1933 onwards, the Nazis acted as self-appointed guardians of the national heritage. And they did this with greater aplomb, audacity, and-in many instances-more skill than conservative propagandists during the before them."

- Hermann Beck

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