Jews From Germany

"...the mathematician uses an indirect definition of congruence, making use of the fact that the axiom of parallels together with an additional condition can replace the definition of congruence."

"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."

"We are frequently faced with the necessity of looking for the picture required for the visualization of an object, not in the perception of this particular object, but in a different perceptual image. ...we can assert the discrepancy between the perceived picture and the objective state. This discrepancy... proves absolutely nothing against the fact that all visualizations are merely sense qualities of the perceptual space. ...If the parallelism is ...to be visualized, we must supplement our assertion by the description of certain qualities with which we are familiar from perceptual space."

"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 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."

"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."

"...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."

"In Memory of Those Who Died Waiting for the Bell"

"Teaching here isn't so bad. Once you accept as one of the ineluctable laws of nature that kids will continue to say "Silas Mariner" and "Ancient Marner" and "between you and I" and "mischievious" and that the administration will continue to use phrases like "egregious conduct" and "ethnic background" you can go on from there."

"We got this jerky sub. she don't know a thing and she's trying to teach it."

"I had used my sense of humor; I had called it proportion, perspective. But perspective is distance."

"My words never reached him; I could almost hear them drop, one by one, like so many pebbles against a closed window."

"One of my students had written wistfully of a dream-school that would have "windows with trees in them.""

"Why did you fail me? I didn't do nothing!" The reply, of course, is: "That's just it."

"Paul asks how I would have handled a love letter from a student. I don't know — by talking, maybe, by listening. I don't know. How sad that we don't hear each other — any of us. Major issues are submerged by minor ones; catastrophes by absurdities."

"Being a female, she spurns him on."

"A teacher is frequently the only adult in the pupil's environment who treats him with respect."

"When I tried to tell McHabe that it would have been more valuable to let Ferone keep his appointment with me than to kick him out, he let me have it: "When you're in the system as long as I," he said (They all say that!) "you'll realize it isn't understanding they need. I understand them all right — they're no good.""

"With me they get a solid foundation, the disciplines of learning. In my class they don't get away with hot air discussions and exchanging their opinions and describing their experiences. What opinions can they have? What have they experienced? What do they know? That's an affront! They learn what I know."

"Frances Egan, the school nurse, left her nutrition charts long enough to tell me there was nothing that could have been done. "Evelyn had a rough time with her father," she said. "Once she came in beaten black and blue." "What did you do for her?" "I gave her a cup of tea." "Tea? Why tea, for heaven's sake?" "Why? Because I know all about it," she said, shaking with anger. "I know more than anyone here what goes on outside — poverty, disease, dope, degeneracy — yet I'm not supposed to give them even a band-aid. I used to plead, bang on my desk, talk myself hoarse arguing with kids, parents, welfare, administration, social agencies. Nobody really heard me. Now I give them tea. At least, that's something." "But you're a nurse," I said helplessly. She showed me the Directive from the Board posted on her wall: THE SCHOOL NURSE MAY NOT TOUCH WOUNDS, GIVE MEDICATION, REMOVE FOREIGN PARTICLES FROM THE EYE... Are we, none of us, then, allowed to touch wounds? What is the teacher's responsibility? And if it begins at all, where does it end?"

"If a teacher wants to know something why doesn't she look it up herself instead of making we students do it? We benefit ourselves more by listening to her, after all she's the teacher!"

"How can I take seriously such mimeographed absurdities as "Lateness due to absence," "High under-achiever," and "Polio consent slips"? —SylDear Syl, I'll match yours any day with "Please disregard the following." —Bea"

"Mythology is studied in the school system because most of us come from it."

"Your lesson plan is excellent — except for the Emily Dickinson line: "There is no frigate like a book." The sentiment is lovely, the quotation is apt — only trouble is the word "frigate." Just try to say it in class — and your lesson is over."

"The ceiling fell? The ink ran dry? A student dared to smile? Of every new disaster I prove myself the master By sending out more circulars, more circulars to file! A missing kid? A kissing kid? A paper on the floor? For every major crisis One remedy suffices: More circulars, more circulars, to put into a drawer!A crowded cafeteria? A substitute's hysteria? A visitor from Syria? A missing Book Receipt?I merely send out circulars To add to other circulars To add to other circulars Numerical and neat!"

"You teachers are all alike, dishing out crap and expecting us to swallow it and then give it back to you, nice and neat, with a place in it for the mark to go in. But you're even phonier than the others because you put on this act — being a dame you know how — and you stand there pretending that you give a damn. Who you kidding? We're dirt to you, just like you're dirt to the fatheads and whistle-blowers who run this jail, and they're dirt to the swindlers and horn-tooters who run the school system."

"Teachers try to make us feel lower than themselves, maybe because this is because they feel lower than outside people. One teacher told me to get out of the room and never come back, which I did."

"Like most chairmen, he teaches only one class of Seniors; the most experienced teachers are frequently promoted right out of the classroom!"

"The cardinal sin, strange as it may seem in an institution of learning, is talking. There are others, of course — sins, I mean, and I seem to have committed a good number. Yesterday I was playing my record of Gielgud reading Shakespeare. I had brought my own phonograph to school (no one could find the Requisition Forms for "Audio-Visual Aids" — that's the name for the school record player) and I had succeeded, I thought, in establishing a mood. I mean, I got them to be quiet, when — enter Admiral Ass, in full regalia, epaulettes quivering with indignation. He snapped his fingers for me to stop the phonograph, waited for the turntable to stop turning, and pronounced: "There will be a series of three bells rung three times indicating Emergency Shelter Drill. Playing records does not encourage the orderly evacuation of the class.""

"During what was presumably my lunch period, Admiral Ass (a Mr. McHabe, who signs himself Adm. Asst.) appeared in my room with Joe Ferone. "This boy is on probation," he said. "Did he show up in homeroom this morning?" "Yes," I said. "Any trouble?" the Admiral asked. There we stood, the three of us, taking each other's measure. Ferone was watching through narrowed eyes. "No. No trouble," I said."

"The building itself is hostile: cracked plaster, broken windows, splintered doors and carved up desks, gloomy corridors and metal stairways, dingy cafeteria (they can eat sitting down only in 20 minute shifts) and an auditorium which has no windows. It does have murals, however, depicting mute, muscular harvesters, faded and immobile under a mustard sun."

"When I had asked why they were taking English, a boy said: "To help us in real life.""

"The clerical work is par for the course. "Keep on file in numerical order" means throw in wastebasket. You'll soon learn the language. "Let it be a challenge to you" means you're stuck with it; "interpersonal relationships" is a fight between kids; "ancillary civic agencies for supportive discipline" means call the cops; "Language Arts Dept." is the English office; "literature based on child's reading level and experiential background" means that's all they've got in the Book Room; "non-academic-minded" is a delinquent; and "It has come to my attention" means you're in trouble."

"I don't allow anyone to talk to me like that. So you're lucky — you're a teacher."

"I will ask of you only the ability to read English and to think logically—no high school mathematics, and certainly no higher mathematics."

"Please don't read the preface for the teacher."

"Wir Mathematiker sind alle ein bißchen meschugge."

"Any book that revisits the foundations of analysis has to reckon with the formidable precedent of Edmund Landau's Grundlagen der Analysis (Foundations of Analysis) of 1930. Indeed, the influence of Landau's book is probably the reason that so few books since 1930 have even been attempted to include the construction of the real numbers in an introduction to analysis. On the other hand, Landau's account is virtually the last word in rigor. The only way to be more rigorous would be to rewrite Landau's proofs in computer-checkable form—which has in fact been done recently. On the other hand Landau's book is almost pathologically reader-unfriendly. ...While memories of Landau still linger, so too does fear of the real numbers. In my opinion, the problem with Landau's book is not so much the rigor (though it is excessive), but the lack of background, history, examples, and explanatory remarks. Also, the fact that he does nothing with the real numbers except construct them. In short, it could be an entirely different story if it were explained that the real numbers are interesting! That is what I have tried to do...."

"Landau was the son of a well-to-do Berlin gynecologist (who invented the myomectomy operation)... his mother was from the banking family of Jacoby, and Landau grew up in a Jacoby house amid other Berlin banking families. ...Landau married Marrianne Erlich, daughter of Paul Ehrlich... Ehrlich had been a fellow student with Landau's father. Thus Landau grew up a well-connected and well-to-do person... he was also something of a prodigy. Legend has it that at age three, when his mother forgot her umbrella in a carriage, he replied, "It was number 354," and the umbrella was quickly reacquired. ...Landau was also something of a cynical snob."

"It is one thing to reflect, as Bieberbach did, for instance on the relative pedagogical merit of different ways to introduce π in a calculus class: geometrically via the circle, or in Landau's way via the zeroes of the cosine, with this function being defined by the power series. And it is quite a different matter to use such reflections as a basis for the forced removal of a distinguished colleague from teaching. Bieberbach's behaviour came all the more as a shock as nothing in his previous biography seemed to prepare one for it..."

"The present work is inspired by Edmund Landau's famous book, Handbuch der Lehre von der Verteilung der Primzahlen, where he posed two extremal questions on cosine polynomials and deduced various estimates on the distribution of primes using known estimates of the extremal quantities. Although since then better theoretical results are available for the error term of the prime number formula, Landau's method is still the best in finding explicit bounds. In particular, Rosser and Schonfeld used the method in their work "Approximate formulas for some functions of prime numbers"."

"The thorough analysis of even simple problems in arithmetic may require the application of advanced mathematics. A striking example is that of the distribution of prime numbers. The solution of this problem lies in finding a general formula which tells us the number of primes that lie in any given numerical interval. ...Edmond Landau ...wrote two large volumes analyzing this problem without solving it, using the most advanced mathematics known at the time. Even in the elementary aspects of mathematics we are thus dealing with complex topics which make great demands on our mathematical skills."

"Hilbert's solution of Waring's Problem was ready to be presented in the joint seminar with Minkowski in the middle of January 1909. After Minkowski's death, Hilbert presented his solution to Göttingen Academy on 6 February 1909, dedicating it to the memory of his friend, who had done so much for number theory. From then on he missed Minkowski, but carried on the seminar with Edmund Landau, Minkowski's successor. In looking for a successor to Minkowski, Klein and Hilbert looked for a young mathematician, whose achievements were still ahead of him. This requirement ruled out Adolf Hurwitz, and the final candidates were Oskar Perron and Edmund Landau. The decision was made by Klein, who said: 'Oh, Perron is such a wonderful person. Everybody loves him. Landau is very disagreeable, very difficult to get along with. But we, being such a group as we are, it is better that we have a man who is not easy'. Landau, though a worthy successor with respect to number theory... showed no interest in geometry and even less in applied mathematics, not to speak of mathematical physics. ...Hilbert knew that in executing his plans concerning physics, he could not count on Landau."

"The principal events... took place in the early months of 1933... By April the Nazis had almost total control of Germany. One of their first decrees, on April 7, was intended to bring about the dismissal of all Jews from the civil service. ...University professors were civil servants ...Of the five professors teaching mathematics at Götingen, three—Edmund Landau, Richard Courant, and Felix Bernstein—were Jewish. A fourth, Hermann Weyl, had a Jewish wife. ...the April decree did not apply to Landau or Courant, since they fell within the Hindenburg exceptions. ...It did not help that Götingen at large was rather strong for Hitler. This was true of both "town" and "gown." ...(That grand house of which Edmund Landau was so proud had been defaced with a painting of the gallows in 1931.) On April 26 the town newspaper... printed an announcement that six professors were being placed on indefinite leave. ...One holdout was Edmund Landau (the only Götingen math professor... who was a member of the town's synagogue). Relying on the integrity of the law, Landau attempted to resume calculus classes in November... but the Science Student's Council... organized a boycott. Uniformed storm troopers prevented Landau's students from entering the lecture hall. With singular courage, Landau asked the Council leader, a 20-year-old student named Oswald Teichmüller, to write out as a letter his reasons... his reasons were ideological. He... felt it improper that German students should be taught by Jews. We are accustomed to think of Nazis activists as thugs, low-lifes, opportunists and failed-artists... which, indeed, most of them were. ...they also included in their ranks some people of the highest intelligence."

"Not only the physical but also the intellectual landscape of German-language mathematics in the early 1930s would be impossible to imagine without German-Jewish mathematicians. Indeed, some fields of mathematics were completely transformed by their contributions. Number theory was transformed by Hermann Minkowski and Edmund Landau, algebra by Ernst Steinitz and Emmy Noether, set theory and general topology by Felix Hausdorff, Abraham Fraenkel and several others—to mention but a few examples."