educators-from-germany

"Philosophy... is that activity by which the meaning of propositions is established or discovered. Philosophy elucidates propositions, science verifies them. In the latter we are concerned with the truth of statements, but in the former with what they actually mean."

"Schlick ( [Wende] p.8 ) interprets Wittgenstein's position as follows: philosophy "is that activity by which the meaning of propositions is established or discovered" ; it is a question of "what the propositions actually mean. The content, soul, and spirit of science naturally consist in what is ultimately meant by its sentences; the philosophical activity of rendering significant is thus the alpha and omega of all scientific knowledge"."

"The 'physical' does not mean any particular kind of reality, but a particular kind of denoting reality, namely a system of concepts in the natural sciences which is necessary for the cognition of reality. 'The physical' should not be interpreted wrongly as an attribute of one part of reality, but not of the other ; it is rather a word denoting a kind of conceptual construction, as, e.g., the markers 'geographical' or 'mathematical', which denote not any distinct properties of real things, but always merely a manner of presenting them by means of ideas."

"Philosophy is not a system of propositions, and not a science."

"The members of the Vienna Circle (Moritz Schlick, Rudolf Carnap, , Hans Hahn, , Fritz Waismann, Kurt Godel, Otto Neurath and others) are working out a ‘Logical Empiricism’. Following Mach and Poincare, but above all Russell and Wittgenstein, all the sciences are treated uniformly. Carnap’s Logischer Aufbau der Welt (1928) shows in which direction future systematic work will move. Wittgenstein’s Tractatus Logico- Philosophicus (1921) clarified, among other things, the position of logic and mathematics; besides the statements that make additions to what is meaningful, there are the ‘tautologies’ that show us which transformations are possible within language. By its syntax the language of science excludes anything that is meaningless from the very beginning."

"The Lord never does anything arbitrarily."

"Infidelity is consumption of the soul."

"I would rather trust my child to a serpent than to a teacher who does not believe in God."

"Be yourself, but always your better self."

"No man shall be more exacting of me or my conduct than I am of myself."

"Every human being is a world in miniature. It has its own centre of observation, its own way of forming concepts and of arriving at conclusions, its own degree of sensibility, its own life's work to do, and its own destiny to reach. All these features may be encompassed by general conditions, governed by general laws, and subject to unforeseen influences and incidents, but within the sphere of their own activity, they constitute that great principle which we call individuality."

"The fireside is an emblem of the future heavenly home...To obtain the highest conception of the calling of a man and a woman in the capacity of parents, one must look upon them from an educational point of view, for from no other does the grandeur of this sacred relationship so well present itself to the mind with all its intricate complexity. The home is the sanctuary of the human race, where each generation is consecrated for its life's mission. The parents are the high priests, responsible to God for the spirit of their ministry"

"He that cheats another is a knave; but he that cheats himself is a fool."

"I have been asked what I mean by 'word of honor.' I will tell you. Place me behind prison walls--walls of stone ever so high, ever so thick, reaching ever so far into the ground--there is a possibility that in some way or another I may escape; but stand me on the floor and draw a chalk line around me and have me give my word of honor never to cross it. Can I get out of the circle? No. Never! I'd die first!"

"The fear of the Lord is the Beginning of Wisdom. This life is one great object lesson to practice on the principles of immortality and eternal life. Man grows with his higher aims. Let naught that is unholy enter here."

"Youth demands recreation, and if it is not provided in high places, they will seek it in low places."

"No righteous rules, however rigid, are too stringent for me; I will live above them."

"Some of the groundbreaking work in the treatment of n-dimensional geometry—was carried out by Hermann Günther Grassmann. ...Grassmann was responsible for the creation of an abstract science of "spaces," inside which the usual geometry was only a special case. Grassmann published his pioneering ideas (originating a branch of mathematics known as linear algebra) in 1844, in a book commonly known as Ausdehnungslehre... Grassmann's suggestion that BA = -AB violates one of the sacrosanct laws of arithmetic... Grassmann faced up squarely to this disturbing possibility and invented a new consistent algebra (known as exterior algebra) that allowed for several processes of multiplication and at the same time could handle geometry in any number of dimensions."

"As the great generality of Grassmann's processes—all results being obtained for n-dimensional space—has been one of the main hindrances to the general cultivation of his system, it has been thought best to restrict the discussion to space of two and three dimensions."

"The wonderful and comprehensive system of Multiple Algebra invented by Hermann Grassmann, and called by him the Ausdehnungslehre or Theory of Extension, though long neglected by the mathematicians even of Germany, is at the present time coming to be more and more appreciated and studied. In order that this system, with its intrinsic naturalness, and adaptability to all the purposes of Geometry and Mechanics, should be generally introduced to the knowledge of the coming generation of English-speaking mathematicians, it is very necessary that a text-book should be provided, suitable for use in colleges and universities, through which students may become acquainted with the principles of the subject and its applications."

"Grassmann's first publication of his new system was made in 1844 in a book entitled "Die Lineale Ausdehnungslehre Ein Neuer Zweig der Mathematik." His novel and fruitful ideas were however presented in a somewhat abstruse and unusual form, with the result, as the author himself states in the preface to the second edition issued in 1878, that scarcely any notice was taken of the book by Mathematicians. He was finally convinced that it would be necessary to treat the subject in an entirely different manner in order to gain the attention of the mathematical world. Accordingly in 1862 he published "Die Ausdehnungslehre vollständig und in strenger Form bearbeitet," in which the treatment is algebraic... Since that time his great work has been more fully appreciated, but not even yet, in the opinion of the writer, at its real value."

"The exchange theorem... is sometimes called the Steinitz exchange theorem after Ernst Steinitz... The result was first proved Hermann Günther Graßmann..."

"I feel entitled to hope that I have found in this new analysis the only natural method according to which mathematics should be applied to nature, and according to which geometry may also be treated, whenever it leads to general and to fruitful results."

"The concept of rotation led to geometrical exponential magnitudes, to the analysis of angles and of trigonometric functions, etc. I was delighted how thorough the analysis thus formed and extended, not only the often very complex and unsymmetric formulae which are fundamental in tidal theory, but also the technique of development parallels the concept."

"The concept of centroid as sum led me to examine Möbius' Barycentrische Calcul, a work of which until then I knew only the title; and I was not little pleased to find here the same concept of the summation of points to which I had been led in the course of the development. This was the first, and... the only point of contact which my new system of analysis had with the one that was already known."

"While I was pursuing the concept of geometrical product, as this idea was established by my father... I concluded that not only rectangles, but also parallelograms, may be viewed as products of two adjacent sides, provided that the sides are viewed not merely as lengths, but rather as directed magnitudes. When I joined this concept of geometrical product with the previously established idea of geometrical sum the most striking harmony resulted. Thus when I multiplied the sum of two vectors by a third coplaner vector, the result coincided (and must always coincide) with the result obtained by multiplying separately each of the two original vectors by the third... and adding together (with due attention to positive and negative values) the two products. [Thus A(B + C) = AB + AC.] From this harmony I came to see a whole new area of analysis was opening up which could lead to important results."

"The first impulse came from the consideration of negatives in geometry; I was accustomed to viewing the distances AB and BA as opposite magnitudes. Arising from this idea was the conclusion that if A, B, and C are points of a straight line, then in all cases AB + BC = AC, this being true whether AB and BC are directed in the same direction or in opposite directions (where C lies between A and B). In the latter case AB and BC were not viewed as merely lengths, but simultaneously their considered since they were oppositely directed, Thus dawned the distinction between the sum of lengths and the sum of distances which were fixed in direction. From this resulted the requirement for establishing this latter concept of sum, not simply for the case where the distances were directed in the same or opposite directions, but also for any other case. This could be done in the most simple manner, since the law that AB + BC = AC remains valid when A, B, and C do not lie on a straight line. This then was the first step which led to a new branch of mathematics... I did not however realize how fruitful and how rich was the field that I had opened up; rather that result seemed scarcely worthy of note until it was combined with a related idea."

"A work on tidal theory... led me to Lagrange's Mécanique analytique and thereby I returned to those ideas of analysis. All the developments in that work were transformed through the principles of the new analysis in such a simple way that the calculations often came out more than ten times shorter than in Lagrange's work."

"I define as a unit any magnitude that can serve for the numerical derivation of a series of magnitudes, and in particular I call such a unit an original unit if it is not derivable from another unit. The unit of numbers, that is one, I call the absolute unit, all others relative. Zero can never be a unit."

"From the imputation of confounding axioms with assumed concepts Euclid himself, however, is free. Euclid incorporated the former among his postulates while he separated the latter as common concepts—a proceeding which even on the part of his commentators was no longer understood, and likewise with modern mathematicians, unfortunately for science, has met with little imitation. As a matter of fact, the abstract methods of mathematical science know no axioms at all."

"As I was reading the extract from your paper in the geometric sum and difference... I was struck by the marvelous similarity between your results and those discoveries which I made even as early as 1832... I conceived the first idea of the geometric sum and difference of two or more lines and also of the geometric product of two or three lines in that year (1832). This idea is in all ways identical to that presented in your paper. But since I was for a long time occupied with entirely different pursuits, I could not develop this idea. It was only in 1839 that I was led back to that idea and pursued this geometrical analysis up to the point where it ought to be applicable to all mechanics. It was possible for me to apply this method of analysis to the theory of tides, and in this I was astounded by the simplicity of the calculations resulting from this method."

"Geometry can in no way be viewed... as a branch of mathematics; instead, geometry relates to something already given in nature, namely, space. I... realized that there must be a branch of mathematics which yields in a purely abstract way laws similar to geometry."

"It was natural that Grassmann chose to introduce his system, not by means of a paper, but rather by means of a long and complicated book. ...such ideas as Grassmann's form of the scaler (dot) and vector (cross) products... have counterparts in modern vector analysis."

"It is clear... that the concept of space can in no wise be generated by thought. ...Whoever maintains the contrary must undertake to derive the dimensions of space from the pure laws of thought—a problem which is at once seen to be impossible of solution."

"One may say without great exaggeration that Grassmann invented linear algebra and, with none at all, that he showed how properly to apply it to geometry. ...He ...anticipated in its most important aspects Peano's treatment of the natural numbers, published 28 years later. ...A feature of Grassmann's work, far in advance of the times, is the tendency towards the use of the implicit definition. ...The definition of a linear space (or vector space) came into mathematics, in the sense of becoming widely known, around 1920, when Hermann Weyl and others published formal definitions. ...Grassmann did not put down a formal definition—again, the language was not available—but there is no doubt that he had the concept."

"The history of geometry may be conveniently divided into five periods. The first extends from the origin of the science to about A. D. 550, followed by a period of about 1,000 years during which it made no advance, and in Europe was enshrouded in the darkness of the middle ages; the second began about 1550, with the revival of the ancient geometry; the third in the first half of the 17th century, with the invention by Descartes of analytical or modern geometry; the fourth in 1684, with the invention of the differential calculus; the fifth with the invention of descriptive geometry by Monge in 1795. The quaternions of Sir William Rowan Hamilton the Ausdehnungslehre of Dr. Hermann Grassmann, and various other publications, indicate the dawn of a new period. Whether they are destined to remain merely monuments of the ingenuity and acuteness of their authors, or are to become mighty instruments in the investigation of old and the discovery of new truths, it is perhaps impossible to predict."

"Miss Fanny Davies, who was studying with Madame Schumann at the same time as myself, is a very good example of easy muscular movement and finely developed finger technique. Leschetizky was a fine teacher; so was Liszt (when he took the trouble). L. Deppe and Caland were the last exponents of this perfectly simple and natural way of playing. For simple and natural it is, as is proved by the fact that all great concert pianists of today play in this way, whether they themselves realise it or not. (I was told that Backhaus, on being asked how he did it, replied that he didn't know.)"

"With the exception of Madame Schumann there is no woman and there will not be any women employed in the Conservatory. As for Madame Schumann, I count her as a man."

"Clara was sort of a modern woman in many ways, suffering the tension between her career and home life, because it was very important for her to keep playing concerts. On the other hand, she was [Robert] Schumann's wife and he wanted her around; he hated it when she traveled. But she was very much his great muse and inspiration, and virtually everything he wrote for the piano, Clara would have been the first to play."

"I’ve been playing the songs by Liszt, with which you so surprised me, with great enthusiasm, especially “Gretchen,” “Erlkönig,” and “Sei mir gegrüßt.” Is Liszt coming to Vienna in the summer? Thalberg as well? Is he still coming to Leipzig as promised? Liszt as well? - What is Mrs. von Cibbini doing? Lickl, Vesque von Püttlingen, Fischhof?"

"How much one has to do to leave town with a few dollars! ...I arrive home, dead tired, at 11 or 12 o'clock [at night], gulp a mouthful of water, lie down and think, "Is an artist much more than a beggar?" Yet, art is a beautiful gift. What, indeed, is more beautiful than to clothe one’s feelings in sound, what a comfort in sad hours, what a pleasure, what a wonderful feeling, to provide an hour of happiness to others. And what a sublime feeling to pursue art so that one gives one’s life for it."

"Music is now quite another thing for me than it used to be. How blissful, how full of longing it sounds; it is indescribable ... I could wear myself out now at the piano, my heart is eased by the tones and what sympathy it offers! ... Oh, how beautiful music is; so often it is my consolation when I would like to cry."

"I stood at the body of my dearly loved husband and was calm; all my feelings were of thankfulness to God that he was finally free, and as I knelt at his bed I had such a holy feeling. It was as if his magnificent spirit hovered above me, oh–if he had only taken me with him!"

"My heart bled as I said goodnight to Felix and went to the concert. The contrast was so dreadful. Throughout the entire concert I saw only him, his emaciated body, his lifeless appearance, and alas, his lack of breath–it was horrible. And yet I played quite well, without even one wrong note!"

"Leibniz ... envisioned real possibility in terms rooted in axiological considerations of evaluative optimality. To be sure, it is clear that one cannot just optimize. ... One has to optimize something, some feature or aspect of things. And if this factor is to be something that is qualified, be accepted as self-validating and self-sustaining, then the clearly most promising candidate would seem to be intelligence itself. ... The value at issue here with "being for the best" is a matter of being so as intelligent creatures see it—that is from the vantage point of intelligence itself. Assuredly, no intelligent being would prefer an alternative that is inferior in this regard. And so, for an intelligent being—a rational creature—intelligence itself must figure high on the scale of values."

"I have never thought for a moment that if you cannot say it with numbers that it just is not worth saying. But all the same, I do firmly believe that where you cannot put numbers to work you will understand the matter better and more clearly for being able to explain why."

"Rescher's work envisions a dialectical tension between our synoptic aspirations for useful knowledge and our human limitations as finite inquirers. The elaboration of this project represents a many-sided approach to fundamental philosophical issues that weaves together threads of thought from the philosophy of science, and from continental idealism and American pragmatism."

"Unfortunately...it was his eyesight that was at fault, not my footnotes. So one was able to skewer him by his own methods. It was horrible, and in retrospect I deeply regret it... Well, I regret that the thing happened at all."

"Elton didn't like that at all. Geoffrey got very cross with me. He wrote an absolutely shocking review of a collection of essays I edited in which he obviously went for me, but he went for very much younger people as well, which I think for somebody who is a knight and a Regius professor is scandalous bullying, and I said so."

"More perhaps than any Briton this century, he exemplified the virtues of the empiricist school of history. Not for Elton the fashionable theory of his continental contemporaries or the anti-imperialist, anti-capitalist posturing of left-wing historians in this country. In his 1967 primer The Practice of History, he argued that laborious work with documents must remain the bedrock of all research into the past. Without this foundation, no analysis or theorising could be taken seriously. National history, he warned, was too important to sacrifice on the altar of intellectual vogue. The survival of traditionalist history in British schools and universities owes much to his reason."