Mathematicians From Germany

"[Up to that time] one would have said that a continuous function is essentially capable of being represented by a curve, and that a curve has always a tangent. Such reasoning has no mathematical value whatever; it is founded on intuition, or rather on a visible representation. But such representation is crude and misleading. We think we can figure to ourselves a curve without thickness; but we only figure a stroke of small thickness. In like manner we see the tangent as a straight band of small thickness, and when we say that it touches the curve, we wish merely to say that these two bands coincide without crossing. If that is what we call a curve and a tangent, it is clear that every curve has a tangent; but this has nothing to do with the theory of functions. We see to what error we are led by a foolish confidence in what we take to be visual evidence. By the discovery of this striking example Weierstrass has accordingly given us a useful reminder, and has taught us better to appreciate the faultless and purely arithmetical methods with which he more than any one has enriched our science."

"Software engineering is the part of computer science which is too difficult for the computer scientist."

"[Software engineering is the] establishment and use of sound engineering principles to obtain economically software that is reliable and works on real machines efficiently."

"The twentieth century return to Middle Age scholastics taught us a lot about formalisms. Probably it is time to look outside again. Meaning is what really matters."

"As I pored over the games of the great masters, two styles appealed to me above all others: Lasker and Steinitz. In Lasker I saw, above all, the supreme tactical genius. Whether a game was won or lost mattered little to him; he fought on to get the most out of every position."

"My chess hero."

"The greatest of the champions was, of course, Emanuel Lasker."

"Lasker also remarked with his detached, penetratingly ironic insight that Dawid Janowski took so much pleasure in a won position that he could not bear to part with it and wind it up to a victorious conclusion."

"Tarrasch teaches knowledge, Lasker teaches wisdom."

"That Lasker was a great fighter is an observation which is common to all studies of his play. Nobody can estimate what enormous will-power went into Lasker's fighting ability; and yet at the core of this quality was his belief that each position is unique, that it has some hidden aspect which the skeptic, the man of resource, will finally unearth."

"Steinitz always looked for the objectively right move. Tarrasch always claimed to have found the objectively right move. Lasker did nothing of the kind. He never bothered about what might or might not be the objectively right move; all he cared for was to find whatever move was likely to be most embarrassing for the specific person sitting on the other side of the board."

"What he really yearned for was some scientific understanding and that beauty peculiar to the process of logical creation, a beauty from whose magic spell no one can escape who has ever felt even its slightest influence."

"He hated chess as much as he loved it, using it chielfy as a means of livelihood while he devoted himself to problems of philosophy and mathematics."

"Spinoza's material life and economic independence were based on the grinding of lenses; in Lasker’s life chess played a similar part. But Spinoza was luckier, for his business was such as to leave his mind free and independent; whereas master-chess grips its exponent, shackling the mind and brain, so that the inner freedom and independence of even the strongest character cannot remain unaffected."

"Without error there can be no brilliancy."

"Put two players against each other who both have perfect technique, who both avoid weaknesses, and what is left?—a sorry caricature of chess.{{cite book"

"Dr. Tarrasch is a thinker, fond of deep and complex speculation. He will accept the efficacy and usefulness of a move if at the same time he considers it beautiful and theoretically right. But I accept that sort of beauty only if and when it happens to be useful. He admires an idea for its depth, I admire it for its efficacy. My opponent believes in beauty, I believe in strength. I think that by being strong, a move is beautiful too."

"You should keep in mind no names, nor numbers, nor isolated incidents, not even results, but only methods. The method is plastic. It is applicable in every situation."

"He who wants to educate himself in Chess must evade what is dead in Chess — artificial theories, supported by few instances and upheld by an excess of human wit; the habit of playing with inferior opponents; the custom of avoiding difficult tasks; the weakness of uncritically taking over variations or rules discovered by others; the vanity which is self-sufficient; the incapacity for admitting mistakes; in brief, everything that leads to a standstill or to anarchy."

"Although the adage "If you find a good move, look for a better one" is often attributed to Lasker, it actually dates earlier."

"What is immobile must suffer violence. The light-winged bird will easily escape the huge dragon, but the firmly rooted big tree must remain where it is and may have to give up its leaves, fruit, perhaps even its life."

"On the chessboard, lies and hypocrisy do not survive long. The creative combination lays bare the presumption of a lie; the merciless fact, culminating in the checkmate, contradicts the hypocrite."

"It is no easy matter to reply correctly to Lasker's bad moves."

"Education in Chess has to be an education in independent thinking and judgement. Chess must not be memorized, simply because it is not important enough. If you load your memory you should know why. Memory is too valuable to be stocked with trifles. Of my fifty-seven years I have applied at least thirty to forgetting most of what I had learned or read, and since I succeeded in this I have acquired a certain ease and cheer which I should never again like to be without. If need be, I can increase my skill in Chess, if need be I can do that of which I have no idea at present. I have stored little in my memory, but I can apply that little, and it is of good use in many and varied emergencies. I keep it in order, but resist every attempt to increase its dead weight."

"Emanuel Lasker was undoubtedly one of the most interesting people I came to know in my later life."

"I rather liked Lasker's stubborn intellectual independence, a most rare quality in a generation whose intellectuals are almost invariably mere camp-followers."

"Now it is time for us to realize that, in his Grundzüge, Kronecker did not merely intend to give his own treatment of the basic problems of ideal-theory which form the main subject of Dedekind's life-work. His aim was a higher own. He was, in fact, attempting to describe and to initiate a new branch of mathematics, which would contain both number-theory and algebraic geometry as special cases. This grandiose conception has been allowed to fade out of our sight, partly because of the intrinsic difficulties of carrying it out, partly owing to historical accidents and to the temporary successes of the partisans of purity and of Dedekind. It will be the main purpose of this lecture to try to rescue it from oblivion, to revive it, and to describe the few modern results which may be considered as belonging to the Kroneckerian program."

"Es handelt sich um meinen liebsten Jugendtraum, nämlich um den Nachweis, dass die Abel’schen Gleichungen mit Quadratwurzeln rationaler Zahlen durch die Transformations-Gleichungen elliptischer Functionen mit singularen Moduln grade so erschöpft werden, wie die ganzzahligen Abel’schen Gleichungen durch die Kreisteilungsgleichungen."

"If one says 'Red' and there are 50 people listening, it can be expected that there will be 50 reds in their minds. And one can be sure that all these reds will be very different."

"Anxiety is dead."

"In visual perception a color is almost never seen as it really is — as it physically is. This fact makes color the most relative medium in art."

"For me, abstraction is real, probably more real than nature. I'll go further and say that abstraction is nearer my heart. I prefer to see with closed eyes."

"In order to use color effectively it is necessary to recognize that color deceives continually."

"A painter works to formulate with or in colors.. .My paintings follow the second option."

"I want color and form to have contradictorily functions."

"Every perception of colour is an illusion.. ..we do not see colours as they really are. In our perception they alter one another."

"THE ORIGIN OF ART: The discrepancy between physical fact and psychic effect. THE CONTENT OF ART: Visual information of our reaction to life. THE MEASURE OF ART: The ratio of effort to effect. THE AIM OF ART: Revelation and evocation of vision."

"The concern of the artist is with the discrepancy between physical fact and psychological effect."

"It is my own personal psychosis that it is only by the background that you can see what is in front of you. Only be accepting all that surrounds you can you be totally self-visualized. And at the same time, your self-visualization is a reflection of your surroundings. Albers was right about that."

"[Josef] Albers' rule is to make order. As for me, I consider myself successful when I do something that resembles the lack of order I sense."

"I don't think he [Joseph Albers] ever realized that it was his discipline that I came for [on Black Mountain College, were Josef Albers was then a leading teacher]. Besides, my response to what I learned from him was just the opposite of what he intended.. .I was very hesitant about arbitrarily designing forms and selecting colors that would achieve some predetermined result, because I didn't have any ideas to support that sort of thing – I didn't want color to serve me, in other words."

"But I've noticed something with other artists who do use the whole range of forms of colours and black - in Albers, for instance, who experiments with yellow, red, blue, the whole scale. Of course I love his colour paintings, but when I see a black-and-white such as 'The Homage to a Black and White Square' [Josef Albers painted large series with this title], I like that best, you know. I think it has something to do with deciding just exactly what you really like best. There is always that wonderful element of doubt."

"[Josef] Albers was a beautiful teacher [at the Black Mountain college ] and an impossible person. He wasn't easy to talk to, and I found his criticism so excruciating and so devastating that I never asked for it. Years later, though, I'm still learning what he taught me, because what he taught me had to do with the entire visual world. He didn't teach you how to 'do painting'. The focus was always on your personal sense of looking.. .I consider Albers the most important teacher I've ever had, and I'm sure that he considers me one of his poorest students."

"When asked later in life about his working methods for the ['Homage to the 'Square' paintings, Albers would often explain that he always began with the center square because his father, who, among other things, painted houses, had instructed him as a young man that when you paint a door you start in the middle and work outwards. [Albers:] 'That way you catch the drips, and don’t get your cuffs dirty'."

"I did not teach painting but seeing. I concentrated on the basic courses for beginners. I taught drawing (purposely without nudes), color (without any painting as such) and design (as 'structural organization'). And so the graduate students came 'down' to the basic courses for beginners."

"Amateurism is an emptiness and I accept it because it has no preconceived ideas or rules to be applied. This is for me [as art teacher] a most welcome situation and I like to keep my students amateurs and dilettantes."

"In 1923, when I had been a student at the 'Bauhaus' in Weimar.. ..Gropius [the director of Bauhaus] asked me to teach the basic course 'Werklehhre'. He wanted me to introduce newcomers to the principles of handicrafts. He knew that I came from that background and had appropriate practice and knowledge."

"The German artist [Albers was from German origine] whose lifelong exploration of shape and color theories influenced a generation of artists and challenged audiences worldwide in addition of being one of the leading artists of the 20th century."

"I have taught – until 10 years ago – for nearly 40 years, that is almost half of my life. And when I think that over – now afterwards -, I come to a surprising conclusion, namely that I did not teach arts as such, but philosophy and psychology of art."

"I say all the time, if I sell that to you, you pay me for 3 colors. And I sell you 4, I betray you. Not to cheat you, but to pet you. You see I betray you in a positive way. I make you see more than there is. And that's in all my art that way. Absolutely something else. And that's what my book is about. You never see what you see. I lead you to see something else. And therefore I direct you. That's help."