The case for my life, then, or for that of any one else who has been a mathematician in the same sense in which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any of the other artists, great or small, who have left some kind of memorial behind them. - G. Hardy

Nothing is more practical than a good theory. - V. Vapnik

I never cease to be amazed at the inverse relationship between how much technical detail a publication has and its impact and citations. Inevitably, it is the simplest ideas, those that look obvious in retrospect, that are the most influential. - E. Sontag

The development of any theoretical field involves the study of technically detailed and special cases, which provide the inspiration for the eventual conceptual synthesis. In turn, once that a concept has been discovered, working out particular cases and proving abstract theorems serves to confirm the strength and depth of the concept. - E. Sontag

Mankind's history has been a struggle against a hostile environment. We finally have reached a point where we can begin to dominate our environment and cease being victims of the vagaries of nature in the form of fire, flood, famine, and pestilence. We are at a time when we wish to cease wagering our existence upon the outcome of a race among the Four Horsemen of the Apocalypse. As soon as we understand this fact, our mathematical interests necessarily shift in many areas from descriptive analysis to control theory. - R. Bellman

In retrospect I was very fortunate to get into methodologies that eventually prospered. Dynamic Programming developed perhaps beyond Bellman's own expectation. He correctly emphasized the curse of dimensionality as a formidable impediment in its use but probably could not have foreseen the tranformational impact of the advances brought about by reinforcement learning, neurodynamic programming, and other approximation methodologies. - D. Bertsekas

I learned two things from Richard Bellman. First, over the years I continue to marvel at the significance of the optimality principle in the form of the verification theorem, which I have used in many contexts. Second and more important, I learned that good theory is very practical. - P. Varaiya

The word programming was used by the military to mean scheduling. Dantzig's linear programming was an abbreviation of "programming with linear models." Bellman has described the origin of the name "dynamic programming" as follows. An Assistant Secretary of the Air Force, who was believed to be strongly anti-mathematics was to visit RAND. So Bellman was concerned that his work on the mathematics of multi-stage decision process would be unappreciated. But "programming" was still OK, and the Air Force was concerned with rescheduling continuously due to uncertainties. Thus "dynamic programming" was chosen a politically wise descriptor. On the other hand, when I asked him the same question, he replied that he was trying to upstage Dantzig's linear programming by adding dynamic. Perhaps both motivations were true. - H. J. Kushner

Reinforcement learning is direct adaptive optimal control (a synthesis of dynamic programming and stochastic approximation methods). - R. Sutton, A. G. Barto, & R. J. Williams in the Proceeding of American Control Conf., and IEEE Control Systems Magazine, Vol. 12, No. 2, 1992.

"Q-learning," by C. Watkins and Peter Dayan in Machine Learning, Vol. 8, pp. 279-292, 1992: Stochastic Approximation (SA) for Proof.
"Asynchronous Stochastic Approximation and Q-Learning" in Machine Learning, Vol. 16, pp. 185-202, 1994 by J. Tsitsiklis: SA proof (more elaborated).
Cf., "Perspectives on Stochastic Optimization Over Time" in INFORMS Journal on Computing, Vol. 22, No. 1, pp. 18-19, 2010, by J. Tsitsiklis

One should clearly distinguish between two aspects of the estimation problem: (1) The theoretical aspect. Here interest centers on: (i) The general form of the solution. (ii) Conditions which guarantee a priori the existence, physical realizability, and stability of the optimal filter. (iii) Characterization of the general results in terms of some simple quantities, such as signal-to-noise ratio, information rate, bandwidth, etc... (2) The computational aspect. The classical (more accurately, old-fashioned) view is that a mathematical problem is solved if the solution is expressed by a formula. It is not a trivial matter, however, to substitute numbers in a formula. The current literature on the Wiener problem is full of semi-rigorously derived formulas which turn out to be unusable for practical computation when the order of the system becomes even moderately large... - R. Kalman

Try to expose yourself to the best things humans have done and then try to bring those things into what you are doing. - S. Jobs

Attempt the impossible to improve your work. - Bette Davis

I don't need the money, dear. I work for art. - M. Callas

It's not who I am underneath, but what I do that defines me. - The Batman

Introduce a little anarchy. Upset the established order, and everything becomes chaos. I'm an agent of chaos. Oh, and you know the thing about chaos? It's fair! - The Joker
Do, or do not. There is no 'try'. - Yoda in The Empire Strikes Back
The number 2 is just an extra for the number 1. - Y. C. Choe
In the practice of computing, where we have so much latitude for making a mess of it, mathematical elegance is not a dispensable luxury, but a matter of life and death. - E. Dijkstra
I guess you have to go back to Einstein who said 'keep it as simple as possible, but no simpler.' It's an illusion to think there's going to be some sort of 'royal road' and everything is going to be easy to understand, but almost never do I find something that can't be simplified if you have the chance to rethink it again. So every once in a while people have to say 'well, let's chuck everything we have and start over again, in view of what we know now.' - D. Knuth
I consider myself a mathematician and the award is for economics. I never, never in my life took a course in economics. - L. Shapley
One man scorned and covered with scars still strove with his last ounce of courage to reach the unreachable stars; and the world will be better for this. -Don Quixote de la Mancha
Only those who risk going too far can possibly find out how far one can go. - T. S. Elliot
Failure is only postponed success as long as courage coaches ambition. The habit of persistence is the habit of victory. - H. Kaufman
I was going to say work on what you love, but that's not really it. It's so easy to convince yourself that you love what you're doing --- who wants to admit that they don't? When I think about it, the happiest and most successful people I know don't just love what they do, they're obsessed with solving an important problem, something that matters to them. They remind me of a dog chasing a tennis ball: their eyes go a little crazy, the leash snaps and they go bounding off, plowing through whatever gets in the way. - D. Houston, commencement address at MIT, 2013.
Nam, Yi
Yi, Sun-sin
- September 15, 1597 [A day before the Battle of Myong-ryang] Kye-sa Clear. With the tide flowing, I entered the sea of Usuyong, leading our ships after me, and there I passed the night. I saw many queer portents in my dream at night. l.v. Clear. By riding the rising tide I led the Captains of all ships to move to the sea off Usuyong [Munnae-myon, Haenam-gun], because it was not right for a small fleet to take a fighting position with its back against Myongnyang (Ultolmok, the Roaring Channel), whose swift current falls like a cataract behind Pyokp'ajong (the Sea-Viewing Pavilion). Calling my Staff Officers and all ships' Captains, I gave the following instruction: "According to the principles of strategy, `He who seeks his death shall live, he who seeks his life shall die.' Again, the strategy says `If one defender stands on watch at a strong gateway he may drive terror deep into the heart of the enemy coming by the ten thousand.' These are golden sayings for us. You Captains are expected to strictly obey my orders. If you do not, even the least error shall not be pardoned, but shall be severely punished by Martial Law." In this way I showed them my firm attitude. In my dream this night a spirit appeared before me and declared, "If you do in this way, you shall win a great victory; if you do in that way you shall suffer a tragic defeat."
If someone asked me what a human being ought to devote the maximum of his life to, I would answer: training. Train more than you sleep. - B. D. Choe
Even if poor, a human being who devotes himself to his one and only goal with risking his own death is really magnificent and beautiful. Such a person looks truly noble. - B. D. Choe
Adrian: It's suicide. You've seen him, you know how strong he is. You can't win. Rocky: Oh, Adrian. Adrian always tells the truth. No, maybe I can't win. Maybe the only thing I can do is just take everything he's got. But to beat me, he's going to have to kill me. And to kill me, he's gonna have to have the heart to stand in front of me. And to do that, he's got to be willing to die himself. I don't know if he's ready to do that. I don't know. - Rocky Balboa
The simplest data structure is an empty set. The simplest algorithm which acts on the empty set is the "bootstrapping" out of the empty set or a recursive/inductive operation (John Von Neumann). This creates "numbers": Let 0 = { } and define S(x) = x union {x} for every set x where S(x) is the successor of x, having that 0 = { }, 1= { {} }, 2 = { {}, { {} } }, 3 = {{}, { {} }, { {}, { {} } }}, .... The whole set of natural numbers is constructed out of sets of sets of sets of nothing. Now the binary represented integers plus the instruction set of a computer are capable of representing any other kind of numbers and simulating and modelling any physical system (Church-Turing-Deutsch principle). Just a few operations are required to equip a general purpose computer with universal physical modelling capabilities - fewer than twenty: SET, MOVE, READ, WRITE, ADD, SUBTRACT, MULTIPLY, DIVIDE, AND, OR, XOR, NOT, SHIFT, ROTATE, COMPARE, JUMP, JUMP-CONDITIONALLY, RETURN. - In the philosophy of Buddhism.
With persistence aroused for the highest goal's attainment, with mind unsmeared, not lazy in action, firm in effort, with steadfastness & strength arisen, wander alone like a rhinoceros. - Sutta Nipata: Khaggavisana Sutta - A Rhinoceros
Perhaps we will recognize then that the thing-in-itself deserves a Homeric laugh, in that it seemed to be so much, indeed everything, and is actually empty, that is, empty of meaning. - F. Nietzsche
Simplicity, Theory, & Harmony - H. Choe
Simplicity is the ultimate sophistication. - L. da Vinci
Simplicity is everything. It is art's final object. One has to have studied a lot, tremendously, to let SIMPLICITY emerge with all its charm, grace and magic. Place the hand at ease, with a maximum of suppleness and flexibility, easily, easily. - F. Chopin
Turing, in Proc. Lond. Math. Soc. 42 (1936), p. 250, gives an argument which is supposed to show that mental procedures cannot carry any farther than mechanical procedures. However, this argument is inconclusive, because it depends on the supposition that that a finite mind is capable of only a finite number of distinguishable states. What Turing disregards completely is the fact that mind, in its use, is not static, but constantly developing. Therefore, although at each stage of the mind's development the number of its possible states is finite, there is no reason why this number should not converge to infinity in the course of its development. Now there may exist systematic methods of accelerating, specializing, and uniquely determining this development, e.g. by asking the right questions on the basis of a mechanical procedure. But it must be admitted that the precise definition of a procedure of this kind would require a substantial deepening of our understanding of the basic operations of the mind. - Kurt Gödel
Rene Magritte
- To equate my painting with symbolism, conscious or unconscious, is to ignore its true nature. People are quite willing to use objects without looking for any symbolic intention in them, but when they look at paintings, they can't find any use for them. So they hunt around for a meaning to get themselves out of the quandary, and because they don't understand what they are supposed to think when they confront the painting. They want something to lean on, so they can be comfortable. They want something secure to hang on to, so they can save themselves from the void. People who look for symbolic meanings fail to grasp the inherent poetry and mystery of the image. No doubt they sense this mystery, but they wish to get rid of it. They are afraid. By asking ''what does this mean?'' they express a wish that everything be understandable. But if one does not reject the mystery, one has quite a different response. One asks other things.
Hermann Hesse
- If I were a composer, I could without difficulty write a melody for two voices, a melody that would consist of two lines, of two rows of tunes and notes that correspond with one another, complement one another, fight with one another, limit one another, but in any case at every instant, at every point in the sequence, have a most profound interrelationship and reciprocal effect. And anyone who can read music could read off my double melody and always see and hear with every tone its counter-tone, its brother, its enemy, its opposite.
  
  Now it is just this, this double voice and constantly advancing antithesis, this double line, that I would like to express in my own medium, in words, and I work myself to the bone trying and do not succeed. I am always attempting it and if anything at all lends tension and weight to my works, it is this intensive concern for something impossible, this wild battling for something unattainable. I would like to find expression for duality, I would like to write chapters and sentences where melody and counter-melody are always simultaneously present, where unity stands beside every multiplicity, seriousness beside every joke. For to me, life consists simply in this, in the fluctuation between two poles, in the hither and thither between the two foundation pillars of the world. - A Guest at the Spa (1924)
- Remember this: one can be a strict logician or grammarian and at the same time full of imagination and music. - The Glass Bead Games
- Siddhartha listened. He was now listening intently, completely absorbed, quite empty, taking in everything. He felt that he had now completely learned the art of listening. He'd often heard all this before, all these numerous voices in the river, but today they sounded different. He could no longer distinguish the different voices - the merry voice from the weeping voice, the childish voice from the manly voice.
  
  The all belonged to each other: the lament of those who yearn, the laughter of the wise, the cry of indignation and the groan of the dying. They're all interwoven and interlocked, entwined in a thousand ways. And all the voices, all the goals, all the yearnings, all the sorrows, all the pleasures, all the good and evil, all of them together was the world. All of them together was the stream of events, the music of life. When Siddhartha listened attentively to this river, to this song of a thousand voices; when he did not listen to the sorrow or laughter, when he did not bind his soul to any one particular voice and absorb it in his Self, but heard them all, the whole, the unity; then the great song of a thousand voices consisted of one word: Om --- perfection. ...
  
  From that hour Siddhartha ceased to fight against his destiny. There shone in his face the serenity of knowledge, of one who is no longer confronted with conflict of desires, who has found salvation, who is in harmony with the string of events, with the stream of life, full of sympathy and compassion, surrendering himself to the stream, belonging to the unity of all things. - Siddhartha
That's what chess is all about. One day you give your opponent a lesson, the next day he gives you one. In chess so much depends on opening theory, so the champions before the last century did not know as much as I do and other players do about opening theory. So if you just brought them back from the dead they wouldn't do well. They'd get bad openings. I think it's almost definite that the game is a draw theoretically. - B. Fischer