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. - Godfrey Harold Hardy
When I am working on a problem I never think about beauty. I only think about how to solve the problem. But when I have finished, if the solution is not beautiful, I know it is wrong. - Buckminster Fuller
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. - Eduardo 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. - Eduardo Sontag
To Betty-Jo, whose decision processes defy analysis. - Richard Bellman
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. - From Some Vistas of Modern Mathematics by Richard Bellman, Univ. of Kentucky Press, 1968.
Try to expose yourself to the best things humans have done and then try to bring those things into what you are doing. - Steve Jobs
Attempt the impossible to improve your work. - Bette Davis
I don't need the money, dear. I work for art. - Maria Callas
The artist is nothing without the gift, but the gift is nothing without work. - Emile Zola
Do, or do not. There is no 'try'. - Yoda ('The Empire Strikes Back')
Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth. - Sherlock Holmes in The Sign of Four
The number 2 is just an extra for the number 1. - My father
If you try to become the number 1, you will be, at least, close to the number 1 in case you cannot become the number 1. - My father
Nothing is more practical than a good theory. - V. Vapnik
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. - Edsger W. Dijkstra
Make everything as simple as possible, but not simpler. - Albert Einstein
Out of clutter, find Simplicity. From discord, find Harmony. In the middle of difficulty lies opportunity. - Albert Einstein
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.' - Donald Knuth
I consider myself a mathematician and the award is for economics. I never, never in my life took a course in economics. - Lloyd Shapley
Opportunities multiply as they are seized. - Sun Tzu
The best way to predict the future is to invent it. - Alan Kay
Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense. - Budda (Siddhartha Gotoma)
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
Nothing shows a man's character more than what he laughs at. - Goethe
Whatever you can do, or dream you can, begin it, Boldness has genius, power and magic in it. - Goethe
I must create a system, or be enslav'd by another Man's; I will not reason and compare; my business is to Create. - William Blake
Laughter is the shortest distance between two people. - Victor Borge
Dream as if you'll live forever. Live as if you'll die tomorrow. - James Dean
Never confuse a single mistake with a final mistake. - F. Scott Fitzgerald
We are apt to forget that a great man is thus not only great, but also a man: that a philosopher, in a life time, spends less hours pondering the destiny of the race than he gives over to wondering if it will rain tomorrow and to meditating upon the toughness of steaks. - H.L. Mencken, The Philosophy of Friedrich Nietzsche
Just because I don't care doesn't mean I don't understand. - Homer J.Simpson
I think I have enough information now. I have one fact (which I made up), by the time I add an introduction, some illustrations, and a conclusion, I will have a thesis. - Calvin in Calvin and Hobbes
However beautiful the strategy, you should occasionally look at the results. - Winston Churchill
Only those who risk going too far can possibly find out how far one can go. - T. S. Elliot
There is a theory which states that if anyone discovers what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened. - Douglas Adams
There are only two mistakes one can make along the road to truth: (1) not going all the way; and (2) not starting. - Budda
What you've done becomes the judge of what you're going to do especially in other people's minds. When you're traveling, you are what you are right there and then. People don't have your past to hold against you. No yesterdays on the road. - William Least Heat Moon, Blue Highways
Failure is only postponed success as long as courage coaches ambition. The habit of persistence is the habit of victory. - Herbert Kaufman
Watch what people are cynical about, and one can often discover what they lack. - General George S. Patton
Steve Jobs from the commencement address at Stanford delivered on June 12, 2005.
- Your work is going to fill a large part of your life, and the only way to be truly satisfied is to do what you believe is great work. And the only way to do great work is to love what you do. If you haven't found it yet, keep looking. Don't settle. As with all matters of the heart, you'll know when you find it. And, like any great relationship, it just gets better and better as the years roll on. So keep looking until you find it. Don't settle.
- When I was 17, I read a quote that went something like: "If you live each day as if it was your last, someday you'll most certainly be right." It made an impression on me, and since then, for the past 33 years, I have looked in the mirror every morning and asked myself: "If today were the last day of my life, would I want to do what I am about to do today?" And whenever the answer has been "No" for too many days in a row, I know I need to change something.
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. - Drew Houston, the commencement address at MIT delivered in 2013.
R. 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.
- If the dream is a translation of waking life, waking life is also a translation of the dream.
Nam, Yi
Yi, Sun-sin
- War Diary: September 15, 1597 [A day before the Battle of Myongnyang] 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. - Bae Dal Choi
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. - Bae Dal Choi
Simplicity is the key to brilliance. - Bruce Lee
To overcome the intelligent by folly is contrary to the natural order of things; to overcome the foolish by intelligence is in accord with the natural order. To overcome the intelligent by intelligence, however, is a matter of opportunity. There are three avenues of opportunity: events, trends, and conditions. When opportunities occur through events but you are unable to respond, you are not smart. When opportunities become active through a trend and yet you cannot make plans, you are not wise. When opportunities emerge through conditions but you cannot act on them, you are not bold. Those skilled in generalship always achieve their victories by taking advantage of opportunities. - Zhuge Liang, circa 200 AD, The Way of the General
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. - Pravin 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. - Harold J. Kushner
Logic and intuition have their necessary role. Each is indispensable. - Henri Poincare
Well, they say reality's stranger than fiction sometimes - Paul Potts, (sings Nessun Dorma)
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. - Dimitri Bertsekas
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. - From the philosophy of Buddhism.
- The foundations of mathematics have parallels with the Buddhist concept of emptiness.
- The development of mathematics from its empty foundations creates computational algorithms and data structures that can simulate all physical phenomena.
- One/Zero can be Everything/Infinity and Everything/Infinity can be One/Zero; Emptiness is "equivalent" to Everything/Infinity.
- A fairly simple mathematics equation, e.g., Bellman's optimality principle, can be a compact representation (algorithmic compression) of a pretty "long" story (a universal set governed by a mathematical law).
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
It's not who I am underneath, but what I do that defines me. - 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
I can respect the artistic aim of a composer if he arrives at the so-called modern idiom after an intense period of preparation. Such composers know what they are doing when they break a law; they know what to react against, because they have had experience in the classical forms and style. Having mastered the rules, they know which can be violated and which should be obeyed. But, I am sorry to say, I have found too often that young composers plunge into the writing of experimental music with their school lessons only half learned. Too much radical music is sheer sham, for this very reason: its composer sets about revolutionizing the laws of music before he learned them himself. - Sergei Rachmaninoff
I have been aware from the outset (end of January 1959, the birthdate of the second paper in the citation) that the deep analysis of something which is now called Kalman filtering were of major importance. But even with this immodesty I did not quite anticipate all the reactions to this work. Up to now there have been some 1000 related publications, at least two Citation Classics, etc. There is something to be explained. To look for an explanation, let me suggest a historical analogy, at the risk of further immodesty. I am thinking of Newton, and specifically his most spectacular achievement, the law of Gravitation. Newton received very ample "recognition" (as it is called today) for this work. it astounded - really floored - all his contemporaries. But I am quite sure, having studied the matter and having added something to it, that nobody then (1700) really understood what Newton's contribution was. Indeed, it seemed an absolute miracle to his contemporaries that someone, an Englishman, actually a human being, in some magic and un-understandable way, could harness mathematics, an impractical and eternal something, and so use mathematics as to discover with it something fundamental about the universe. - Rudolf E. Kalman
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... - Rudolf E. Kalman
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. - Frederic Chopin
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. - In Rocky IV
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. - Friedrich Nietzsche in Human, All Too Human
Simplicity, Theory, & Harmony - My motto
Kurt Gödel
- There is a difference between a thing and talking about a thing.
- Intuition is not proof; it is the opposite of proof. We do not analyze intuition to see a proof but by intuition we see something without a proof.
- The brain is a computing machine connected with a spirit.
- 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.
- In the middle of January 1978, I dreamed I was at Gödel's bedside. There was a chess board on the covers in front of him. Gödel reached his hand out and knocked the board over, tipping the men onto the floor. The chessboard expanded to an infinite mathematical plane. And then that, too, vanished. There was a brief play of symbols, and then emptiness, an emptiness flooded with even white light. The next day I learned that he passed away. from R. Rucker's Infinity and the Mind