Mathematicians From The United States

"In a lifetime of many, many independent choices, small and large, a career with a vision will get you a distance proportional to n, while no vision will get you only the distance √n. ...the accuracy of the vision matters less than you suppose, getting anywhere is better than drifting, there are potentially many paths to greatness for you... No vision, not much of a future."

"Unforeseen technological inventions can completely upset the most careful predictions."

"It is probable that the future will be more limited by the slow evolution of the human animal and the corresponding human laws, social institutions, and organizations than it will be by the rapid evolution of technology."

"Often it is not physical limitations... but rather it is human made laws, habits, and organizational rules, regulations, personal egos, and inertia, which dominate the evolution of the future."

"It is obvious: The past was once the future and the future will become the past."

"Much of present science rests on engineering tools, and as time goes on, engineering seems to involve more and more of the science part. ...the two fields are growing together! ...and now there is not enough time to allow us the leisure which comes from separating the two fields."

"All of engineering involves some creativity to cover the parts not known, and almost all of science includes some practical engineering to translate the abstractions into practice."

"In science if you know what you are doing you should not be doing it. In engineering if you do not know what you are doing you should not be doing it. Of course, you seldom, if ever, see either pure state."

"Either you will be a leader, or a follower, and my goal is for you to be a leader."

"I am concerned with educating and not training you. ...Education is what, when, and why to do things. Training is how to do it. Either one without the other is not of much use. You might think education should precede training, but the kind of educating I am trying to do must be based on your past experiences and technical knowledge."

"The past is... much more uncertain—or even falsely reported—than is usually recognized."

"Apparently an "art"—which almost by definition cannot be put into words—is probably best communicated by approaching it from many sides and doing so repeatedly, hoping thereby students will finally master enough of the art, or if you wish, style, to significantly increase their future contributions to society."

"Teachers should prepare the student for the student's future, not for the teacher's past."

"Great results in science and engineering are "bunched" in the same person too often for success to be a matter of random luck."

"Since I was trying to teach "style" of thinking in science and engineering, and "style" is an art, I should therefore copy the methods of teaching used for the other arts—once the fundamentals have been learned. How to become a great painter cannot be taught in words... Art teachers usually let the advanced student paint, and then make suggestions... more or less as the points arise in the student's head—which is where learning is supposed to occur!"

"If the prior distribution, at which I am frankly guessing, has little or no effect on the result, then why bother; and if it has a large effect, then since I do not know what I am doing how would I dare act on the conclusions drawn?"

"Probability is too important to be left to the experts. [...] The experts, by their very expert training and practice, often miss the obvious and distort reality seriously. [...] The desire of the experts to publish and gain credit in the eyes of their peers has distorted the development of probability theory from the needs of the average user. The comparatively late rise of the theory of probability shows how hard it is to grasp, and the many paradoxes show clearly that we, as humans, lack a well grounded intuition in the matter. Neither the intuition of the man in the street, nor the sophisticated results of the experts provides a safe basis for important actions in the world we live in."

"One of the characteristics of successful scientists is having courage. Once you get your courage up and believe that you can do important problems, then you can. If you think you can't, almost surely you are not going to. [...] The average scientist, so far as I can make out, spends almost all his time working on problems which he believes will not be important and he also doesn't believe that they will lead to important problems. [...] In summary, I claim that some of the reasons why so many people who have greatness within their grasp don't succeed are: they don't work on important problems, they don't become emotionally involved, they don't try and change what is difficult to some other situation which is easily done but is still important, and they keep giving themselves alibis why they don't."

"I noticed the following facts about people who work with the door open or the door closed. I notice that if you have the door to your office closed, you get more work done today and tomorrow, and you are more productive than most. But 10 years later somehow you don't quite know what problems are worth working on; all the hard work you do is sort of tangential in importance. He who works with the door open gets all kinds of interruptions, but he also occasionally gets clues as to what the world is and what might be important."

"Most people like to believe something is or is not true. Great scientists tolerate ambiguity very well. They believe the theory enough to go ahead; they doubt it enough to notice the errors and faults so they can step forward and create the new replacement theory. If you believe too much you'll never notice the flaws; if you doubt too much you won't get started. It requires a lovely balance."

"When you are famous it is hard to work on small problems. [...] The great scientists often make this error. They fail to continue to plant the little acorns from which the mighty oak trees grow. They try to get the big thing right off. And that isn't the way things go. So that is another reason why you find that when you get early recognition it seems to sterilize you. [...] The Institute for Advanced Study in Princeton, in my opinion, has ruined more good scientists than any institution has created, judged by what they did before they came and judged by what they did after."

"It is not easy to become an educated person."

"Although textbooks (and professors) like to make definite statements indicating that they know what they are talking about, there is in fact a great deal of uncertainty and ambiguity in the world. ...we will not evade this question but rather explore (overexplore?) it. ...great progress is often made when what was long believed to be true is now seen to be perhaps not the whole truth. Thus the text often uses words... to cause you to think about the uncertainess and even the arbitrariness of much of our current conventions and definitions, to ponder about your acceptance of them."

"There are so many ways of being wrong and so few ways of being right that it is much more economical to study successes."

"Besides the theory there are a lot of small technical details that must be learned so well that you can recall them almost instantaneously, such as the trigonometric identities... put one part of the identity on one side of a 3 x 5 card and the other part on the other side. Using these flash cards you can, in the odd moments of your daily life, learn the mechanical parts of the course. ...for this kind of low-level material many short learning sessions are much more efficient than a few long, intense ones; but this is not necessarily true for larger ideas. ...most students will not use such trivial devices as flash cards; it seems to be beneath their dignity. They suffer accordingly."

"This text is organized in the "spiral" for learning. A topic... is returned to again and again, each time higher up in the spiral. The first time around you may not be completely sure of what is going on, but on the repeated returns to the topic it should gradually become clear. This is necessary when the ideas are not simple but require a depth of understanding..."

"The applications of knowledge, especially mathematics, reveal the unity of all knowledge. In a new situation almost anything and everything you ever learned might be applicable, and the artificial divisions seem to vanish."

"In the long run, the methods are the important part of the course. It is not enough to know the theory; you should be able to apply it."

"It is easy to measure your mastery of the results via a conventional examination; it is less easy to measure your mastery of doing mathematics, of creating new (to you) results, and of your ability to surmount the almost infinite details to see the general situation."

"We intend to teach the doing of mathematics. The applications of these methods produce the results of mathematics (which usually is only what is taught)... There is also a deliberate policy to force you to think abstractly...it is only through abstraction that any reasonable amount of useful mathematics can be covered. There is simply too much known to continue the older approach of giving detailed results."

"We do not always know what we are talking about. ...Troubles... can be made to arise whenever what is being said includes itself—a self-referral situation."

"When you yourself are responsible for some new application in mathematics... then your reputation... and possibly even human lives, may depend on the results you predict. It is then the need for mathematical rigor will become painfully obvious to you. ...Mathematical rigor is the clarification of the reasoning used in mathematics. ...a closer examination of the numerous "hidden assumptions" is made. ...Over the years there has been a gradually rising standard of rigor; proofs that satisfied the best mathematicians of one generation have been found inadequate by the next generation. Rigor is not a yes-no property of a proof... it is a vague standard of careful treatment that is currently acceptable to a particular group."

"A central problem in teaching mathematics is to communicate a reasonable sense of taste—meaning often when to, or not to, generalize, abstract, or extend something you have just done."

"There is no unique, correct answer in most cases. It is a matter of taste, depending on the circumstances... and the particular age you live in. ...Gradually, you will develop your own taste, and along the way you may occasionally recognize that your taste may be the best one! It is the same as an art course."

"Faced with almost an infinity of details you cannot afford to deal constantly with the specific; you must learn to embrace more and more detail under the cover of generality."

"Calculus systematically evades a great deal of numerical calculation."

"Theorems... record more complex patterns of thinking that once shown to be valid need not be repeated every time they are needed."

"Mathematics, being very different from the natural languages, has its corresponding patterns of thought. Learning these patterns is much more important than any particular result... They are learned by the constant use of the language and cannot be taught in any other fashion."

"Science and mathematics... have added little to our understanding of such things as Truth, Beauty, and Justice. There may be definite limits to the applicability of the scientific method."

"We are mainly interested in the processes... not... in presenting mathematics in its most abstract form. ...we will often begin with concrete forms and then exhibit the process of abstraction."

"There is no agreed upon definition of mathematics, but there is widespread agreement that the essence of mathematics is extension, generalization, and abstraction... [which] often bring increased confidence in the results of a specific application, as well as new viewpoints."

"When a theory is sufficiently general to cover many fields of application, it acquires some "truth" from each of them. Thus... a positive value for generalization in mathematics."

"The beauty of mathematics often makes the subject matter much more attractive and easier to master."

"If you expect to continue learning all your life, you will be teaching yourself much of the time. You must learn to learn, especially the difficult topic of mathematics."

"The assumptions and definitions of mathematics and science come from our intuition, which is based ultimately on experience. They then get shaped by further experience in using them and are occasionally revised. They are not fixed for all eternity."

"In the face of almost infinite useful knowledge, we have adopted the strategy of "information regeneration rather than information retrieval." ...most importantly, you should be able to generate the result you need even if no one has ever done it before you—you will not be dependent on the past to have done everything you will ever need in mathematics."

"Most mathematics books are filled with finished theorems and polished proofs, and to a surprising extent they ignore the methods used to create mathematics. It is as if you merely walked through a picture gallery and never told how to mix paints, how to compose pictures, or all the other "tricks of the trade.""

"Probability is the mathematics of uncertainty. ...many modern theories have uncertainty built into their foundations. Thus learning to think in terms of probability is essential."

"Calculus is the mathematics of change. ...Change is characteristic of the world."

"Probability and statistics are now so obviously necessary tools for understanding many diverse things that we must not ignore them even for the average student."