Richard Hamming

"The purpose of computing is insight, not numbers."

"As a practicing computer veteran, this reviewer has the habit of looking at the hypothesis of a theorem and asking:"

"Typing is no substitute for thinking."

"The only generally agreed upon definition of mathematics is "Mathematics is what mathematicians do." [...] In the face of this difficulty [of defining "computer science"] many people, including myself at times, feel that we should ignore the discussion and get on with doing it. But as George Forsythe points out so well in a recent article*, it does matter what people in Washington D.C. think computer science is. According to him, they tend to feel that it is a part of applied mathematics and therefore turn to the mathematicians for advice in the granting of funds. And it is not greatly different elsewhere; in both industry and the universities you can often still see traces of where computing first started, whether in electrical engineering, physics, mathematics, or even business. Evidently the picture which people have of a subject can significantly affect its subsequent development. Therefore, although we cannot hope to settle the question definitively, we need frequently to examine and to air our views on what our subject is and should become."

"Without real experience in using the computer to get useful results the computer science major is apt to know all about the marvelous tool except how to use it. Such a person is a mere technician, skilled in manipulating the tool but with little sense of how and when to use it for its basic purposes."

"Indeed, one of my major complaints about the computer field is that whereas Newton could say, "If I have seen a little farther than others, it is because I have stood on the shoulders of giants," I am forced to say, "Today we stand on each other's feet." Perhaps the central problem we face in all of computer science is how we are to get to the situation where we build on top of the work of others rather than redoing so much of it in a trivially different way. Science is supposed to be cumulative, not almost endless duplication of the same kind of things."

"The Postulates of Mathematics Were Not on the Stone Tablets that Moses Brought Down from Mt. Sinai."

"The idea that theorems follow from the postulates does not correspond to simple observation. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid's postulates came from the Pythagorean theorem, not the other way around."

"Just as there are odors that dogs can smell and we cannot, as well as sounds that dogs can hear and we cannot, so too there are wavelengths of light we cannot see and flavors we cannot taste. Why then, given our brains wired the way they are, does the remark, "Perhaps there are thoughts we cannot think," surprise you?"

"The calculus is probably the most useful single branch of mathematics. ...I have found the ability to do simple calculus, easily and reliably, was the most valuable part of mathematics I ever learned."

"Understanding the methods of calculus is vital to the creative use of mathematics... Without this mastery the average scientist or engineer, or any other user of mathematics, will be perpetually stunted in development, and will at best be able to follow only what the textbooks say; with mastery, new things can be done, even in old, well-established fields."

"Probability plays a central role in many fields, from quantum mechanics to information theory, and even older fields use probability now that the presence of "noise" is officially admitted. The newer aspects of many fields start with the admission of uncertainty."

"Continuous distributions are basic to the theory of probability and statistics, and the calculus is necessary to handle them with any ease."

"Statistics should be taught early so that the concepts are absorbed by the student's flexible, adaptable mind before it is too late."

"Increasingly... the application of mathematics to the real world involves discrete mathematics... the nature of the discrete is often most clearly revealed through the continuous models of both calculus and probability. Without continuous mathematics, the study of discrete mathematics soon becomes trivial and very limited. ...The two topics, discrete and continuous mathematics, are both ill served by being rigidly separated."

"The methods of mathematics are the main topic of the course, not a long list of finished mathematical results with such highly polished proofs that the poor student can only marvel at the results, with no hope of understanding how mathematics is actually created by practicing mathematicians."

"You live in an age that is dominated by science and engineering. ...Thus if you wish to be effective in the world and to achieve the things that you want, it is necessary to understand both science and engineering (and those require mathematics)."

"Any unwillingness to learn mathematics today can greatly restrict your possibilities tomorrow."

"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."

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

"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."

"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.""

"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."

"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."

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

"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."

"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."

"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."

"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."

"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."

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

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

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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..."

"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."

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

"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."

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

"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."

"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."

"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."