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."

"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."

"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."

"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?"

"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!"

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

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

"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."

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

"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."

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

"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."

"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."

"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."

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

"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 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."

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

"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."

"In forming your plan for your future you need to distinguish three different questions: What is possible? What is likely to happen? What is desirable to have happen? In a sense the first is Science... The second is Engineering.. The third is ethics, morals, or... value judgements. ...you will probably have an idea of how to alter things to make the more desirable future occur ...having a vision is what tends to separate the leaders from the followers."

"The standard process of organizing knowledge into departments, and subderpartments, and further breaking it up into separate courses, tends to conceal the homogeneity of knowledge, and at the same time to omit much which falls between the courses."

"You ought to try to make significant contributions to humanity rather than just get along through life comfortably... the life of trying to achieve excellence in some area is in itself a worthy goal... A life without a struggle on your part to make yourself excellent is hardly a life worth living. ...a life without such a goal... is merely existing..."

"Transmission through space (typically signaling) is the same as transmission through time (storage)."

"The more complex the designed system the more field maintenance must be central to the final design. Only when field maintenance is part of the original design can it be safely controlled... This applies to both mechanical things and to human organizations."

"Somewhere in the mid-to-late 1950s in an address to the President and V.Ps of Bell Laboratories I said, "At present we are doing 1 out of 10 experiments on the computers and 9 in the labs, but before I leave it will be 9 out of 10 on the machines". They did not believe me... by now we do somewhere between 90% to 99% on the machines... And this trend will go on!"

"We must not forget, in all the enthusiasm for computer simulations, occasionally we must look at Nature as She is."

"The people at the bottom do not have the larger, global view, but at the top they do not have the local view of all the details, many of which can often be very important, so either extreme gets poor results."

"An idea which arises in the field, based on the direct experience of the people doing the job, cannot get going in a centrally controlled system... The not invented here (NIH) syndrome is one of the major curses of our society..."

"The Buddha told his disciples, "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". I say the same to you—you must assume the responsibility for what you believe."

"Unfortunately... the ADA language was designed by experts, and it shows all the non-humane features you can expect from them. It is... a typical Computer Science hacking job—do not try and understand what you are doing, just get it running. As a result of this poor psychological design... although a government contract may specify the programming be in ADA, probably over 90% will be done in FORTRAN, debugged, tested, and then painfully, by hand, be converted to a poor ADA program, with a high probability of errors!"

"The fundamentals of language are not understood to this day. ...Until we understand languages of communication involving humans as they are then it is unlikely many of our software problems will vanish."

"When you take a course in Euclidean geometry is not the teacher putting a... learning program into you? ...You enter the course and cannot do problems; the teacher puts into you a program and at the end of the course you can solve such problems. ...Are you sure you are not merely "programmed" in life by what by chance events happens to you?"

"We are beginning to find not only is intelligence not adequately defined so arguments can be settled scientifically, but a lot of other associated words like, computer, learning, information, ideas, decisions, expert behavior—all are a bit fuzzy..."

"This is the type of AI I am interested in—what can the human and machine do together, and not in the competition which can arise. ...There are doubts as to what fraction of the population can compete with computers, even with nice interactive prompting menus."

"The feeling of having free will is deep in us and we are reluctant to give it up for ourselves—but we are often willing to deny it to others!"

"We constantly use the word "simplify", but its meaning depends on what you are going to do next, and there is no uniform definition."

"Perhaps thinking should be measured not by what you do but by how you do it."

"I know that the great Hilbert said, "We will not be driven out of the paradise Cantor has created for us," and I reply, "I see no reason for walking in!" Indeed, in time, as more and more people get used to computers, I am inclined to believe that we here on this Earth will decide that the computable numbers are enough."

"Formal proofs, where there is deliberately no meaning, can convince only formalists, and of the results they themselves seem to deny any meaning. Is that to be the mathematics we are to use to understand the world we live in?"