"Perhaps the best way to approach the question of what mathematics is, is to start at the beginning. In the far distant prehistoric past, where we must look for the beginnings of mathematics, there were already four major faces of mathematics. First, there was the ability to carry on the long chains of close reasoning that to this day characterize much of mathematics. Second, there was geometry, leading through the concept of continuity to topology and beyond. Third, there was number, leading to arithmetic, algebra, and beyond. Finally there was artistic taste, which plays so large a role in modern mathematics. There are, of course, many different kinds of beauty in mathematics. In number theory it seems to be mainly the beauty of the almost infinite detail; in abstract algebra the beauty is mainly in the generality. Various areas of mathematics thus have various standards of aesthetics." (Richard Hamming, "The Unreasonable Effectiveness of Mathematics", The American Mathematical Monthly Vol. 87 (2), 1980)
"Some people believe that a theorem is proved when a logically correct proof is given; but some people believe a theorem is proved only when the student sees why it is inevitably true. The author tends to belong to this second school of thought." (Richard W Hamming, "Coding and Information Theory", 1980)"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." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"Any unwillingness to learn mathematics today can greatly restrict your possibilities tomorrow." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"Calculus systematically evades a great deal of numerical calculation." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"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."(Richard Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"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." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)"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." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"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."
"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." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"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’." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"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." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"Probability is the mathematics of uncertainty. Not only do we constantly face situations in which there is neither adequate data nor an adequate theory, but many modem theories have uncertainty built into their foundations. Thus learning to think in terms of probability is essential. Statistics is the reverse of probability (glibly speaking). In probability you go from the model of the situation to what you expect to see; in statistics you have the observations and you wish to estimate features of the underlying model." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)"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." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"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." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"Statistics should be taught early so that the concepts are absorbed by the student's flexible, adaptable mind before it is too late." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"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." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"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." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"The beauty of mathematics often makes the subject matter much more attractive and easier to master." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"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." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"Theorems […] record more complex patterns of thinking that once shown to be valid need not be repeated every time they are needed." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"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." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"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."
"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." (Richard W Hamming, "You and Your Research", 1986)
"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." (Richard W Hamming, "You and Your Research", 1986)
"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." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)
"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." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)
"Every field of knowledge has its subject matter and its
methods, along with a style for handling them. The field of Probability has a
great deal of the Art component in it-not only is the subject matter rather
different from that of other fields, but at present the techniques are not well
organized into systematic methods. As a result each problem has to be
"looked at in the right way" to make it easy to solve. Thus in
probability theory there is a great deal of art in setting up the model, in
solving the problem, and in applying the results back to the real world actions
that will follow.
"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?" (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)
"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." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)
"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." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)
"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."
"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." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)
"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."
"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."
"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."
"Unforeseen technological inventions can completely upset the most careful predictions." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)
"We constantly use the word 'simplify' , but its meaning depends on what you are going to do next, and there is no uniform definition." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)
"We will use the convenient expression 'chosen at random' to mean that the probabilities of the events in the sample space are all the same unless some modifying words are near to the words 'at random'. Usually we will compute the probability of the outcome based on the uniform probability model since that is very common in modeling simple situations. However, a uniform distribution does not imply that it comes from a random source; […]" (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)
"All things which are proved to be impossible must obviously rest on some assumptions, and when one or more of these assumptions are not true then the impossibility proof fails - but the expert seldom remembers to carefully inspect the assumptions before making their 'impossible' statements." (Richard Hamming, "The Art of Doing Science and Engineering: Learning to Learn", 1997)
"Another curious phenomenon you may meet is in fitting data to a model there are errors in both the data and the model. For example, a normal distribution may be assumed, but the tails may in fact be larger or smaller than the model predicts, and possibly no negative values can occur although the normal distribution allows them. Thus there are two sources of error. As your ability to make more accurate measurements increases the error due to the model becomes an increasing part of the error." (Richard Hamming, "The Art of Doing Science and Engineering: Learning to Learn", 1997)
"If you want to be certain then you are apt to be obsolete." (Richard Hamming, "The Art of Doing Science and Engineering: Learning to Learn", 1997)
"It has been my experience, as well as many others who have looked, data is generally much less accurate than it is advertised to be. This is not a trivial point - we depend on initial data for many decisions, as well as for the input data for simulations which result in decisions." (Richard Hamming, "The Art of Doing Science and Engineering: Learning to Learn", 1997)
"Probably the most important tool in creativity is the use of an analogy. Something seems like something else which we knew in the past. Wide acquaintance with various fields of knowledge is thus a help - provided you have the knowledge filed away so it is available when needed, rather than to be found only when led directly to it. This flexible access to pieces of knowledge seems to come from looking at knowledge while you are acquiring it from many different angles, turning over any new idea to see its many sides before filing it away." (Richard Hamming, "The Art of Doing Science and Engineering: Learning to Learn", 1997)
"Sometimes the mathematician can accurately estimate the frequency content of the signal (possibly from the answer being computed), but usually you have to go to the designers and get their best estimates. A competent designer should be able to deliver such estimates, and if they cannot then you need to do a lot of exploring of the solutions to estimate this critical number, the sampling rate of the digital solution. The step by step solution of a problem is actually sampling the function, and you can use adaptive methods of step by step solution if you wish." (Richard Hamming, "The Art of Doing Science and Engineering: Learning to Learn", 1997)
"Systems engineering is the attempt to keep at all times the larger goals in mind and to translate local actions into global results. But there is no single larger picture. [...] Systems engineering is a hard trade to follow; it is so easy to get lost in the details! Easy to say; hard to do." (Richard Hamming, "The Art of Doing Science and Engineering: Learning to Learn", 1997)
"The desire for excellence is an essential feature for doing great work. Without such a goal you will tend to wander like a drunken sailor. The sailor takes one step in one direction and the next in some independent direction. As a result the steps tend to cancel each other, and the expected distance from the starting point is proportional to the square root of the number of steps taken. With a vision of excellence, and with the goal of doing significant work, there is tendency for the steps to go in the same direction and thus go a distance proportional to the number of steps taken, which in a lifetime is a large number indeed." (Richard Hamming, "The Art of Doing Science and Engineering: Learning to Learn", 1997)
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