Richard Hamming - Mathematics Quotes

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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.

Richard Hamming
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 Hamming
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 Hamming
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.

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

Richard Hamming
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.

Richard Hamming
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.

Richard Hamming
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 Hamming
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 Hamming
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.

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

Richard Hamming
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.

Richard Hamming
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.

Richard Hamming
Calculus is the mathematics of change.... Change is characteristic of the world.

Richard Hamming
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.

Richard Hamming
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).

Richard Hamming
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 Hamming
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 Hamming
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.

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

Richard Hamming
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.

Richard Hamming
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 Hamming