Richard Hamming: Quotes about mathematics

Richard Hamming was American mathematician and information theorist. Explore interesting quotes on mathematics.
Richard Hamming: 180 quotes1 like

“Without continuous mathematics, the study of discrete mathematics soon becomes trivial and very limited. …The two topics”

Richard Hamming

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)
Context: 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.

“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”

Richard Hamming

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)
Context: 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.

“The assumptions and definitions of mathematics and science come from our intuition”

Richard Hamming

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)
Context: 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.

“Calculus is the mathematics of change. …Change is characteristic of the world.”

Richard Hamming

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

“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

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

“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

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

“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.”

Richard Hamming

Hamming cites Forsythe, G.E., "What to do until the computer scientist comes", Am. Math. Monthly 75 (5), May 1968, p. 454-461.
One Man's View of Computer Science (1969)

“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

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

“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

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 Hamming

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

“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

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

“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

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 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 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 Hamming

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

“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”

Richard Hamming

and those require mathematics
Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

“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

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

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

Richard Hamming

Emphatic capitalization in original.
The Unreasonable Effectiveness of Mathematics (1980)

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

Richard Hamming

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

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

Richard Hamming

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)