Richard Hamming on Knowledge

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

"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."

"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."

"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."

"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 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."

"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?"

"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?"

"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."

"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."

"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."

"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..."