"The properties of executability and universality associated with programming languages can be combined, in a single language, with the well-known properties of mathematical notation which make it such an effective tool of thought."

“If it is to be effective as a tool of thought, a notation must allow convenient expression not only of notions arising directly from a problem, but also of those arising in subsequent analysis, generalization, and specialization.”

“Overemphasis of efficiency leads to an unfortunate circularity in design: for reasons of efficiency early programming languages reflected the characteristics of the early computers, and each generation of computers reflects the needs of the programming languages of the preceding generation.”

“I was appalled to find that the mathematical notation on which I had been raised failed to fill the needs of the courses I was assigned, and I began work on extensions to notation that might serve. In particular, I adopted the matrix algebra used in my thesis work, the systematic use of matrices and higher-dimensional arrays learned in a course in Tensor Analysis rashly taken in my third year at Queen’s, and the notion of Operators in the sense introduced by Heaviside in his treatment of Maxwell’s equations.”

“With the computer and programming languages, mathematics has newly-acquired tools, and its notation should be reviewed in the light of them. The computer may, in effect, be used as a patient, precise, and knowledgeable "native speaker" of mathematical notation.”

“ACM Turing Award citation”

“It is important to distinguish the difficulty of describing and learning a piece of notation from the difficulty of mastering its implications. Indeed, the very suggestiveness of a notation may make it seem harder to learn because of the many properties it suggests for exploration.”