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.
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§5Kenneth E. Iverson
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Although mathematical notation undoubtedly possesses parsing rules, they are rather loose, sometimes contradictory, and seldom clearly stated. [...] The proliferation of programming languages shows no more uniformity than mathematics. Nevertheless, programming languages do bring a different perspective. [...] Because of their application to a broad range of topics, their strict grammar, and their strict interpretation, programming languages can provide new insights into mathematical notation.
Kenneth E. Iverson
Von Neumann languages do not have useful properties for reasoning about programs. Axiomatic and denotational semantics are precise tools for describing and understanding conventional programs, but they only talk about them and cannot alter their ungainly properties. Unlike von Neumann languages, the language of ordinary algebra is suitable both for stating its laws and for transforming an equation into its solution, all within the "language."
John Backus
Most programming languages are decidedly inferior to mathematical notation and are little used as tools of thought in ways that would be considered significant by, say, an applied mathematician.
Kenneth E. Iverson
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.
Kenneth E. Iverson
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.
Kenneth E. Iverson
Iverson, Kenneth E.
Ives, Charles
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