"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, "Notation as a Tool of Thought", 1979)
"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." (Kenneth E Iverson, "Notation as a Tool of Thought",
1979)
"The practice of first developing a clear and precise definition of a process without regard for efficiency, and then using it as a guide and a test in exploring equivalent processes possessing other characteristics, such as greater efficiency, is very common in mathematics. It is a very fruitful practice which should not be blighted by premature emphasis on efficiency in computer execution." (Kenneth E Iverson, "Notation as a Tool of Thought", 1979)
"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." (Kenneth E Iverson, "Notation as a Tool of Thought", 1979)
"The utility of a language as a tool of thought increases with the range of topics it can treat, but decreases with the amount of vocabulary and the complexity of grammatical rules which the user must keep in mind. Economy of notation is therefore important." (Kenneth E Iverson, "Notation as a Tool of Thought", 1979)
"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, "Math for the Layman", 1999)
"The precision provided (or enforced) by programming languages and their execution can identify lacunas, ambiguities, and other areas of potential confusion in conventional [mathematical] notation." (Kenneth E Iverson, "Math for the Layman", 1999)
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