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Quis costodiet ipsos custodies? (Who will watch the watchers?)

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Kenneth E. Iverson

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.

Applied, Littles, Mathematical, Tools, Used, Would Be
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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.

Confusion, Execution, Language, Potential, Precision, Provided

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.

Difficulty, Important, May, Pieces, Suggestiveness, Suggests

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.

Different, Interpretation, Notation, Perspective, Possesses, Topics

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.

Circularity, Design, Needs, Reflected

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.

Definitions, Greater, Mathematics, Possessing, Practice, Tests
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