On Notation (1975-1999)

"Mechanical drawings and blueprints are not mere pictures, but a complete and rich language. In blueprint language, scientific, mathematical, and geometric formulations, notations, mensurations, and naming do not merely describe an object or process, they actually model it. Because of broad differences in subject, purpose, roles, and the needs of the people who use them, many forms of blueprint have evolved, but all rigorously present well structured information in understandable form." (Douglas T Ross, "Structured analysis (SA): A language for communicating ideas", 1977)

"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, "Notation as a Tool of Thought", 1979)

"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." (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)

"This difficulty lead very gradually to the recognition of the need for a shorthand to make the sequence of operations easily comprehensible: here we have the problem of notation, which crops up again after every introduction of new objects, and which will probably never cease to torment mathematicians." (Jean Dieudonné, "Mathematics: The Music of Reason", 1992)

"This ambiguity is another example of a growing problem with mathematical notation: There aren't enough squiggles to go around." (Jim Blinn, Jim Blinn's Corner: Dirty Pixels", 1996)

"Mathematical notation is for the scientist what musical notation is for the composer." (John Holland, "Emergence: From Chaos to Order", 1998)

"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)

"Mathematicians have always appreciated clever notations; but symbolism is usually seen as a tool - it's what the tool does that we really care about. Fair enough. But if we want a richer appreciation of mathematics, we should focus some of our energy on this remarkable tool - notation. Besides mathematics, poetry alone works wonders with it." (James R Brown, "Philosophy of Mathematics", 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)

"Whatever the ins and outs of poetry, one thing is clear: the manner of expression - notation - is fundamental. It is the same with mathematics - not in the aesthetic sense that the beauty of mathematics is tied up with how it is expressed - but in the sense that mathematical truths are revealed, exploited and developed by various notational innovations." (James R Brown, "Philosophy of Mathematics", 1999)