“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

Source: Math for the Layman (1999), Ch. 10, §D

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Kenneth E. Iverson 12

Canadian computer scientist 1920–2004

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“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 (1920–2004) Canadian computer scientist

§5
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 (1920–2004) Canadian computer scientist

Source: Math for the Layman (1999), Ch. 10, §D

“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 (1920–2004) Canadian computer scientist

Source: Math for the Layman (1999), Ch. 10, §D

“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 (1920–2004) Canadian computer scientist

§0
Notation as a Tool of Thought (1979)

“The initial motive for developing APL was to provide a tool for writing and teaching. Although APL has been exploited mostly in commercial programming, I continue to believe that its most important use remains to be exploited: as a simple, precise, executable notation for the teaching of a wide range of subjects.”

Kenneth E. Iverson (1920–2004) Canadian computer scientist

"A Personal View of APL", IBM Systems Journal, 30 (4), 1991

“The calculated ambiguity of expression is based on the confusion of experience as reflected in the architectural program.”

Robert Venturi book Complexity and Contradiction in Architecture

3. Ambiguity
Complexity and Contradiction in Architecture (1966)

“The real problem in speech is not precise language. The problem is clear language. The desire is to have the idea clearly communicated to the other person. It is only necessary to be precise when there is some doubt as to the meaning of a phrase, and then the precision should be put in the place where the doubt exists. It is really quite impossible to say anything with absolute precision, unless that thing is so abstracted from the real world as to not represent any real thing.Pure mathematics is just such an abstraction from the real world, and pure mathematics does have a special precise language for dealing with its own special and technical subjects. But this precise language is not precise in any sense if you deal with real objects of the world, and it is only pedantic and quite confusing to use it unless there are some special subtleties which have to be carefully distinguished.”

Richard Feynman (1918–1988) American theoretical physicist

" New Textbooks for the "New" Mathematics http://calteches.library.caltech.edu/2362/1/feynman.pdf", Engineering and Science volume 28, number 6 (March 1965) p. 9-15 at p. 14
Paraphrased as "Precise language is not the problem. Clear language is the problem."

“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 (1924–2007) American computer scientist

"Can Programming Be Liberated From the von Neumann Style?" http://dl.acm.org/ft_gateway.cfm?id=1283933&type=pdf, 1977 Turing Award Lecture, Communications of the ACM 21 (8), (August 1978): pp. 639-640

“I know that I disagree with many other UML experts, but there is no magic about UML. If you can generate code from a model, then it is programming language. And UML is not a well-designed programming language.
The most important reason is that it lacks a well-defined point of view, partly by intent and partly because of the tyranny of the OMG standardization process that tries to provide everything to everybody. It doesn't have a well-defined underlying set of assumptions about memory, storage, concurrency, or almost anything else. How can you program in such a language?
The fact is that UML and other modelling language are not meant to be executable. The point of models is that they are imprecise and ambiguous. This drove many theoreticians crazy so they tried to make UML "precise", but models are imprecise for a reason: we leave out things that have a small effect so we can concentrate on the things that have big or global effects. That's how it works in physics models: you model the big effect (such as the gravitation from the sun) and then you treat the smaller effects as perturbation to the basic model (such as the effects of the planets on each other). If you tried to solve the entire set of equations directly in full detail, you couldn't do anything.”

James Rumbaugh (1947) Computer scientist, software engineer

James Rumbaugh in Federico Biancuzzi and Shane Warden eds. (2009) Masterminds of Programming. p. 339; cited in " Quote by James Rumbaugh http://www.ptidej.net/course/cse3009/winter13/resources/james" on ptidej.net. Last updated 2013-04-09 by guehene; Rumbaugh is responding to the question: "What do you think of using UML to generate implementation code?"

“The musical scale is a convention which circumscribes the area of potentiality and permits construction within those limits in its own particular symmetry.”

Iannis Xenakis (1922–2001) Greek composer

Formalized Music

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