“No mathematical idea has ever been published in the way it was discovered. Techniques have been developed and are used, if a problem has been solved, to turn the solution procedure upside down, or if it is a larger complex of statements and theories, to turn definitions into propositions, and propositions into definitions, the hot invention into icy beauty. This then if it has affected teaching matter, is the didactical inversion, which as it happens may be anti-didactical.”

— Hans Freudenthal

Rather than behaving anti-didactically, one should recognise that the learner is entitled to recapitulate in a fashion of mankind. Not in the trivial matter of an abridged version, but equally we cannot require the new generation to start at the point where their predecessors left off.
Source: The Concept and the Role of the Model in Mathematics and Natural and Social Sciences (1961), p. ix

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Hans Freudenthal 27

Dutch mathematician 1905–1990

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“No mathematical idea has ever been published in the way it was discovered. Techniques have been developed and are used, if a problem has been solved, to turn the solution procedure upside down, or if it is a larger complex of statements and theories, to turn definitions into propositions, and propositions into definitions, the hot invention into icy beauty. This then if it has affected teaching matter, is the didactical inversion, which as it happens may be anti-didactical. Rather than behaving anti-didactically, one should recognise that the learner is entitled to recapitulate in a fashion of mankind. Not in the trivial matter of an abridged version, but equally we cannot require the new generation to start at the point where their predecessors left off.”

Hans Freudenthal (1905–1990) Dutch mathematician

Source: The Concept and the Role of the Model in Mathematics and Natural and Social Sciences (1961), p. ix

“Our mathematical concepts, structures, ideas have been invented as tools to organise the phenomena of the physical, social and mental world. Phenomenology of a mathematical concept, structure, or idea means describing it in its relation to the phenomena for which it was created, and to which it has extended in the learning process of mankind, and, as far as this description is concerned with the learning process of the young generation, it is didactical phenomenology, a way to show the teacher the places where the learner might step into the learning process of mankind.”

Hans Freudenthal (1905–1990) Dutch mathematician

Source: The Concept and the Role of the Model in Mathematics and Natural and Social Sciences (1961), p. ix

“The definition I use for pattern is an idea that has been useful in one practical context and will probably be useful in others”

Martin Fowler (1963) British programmer

Source: Analysis Patterns: Reusable Object Models, 1997, p. 8

“Complexity has always been difficult to resolve and to understand. Evolutionary biologists were among the first scientists to recognize this problem, when they dug their way toward new theories about evolution. They discovered that matter does not lack purpose.”

L. K. Samuels (1951) American writer

Source: In Defense of Chaos: The Chaology of Politics, Economics and Human Action, (2013), p. 129

“The " general" definition may give you no hint as to the way in which a particular assertion is meant to be interpreted. When, therefore, as in formal debate, a proposition is advanced by persons other than those who discuss it, clearness in defining is not sufficient. In such cases a definition must have two qualities: it must be clear, and it must be satisfactory to the persons addressed. They must know what the speaker means, and they must be convinced that his meaning is the correct one, — correct for that particular proposition at the time and place and under the conditions of that particular debate. Otherwise, they may object in the end that he has proved, not the given proposition, but another which he has substituted for it by means of arbitrary definitions.”

William Trufant Foster (1879–1950) American economist

Source: Argumentation and debating, 1908, p. 27; partly cited in: Branham (2013, p. 39)

“It is commonly said that a pattern, however it is written, has four essential parts: a statement of the context where the pattern is useful, the problem that the pattern addresses, the forces that play in forming a solution, and the solution that resolves those forces. … it supports the definition of a pattern as "a solution to a problem in a context", a definition that [unfortunately] fixes the bounds of the pattern to a single problem-solution pair”

Martin Fowler (1963) British programmer

Source: Analysis Patterns: Reusable Object Models, 1997, p. 6

“In spite the mountains of books written about art, no precise definition of art has been constructed. And the reason for this is that the conception of art has been based on the conception of beauty.”

Leo Tolstoy (1828–1910) Russian writer

What is Art? (1897)

“Mankind has never been renowned for accepting logical solutions to its most serious problems. As a rule, the degre of logicality of solutions seemed to vary inversely with the urgency or seriousness of the problems.”

Edmund Cooper book Five to Twelve

Five to Twelve (1968)

“A theorem is a proposition which is a strict logical consequence of certain definitions and other propositions”

Anatol Rapoport (1911–2007) Russian-born American mathematical psychologist

Anatol Rapoport. " Various meanings of “theory”." http://www.acsu.buffalo.edu/~fczagare/PSC%20504/Rapoport%20(1958).pdf American Political Science Review 52.04 (1958): 972-988.
1950s

“The continuity of our science has not been affected by all these turbulent happenings, as the older theories have always been included as limiting cases in the new ones.”

Max Born (1882–1970) physicist

As quoted in Beyond Positivism and Relativism : Theory, Method, and Evidence (1996) by Larry Laudan, p. 259

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