“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

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

Adopted from Wikiquote. Last update June 3, 2021. History

Help us to complete the source, original and additional information

Do you have more details about the quote "Our mathematical concepts, structures, ideas have been invented as tools to organise the phenomena of the physical, soc…" by Hans Freudenthal?
Hans Freudenthal photo
Hans Freudenthal 27
Dutch mathematician 1905–1990

Related quotes

Eduardo Torroja photo

“Mathematics is merely a convenient tool by which the designer determines the physical proportions and details of a planned structure in order to transform his ideas… to the actuality of a finished structure.”

Eduardo Torroja (1899–1961) Spanish architect

p, 125
Philosophy of Structures (1958)

Eduard Jan Dijksterhuis photo

“[The mathematical character of Descartes' physics lies in its methodological nature, namely, the] axiomatic structure of the whole system, in the establishment of indubitable foundations and the deduction of the phenomena.”

Eduard Jan Dijksterhuis (1892–1965) Dutch historian

Source: The mechanization of the world picture, 1961, p. 414; as cited in: ‎Marleen Rozemond (2009), Descartes's Dualism. p. 235

Kurt Lewin photo

“The attitude of Aristotelian physics toward lawfulness takes a new direction. So long as lawfulness remained limited to such processes as occurred repeatedly in the same way, it is evident, not only that the young physics still lacked the courage to extend the principle to all physical phenomena, but also that the concept of lawfulness still had a fundamentally historic, a temporally particular, significance. Stress was laid not upon the “general validity” which modem physics understands by lawfulness, but upon the events in the historically given world which displayed the required stability. The highest degree of lawfulness, beyond mere frequency, was characterized by the idea of the always eternal.”

Kurt Lewin (1890–1947) German-American psychologist

Source: 1930s, The conflict between Aristotelian and Galileian modes of thought in contemporary psychology, 1931, p. 147.

“Concepts of uncertainty and information processing are used to integrate the diverse organization design/structure literatures. This approach more fully explicates the concept of congruence which lies at the heart of contingency ideas. The review suggests a contingency approach to design which develops a feasible set of structural alternatives from which the organization can choose.”

David A. Nadler (1948–2015) American organizational theorist

Source: "Information Processing as an Integrating Concept in Organizational Design." 1978, p. 613: Abstract

Didier Sornette photo

“The concept of a random walk is simple but rich for its many applications, not only in finance but also in physics and the description of natural phenomena. It is arguably one of the most founding concepts in modern physics as well as in finance, as it underlies the theories of elementary particles, which are the building blocks of our universe, as well as those describing the complex organization of matter around us.”

Didier Sornette (1957) French scientist

Source: Why Stock Markets Crash - Critical Events in Complex Systems (2003), Chapter 2, Fundamentals Of Financial Markets, p. 38.

“As a science mathematics has been adapted to the description of natural phenomena, and the great practitioners in this field… have never concerned themselves with the logical foundations of mathematics, but have boldly taken a pragmatic view of mathematics as an intellectual machine which works successfully. Description has been verified by further observation, still more strikingly be prediction, and sometimes, more ominously, by control of natural forces. Happily, unresolved problems… still remain as challenges.”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

“Mathematics has also been developed as a philosophy, in the sense in which this term is defined by A. N. Whitehead as 'the endeavor to frame a coherent, logical and necessary system of general ideas in terms of which every element of our experience can be interpreted'. Substitute 'mathematics' for 'experience' and we have an admirable description of its speculative and philosophic development…. Philosophy of mathematics… has its paradoxes and antimonies, and also diverse schools of thought…”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

Eduardo Torroja photo

“Before a man can successfully plan a structure… he must study, from every possible angle, the ultimate purpose of his building. Attention must be directed to the basic structural concept before the mathematical process of calculation is undertaken.”

Eduardo Torroja (1899–1961) Spanish architect

p, 125
Philosophy of Structures (1958)

Hans Freudenthal photo

“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 (1905–1990) Dutch mathematician

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

Henry Flynt photo

“Concept art was meant to replace all of mathematics with an endeavor which involved a Rorschach-blot semantics; and which did not claim to be cognitive, at least not in the inherited sense. Mathematics had already been disconnected from claims of realism; and I was extending that disavowal to a disconnection from claims of a priori truth. Concept art's value consisted in beauty, a beauty which was non-sentimental. Later I would say that its value consisted in "the invention of new mental abilities."”

Henry Flynt (1940) American musician

Popularity had nothing to do with whether this avenue was worth taking.
Henry Flynt. " The Crystallization of Concept Art in 1961 http://www.henryflynt.org/meta_tech/crystal.html," at henryflynt.org, 1994.

Related topics