„It is with Bernhard Riemann's work that we finally have the mathematical glasses to explore such worlds of the mind. And now my journey through the abstract world of 20th century mathematics has revealed that maths is the true language that the universe is written in. They key to understanding the world around us. Mathematicians aren't motivated by money and material gain, or even by practical applications of their work. For us it's the glory of solving one of the great unsolved problems that have outwitted previous generations of mathematicians. David Hilbert was right; it’s the unsolved problems of mathematics which make it a living subject. Which obsess each new generation of mathematicians. Despite all the things we've discovered over the last 7 millennia, there are still many things we don't understand. And its Hilbert’s call of "We must know, we will know" which drives mathematics.“

—  Marcus du Sautoy, Conclusion in BBC's The Story of Maths, episode 4

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Nicolaus Copernicus photo

„Mathematics is written for mathematicians“

—  Nicolaus Copernicus Renaissance mathematician, Polish astronomer, physician 1473 - 1543
Context: If perchance there should be foolish speakers who, together with those ignorant of all mathematics, will take it upon themselves to decide concerning these things, and because of some place in the Scriptures wickedly distorted to their purpose, should dare to assail this my work, they are of no importance to me, to such an extent do I despise their judgment as rash. For it is not unknown that Lactantius, the writer celebrated in other ways but very little in mathematics, spoke somewhat childishly of the shape of the earth when he derided those who declared the earth had the shape of a ball. So it ought not to surprise students if such should laugh at us also. Mathematics is written for mathematicians to whom these our labors, if I am not mistaken, will appear to contribute something even to the ecclesiastical state the headship of which your Holiness now occupies. (Author's preface to de revolutionibus) http://la.wikisource.org/wiki/Pagina:Nicolai_Copernici_torinensis_De_revolutionibus_orbium_coelestium.djvu/8 Translation as quoted in The Gradual Acceptance of the Copernican Theory of the Universe (1917) by Dorothy Stimson, p. 115

R. G. Collingwood photo

„The theory of the nature of mathematics is extremely reactionary. We do not subscribe to the fairly recent notion that mathematics is an abstract language based, say, on set theory. In many ways, it is unfortunate that philosophers and mathematicians like Russell and Hilbert were able to tell such a convincing story about the meaning-free formalism of mathematics. In Greek, mathematics simply meant learning, and we have adapted this... to define the term as "learing to decide."“

—  C. West Churchman American philosopher and systems scientist 1913 - 2004
1960s - 1970s, Mathematics is a way of preparing for decisions through thinking. Sets and classes provide one way to subdivide a problem for decision preparation; a set derives its meaning from decision making, and not vice versa. C. West Churchman, Leonard Auerbach, Simcha Sadan, Thinking for Decisions: Deductive Quantitative Methods (1975) Preface.

Nicolaus Copernicus photo

„Mathematics is written for mathematicians, to whom these my labours“

—  Nicolaus Copernicus, livro De revolutionibus orbium coelestium
De revolutionibus orbium coelestium (1543), Context: Nor do I doubt that skilled and scholarly mathematicians will agree with me if, what philosophy requires from the beginning, they will examine and judge, not casually but deeply, what I have gathered together in this book to prove these things.... Mathematics is written for mathematicians, to whom these my labours, if I am not mistaken, will appear to contribute something.... What... I may have achieved in this, I leave to the decision of your Holiness especially, and to all other learned mathematicians.... If perchance there should be foolish speakers who, together with those ignorant of all mathematics, will take it upon themselves to decide concerning these things, and because of some place in the Scriptures wickedly distorted to their purpose, should dare to assail this my work, they are of no importance to me, to such an extent do I despise their judgment as rash. Preface Letter to Pope Paul III as quoted by Edwin Arthur Burtt in The Metaphysical Foundations of Modern Physical Science (1925)

„For the great majority of mathematicians, mathematics is“

—  George Frederick James Temple British mathematician 1901 - 1992
100 Years of Mathematics: a Personal Viewpoint (1981), Context: For the great majority of mathematicians, mathematics is... a whole world of invention and discovery—an art. The construction of a new theorem, the intuition of some new principle, or the creation of a new branch of mathematics is the triumph of the creative imagination of the mathematician, which can be compared to that of a poet, the painter and the sculptor.

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„We wish to see… the typical attitude of the scientist who uses mathematics to understand the world around us. …In the solution of a problem …there are typically three phases.“

—  George Pólya Hungarian mathematician 1887 - 1985
Mathematical Methods in Science (1977), Context: We wish to see... the typical attitude of the scientist who uses mathematics to understand the world around us.... In the solution of a problem... there are typically three phases. The first phase is entirely or almost entirely a matter of physics; the third, a matter of mathematics; and the intermediate phase, a transition from physics to mathematics. The first phase is the formulation of the physical hypothesis or conjecture; the second, its translation into equations; the third, the solution of the equations. Each phase calls for a different kind of work and demands a different attitude.<!--p.164

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Vladimir I. Arnold photo

„At the beginning of this century a self-destructive democratic principle was advanced in mathematics (especially by Hilbert), according to which all axiom systems have equal right to be analyzed, and the value of a mathematical achievement is determined, not by its significance and usefulness as in other sciences, but by its difficulty alone, as in mountaineering.“

—  Vladimir I. Arnold Russian mathematician 1937 - 2010
Context: At the beginning of this century a self-destructive democratic principle was advanced in mathematics (especially by Hilbert), according to which all axiom systems have equal right to be analyzed, and the value of a mathematical achievement is determined, not by its significance and usefulness as in other sciences, but by its difficulty alone, as in mountaineering. This principle quickly led mathematicians to break from physics and to separate from all other sciences. In the eyes of all normal people, they were transformed into a sinister priestly caste... Bizarre questions like Fermat's problem or problems on sums of prime numbers were elevated to supposedly central problems of mathematics. "Will Mathematics Survive? Report on the Zurich Congress" in The Mathematical Intelligencer, Vol. 17, no. 3 (1995), pp. 6–10.

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„Since Leibniz there has perhaps been no man who has had a full command of all the intellectual activity of his day. Since that time, science has been increasingly the task of specialists, in fields which show a tendency to grow progressively narrower… Today there are few scholars who can call themselves mathematicians or physicists or biologists without restriction. A man may be a topologist or a coleopterist. He will be filled with the jargon of his field, and will know all its literature and all its ramifications, but, more frequently than not, he will regard the next subject as something belonging to his colleague three doors down the corridor, and will consider any interest in it on his own part as an unwarrantable breach of privacy… There are fields of scientific work, as we shall see in the body of this book, which have been explored from the different sides of pure mathematics, statistics, electrical engineering, and neurophysiology; in which every single notion receives a separate name from each group, and in which important work has been triplicated or quadruplicated, while still other important work is delayed by the unavailability in one field of results that may have already become classical in the next field.
It is these boundary regions which offer the richest opportunities to the qualified investigator. They are at the same time the most refractory to the accepted techniques of mass attack and the division of labor. If the difficulty of a physiological problem is mathematical in essence, then physiologists ignorant of mathematics will get precisely as far as one physiologists ignorant of mathematics, and no further. If a physiologist who knows no mathematics works together with a mathematician who knows no physiology, the one will be unable to state his problem in terms that the other can manipulate, and the second will be unable to put the answers in any form that the first can understand… A proper exploration of these blank spaces on the map of science could only be made by a team of scientists, each a specialist in his own field but each possessing a thoroughly sound and trained acquaintance with the fields of his neighbors; all in the habit of working together, of knowing one another's intellectual customs, and of recognizing the significance of a colleague's new suggestion before it has taken on a full formal expression. The mathematician need not have the skill to conduct a physiological experiment, but he must have the skill to understand one, to criticize one, and to suggest one. The physiologist need not be able to prove a certain mathematical theorem, but he must be able to grasp its physiological significance and to tell the mathematician for what he should look. We had dreamed for years of an institution of independent scientists, working together in one of these backwoods of science, not as subordinates of some great executive officer, but joined by the desire, indeed by the spiritual necessity, to understand the region as a whole, and to lend one another the strength of that understanding.“

—  Norbert Wiener, livro Cybernetics: Or Control and Communication in the Animal and the Machine
Cybernetics: Or Control and Communication in the Animal and the Machine (1948), p. 2-4; As cited in: George Klir (2001) Facets of Systems Science, p. 47-48

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