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I remember reading somewhere (perhaps in The Mathematical Experience) that Richard Courant wrote something to the effect that, without applications to guide the river of mathematical discovery, mathematical research is doomed to wander aimlessly. There was indeed some river imagery, if I remember correctly. What is this quotation exactly, and where did it come from?

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There is river imagery in a passage from the preface (written by Courant) to Courant-Hilbert's Methods of Mathematical Physics, vol. 1, but the point of the metaphor is somewhat different. It is excessive specialization rather than disconnect from applications that is seen as the danger, and it is drying out research rather then aimlessness that is feared:

"Since the seventeenth century, physical intuition has served as a vital source for mathematical porblems and methods. Recent trends and fashions have, however, weakened the connection between mathematics and physics; mathematicians, turning away from their roots of mathematics in intuition, have concentrated on refinement and emphasized the postulated side of mathematics, and at other times have overlooked the unity of their science with physics and other fields. In many cases, physicists have ceased to appreciate the attitudes of mathematicians. This rift is unquestionably a serious threat to science as a whole; the broad stream of scientific development may split into smaller and smaller rivulets and dry out. It seems therefore important to direct our efforts towards reuniting divergent trends by classifying the common features and interconnections of many distinct and diverse scientific facts. Only thus can the student attain some mastery of the material and the basis be prepared for further organic development of research."

That is not so say that Courant did not see applications as vital to mathematics. I looked through Reid's biography of Courant, and there is plenty of material on that, but I did not find any similar sounding quote. You may also want to look at Courant's article Mathematics in the Modern World in Scientific American (it is behind paywall).

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I quote from the book Physically Speaking A Dictionary of Quotations on Physics and Astronomy by Carl C Gaither. In the entry for Courant, there is discussion of rivulets drying up...

Courant, Richard

Since the seventeenth century, physical intuition has served as a vital source for mathematical problems and methods. Recent trends and fashions have, however, weakened the connection between mathematics and physics; mathematicians, turning away from the roots of mathematics in intuition, have concentrated on refinement and emphasized the postulated side of mathematics, and at times have overlooked the unity of their science with physics and other fields. In many cases, physicists have ceased to appreciate the attitudes of mathematicians. This rift is unquestionably a serious threat to science as a whole; the broad stream of scientific development may split into smaller and smaller rivulets and dry out. It seems therefore important to direct our efforts toward reuniting divergent trends by classifying the common features and interconnections of many distinct and diverse scientific facts.

Methods of Mathematical Physics (pp. v-vi)

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Imagery slightly different from what is described in the question appeared in Richard Courant, David Hilbert, "Methoden der mathematischen Physik. Erster Band," Berlin: Julius Springer 1924.

In the preface we read (emphasis mine):

Von jeher hat die Mathematik mächtige Antriebe aus den engen Beziehungen gewonnen, welche zwischen den Problemen und Methoden der Analysis und den anschaulichen Vorstellungen der Physik bestehen. Erst die letzten Jahrzehnte brachten eine Lockerung dieses Zusammenhanges, indem sich die mathematische Forschung vielfach von ihren anschaulichen Ausgangspunkten ablöste und insbesondere in der Analysis manchmal allzu ausschließlich um Verfeinerung ihrer Methoden und Zuspitzung ihrer Begriffe bemühte. So kommt es, dass viele Vertreter der Analysis das Bewußtsein der Zusammengehörigkeit ihrer Wissenschaft mit der Physik und anderen Gebieten verloren haben, während auf der anderen Seite oft den Physikern das Verständnis für die Probleme und Methoden der Mathematiker, ja sogar für deren ganze Interessensphäre und Sprache abhandengekommen ist. Ohne Zweifel liegt in dieser Tendenz eine Bedrohung für die Wissenschaft überhaupt; der Strom der wissenschaftlichen Entwicklung ist in Gefahr sich weiter und weiter zu verästeln, zu versickern und auszutrocknen. Soll er diesem Geschick entgehen, so müssen wir einen guten Teil unserer Kräfte darauf richten, Getrenntes wieder zu vereinigen, indem wir unter zusammenfassenden Gesichtspunkten die innere Zusammenhänge der mannigfaltigen Tatsachen klarlegen.

My translation: "Mathematics has always derived powerful impulses from the close relationship that exists between the problems and methods of analysis and the concrete ideas of physics. Only recent decades have brought a relaxation of this connection, in that mathematical research detached itself from its concrete starting points and in analysis in particular became at times far too exclusively concerned with its methods and the refinement of its terminology. As a result, many representatives of analysis have lost their awareness of the connection between their science and physics and other fields, while on the other hand physicists have often lost the understanding for the problems and methods of mathematicians, even for their entire sphere of interest and language. Without doubt this development represents a threat to the entirety of science; the current of scientific development is in danger of splitting itself up further and further, of draining away and drying up. If it is to escape this fate, me must concentrate a good deal of our efforts on reconnecting what has been split apart by clarifying, from a unifying perspective, the inner connection of the diverse facts."

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