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The Unreasonable Effectiveness of Mathematics in the Natural Sciences
Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959.
Communications on Pure and Applied Mathematics. 13 (1960) 1–14
The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories. […]
The statement that the laws of nature are written in the language of mathematics was properly made three hundred years ago ; it is now more true than ever before.
Philosophyi.e. physics is written in this grand book — I mean the universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth. (Galilei, Il Saggiatore, 1623).
See also:
Immanuel Kant: Metaphysische Anfangsgründe der Naturwissenschaft (1786), Vorrede: I maintain that…]
David Hilbert, Naturerkennen und Logik, Lecture at the Kongress der Gesellschaft Deutscher Naturforscher und Ärtze, 1930 (pdf, audio)
Michael Atiyah: The Nature of Space, KITP Public Lecture (Oct 2005) [webpage, slides: pdf, pdf]
Michael Atiyah, Robbert Dijkgraaf, Nigel Hitchin, Geometry and physics, Phil. Trans. R. Soc. A 368 1914 (2010) 913-926 [doi:10.1098/rsta.2009.0227, pdf]
on, conversely: The Unreasonable Effectiveness of Physics in the Mathematical Sciences
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