Music and Mathematics VI

"Mathematics is the instrument by which the engineer tunnels out mountains, bridges our rivers, constructs our aqueducts, erects out factories and makes them musical by the busy hum of spindles. Take away the results of the reasoning of mathematics, and there would go with it nearly all the material achievements which give convenience and glory to modern civilization." (Edward Brooks, "Mental Science and Culture", 1891)

"The best proofs in mathematics are short and crisp like epigrams, and the longest have swings and rhythms that are like music." (Scott Buchanan, Poetry and Mathematics, 1929)

"Mathematics, like music and poetry, is a creation of the mind; [...] the primary task of the mathematician, like that of any other artist, is to extend man's mental horizon by representation and interpretation." (Graham Sutton, "Mathematics in Action", 1954)

"The question ‘What is mathematics?’ cannot be answered meaningfully by philosophical generalities, semantic definitions or journalistic circumlocutions. Such characterizations also fail to do justice to music or painting. No one can form an appreciation of these arts without some experience with rhythm, harmony and structure, or with form, color and composition. For the appreciation of mathematics actual contact with its substance is even more necessary." (Richard Courant, "Mathematics in the Modern World", Scientific American Vol. 211 (3), 1964)

"The structures with which mathematics deals are more like lace, the leaves of trees, and the play of light and shadow on a human face, than they are like buildings and machines, the least of their representatives. The best proofs in mathematics are short and crisp like epigrams, and the longest have swings and rhythms that are like music. The structures of mathematics and the propositions about them are ways for the imagination to travel and the wings, or legs, or vehicles to take you where you want to go." (Scott Buchanan, "Poetry and Mathematics", 1975)

"There are three reasons for the study of inequalities: practical, theoretical and aesthetic. On the aesthetic aspects, as has been pointed out, beauty is in the eyes of the beholder. However, it is generally agreed that certain pieces of music, art, or mathematics are beautiful. There is an elegance to inequalities that makes them very attractive." (Richard E Bellman, 1978)

"The way the mathematics game is played, most variations lie outside the rules, while music can insist on perfect canon or tolerate a casual accompaniment." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"Music is mathematics, the mathematics of listening, mathematics for the ears." (Karlheinz Stockhausen, "Conversations with Stockhausen", 1987)

"Mathematical notation is for the scientist what musical notation is for the composer." (John Holland, "Emergence: From Chaos to Order", 1998)

"Music is a science which should have definite rules; these rules should be drawn from an evident principle; and this principle cannot really be known to us without the aid of mathematics." (Jean-Philippe Rameau, Treatise on Harmony, 2012)