Music and Mathematics VI

"But mathematics, certainly, does not play the smallest part in the charm and movement of the mind produced by music. Rather is it only the indispensable condition (conditio sine qua non) of that proportion of the combining as well as changing impressions which makes it possible to grasp them all in one and prevent them from destroying one another, and to let them, rather, conspire towards the production of a continuous movement and quickening of the mind by affections that are in unison with it, and thus towards a serene self-enjoyment." (Immanuel Kant, "The Critique of Judgment", 1790)

"May not Music be described as the Mathematic of sense, Mathematic as the Music of the reason? the soul of each the same! Thus the musician feels Mathematic, the mathematician thinks Music - Music the dream, Mathematic the working life - each to receive its consummation from the other when the human intelligence, elevated to its perfect type […]" (James J Sylvester, "On Newton’s Rule for the Discovery of Imaginary Roots", 1865)

"Mathematics and music! the most glaring possible opposites of human thought! and yet connected, mutually sustained!" (Hermann von Helmholtz, "Popular Lectures on Scientific Subjects", 1885)

"Mathematics has a triple end. It is to furnish an instrument for the study of nature. But that is not all. It has a philosophic end, and I dare say it, an esthetic end. […] Those skilled in mathematics find in it pleasure akin to those which painting and music give. They admire the delicate harmony of numbers and of forms; they marvel when a new discovery opens an unexpected perspective; and is this pleasure not esthetic, even though the senses have no part in it?" (Henri Poincaré, "Sur les rapports de l’analyse pur et de la physique mathématique", [Report to the Zurich International Congress of Mathathematics], 1897)

"[…] mathematics, accessible in its full depth only to the very few, holds a quite peculiar position amongst the creation of the mind. It is a science of the most rigorous kind, like logic but more comprehensive and very much fuller; it is a true art, along with sculpture and music, as needing the guidance of inspiration and as developing under great conventions of form […]" (Oswald Spengler, "The Decline of the West" Vol. 1, 1926)

"If all the arts aspire to the condition of music, all the sciences aspire to the condition of mathematics." (George Santayana, Some Turns of Thought in Modern Philosophy: Five Essays, 1933)

"[…] mathematics is like music, freely exploring the possibilities of form. And yet, notoriously, mathematics holds true of things; hugs and permeates them far more closely than does confused and inconstant human perception; so that the dream of many exasperated critics of human error has been to assimilate all science to mathematics, so as to make knowledge safe by making it, as Locke wished, direct perception of the relations between ideas […]" (George Santayana, "The Realm of Truth: Book Third of Realms of Being", 1937)

"Mathematizing may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalizations." (Hermann Weyl, “Obituary for David Hilbert”, Royal Society Biographies Vol. 4, 1944)

"The fact is that there are few more ‘popular’ subjects than mathematics. Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics than in music. Appearances may suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognized (no doubt rightly) as mildly discreditable [...]" (Godfrey H Hardy, "A Mathematician’s Apology", 1967)

"I see some parallels between the shifts of fashion in mathematics and in music. In music, the popular new styles of jazz and rock became fashionable a little earlier than the new mathematical styles of chaos and complexity theory. Jazz and rock were long despised by classical musicians, but have emerged as art-forms more accessible than classical music to a wide section of the public. Jazz and rock are no longer to be despised as passing fads. Neither are chaos and complexity theory. But still, classical music and classical mathematics are not dead. Mozart lives, and so does Euler. When the wheel of fashion turns once more, quantum mechanics and hard analysis will once again be in style." (Freeman J Dyson, "Book Review of ‘Nature’s Numbers’", The American Mathematical Monthly, Vol. 103 (7), 1996)