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“Music is the pleasure the human soul experiences from counting without being aware that it is counting.”
German mathematician and philosopher Gottfried Wilhelm von Leibniz (1646-1716) who co-discovered calculus

“Mathematics and music, the most sharply contrasted fields of scientific activity which can be found,
and yet related, supporting each other, as if to show forth the secret connection which ties together
all the activities of our mind, and which leads us to surmise that the manifestations of the artist's genius are
but the unconscious expressions of a mysteriously acting rationality.”19th century German physician and physicist Hermann von Helmholtz, Vorträgeund Reden, Bd. 1 (Braunschweig, 1884), p. 82

“May not music be described as mathematics of the sense, mathematics as music of the reason?”19th century English mathematician James Joseph Sylvester, On Newton's Rule for the Discovery of Imaginary Roots; Collected Mathematical Papers, Vol. 2, p. 419

“If all art aspires to the condition of music, all the sciences aspire to the condition of mathematics.” – Spanish philosopher/writer George Santayana (1863-1952)

“Musical training is a more potent instrument than any other,
because rhythm and harmony find their way into the inward places of soul,
on which they mightily fasten, imparting grace,
and making the soul of him who is rightly educated graceful...”
-- classical Greek philosopher Plato (428-348 BCE), The Republic, III

“Educators have always known that learning and life are maximal
where play and work coincide.” -- L. W. Gibbs

“In the future, we can expect that not much difference will exist between education and entertainment.
We just have to put intelligence behind the entertainment.”
--  North Carolina State University's James Lester,
quoted at the 12th International Conference on College Teaching and Learning

“It is harmony which restores unity to the contrasting parts and which moulds them into a cosmos.
Harmony is divine, it consists of numerical ratios.
Whosoever acquires full understanding of this number harmony, he becomes himself divine and immortal.”
--20th century Dutch mathematician B. L. van der Waerden, describing the beliefs of the followers of Pythagoras

“We do not listen with the best regard to the verses of a man who is only a poet,
nor to his problems if he is only an algebraist;
but if a man is at once acquainted with the geometric foundation of things and with their festal splendor,
his poetry is exact and his arithmetic music.”
--19th century American philosopher/writer Ralph Waldo Emerson, Society and Solitude, Ch. 7, Works and Days

“Mathematics is on the artistic side a creation of new rhythms, orders, designs, harmonies,
and on the knowledge side, is a systematic study of various rhythms, orders, designs and harmonies.”
-- William L. Schaaff
, author and mathematics education professor

“A mathematician, like a painter or a poet, is a maker of patterns.
If his patterns are more permanent than theirs, it is because they are made of ideas.
His patterns, like the painter's or the poet's must be beautiful;
the ideas, like the colors or the words, must fit together in a harmonious way.”  -- English mathematician G. H. Hardy

“Music and math together satisfied a sort of abstract 'appetite',
a desire that was partly intellectual, partly aesthetic, partly emotional, partly, even, physical.”
--music critic/composer Edward Rothstein (p. xv of his 1995 book Emblems of Mind: The Inner Life of Music and Mathematics)

“We can no more come to understand mathematics by examining its final product than we can understand the experience of music through simply looking at a score or an analysis of one; there is an experience that lies underneath and behind the systematic organization of the material.”
--music critic/composer Edward Rothstein (p. 38 of his 2006 book Emblems of Mind: The Inner Life of Music and Mathematics)

“Each time I've learned a little bit more about the inner structures of music,
the math of it and the shape of it,
my joy in it has increased.”
-- recording artist Peter Mulvey,  in the Nov. 2000 Performing Songwriter

“Music is true.  An octave is a mathematical reality.  So is a 5th. So is a major 7th chord.
And I have the feeling that these have emotional meanings to us,
not only because we're taught that a major 7th is warm and fuzzy and a diminished is sort of threatening and dark,
but also because they actually do have these meanings.  It's almost like it's a language that's not a matter of our choosing.
It's a truth.  The laws of physics apply to music, and music follows that.
So it really lifts us out of this subjective, opinionated human position and drops us into the cosmic picture just like that.”
-- recording artist James Taylor,  in the May 2002 Performing Songwriter

“The syntax and the grammar of the language of music are not capricious;
they are dictated by the texture and organization of the deep levels of the mind, so with mathematics.”mathematician H. E. Huntley

“The most distinct and beautiful statement of any truth (as of music)
must take at last the mathematical form.”19th century American writer/philosopher Henry David Thoreau, A Week on the Concord and Merrimac Rivers (Boston, 1893), p. 477

“You cannot evade quantity.
You may fly to poetry and music, and quantity and number will face you in your rhythms and your octaves.”
-- Alfred North Whitehead (1861-1947), English mathematician

“Musical form is close to mathematics -- not perhaps to mathematics itself,
but certainly to something like mathematical thinking and relationship.”  -- 20th century Russian composer Igor Stravinsky



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