Not too long ago, I happened to listen to Bartók’s Piano Concerto No. 2; I think there was more emphasis on creating elaborate patterns of sounds than on producing what some call “nice” music. As a result, enjoyment from listening such a piece seems to come not because one wants to hum and remember any easily recognizable patterns but because one can mentally grasp the intricacy of the work and the sophistication of the composer. Classical music of such a feature seems to be of interest of only few people, many of whom are professional musicians (I am not). Maybe one could say that classical music has lost its intuition. I have the impression that a similar phenomenon happens with mathematics: early mathematics, such as arithmetic, might be seen as more intuitive than modern mathematics, such as mathematical logic or abstract algebra. Yet, I have heard people refer to both as art. This then has led me to think that they could be categorized as “academic art,” which I define as a highly codified activity with the idea that to produce something new is still possible.
With this definition, many similarities could be found between mathematics and classical music. For example, I have seen mathematical proofs that seem to be very complicated by the amount of work it takes to get through it because it is either very long or involves many difficult notions or both; I have also seen music scores in which a rather wild mix of notes and symbols fills the page, which suggests to me that a musician might have a hard time to go through such piece (If you are a musician, can you think of any example?). Another similarity is that both mathematics and classical music are taught in schools, where such an education can lead to a degree. In addition, what perhaps makes both activities alike is the heavy amount of work that both tend to demand for their mastery, where more or less constant practice appears to be the norm; I have heard some musicians and mathematicians claim they spend most of their days to practice, and they seem to always need to practice more and more.
Of course, there are differences that might suggest that music and mathematics have nothing to do with each other such as the symbols that each discipline uses and the apparent obsessive need for justification of mathematical statements and the apparent lack of it in music (If you are a musician, do you have to know why a composer has chosen to write a certain line?). I don’t really see how the deductive reasoning used in mathematics is applied to music performance.
One question that might arise from the codified nature of both activities is: where does the creativity come from if all what musicians and mathematicians do is following rules? If, by creativity, one means the ability to come up with something new, I then think that there might be different levels of it, such as creativity within the rules and creativity inspired by the rule. The former would be when the existing rules are used in unexpected ways, whether to use proof techniques from one area and cleverly apply to some apparently unrelated area or to use centuries old technique to compose music that Bach or Mozart would not recognize. The latter would be when new rules are created by questioning existing rules; for example, non-Euclidean geometry was born by questioning Euclid’s Fifth Postulate, and I think the twelve-tone technique was created as a reaction to romantic or earlier classical music.
So, do you think that mathematics and classical musical are total strangers?