Mathematics and music: some educational considerations

by Philip Maher

§1.   

          There is the widely held view that mathematics (in which so few are confident) and music (which moves so many) are somehow, mysteriously, alike. In this essay I hope to illuminate this mystery by articulating the affinities and - to coin a neologism - the anti-affinities between mathematics and music.

          To begin, consider the claim that many mathematicians are musical. In my own experience, some are highly musical, some not at all; and to go beyond the purely anecdotal one would need to administer a very carefully formulated survey (designed to explore not only the extent of mathematicians' musicality but also, more interestingly, its kind).

          It's often mentioned, in this context, that, statistically speaking, performing musicians and mathematicians tend to mature young (For example, music and mathematics both tend to exhibit prodigies more so than other disciplines). Psychological factors apart, the facts of human physical development imply the necessity of early training for a performing musician; and, in any case, the evidence is only statistical: there are examples of mathematicians who did not embark on a creative career in mathematics until their mid forties. (For interesting comments on the last two paragraphs see Gowers' marvellous book [1]).

          Does acoustics - which involves mathematics - yield a possible link? Certainly acoustics is connected - in a complex way - with music ("in a complex way" because of the musical context involved: it is false to argue (as the theory attributed to Pythagoras does) that intervals having simple relative frequencies are necessarily aesthetically preferable to those having complex ones): yet it is ridiculous to argue that because acoustics has a mathematical formulation, therefore mathematics and music are connected: it does not explain the feeling we have that there is some intrinsic affinity between music and mathematics.

          At a somewhat deeper level consider the various conceptual models used to explicate our non-conceptual understanding of music (By "conceptual" I mean that which can be expressed in words (or some symbolic system, like mathematics); by "non-conceptual" that which can't. This distinction is central to Wittgenstein's philosophy and crucial in much writing on music notably the work of Hans Keller [5], [6] and Alan Walker [13], [14]. Examples of such conceptual models are the theories of diatonic harmony and counterpoint and serial technique. These theoretic models are (perhaps as a consequence of twentieth century cultural history, as I shall argue later) sometimes presented so as to have (to coin another neologism) a mathematicalish air. Consider the pages of that highly respected journal "Musical Analysis" or, say, the writings of Milton Babbitt who frequently uses mathematical concepts (e.g., equivalence class) in his work.

          Most striking, as an example of an actual theory of music which intimately involves mathematical concepts - specifically the golden section and the mathematically related Fibonacci sequence - is Bartók's own theory of composition which he evolved in his early thirties (about the same time, coincidentally, that Schönberg was evolving his twelve-tone technique). Bartók's theory comprehensively integrates vertical (pitch) and horizontal (temporal) elements both locally (e.g., individual chord structures) and globally (e.g., proportioning movements). A riveting exposition of this theory is in Erno Lendvai's book [8]. (Significantly, Bartók never expounded his theory in his lifetime.)

          For two concrete examples, consider Bartók's "Music for strings, percussion and celesta" and the "Sonata for two pianos and percussion". The extremely powerful opening fugue of the "Music for strings, percussion and celesta" has a plan based rigorously on the Fibonacci sequence: 89 bars long, the big climax occurs at bar 55, with the subsidiary climaxes all occurring at junctures predicated by the Fibonacci sequence; and the "Sonata for two pianos and percussion" uses the golden section itself (or, rather, rational approximations to it, of course) to construct the plan of this extraordinary work.

          Bartók's practice in having to construct his own theory is characteristic of the high modernism of the arts in the early twentieth century. Many of the arts were, in the first decade or so of that century, concerned with the abandonment of the belief that a fixed and priviledged viewpoint existed - and had, perforce, to formulate a fresh modus operandi. Examples abound. One thinks of the abandonment of key centres and hence of tonal perspective in the atonal (pre-twelve-tone technique) of Schönberg and Webern (or, in painting, of the abandonment of visual perspective in cubism). Given this, it is not surprising that many of the proseletyzers of modernism looked to mathematics as an exemplar of that which is objectively and ineluctably true. This was accentuated by the fact that a crucial, then contemporary development in mathematical physics, namely the theory of special relativity, had, as its starting point, the axiom that there was no one single, priviledged viewpoint, no one priviledged (Newtonian) frame of reference.

          Of course, composers have for a very long time used mathematical structures. Apart from the arithmetic, rhythmic complexity of some mediaeval music, much classical music is organized along symmetric lines. For instance, in the first movement of Mozart's 40th Symphony, the development begins exactly half way through the movement and the recapitulation exactly two thirds of the way through. The first movement of Beethoven's Fifth Symphony both shows great symmetry (The development and recapitulation are the same length) plus - arguably - the use of the golden section. (These appearances of mathematics as a principle are quite different to numerology (Numbers having extra-musical significance). Whilst numerology is well-established in Bach [12], Mozart [2] and Beethoven [3] - and can be seen as an expresion of their religious and philosophical beliefs - it is never clear what aesthetic significance numerology can have for us.)

          Yet it is doubtful if we are aware aurally, by our ears alone, of the satisfying mathematical aesthetic the structures cited above make. How is the affinity (assuming there is one) between mathematics and music revealed in the actual experience of mathematics and music? I am seeking to name some concept(s?) embedded in our actual experience of mathematics and music.

          One concept so embedded is to do with time: to do with the fact that, on the one hand, a piece of music unfolds in time; and, on the other, one's experience of a piece of mathematics takes place in time. Consider the case of mathematics first. When one is studying a good piece of mathematics (pure or applied) one is struck by the inexorability of the logic coupled - simultaneously - to one's astonishment at where the logic has, ineluctably, taken one. This is, if anything, accentuated in the creation/discovery of new mathematics: here, you, the mathematician, are not only surprised at the steps you find yourself taking but also by how, on reflection, each such step is seen to follow inexorably from the previous ones, thus leading to the mathematical conclusion of your argument which, whilst now seen as inevitable, may seem surprising to state in view of your initial premises. The process I have just outlined is not restricted to constructing proofs at the research level (although proof is paradigmatic of this process) but also operates, at least with non-routine problems, at the elementary level. The process I have been describing, is, to use the appropriate philosophical nomenclature, an example of a dialectical process.

          The process of listening to music - to tonal music, at least - is strikingly similar. The composer of a piece of music creates certain well-defined expectations (many of which the listener need not be consciously aware of) which s/he the composer, then proceeds to meaninglfully contradict, thus setting up new expectations which may themselves - probably will - be meaningfully contradicted in turn. Thus goes a brutally compressed summary of what is in common to the theories of music associated with Keller, Schenker, Reti and Meyer (See [5], [6], [11], [10]) (who tend to exemplify their theories by examples of tonal music: hence my reference to tonal music although I think the theory would work for most Western art music). The above theory of music (and the theory of listening it implies) reveals how the experience of listening to music, like the experience offered by mathematics, is dialectical.

          But if mathematics and music are, as popularly supposed, somehow alike there must be some elemental property, understandable to those with little experience of mathematics, which is possessed by both music and mathematics; and such an elemental property should evince itself in the most elementary of mathematics.

          The common property I am thinking of is non-representationality. To take mathematics first, if you consider any mathematical statement, however elementary, say  you will be struck by its non-representationality: it does not of itself represent or refer to the world outside mathematics and in order to understand it no such reference need be made. This is not to say that physicalist interpretations are not used in the teaching of elementary arithmetic: one can motivate small children by explaining that 3 bananas plus 5 bananas makes 8 bananas (or 3 boats plus 5 boats make 8 boats). But the statement that  is entirely independent of bananas, boats or whatever was used in the motivation. (Exactly the same argument holds for applied mathematics - at least to the mathematics (as distinct from the modelling) performed in that subject.)

          Music, too, is a non-representational art (like architecture and unlike literature or figurative painting). For consider: confronted with a representational work of art (say, a landscape) you recognise its subject (a landscape) whose recognition (as a landscape) is necessary for understanding the work and about which you can frame propositions. None of this is ever possible with a piece of music, not even those with literary or descriptive titles (Listening to Beethoven's Pastoral Symphony, the most you can say about the 4th movement, meant to depict a storm, is that it's "stormy", nothing about the storm so depicted).

          In arguing above that music is non-representational I am not arguing for some formalist aesthetic or claiming that music cannot have powerful psychological affects - the difficulty is that we cannot use words to identify these affects. Indeed, despite the existence I have demonstrated of affinities between music and mathematics I submit there is a diametric difference.

          As well as music being able to have powerful psychological affects, which cannot be named, one's understanding of music is direct and non-conceptual. By contrast, mathematics can only be appreciated once it is conceptually understood. Let us explore the differences between the conceptual and the non-conceptual.

          Conceptual logic is so much a part of our mental furniture we take so for granted that it requires effort even to identify it and its three laws. The three laws of conceptual logic are: the law of identity that says that A is A; the law of contradiction that says that if A is B then it cannot be not B; and the law of the excluded middle that says that A is, or is not, B. These three laws govern the way we use language, verbal and written, and - a fortiori - the practice of mathematics.

          What is remarkable is that these three laws are diametrically contradicted by music. Thus, in music: A must not remain A but must become more than A (otherwise nothing will be said); that if A is B then it is, equally, not B; and that A must be both B and not B. The reader familiar with psychoanalytic theory will recognize that the three processes I have just listed - the three diametric opposites of the laws of conceptual logic - are precisely the primary processes (of condensation, displacement and representation through the opposite, respectively that Freud posited as governing the working of the subconscious, in general, and dream-life, in particular (I should mention here that the audaciously brilliant theory I have just outlined is not mine but Keller's [7]).

          That music thus appropriates the primary processes of the unconscious explains, at least to some extent, its awesome affective power and why, moreover, the psychological states it causes cannot be identified verbally. But whilst musical logic operates oppositely to conceptual logic, a musical masterpiece (as one can be aware of if one studies it analytically, i.e., conceptually, after the musical event) manages, simultaneously, to be a triumph of conceptual logic, too.

§2.

          Which brings us to the educational implications, whatever they may be of the above. The arguments of the above show that music, whilst being crucially different from mathematics, mirrors some of its features; and that this, in turn, emphasizes how mathematics arises from the world (I am not here necessarily advocating that mathematics should be taught through its applications; rather, that throughout history, mathematics has been precipated by developments outside it: for instance, operator theory by quantum theory in the twentieth century; the calculus by dynamics in the seventeenth century).

          This suggests that a healthy and realistic way to present mathematics is to embed it in the world; and this, in turn, both reflects and has implications for one's philosophy of mathematical practice (my phrase: this has, arguably, slightly wider connotations than the phrase "philosophy of mathematical education"). I must add that I am speaking as a mathematican and not as someone professionally involved in mathematical education. Nevertheless, it seems to me that there are, roughly speaking, two competing philosophical models of mathematics practice.

          One is Platonism, the belief that there exists a mathematical reality, which is neither psychological or physical and which exists independently of us, the duty of the mathematician being to observe it. A famous defence of this is Hardy's "A mathematician's apology"[4]; and no less than Gödel was an adherent of this viewpoint.

          Nevertheless, it would seem that Platonism suffers glaring philosophical problems; and, as I have argued elsewhere [9, Chapter 11], its great affective power for mathematicians needs to be accounted for in psychoanalytical, rather than philosophical, terms (specifically, as being a description of the potential space (Winnicot's concept) of the mathematician). One problem of Platonism is that whilst giving an emotionally beneficent (if essentially conservative) description of being a mathematician, it doesn't give any idea about how you go about making mathematics or how you become one: mathematics is just there.

          Social constructivism - the alternative philosophical viewpoint - engages with this issue. Influenced by the later Wittgenstein who wrote "The meaning of a word is its use in language" [15], it adopts, in the words of that top-flight research mathematician, Gowers, the viewpoint that " a mathematical object is what it does" [1, p.18]. Thus, social constructivism stresses the making of mathematics by human beings, and consequently the making of meanings, and their communication, by human beings.

          Social constructivism, thus, by placing mathematics in the world places it centrally in culture; and thus provides the philosophical framework for the relationships that I have been exploring in §1 of this essay.

BIBLIOGRAPHY

1.      Gowers, T. Mathematics: a very short introduction, Oxford University Press,
          (2002).

2.      Grattan-Guiness, I. O. Counting the notes: Numerology in the Works of
          Mozart, Especially Die Zauberflöte, Annals of Science 49 (1992), pp. 201-232.

3.      Grattan-Guiness, I. O. Some Numerological Features of Beethoven's Output,
          Annals of Science 51 (1994), pp. 103-135.

4.      Hardy, G. H. A mathematician's apology, Cambridge University Press, (1940).

5.      Keller, H. Towards a Theory of Music, Listener 83/2150 (11 June 1970), pp.
          795-796.

6.      Keller, H. Closer Towards a Theory of Music, Listener 85/2186 (18 February
          1971), pp. 218-219.

7.      Keller, H. Why this piece is about "Billy Budd", Listener 88/2270 (28
          September 1972), p. 419.

8.      Lendvai, E. Bela Bartók: an Analysis of His Music, Kahn and Averill, (1971).

9.      Maher, P. J. Potential space and mathematical reality in Constructing
          Mathematical Knowledge: Epistemology and mathematical education ed.
          Ernest, P., The Falmer Press, (1994)

10.    Meyer, L. B. Music, the Arts and Ideas, University of Chicago Press, (1967).

11.    Reti, R. The Thematic Process in Music, Faber and Faber, (1961).

12.    Tatlow, R. Bach and the riddle of the number alphabet, Cambridge University
          Press, (1991).

13.    Walker, A. A Study in Musical Analysis, Barrie and Rockliff, (1962).

14.    Walker, A. An Anatomy of Music Criticism, Barrie and Rockliff, (1966).

15.    Wittgenstein, L. Philosophical investigations, Blackwell (3rd edition), (2001).

PHILIP MAHER

Mathematics and Statistics Academic Group, Middlesex University, Hendon Campus, The Burroughs, LONDON NW4. E-mail: p.maher@mdx.ac.uk