However, just as political success awaits the candidate who can appeal to
the majority, fame and fortune undoubtedly await the individual who dis-
covers the key to creating widely appreciated art, music—or food. Mathe-
matics has met with little success in this area.
Garrett Birkhoff was one of the preeminent American mathematicians
of the first half of the twentieth century. He made noteworthy contribu-
tions to celestial mechanics, statistical mechanics, and quantum me-
chanics, in addition to his work in pure mathematics. Generations of
college students—including mine—learned the theories of groups, rings,
and fields from his landmark text on abstract algebra,^7 coauthored with
the equally eminent Saunders Mac Lane.
Birkhoff also had a keen interest in aesthetics, and attempted to apply
mathematics to the evaluation of art, music, and poetry. To be fair, his
efforts were nowhere near as laughable as the reaction of Charles Bab-
bage, one of the pioneers in the construction of mechanical computa-
tional devices. On reading a poem by Tennyson that included the line
“Every moment dies a man, / Every moment one is born,” Babbage sent a
note to Tennyson pointing out that, to be strictly accurate, Tennyson
should have written, “Every moment dies a man, / Every moment one and
one-sixteenth is born.”
Birkhoff’s basic formula for computing aesthetic value was that the
aesthetic measure of a work of art was equal to the quotient of its aes-
thetic order divided by its complexity—orderly things were beautiful,
complex things were not. The mathematicians whose musical tastes I
have ascertained generally seem to conform to this rule; Bach generally
receives a better reception among mathematicians than does Shostako-
vitch. In fact, when a friend introduced me to a Bach chaconne, he
started by describing it by saying that it has 256 measures (256 28 ) di-
vided into 4 sections of 64 measures (64 26 ), and I liked it even before I
heard a single note.
To some extent, the idea that order is more attractive accords with statisti-
cal surveys that have determined some fairly obvious broad generalities in
aesthetics: the majority prefers symmetry to asymmetry, pattern to absence
of pattern. However, some of Birkhoff’s subsidiary formulas are almost
painful to read. For example, to compute the aesthetic order of a poem,
Birkhoff devised the formula Oaa 2 r 2 m 2 ae 2 ce, where aa stands
for alliteration and assonance, r for rhyme, m for musical sounds, ae for al-
literative excess, and ce for e xcess of conson a nt sou nd s. To b e fa i r to Bi rk hof f,
his efforts antedate Arrow’s theorem by decades, and he admitted that in-
tuitive appreciation outweighed mathematical calculation. Nonetheless,
246 How Math Explains the World