Number lies behind scientific theory, ideas of artistic proportion and rules of musical harmony. HILDI HAWKINS shows how artists and scientists have seen meaning in number, and how patterns of numbers have guided them in their creative work
THE BELIEF THAT NUMBER is the key to the secrets of the Universe- a belief verging on the mystical - lay at the heart of many arts and branches of learning up to, and even after, the scientific revolution of the 17th century. It was the inspiration for some of their most spectacular achievements. Diverse disciplines were brought together by the common language of number: music, astronomy, architecture, poetry and theology reflected the harmonia mundi, the harmony of the world, by means of number.
Number was in everything; a typical
expression of this idea is St Augustine's remark
Number was the essence of the harmonia
mundi. And so the way to create a perfect work of art was to use number
in the correct way. This belief can be traced to
Plato, who states in his philosophical dialogue
The large number of handbooks on architectural proportion that
appeared during the Renaissance are testimony to the seriousness with which
this idea was regarded. And in the 20th century a major attempt at constructing
a harmonious system of design by proportion was made by the great architect
A simpler example of such a series they
are called Fibonacci series after Leonardo
Fibonacci, a mathematician who worked in Pisa about 1200 - is the sequence:
1, 2, 3, 5, 8.... Fibonacci series are found in the proportions and ratios
of many natural patterns: the pads on a cat's foot, the arrangement of leaves
on a plant, the spirals of a snail's shell.
Building by numbers
Are these buildings more beautiful than others not constructed
on any particular system of proportion? The question is almost impossible
to answer, since the buildings named are the work of a great architect and,
as such, are likely to be better than those of an indifferent architect,
whether designed according to a system or not.
It is even doubtful that the golden section leads to more beautiful
proportions than any other ratio. A rectangle with length and breadth in
the ratio of the golden section has long been regarded as in some sense ideal.
Yet as long ago as 1876 it was found by experiment that, although subjects
preferred golden-section rectangles in the laboratory (35 per cent chose
them when offered a choice of 10 rectangles), a much shorter rectangle was
preferred for pictures in a gallery - 5:4 for upright shapes 4:3 for horizontal.
If it were the case that certain proportions were preferred, then, as the
architectural writer P. H. Scholfield points out:
What role is played by number relationships
in science? The idea that there was an underlying numerical harmony in the
world made it natural for the ancients to seek scientific explanations in
terms of number. Reasoning of this sort led them to think that God must have
made the world in six days because six is the first
'perfect' number - a number equal
to the sum of its factors (see page
Another example of scientific discovery guided by numerical patterns comes from chemistry. In the 19th century the Russian chemist Dmitri Mendeleev noticed that when the elements are listed in order of their atomic weights (which are literally, the relative weights of their atoms), patterns emerge: the list can be arranged in rows and columns so that the chemical properties change systematically along the rows and down the columns. To make this system work, it was sometimes necessary to revise the atomic weights assigned to certain elements, or to suppose that there were gaps, corresponding to as yet undiscovered elements. Subsequent discoveries vindicated Mendeleev's ideas triumphantly. All known elements fit neatly into this 'periodic' table, and modern knowledge of atomic structure explains why these patterns should exist.
The modern view
The desire to see the world as a place ruled by harmony and
governed by number may seem naïve today, but it is an expression ofthe
perennial tendency - and desire - of human beings to see order and pattern
in everything around them. Number may not be everything, but the unprovable
assumption that nature is rational and intelligible is the basis of science
- and of the practical reasoning of everyday life.
A radical attempt to account for this
intelligibility of the Universe was made by the German philosopher
Immanuel Kant in the 18th century.
He saw the source of the orderliness of the world as
being the human mind itself, and the concepts with which it does its thinking.
He went so far as to say: 'Our intellect does not draw its laws from nature
. . . but imposes them on nature.' Cause and effect,
time and space, the
laws of mathematics and
logic, are the result of the constitution of our
Eddington, an English astronomer, pushed a form of this idea even further
in the early 20th century. He believed that many highly detailed facts about
the Universe - namely, the value of certain numerical
constants - could be calculated without any appeal to experiment or
observation, but by pure mathematics alone.
Even more remarkable: nature has (so far) proved to be not only
intelligible by rational laws but intelligible in a way that is frequently
elegant and even beautiful. When Copernicus put forward his idea that the
Earth revolved around the Sun, rather than vice versa, it was not so much
the experimental evidence as the theory's elegance, or aesthetic appeal,
that was persuasive to contemporary thinkers.
Similarly, Einstein developed the
theory of relativity on the premise that 'absolute'
motion does not exist; that is, there can be no justification for saying
that, of two scientific observers moving relatively to each other, one is
'really' at rest and the other is not. There is no absolute standard of rest
in the Universe. Einstein was very little influenced by the problems raised
by such experiments as that of Michelson
and Morley (see page 803). The elegance and intrinsic power of
the theory played a large part in the theory's quick acceptance and the
conviction of scientists that it must be true.
That the laws of nature should be not only rational but elegant
seems almost too much to expect. Yet this astonishing metaphysical hypothesis
is constantly being borne out by science. Time and again physical reality
is found to coincide with the speculations of scientists, evolved from a
few basic facts and couched in the subtle language of mathematics. The
existence of many subatomic particles has
been suggested by 'gaps' in mathematical patterns somewhat analogous to
Mendeleev's table of the elements. The branch of mathematics
responsible for these achievements, called 'group
theory', was evolved between the World wars as part of purely abstract
algebra; yet it seemed tailor-made for the understanding of particles that
were to be discovered decades later.
In view of these amazing anticipations of experimental results
by the mathematical speculations of theoretical physicists, how does modern
science differ from numerology? It differs
in that it makes no assumptions about the
symbolic meaning of numbers: numbers do not give insight into some
divine master plan of the Universe. But the assumption on which science
rests - that nature is regular and comprehensible by human reason - is every
bit as mystical as the idea that 'number is all'.