252 A History ofMathematics
All this was familiar to me from my research in high-energy physics, but until that moment I had only experienced
it through graphs, diagrams, and mathematical theories. As I sat on that beach my former experiences came to life;
I ‘saw’ cascades of energy coming down from outer space, in which particles were created and destroyed in rhythmic
pulses; I ‘saw’ the atoms of the elements and those of my body participating in this cosmic dance of energy; I felt its
rhythm and I ‘heard’ its sound, and at that moment I knew that this was the Dance of Shiva, the Lord of Dancers
worshiped by the Hindus. (Capra 1983, p. 9)One quite unexpected development, particularly of the mid to late twentieth century, has been that
physics has become detached from mathematics. This does not mean that physicists in general see
their subject in terms of the dance of Shiva; Capra is to be seen as a symptom, an index of the kind
of statement which some people on the margins of physics now think they can getawaywith. The
relation between physics and mathematics was close, almost incestuous until fairly recently. In the
golden age of the eighteenth century, following Newton, physics or ‘rational mechanics’ could be
seen as a particular kind of mathematics; the study of certain principles—first those laid down in
thePrincipia, and later more abstract versions such as the Principle of Least Action. While physics
necessarily did need experimentalists to advance (in new fields like electricity and magnetism, for
example), it was possible to be a mathematician and study the diffusion of heat, or the vibrations
of a drum, without leaving one’s desk—as Kovalevskaya, in the late nineteenth century, solved the
problem of the rotating top. Such, in the early twentieth century, was Musil’s hero Ulrich, using
formulae and symbols to describe an equation of state of water. We can see this happy situation
as an outcome of the Scientific Revolution (Chapter 6), which saw the behaviour of the physical
world as ordered by mathematical laws.
Indeed, which laws they were was not immediately important, and both the special and the
general theories of relativity provided more employment for mathematicians, by replacing one
mathematical description by another. However, by the 1920s it appeared that the physicists were
becoming more impatient. Early formulations of the quantum theory were investigated and seen to
work before their often ad hoc mathematical underpinnings were certified legal. Further, a combin-
ation of doubt about the continuum (Chapter 9) and uncertainty about measuring the extremely
small raised some questions—which had been left in suspension—about whether traditional math-
ematics was as well-adapted to the universe as one had hoped. Nevertheless, the new quantum
theory remained completely dependent on quite complex mathematical ideas. In his classic article
‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’, Eugene Wigner cites the
example of the helium atom.
The miracle occurred only when matrix mechanics, or a mathematically equivalent theory, was applied to problems for
which Heisenberg’s calculating rules were meaningless. Heisenberg’s rules presupposed that the classical equations
of motion had solutions with certain periodicity properties; and the equations of motion of the two electrons of the
helium atom, or of the even greater number of electrons of heavier atoms, simply do not have these properties, so that
Heisenberg’s rules cannot be applied to these cases. Nevertheless, the calculation of the lowest energy level of helium,
as carried out a few months ago by Kinoshita at Cornell and by Bazley at the Bureau of Standards, agrees with the
experimental data within the accuracy of the observations, which is one part in ten million. Surely in this case we ‘got
something out’ of the equations that we did not put in. (Wigner, 1960)Simplified, the helium atom is a three-body problem (nucleus and two orbiting electrons (Fig. 7)),
and while experimenters could observe the energy levels, mathematical physicists—again working
on paper—did calculations as Wigner describes to check whether theory and observation agree.
They were, however, increasingly growing apart from the dominant trends in pure mathematics, in
spite of a serious engagement in the field by, among others, Weyl, Noether, and van der Waerden,