would be to consider the speed of a car—it should be able to travel at all
velocities up to its theoretical limit. If a car cannot go faster than 100
miles per hour, for instance, it should be able to move at 30 miles per
hour, or 40 miles per hour, or 56.4281 miles per hour. However, writing
down a few numbers is somewhat deceptive, because they are all rational
numbers. As we know from the previous chapter, there are uncountably
many real numbers less than 100.
One day in 1900, the German physicist Max Planck made a bizarre as-
sumption in an attempt to escape the ultraviolet catastrophe. Instead of
assuming that energy could be radiated at all frequencies, he assumed
that only a finite number of frequencies were possible, and these were all
multiples of some minimum frequency. Continuing the analogy with the
speed of the car, Planck’s hypothesis would be that something like only
speeds that were multiples of 5—25 miles per hour, 40 miles per hour,
etc.—would be possible. He was able to show almost immediately that
this counterintuitive hypothesis resolved the dilemma, and the radiation
curves he obtained from making this assumption matched the ones re-
corded by experiment. That day, while walking with his young son after
lunch, he said, “I have had a conception today as revolutionary and as
great as the kind of thought that Newton had.”^3
His colleagues did not immediately think so. Planck was a respected
physicist, but the idea of the quantum—energy existing only at certain
levels—was at first not taken seriously. It was viewed as a kind of mathe-
matical trickery that resolved the ultraviolet catastrophe, but did so by
using rules that the real world did not obey. Ever since Isaac Newton in-
corporated mathematics as an essential part of a description of natural
phenomena, it has generally been easier for a theoretician to sit down
with pencil and paper, and derive mathematical consequences, than it
has for an experimenter to devise and carry out a successful experiment.
As a result, there is sometimes the feeling that mathematics is merely a
convenient language to describe phenomena, but it does not give us an
intuitive insight into the nature of the phenomena.
Planck’s idea languished for five years, until Einstein used it in 1905 to
explain the photoelectric effect. Eight years later, Niels Bohr used it to
explain the spectrum of the hydrogen atom. Within another twenty years,
Planck had won a Nobel Prize, and quantum mechanics had become one
of the fundamental theories of physics, explaining the behavior of the
world of the atom and making possible many of the high-tech industries
of today.
With the coming of the Nazis, German science suffered severely. Many
of the leading scientists were either Jewish or had Jewish relatives, and
All Things Great and Small 45