mechanics, including Einstein, but there was no previous Isaac Newton in
this branch of physics to be knocked off his or her pedestal. Yet quantum
mechanics has changed our lives, perhaps more than any single branch of
physics has ever done—although the classical theory of electromagnetism
would be a strong contender. But quantum mechanics has been much more
than a generator of technology; it has greatly changed and challenged our
understanding of the nature of reality.
What Does It All Mean?
Ever since Pythagoras proved what is arguably the most important theo-
rem in mathematics, mathematics has generally had a very clear view of
what it is trying to accomplish. Pythagoras knew, as it had been known
since Egyptian times, that some of the classic triangles were right trian-
gles, such as the triangle with sides 3, 4, and 5. Noticing that 3^2 42 52 ,
he was able to generalize this to show that in a right triangle, the square
of the hypotenuse was equal to the sum of the squares of the other two
sides. He knew what he wanted to prove, and when he proved it he knew
what he had—a theorem so important that he ordered a hundred oxen to
be barbecued in celebration. I sometimes tell my students this tale, add-
ing that this provides a measuring rod for the importance of mathemati-
cal theorems. The fundamental theorem of arithmetic (that every number
can be uniquely expressed as a product of primes), the fundamental theo-
rem of algebra (that every nth-degree polynomial with real coefficients
has n complex roots), and the fundamental theorem of calculus (that inte-
grals can be evaluated via anti-differentiation) are all sixty-oxen theorems,
and to my mind, no other theorems come close to those.
It’s different in physics—especially in quantum mechanics. Both physi-
cists and mathematicians “play” with what they have in an attempt to de-
duce new and interesting results, but when mathematicians deduce such
a result, they almost never have to worry about what it means. It is what
it is, and the next step is to find applications of the result, or deduce new
consequences from it.
Physicists, on the other hand, have to decide what the result means—
what the mathematics actually represents in the real world. Quantum
mechanics is such an incredibly rich and profound area that physicists
are debating the meaning of results nearly a century old. Niels Bohr, one
of the architects of the theory, expressed this sentiment perfectly when he
declared, “If quantum mechanics hasn’t profoundly shocked you, you
haven’t understood it yet.”^1
42 How Math Explains the World