the two theories were later shown to be equivalent, generating the same
results using different ideas.
In 1927, Heisenberg was to make the discovery that would not only win a
Nobel Prize, but would forever change the philosophical landscape. Recall
that in the late eighteenth century, the French mathematician Pierre La-
place enunciated the quintessence of scientific determinism by stating that
if one knew the position and momentum of every object in the universe,
one could calculate exactly where every object would be at all future times.
Heisenberg’s uncertainty principle^6 states that it is impossible to know
both exactly where anything is and where it is going at any given moment.
These difficulties do not really manifest themselves in the macroscopic
world—if someone throws a snowball at you, you can usually extrapolate
the future position of the snowball and possibly maneuver to get out of
the way. On the other hand, if both you and the snowball are the size of
electrons, you’re going to have a problem figuring out which way to move,
because you will not know where the snowball will go.
We can get a sense of the underlying idea behind Heisenberg’s uncer-
tainty principle by looking at an everyday occurrence—the purchase of
gasoline at a service station. The cost of the transaction is a number of
dollars and cents—the penny is the quantum of our monetary system,
the smallest irreducible unit of currency. The cost of the transaction is
computed to the nearest penny, and this makes it impossible for us to
determine precisely how much gasoline was actually purchased even if
we know the exact price per gallon.
If gasoline costs $2.00 per gallon (as it did in the good old days), round-
ing the cost of the purchase to a penny can result in a difference of^1 / 200
of a gallon of gasoline (yes, if you adopt a reasonable rule for rounding,
you can cut this to half of^1 / 200 of a gallon of gasoline, but the meters in a
service station probably round a purchase of $12.5300001 to $12.54). If
you start driving from a known position on a straight road and your car
gets 30 miles to a gallon,^1 / 200 of a gallon of gasoline will take you 0.15 of
a mile—792 feet. So the fact that cost is computed in pennies results in a
positional uncertainty of 792 feet. I can remember the first time I had the
use of a car of my own in the summer of 1961; I used to leave two quar-
ters in the glove compartment for gas in case of emergencies. Gas was
about 25 cents a gallon then—at 30 miles per gallon, the cost computed
in pennies will result in a positional uncertainty of 1.2 miles. The lower
the cost of gasoline, the greater the positional uncertainty. In fact, if gaso-
line were free, you wouldn’t have to pay anything—and you’d have no idea
where the car was.
The uncertainty principle operates along similar lines. It states that the
56 How Math Explains the World