his or her seaside villa. No wonder they renamed the country in his
honor.
The intellectual resolution of this particular Ponzi scheme involves re-
arrangements of infinite series, a topic generally not covered until a math
major takes a course in real analysis. Suffice it to say that there are prob-
lems, which go to the heart of how infinite arithmetic processes differ
from finite ones—when we tallied the total assets of the country by look-
ing at the total assets of I1 through I10 (minus $9) and adding them to the
total assets of I11 through I20 (minus $9), and so on, we get a different
outcome from when we total the assets by adding (I10 I20 I1) (I30
I40 I2) (I50 I60 I3)... (1 1 1)(1 1 1)(1 1 1)
... 1 1 1 .... The two different ways of collecting money (doing
arithmetic) yield different results. Unlike bookkeeping in the real world,
in which no matter how you rearrange assets the total is always the same,
a good bookkeeper in Ponzylvania can spin gold from straw.
Georg Cantor (1845–1918)
Until Georg Cantor, mathematicians had never conducted a successful
assault on the nature of infinity. In fact, they hadn’t really tried—so great
a mathematician as Carl Friedrich Gauss had once declared that infinity,
in mathematics, could never describe a completed quantity, and was only
a manner of speaking. Gauss meant that infinity could be approached by
going through larger and larger numbers, but was not to be viewed as a
viable mathematical entity in its own right.
Perhaps Cantor’s interest in the infinite might have been predicted,
given his unusual upbringing—he was born a Jew, converted to Protes-
tantism, and married a Roman Catholic. Additionally, there was a sub-
stantial amount of artistic talent in the family, as several family members
played in major orchestras, and Cantor himself left a number of drawings
that were sufficient to show that he possessed artistic talent as well.
Cantor took his degree in mathematics under the noted analyst Karl
Theodor Wilhelm Weierstrass, and Cantor’s early work traveled along the
path marked out by his thesis adviser—a common trait among mathema-
ticians. However, Cantor’s interest in the nature of infinity persuaded
him to study this topic intensely. His work generated considerable inter-
est in the mathematical community—as well as considerable controversy.
Cantor’s work f lew in the face of Gauss, as it dealt with infinities as com-
pleted quantities in a manner analogous to finite ones.
Among the mathematicians who had a great deal of difficulty accepting
this viewpoint was Leopold Kronecker, a talented but autocratic German
18 How Math Explains the World