Perhaps we should say there is no fair way to do this in the short run.
What we mean by this is that there is no way to allocate representatives to
states in such a way that every time a census occurs, each state is allo-
cated representatives via a quota method that does not run afoul of the
Alabama and population paradoxes. However, there is a method of doing
this that will give each state its fair share in the long run. Simply compute
the quota for each state and use a randomized rounding procedure to de-
termine whether the number of representatives allocated to that state is
the lower or higher of the two possibilities. For instance, if a state has a
quota of 14.37 representatives, put 100 balls with numbers 1 through 100
in a jar, blindfold the governor of the state, and have him or her pull out a
number. If it is 1 through 37, the state receives 14 representatives; other-
wise, it receives 15. In the long run, each state will receive its quota of
representatives.
The immediate problem, though, is that this procedure produces Houses
of Representatives with varying numbers of representatives. There are
fifty states; if each state is awarded the number of representatives equal to
the lower of the two possible integers, there would be only 385 representa-
tives. Similarly, there could be as many as 485 representatives. In the long
run, of course, there will be an average of 435 representatives.
Recent Developments
Mathematical research is a lot more efficient than when I entered the
field in the 1960s. Back then, the institution at which you were teaching
subscribed to a number of journals; and most mathematicians had indi-
vidual subscriptions to the Notices of the American Mathematical Society,
which printed abstracts of papers published, about to be published or
delivered at conferences. If you saw something that interested you, you
asked the author for a preprint if the paper wasn’t readily available. You
read the article, then looked at the bibliography and found other articles
of interest, which you xeroxed if they were in your library, or which you
obtained by writing the author. Collaboration was still a key aspect of
mathematical activity, but it was generally done with colleagues you knew
locally or people you had met at conferences.
The Internet completely reshaped the way mathematics is done. The
American Mathematical Society maintains MathSciNet,^7 a searchable
database of practically every article that has been published in the last
fifty years. If you are interested in a particular theorem, such as the
Gibbard-Satterthwaite theorem, you simply type it into the MathSciNet
234 How Math Explains the World