mathematics) in areas vastly different from the social preference setting
for which it was originally formulated.
I can’t resist the opportunity to insert an idea that has occurred to me
(and probably others). One of the consequences of the theory of relativity
is that there is no “absolute time”; one observer may see event A as pre-
ceding event B, but another observer may see event B as preceding event
A. For each observer, the temporal ordering of events is a rank ordering,
but the theory of relativity shows that there is no way that the rank order-
ing of events can be incorporated into a definitive ordering of events on
which all observers can agree. Does this sound familiar? It seems a lot
like Arrow’s theorem to me.
That Reminds Me of a Problem I’ve Been Working On
Whenever a new result appears in mathematics, especially a break-
through result such as Arrow’s theorem, mathematicians look at it to
see whether there is anything about it they can use. Possibly the conclu-
sion of the theorem will supply the vital missing step for a proof, or pos-
sibly the proof technique can be adapted to suit their particular needs.
A third possibility is a little more indirect: something about the theo-
rem will look familiar. It’s not exactly the same problem that is cur-
rently stumping them, but there are enough similarities to make them
think that with a little tweaking, the theorem is something they can use
in one way or another in their own research. And it was a little tweaking
of Arrow’s theorem that led researchers directly into politics’ smoky
back rooms.
NOTES
- See http:// www .hoover .org/ multimedia/ uk/ 2933921 .html.
- S e e h t t p : / / w w w. c s. u n c. e d u / ~ l i v i n g s t / B a n z h a f /.
- See http:// lorrie .cranor .org/ pubs/ diss/ node4 .html.
- See http:// www .cooperativeindividualism .org/ condorcetbio .html.
- See http:// en .wikipedia .org/ wiki/ Nicholas _of _Cusa.
- COMAP, For All Practical Purposes (New York: COMAP, 1988). The examples
used in this chapter are based on the wonderful textbook For All Practical Pur-
poses. If you are going to buy one book from which to continue examining some
of the topics in this book, as well as others you might find interesting, I would
recommend this one. It was originally constructed by a consortium of teachers
of courses with the intention of creating a book that would enable students with
a minimal background in mathematics to learn some mathematics that relates
to the modern world. It succeeded admirably. Early editions of this book can be
purchased on eBay for less than $10.
220 How Math Explains the World