NEW USES OF THE ABSTRACT—BOEHM Sab)! 
tion of radio waves through the ionosphere. The same topological 
network may be a mathematical model of wires carrying current in 
an electric circuit and of gossips spreading rumors at a tea party. 
Because applied mathematics is inextricably tied to the problems it 
solves, the applied mathematician must be familar with at least one 
other field—e.g., aerodynamics, electronics, or genetics. 
The pure mathematician judges his work largely by esthetic 
standards; the applied mathematician is a pragmatist. His Job is to 
make abstract mathematical models of the real world, and if they 
work, he is satisfied. Often his abstractions are outlandishly far- 
fetched. He may, for example, consider the sun as a mass concen- 
trated at a point of zero volume, or he may treat it as a perfectly 
round and homogeneous sphere. Either model is acceptable if it 
leads to predictions that jibe with experiment and observation. 
This matter-of-fact attitude helps to explain the radical changes 
in the long-established field of probability theory. Italian and 
French mathematicians broached the subject about three centuries 
ago to analyze betting odds for dice. Since then philosophers inter- 
ested in mathematics have been seriously concerned about the nature 
of a mysterious “agency of chance.” Working mathematicians, 
however, do not worry about the philosophic notion of chance. They 
consider probability as an abstract and undefined property—much as 
physicists consider mass or energy. In so doing, mathematicians 
have extended the techniques of probability theory to many problems 
that do not obviously involve the element of chance. 
Probability today is almost like a branch of geometry. Each out- 
come of a particular experiment is treated as the location of a point 
on a line. And each repetition of the experiment is the coordinate 
of the point in another dimension. The probability of an outcome 
is a measure very much like the geometric measure of volume. Many 
problems in probability boil down to a geometric analysis of points 
scattered throughout a space of many dimensions. 
One of the most fertile topics of modern probability theory is the 
so-called “random walk.” A simple illustration is the gambler’s ruin 
problem, in which two men play a game until one of them is bankrupt. 
If one starts with $100 and the other with $200 and they play for 
$1 a game, the progress of their gambling can be graphed as a point 
on a line 300 units (ie, dollars) long. The point jumps one unit, 
right or left, each time the game is played, and when it reaches either 
end of the line, one gambler is broke. The problem is to calculate 
how long the game is likely to last and what chance each gambler has 
of winning. 
Mathematicians have recently discovered some surprising facts 
about such games. When both players have unlimited capital and 
