NEW USES OF THE ABSTRACT—BOEHM 317 
lie about what they wanted to get from the game and how much they 
valued it. 
While game theory has already contributed a great deal to decision 
theory in modern statistics, practical applications to complex human 
situations have not been strikingly successful. The chief troubles 
seem to be that there are no objective mathematical ways to formulate 
“rational” behavior or to measure the value of a given outcome to a 
particular player. At the very least, however, game theory has got 
mathematicians interested in analyzing human affairs and has stimu- 
lated more economists and social scientists to study higher mathe- 
matics. Game theory may be a forerunner of still more penetrating 
mathematical approaches that will someday help man to interpret 
more accurately what he observes about human behavior. 
UNIVERSAL TOOL 
The backbone of mathematics, pure as well as applied, is a con- 
glomeration of techniques known as “analysis.” Analysis used to be 
virtually synonymous with the applications of differential and inte- 
gral calculus. Modern analysts, however, use theorems and tech- 
niques from almost every other branch of mathematics, including 
topology, the theory of numbers, and abstract algebra. 
In the last 20 or 30 years mathematical analysts have made rapid 
progress with differential equations, which serve as mathematical 
models for almost every physical phenomenon involving any sort of 
change. ‘Today mathematicians know relatively simple routines for 
solving many types of differential equations on computers. But 
there are still no straightforward methods for solving most nonlinear 
differential equations—the kind that usually crop up when large or 
abrupt changes occur. Typical are the equations that describe the 
aerodynamic shock waves produced when an airplane accelerates 
through the speed of sound. 
Russian mathematicians have concentrated enormous effort on the 
theory of nonlinear differential equations. One consequence is that 
the Russians are now ahead of the rest of the world in the study of 
automatic control, and this may account for much of their success with 
missiles. 
Tt is in the field of analysis that electronic computers have made 
perhaps their most important contributions to applied mathematics. 
Tt still takes a skillful mathematician to set up a differential equation 
and interpret the solution. But in the final stages he can usually re- 
duce the work to a numerical procedure—long and tedious, perhaps, 
but straightforward enough for a computer to carry out in a few 
minutes or at most a few hours. The very fact that computers are 
available makes it feasible to analyze mathematically a great many 
