The New Uses of the Abstract! 
By Grorce A. W. BorHM 
NEVER BEFORE have so many people applied such abstract mathe- 
matics to so great a variety of problems. To meet the demands of 
industry, technology, and other sciences, mathematicians have had to 
invent new branches of mathematics and expand old ones. They have 
built a superstructure of fresh ideas that people trained in the classi- 
cal branches of the subject would hardly recognize as mathematics 
at all. 
Appled mathematicians have been grappling successfully with the 
world’s problems at a time, curiously enough, when pure mathemati- 
cians seem almost to have lost touch with the real world. Mathe- 
matics has always been abstract, but pure mathematicians are pushing 
abstraction to new limits. To them mathematics is an art they pursue 
for art’s sake, and they do not much care whether it will ever have 
any practical use. 
Yet the very abstractness of mathematics makes it useful. By ap- 
plying its concepts to worldly problems the mathematician can often 
brush away the obscuring details and reveal simple patterns. Celes- 
tial mechanics, for example, enables astronomers to calculate the posi- 
tions of the planets at any time in the past or future and to predict the 
comings and goings of comets. Now this ancient and abstruse branch 
of mathematics has suddenly become impressively practical for cal- 
culating orbits of earth satellites. 
Even mathematical puzzles may have important applications. 
Mathematicians are still trying to find a general rule for calculating 
the number of ways a particle can travel from one corner of a rectan- 
gular net to another corner without crossing its own path. When they 
solve this seemingly simple problem, they will be able to tell chemists 
something about the buildup of the long-chain molecules of polymers. 
Mathematicians who are interested in down-to-earth problems have 
learned to solve many that were beyond the scope of mathematics 
only a decade or two ago. They have developed new statistical 
methods for controlling quality in high-speed industrial mass produc- 
1Reprinted by permission from the July 1958 issue of Fortune Magazine; copyright 
1958, Time Ine. 
309 
