INDUSTRIAL MATHEMATICS 257 



physicist would resort to the laboratory. For the same reason the mathe- 

 matician in his "pure" form delights in building logical structures, such as 

 topology or abstract algebra, which have no apparent connection with the 

 world of physical reality and which would not interest the typical engineer; 

 while conversely the engineer or physicist in his "pure" form takes great 

 interest in such useful information as a table of hardness data which may, 

 so far as he is aware, be totally unrelated to any theory, and which the 

 typical mathematician would find quite boring. 



A second characteristic of the typical mathematician is his highly critical 

 attitude toward the details of a demonstration. For almost any other 

 class of men an argument may be good enough, even though some minor 

 question remains open. For the mathematician an argument is either 

 perfect in every detail, in form as well as in substance, or else it is wrong. 

 There are no intermediate classes. He calls this "rigorous thinking," and 

 says it is necessary if his conclusions are to be of permanent value. The 

 typical engineer calls it "hair splitting," and says that if he indulged in it he 

 would never get anything done. 



The mathematician also tends to idealize any situation with which 

 he is confronted. His gases are "ideal," his conductors "perfect," his 

 surfaces "smooth." He admires this process and calls it "getting down to 

 essentials"; the engineer or physicist is likely to dub it somewhat con- 

 temptuously "ignoring the facts." 



A fourth and closely related characteristic is the desire for generality. 

 Confronted with the problem of solving the simple equation o^ — \ = 0, 

 he solves x" — 1 = instead. Or asked about the torsional vibration 

 of a galvanometer suspension, he studies a fiber loaded with any number 

 of mirrors at arbitrary points along its length. He calls this "conserving 

 his energy"; he is solving a whole class of problems at once instead of dealing 

 with them piecemeal. The engineer calls it "wasting his time"; of what 

 use is a galvanometer with more than one mirror? 



In the vast army of scientific workers who cannot be tagged so easily 

 with the badge of some one profession, those may properly be called "mathe- 

 maticians" whose work is dominated by these four characteristics of greater 

 confidence in logical than experimental proof, severe criticism of details, 

 idealization, and generalization. The boundaries of the profession are 

 perhaps not made sharper by this definition, but it has the merit of being 

 based upon type of mind, which is an attribute of the man himself, and not 

 upon such superficial and frequently accidental matters as the courses he 

 took in college or the sort of job he holds. 



It is, moreover, a more fundamental distinction than can be drawn 

 between, say, physicist, chemist and astronomer. That is why the mathe- 



