382 THE REAL WORLD 



imagination unaided by mathematical considerations generally breaks 

 down. 



In one minute I can make a Mobius strip which, in a few minutes, I 

 can cut repeatedly. Words just will not convey— but mathematics 

 will— the real essence of what is done by these simple operations. 



A Mobius strip 



The advent of physical theories that demand use of mathematics 

 in place of words indicates neither the unreality of the things and 

 situations only so describable, nor even the ultimate defeat of the 

 quest for understanding. Quite the contrary, it represents a trium- 

 phant transcendence of the bounds to human understanding set by 

 everyday language. Consider the statement with which Dirac intro- 

 duces his quantum mechanics. 



Her [natm-e's] fundamental laws do not govern the world as it ap- 

 pears in our mental picture [of particles, fluids, fields, etc.] in any 

 verv direct wav, but instead thev control a substratum of which we 

 cannot form a mental picture without introducing irrelevancies. The 

 foiTnulation of these laws requires the use of the mathematics of 

 transformations. . . . 



. . . The new theories, if one looks apart from their mathematical 

 setting, are built up from physical concepts which cannot be ex- 

 plained in terms of things previously known to the student, which 

 cannot even be explained adequately in words at all. Like the funda- 

 mental concepts (e.g., proximity, identity) which every one must 

 learn on his arrival into the world, the newer concepts of physics can 

 be mastered only by long familiarity with their properties and uses. 



Note carefully that Dirac does not say that the new concepts are in- 

 comprehensible. He likens them to such more familiar concepts as 

 proximity and identity— which we all find conceivable, though they 

 cannot be defined verbally without circularity. The newer concepts 

 can be mastered, though they are difficult and verbally inexpressible; 

 can be mastered, though diflScult to imagine without introducing 



