298 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1960 



is parallel to (its) base." Otherwise, the proof involves the definition 

 of a midpoint and on assumptions based on the diagram. 



It will be noted that the information must be specified to the com- 

 puter in a degree of detail that may not be required in a human proof 

 of theorems. In this example, the term "precedes DGC" means that 

 points D, G, and C are collinear in that order. This may appear to 

 be obvious, but it is needed for the proof. Similarly, much of what is 

 given as "syntactic symmetries" appears to be "obviously" implied by 

 the diagram. The usual proof of this theorem assumes these sym- 

 metries but does not necessarily consider them as formally as the 

 machine must. 



This example is a relatively simple one; much more complicated 

 theorems have been proved. Furthermore, the brief description given 

 here does justice neither to the magnitude nor the significance of the 

 work being done in using computers to "prove" as well as compute. 

 Obviously, the ability to prove geometrical or other theorems is not 

 significant in itself ; the important investigation is to show how these 

 significant intellectual endeavors can be performed in terms of the sim- 

 ple operations which a computer can perform. Knowing this, it may 

 be possible to extend these tecliniques to more useful intellectual 

 activities. 



FUTURE OF COMPUTERS 



It is apparent that computers are acquiring much faster operating 

 speeds and that their storage capacity is increasing while at the same 

 time their physical size is decreasing. The cost per operation is going 

 down, and it is certain that computers are going to be much more 

 widely used than they are now. Many more thousands of people in 

 the next few years will find that digital computers will play an essen- 

 tial part in their activities. 



"Wliile computers will be increasingly used for arithmetic problems, 

 it is also to be expected that they Avill find more and more uses of a 

 logical nature. The proving of geometry theorems is only a step in 

 the direction of using computers for nonarithmetic operations. It 

 does illustrate the use of computers in situations in which the pro- 

 gramer cannot possibly anticipate all the possible courses of action. 

 The computer is given very general instiiictions for determining its 

 sequence of operations and will be able to adapt or "learn" as necessary 

 to solve the problem presented to it. This should open new vistas for 

 application of computers, and it has even been suggested that tliis use 

 of computers has significant sociological implications. 



Digital computers by their nature will also produce other indirect 

 benefits. Lacking a tool that would permit doing a large number of 

 operations to solve a problem, man has characteristically developed 

 techniques using relatively few but necessarily complex operations. 



