Naturally, the most important applications of a science cannot be foreseen when it is just beginning. However, the following applications can be foreseen.
1. At present, programming languages are constructed in a very unsystematic way. A number of proposed features are invented, and then we argue about whether each feature is worth its cost. A better understanding of the structure of computations and of data spaces will make it easier to see what features are really desirable.
2. It should be possible almost to eliminate debugging. Debugging is the testing of a program on cases one hopes are typical, until it seems to work. This hope is frequently vain.
Instead of debugging a program, one should prove that it meets its specifications, and this proof should be checked by a computer program. For this to be possible, formal systems are required in which it is easy to write proofs. There is a good prospect of doing this, because we can require the computer to do much more work in checking each step than a human is willing to do. Therefore, the steps can be bigger than with present formal systems. The prospects for this are discussed in [McC62].