In the earlier sections of this paper I have tried to lay a basis for a theory of how computations are built up from elementary operations and also of how data spaces are built up. The formalism differs from those heretofore used in the theory of computability in its emphasis on cases of proving statements within the system rather than metatheorems about it. This seems to be a very fruitful field for further work by logicians.
It is reasonable to hope that the relationship between computation and mathematical logic will be as fruitful in the next century as that between analysis and physics in the last. The development of this relationship demands a concern for both applications and for mathematical elegance.