Some basic theorems on the foundations of mathematics and their philosophical implications

Research in the foundations of mathematics during the past few decades has produced some results, which seem to me of interest, not only in themselves, but also with regard to their implications for the traditional philosophical problems about the nature of mathematics. The results themselves, I believe, are fairly widely known, but nevertheless, I think, it will be useful to present them in outline once again, especially in view of the fact that, due to the work of various mathematicians, they have taken on a much more satisfactory form, than they had had originally. The greatest improvement was made possible through the precise definition of the concept of finite procedure, which plays a decisive role in these results. There are several different ways of arriving at such a definition, which however all lead to exactly the same concept. The most satisfactory way, in my opinion, is that of reducing the concept of finite procedure to that of a machine with a finite number of parts, as has been done by the British mathematician Turing. As for the philosophical consequences of the results under consideration, I don’t think they have ever been adequately discussed or only taken notice of.