The approach represented in this book has a strong spiritual kinship with constructivistic ideas. This kinship can be illustrated in a variety of ways. One of the basic ideas of constructivists like Michael Dummett (1978, 1993) is that meaning has to be mediated by teachable, learnable, and practicable human activities. This is precisely the job which semantical games do in game-theoretical semantics. These games can be thought of as being a variety of Wittgensteinian language games. Now these very same Wittgensteinian ideas have been one of the main sources of inspiration to contemporary constructivists. In view of this close relationship of my ideas to those of the constructivists, it is in order to ask what relevance the concepts and results reached here might have to the prospects of a constructivistic theory of the foundations of mathematics.

The answer to this question is not immediately obvious. It might seem that the results reached in the earlier chapters of this book entail a virtual Aufhebung of all constructivistic approaches to the foundations of mathematics. This loaded Hegelian term is appropriate because it almost looks as if I had perhaps vindicated the constructivistic approach to logic and mathematics by refuting it. As was mentioned, the basic ideas of my approach are very much in the spirit of constructivistic ways of thinking. Yet I have ended up apparently rejecting many of the characteristic tenets of constructivism.