Intuitionism { Counting its Blessings . Preliminary

A brief survey of the impact of intuitionistic logic and mathematics on modern practice is presented. The main in uence is via the so-called \BHK interpretation" (or \proof interpretation"). By somewhat relaxing the conditions on the notion of proof the familiar notions of `realizability', `Curry-Howard isomorphism' etc. are motivated. Also some attention is paid to the characteristic process of replacing traditional notions by strong positive ones. 1 The paradox of success The history of intuitionism a rms the general rule of success in science: once a topic or notion has been successfully adopted by the scienti c community, its founder and origins become almost irrelevant to the user. Examples are all around us: nobody quotes Euclid when doing geometry, or Newton or Leibniz when di erentiating a function. Even notions or areas called after their creators loose the association with the persons, e.g. hardly anybody thinks of Euclid when mentioning euclidean geometry, or of Turing when writing down a Turing machine. Perhaps the real mark of success is a dissociation of the notion from its creator (In modern terms: the ultimate success of a scientist consists of a gradual withdrawal from the citation index). In the case of intuitionism this kind of success has been attained. Few people will still think of Brouwer when they use bar-induction, or appeal to the continuity theorem, and even fewer will cite Brouwer. After the boom of the sixties-seventies in the foundations of mathematics, intuitionism has lost its controversial character and became accepted within the larger framework of constructivism. Its methods and insights have been so widely recognized, that it mostly seems to live on as an adjective (as in intuitionistic logic). Its success has killed the topic as an independent area of research in mathematics or logic. It has become part of a wider tradition of constructivism. Contrary to Brouwer's expectations and hopes, intuitionism has found its acceptation by way of logic and computer science, rather than in mathematics. As a consequence of the present position of intuitionism in the wider context this talk will deal with general issues in constructivity and not just in intuitionism. To make the distinction clear: intuitionism is based on the mental, intellectual activity of the individual, and it derives its objects and principles c 1996 Elsevier Science B. V.