The Hilbert-Brouwer controversy resolved?

Abstract

Since this lecture is called the Beth lecture, It is natural to ask oneself what connection is there between Beth and intuitionism? and, to my mind, the most obvious answer is: the Beth models for intuitionistic propositional and predicate logic. I would like to remind you of a certain picture that we all use when thinking in terms of possible worlds, namely the picture in which we are always at a certain stage, considering a number of possible future alternatives, and then one of those alternatives is realized and we get to a new stage, at which we are faced with some other alternatives, one of those alternatives materializes, and then the whole process continues in the same way. That means that, at each moment, we are at the end of a certain path, which I will draw by a thick line, symbolizing that which has already materialized, and we are considering all possible future alternatives, one of those alternatives materializes, as a result of which the thick path gets extended, and then the pattern repeats itself:

We all think in terms of this picture: it underlies possible world semantics, and it is natural to ask: where does it really come from? I mean: who had this picture for the first time? Most of us have probably met it in connection with Kripke semantics (Kripke 1965), but actually Kripke semantics was predated by Beth semantics by a couple of years: Beth’ papers were published in 1956 and 1959 (Beth 1956a, 1959), and Beth in turn drew on Brouwer here, because he took the nodes, or stages, to form a spread in Brouwer’s sense (Brouwer 1918B).