Alonzo Church's mathematical work on computability and undecidability is well-known indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of G odel's general recursiveness, not his own #-definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of e#ective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the char...

. In order to extend classical models of computing with symbols we introduce a model for quantitative computation which is based on infinite-dimensional topological linear structures. In particular machines which operate on data taken from Hilbert spaces will be looked at. These Hilbert machines (and other topological linear machines) allow the adequate treatment of concepts like infiniteness and similarity as they are based on a combination of a simple algebraic structure together with a topological one. We first discuss the differences between qualitative data representation using symbols and quantitative descriptions. Hilbert machines are then introduced on a concrete as well as on a purely abstract -- C -algebraic -- level. Furthermore, the relation to other computational models will be investigated. It will also be shown that various novel computational models like Quantum Computation, Girard's Geometry of Interaction, or Neural Networks are actually instances of Hilbert mach...

In order to extend classical models of computing with symbols we introduce a model for quantitative computation which is based on infinite-dimensional topological linear structures. In particular machines which operate on data taken from Hilbert spaces will be looked at. These Hilbert machines (and other topological linear machines) allow the adequate treatment of concepts like infiniteness and similarity as they are based on a combination of a simple algebraic structure together with a topological one. We first discuss the differences between qualitative data representation using symbols and quantitative descriptions. Hilbert machines are then introduced on a concrete as well as on a purely abstract -- C -algebraic -- level. Furthermore, the relation to other computational models will be investigated. It will also be shown that various novel computational m...