This is a three part proposal revolving around the assessment of the limits of mathematical knowledge primarily in terms of the concept of open-ended axiomatic systems. The first part is to complete the research on and systematic exposition in book form of systems of explicit mathematics, comprehending a very extensive literature. The second part is to develop in detail and study a new global open-ended axiomatization of mathematics with a pluralistic ontology within which mathematical practice can be faithfully represented. The third part is concerned with the elaboration of a new philosophy of mathematics, called conceptual structuralism, underlying these directions of technical work. The first part is to be carried out in continued active collaboration with Gerhard Jaeger and Thomas Strahm of the University of Bern. The three parts are to be pursued concurrently over three years. The activities will include research, writing, and several exchange visits; the outcomes will include a number of research articles and a book. The enduring impact of the book on explicit mathematics will be to set the stage for new developments extending its approach to other frameworks, as has already been done with operational set theory. The aim for the second and third parts is to open up new perspectives on the foundations of mathematics more in accord with the experience of working mathematicians.