home | mathematics ::... 1 | 2

Since the end of 1995, I have been developing what I now call the Basic Picture (BP). This is a deep structure which underlies and generalizes topology. It emphasizes the symmetries and logical dualities involved in the notions of open, closed, continuity, pointwise-pointfree, etc.

The role of intuitionistic logic is essential, also for the discovery of BP, since classical logic would reduce all to complementation.

The duality between open and closed is extended to pointfree topology: in a basic topology, the covering relation is accompanied by its dual, the new notion of positivity relation.

Topology in the common sense is obtained by adding a condition expressing convergence, that is summability of approximations. Formal topology and locale theory then become particular cases.

BP starts an abstract mathematical treatment of existential statements. Crucial is the systematic use of the relation of "overlap" between two subsets, followed by the notions of closed subset, the positivity relation and its generation by co-induction, etc.

I feel the Basic Picture as my most personal contribution to constructive mathematics. More on Basic Picture ::...

Maths, i.e. the force of (earth's) content. Picture of Vulcano, Sicily