This paper summarizes my findings in the quest for a suitable mathematics for the production function of economists. It is ment as a proposal for the direction of a long run reasearch project uniting various sciences for the benefit of human societies. The information revolution is yet to arrive in the realm of economics and this paper is my point of view for a suitable framework for a positive and normative analysis and syntesis of economic policy in social systems.
On the individualistic level the economic situation can be discribed by multi modal logic, with at least temporal, epistemic, doxastic and deontic modal operators. The production functions in economics are to be modelled within a constructive mathematics taking into account the algorithmic nature of economic value generation. Here the monadic treatment of external effects in economics may capture ecological problems of the depletion of natural ressources. On the level in between the micro and macro structures of economies the suitable framework are process logics taking into account concurrency, hierarchies and spatial and relational links in decentralized mixed market economies.
The recent bigraphs of Milner are at the heart of my proposal for a modern mathematical framework of economics. Especially the treatment of money and budgeting is to be formalized here giving the necessary technological base in the cry out for a new world financial market architecture. The aggregation and the emergent structures at the macrolevel can be captured with analogous discussions as in systems biology where colimits, adjoints and other constructions from category theory are about to revolutionize the modelling of complex systems capable to reflect themselves and the world outside necessary for anticipative behaviour. These remarkable constructions are for the first time able to formally represent the difference between a reductionistic and holistic scientific approach.
Category theory is the glue for all these mathematical branches and all of them mentioned so far have a categorical formulation. Finally the only kind of software framework I see currently to be able to represent and model all the intertwined mathematics is Maude which is a logical framework with reflective metaprogramming capabilities build on rewrite logics and firmly grounded in category theory allowing to program, analyse and simulate directly in algebras.
The overall approach has to unite artificial intelligence, cognitive and theoretical computer science, mathematics, biology, philosophy to mention a few hotspots of the modern paradigmatic revolution of sciences which characterizes the road to a knowledge based life on earth. On the political side the drop of information processing costs questions the level of representation in modern democracies as well as the relative costs of the organization of interests by lobbying groups.
The first step is to translate and debug the current differential equation based equilibrium obsessed ahistorical and mechanistic knowledge representation and reasoning scheme of economics into a variety of modern and nonreductionistic knowledge bases with natural language interfaces such that the principals of economists are able to understand the policy derivations imposed on them by a now miserable coalition of reality blind economists and politicians.