Vision by Viktor Winschel

The Mathematics of Yin and Yang: Category Theory

This page describes the development of OiCOS from a vision into a company along the lines of my research in economics.

The Vision of OiCOS is to find the principle of the categorical imperative of human organisations: how to setup human organisations such that the principle of organisation could be made the foundation of all human organisations?

The principle we found are our open games and their compositions. Open games capture the least unit of economic modelling as the teleological pattern underlying any forward looking economic model. Composed forward looking models in turn are how we describe human organisations. The practical task now is to create an organisation for developing the tools of creation, exchange, reuse and optimal design of organisations. It might be obvious to the involved but not to someone new to this field: We are talking about the oldest and most difficult problem facing human epistemology: what is complexity, self-organisation and consciousness?

"... consciousness is synonymous with complex systems theory. Category theory can encompass mathematically such an ubiquity of complexity in a general manner. This is not a flaw or a Godelian recursion, but a requirement to sufficiently envelop the full universal nature of mind in a complex conscious universe with inherent self-organized teleological principles governing its evolution." from "Consciousness: ‘A Thousand Points Of Light’, The Emergence Of Self-Organization From The Noumenon Of The Conscious Universe" by Richard Amoroso and Barry Martin

Obviously, there is still a wide gap from a theory of mind and consciousness to human organisations be them companies, cooperatives, economies or monetary systems. However, I feel that complementing the research with the practical doing of organising a company that has as its product the organisation of economic organisations is now of paramount importance. Here is the path towards the current state of OiCOS.

Timeline of Business and Research:

1993-1996: Starting up media mutant

Learning economics by doing as a producer and entrepreneur in the film industry.

1994-2000: Economic Studies in Germany and Belgium

Studying monetary theory, econometrics and political economics.

1996-1998:  MAMBA at the ZEW (Center for European Economic Research)

Programming option pricing and derivative portfolio simulations for an investment bank.

2000-2005: PhD , Exchange Rate Regimes, Computational Econometrics

Evaluating the suitability of groups of European countries for a monetary union.

2005-2010: Reflexive Turn in Economics

Interdisciplinary research to improve the shortcomings of economic theory.

Accomplished: Understanding that reflexivity is the core unresolved modelling problem of economics and identification of the mathematics of computer science as a tool to solve it.

Next steps: Applying reflexive structures from computer science to economics.

2010-2019: Reflexive Game Theory and Compositionality

Putting the foundations of economics on the reflexive foundation of computer science.

Accomplished: Defining open games in category theory as a reflexive foundation of economics.

Next steps: Programming software for real world applications.

2015 -2017: Sensitivity Analysis

Applying sophisticated sensitivity analytical methods from engineering to economic models.

2017-2020: OiCOS

Founding OiCOS with a first product in smart energy networks in need of many intelligent agents to be coordinated by complex institutions.

Accomplished: Software for optimal configuration and control of prosumers in energy networks. Identified invariance principles reigning monetary systems of a bank of central banks.

Next steps:  Scaling the software to a smart energy network. Implement the business model of a bank of central banks.


Summary of OiCOS Technology:

The Vision of OiCOS is to create theories, technologies, tools and an organisation (the OiCOS company) for the description, creation, exchange, reuse and optimal design of organisations.


We define organisations as self-maintaining objects. Such objects can repair and recreate themselves or create other objects. Organisations are for example calculations, communications, cognitive processes, living beings, humans, households, companies, cooperatives, governments, national states, exchange rate regimes, institutions, laws, constitutions or societies. We want tools to force organisations to obey unavoidable  or decided upon invariant principles of any science like economics, biology, physics, computer science or mathematics. These tools of organisation should enable us to construct, simulate and reason about digital twins of organisations to the help of citizens, management or governments.

OiCOS is based on the principle that the context and the content has to be united for a suitable representation of organisations. This would be the case if for example the descendants of the word oikos: ecology and economy are united in consistency. Hence the name OiCOS for the vision of OiCOS. But OiCOS is also a company that markets the tools of the creation of organisations to the benefits of humans and living beings. Ultimately OiCOS wants to create a market place for organisations where organisations are created, exchanged and reused. The steps towards these goals are

- Models of organisations

- Tools to model organisations

- Market for organisations

OiCOS is now in the first stage of creating models for clients. At the same time we are now creating the modelling tools for us. In the next stage we will market these modelling tools for organisation. It is now developed as an integrated modelling environment (IME) just like programmers use integrated development environments for creating software. In the last stage OiCOS wants to be a market for organisations. Here modellers using the IME will create organisations which they supply to the demanders of organisations. This is economically useful since it is a waste of resources if let's say call centres are repeatedly invented with all the mistakes it takes until a useful organisation is found. The market is to become a discovery mechanism for optimal organisations.

The three stages of OiCOS can be described by three slogan

- we model you

- you model you

- we model us

The origin of this program is the insight that none of the available theories and tools from physics or economics like system dynamics, differential equations or similar approaches are able to provide the necessary tools to account for the existence of objects. The usual mathematical tools of phase spaces in physics talk about the dynamics of properties or actions of objects only but not about the objects themselves. The tools of physics are more often than not applied to other fields with doubtful results. There is for example no symbol for households in household theory in economics but only utilities or demands given a context for example with prices. Accordingly our mathematical theories in economics do not talk about households creating other households or repairing themselves.

This is not a mere academic exercise but the solution for many practical problems. Unsuitable tools considerably reduce the scope where we can apply traditional economics. The theoretical shortcomings results in ever repeating problems with unstable institutions, ecological disasters, failed states and bankruptcies, reorganisation of companies and markets. A typical human way to cope with instability and inconsistency of context and content is to fight wars. These examples involve objects and their dynamics featuring their creation and destruction. Accordingly, adaptation to changed contexts, innovations which change contexts and similar dynamical patterns are causing exorbitant costs in economies and societies. Management science reacts with flowery creations of words like change management without a suitable scientific foundation, tools and appropriate software.

After searching the solution to self-maintaining and reflexive phenomena in philosophy, sociology, economics, biology, computer science and mathematics I have found theoretical computer science and the theory of computation to be able to represent objects, their behaviours and the composed objects with their composed behaviours. By that we may hope for a suitable foundation of economics and a science of organisation. Beside this theoretical commonality computer science of course also helps in implementing the necessary software tools for organisations.

The explicit scandal, after my ever existing suspicions with mathematical economics, happened long time ago in the 2000 somethings. To my dismay it occurred to me during the PhD on optimal currency areas that economics lacks an understanding of what money is. Similarly, we were not able to program economic intelligence of an agent taking into account more than about four decisions factors or basic physical invariances like the second thermodynamic law. Having programmed economic intelligence for one agent with 20 factors in my thesis it happened to define the state of the art in econometrics. However, I was thinking that this was way not enough for understanding what 8 billion intelligent humans do in our monetary systems. Trying to program systems of that size may help to understand existing ones as well as help to design properly operational organisations and deliver intelligent decision making algorithms for humans and machines interacting therein.


The underlying problem of course was that I had to think about the creation of a currency union, an organisation even more complex as usual ones since institutions like a currency union is providing the context of economic activities and by that a suitable theory of a currency union was out of scope in the existing mathematical economics.

A technology for a world wide software system of a nondestabilisable accounting system for 8 billions intelligent agents and even more goods and financial contracts was unthinkable in the nineties. Accordingly my colleagues believed that I turned mad in trying to equip monetary systems with an intelligence that is governed by a principled approach from software design. The assessment of madness in such an approach was shared by my PhD committee but at least my thesis was rated by a summa cum laude. I was thinking that pursing the goal of programming intelligent monetary systems is a natural one for a monetary economist. My understanding is that researchers as well as entrepreneurs have to think the unthinkable in order to turn reality into what before was called science fiction.

After my PhD in 2005 until 2020 I actually researched how to implement monetary institutions by suitable software systems. The starting point was Schrödinger's famous question "What is life?" in order to ensure that social institutions do respect the nature of life. Biology and economics share a strange reflexive pattern of any cognitive and conscious living beings. They carry a model of themselves in themselves, theories change reality, living beings talk about themselves or repair systems repair metabolisms as well as themselves. There are many more patterns like this including the Lucas critique of econometrics taking into account that parameter estimation from data has to model cognitive processes of agents adapting to the historical contexts and system designs when new systems are to be designed in politics and management. Explainable AI is about to learn that context dependency is key to understand and design intelligence. I found the connection to computer science via Lisp, lambda calculus and compiler technology where a compiler is a software that processes and produces software in a genuinely reflexive way. Hence, economics had to be put on the foundations of computer science which is not a random coincidence since computer science has to and does help to program our economic systems in the real world. Lambda calculus captures the reflexive structure of higher order functions and by that is able to represent self-maintaining organisations and provides a mathematical foundation of synthetic biology.

By now we have developed within our research network or identified the tools available in the sciences for actually starting OiCOS as a company. Research alone is not enough by now anymore but concrete projects in real life is a prerequisite for further development. Interesting enough in 2020 the madness as it appeared after my PhD is now shared by world wide attempts towards crypto currencies or decentralised autonomous organisations. However, I believe that "crypto currencies" are not currencies and "decentralised organisation" is a contradiction. The concept of organisation is a centralistic one but of course decentralisation is a key problem that needs to be balanced with centralisation. I also believe that understanding and implementing the logic of money especially the unavoidable invariances at the macroeconomic and in fact planetary level is a necessary condition for the survival for human kind by a resource aware setup of human-machine-nature systems. In this sense Leibniz's "calculemus" in his famous saying "Let us calculate [calculemus], without further ado, to see who is right" must be applied as well to questions of money, reserve currencies or debt cuts. Right now our economic and money systems are not in line with  a proper resource, energy and goods allocation on earth let alone a fair devision of income and wealth derivable from accounting systems. Such systems of man kind are not self-maintaining organisations.

At OiCOS we are now building an operating system for prosumers in energy networks. These organisations need an underlying accounting and monetary system as well. I think that OiCOS technology will need to be thought as an operating system for smart cities and human economies. Economic models and IT systems implementing macroeconomic human-machine-nature systems are what I am building OiCOS for. I even believe that what we need to build is not an operating systems but a consciousness or self-maintaining organisation.

Detailed Timeline of Research:

1992-1996: Starting up media mutant

Enthusiasm is key.

During the first years of my studies of economics I had the chance to learn in parallel how to set up an own company. We pioneered the film industry in Stuttgart in Germany with three film directors graduated from the new Film Academy of Ludwigsburg. This Academy is now a world renowned institution regularly nominated and decorated for Oscars. We have set up the company media mutant for music clips and advertisement movies and I was responsible for everything non-movie related as the producer and unit manager at the film set. This work made me talking on three mobile phones, sending around best boys and calming down freaking out actors at the same time. The usual time schedule was starting at 7am and leaving at 3am for about ten weeks followed by non-stop sleeping breaks for one week. This heavy work load was due to the fact that at that time renting the film equipment costed about 30K German Mark a day. We were working essentially all around the clock. This time taught me that enthusiasm for a topic is able to set free otherwise unimaginable quantities of energy.

1994-2000: Economic Studies in Mannheim and Belgium

Money is not understood and econometrics is key.

Economics is by now a rather mathematised field with quantum physicists programming investment algorithms at banks. Econometrics is the data analytical field of economics and a playing ground for most sophisticated statistics. Macroeconometric models challenge complex methods in mathematics and computer science. However, the usual mathematical approach taught in economics was not enough for my purposes. In fact I was clear about the shortcomings since the very first lecture when I realised that in economics the units are missing. Even more troublesome was that the household in household theory in economics was missing. There have been no households but only properties and actions of them like utilities and demand given some prices. The creation of a household or the change of a household can not be made the subject of a rigorous calculus. Economics is a naked emperor.

Being a major in physics and mathematics at school this insight made me studying the most sophisticated economics with a professor who is not relying on mathematics, my later PhD advisor, while getting into mathematical sophistication by an education in econometrics in Mannheim and in one of the best schools for econometrics at the CORE Institute in Louvain-la-Neuve in Belgium.

The most fascinating topic in economics I found was money theory since money is said to make the world go around. An even more astonishing issue is that we lack a convincing theory of money. What is money? Economics as a field has not yet arrived at an understanding what money is. I have studied monetary policy with my PhD advisor Prof. Roland Vaubel and money theory with the world renowned money theorist Prof. Martin Hellwig. He taught us that we are missing a mathematical language to even talk about money. From this point of view I have researched from 2005 to 2020 to be able to design a language of money in economic theory by the tools of programming language design and category theory. This allows OiCOS to put economics on the suitable mathematical foundation of the reflexive computational theory of compiler design. This technology at the same time allows us to implement real world economic systems accompanied by large scale software systems.


1996-1998: MAMBA at the ZEW (Center for European Economic Research)

Self-referentiality can feed you.

After I got my bachelor degree in economics I programmed at the ZEW the software MAMBA for simulating derivative portfolios for the Investment Bank Metzler in Frankfurt. The goal was to show to the clients of Metzler how derivatives skew return distributions and how to measure the risk exposure of actual portfolios of Metzler. Again, reflexivity came to my help as self-referential Excel cells solving the Black-Scholes formula of option pricing. The investment bankers have been very astonished that their Excel sheets are able to do so. Accordingly, they paid me double the wage of a student such that I have stopped studying economics for about a year, programmed day-and-night and by that financed my master studies in economics until the PhD. Programming, a fascinating hobby since I was 12, turned into an inflow of money. Being not a rich families kid, my first computer, the Atari 600 XL, was bought to me without an external storage device. Therefore, I needed to type in my code every time anew when I switched on the computer. This constraint reduced my first AI for the computer opponents of human players from an iterative one-pager to a recursive one-liner. The lesson to learn was that brain substitutes money. I got the computer and was allowed to play only those games that I coded myself. This pedagogical trick taught me that programming games adds a lot of fun to playing games only. At the bank of Metzler, I needed to play the game of monetary systems. Accordingly, following my mother's pedagogical trick, I thought that I should better program monetary systems rather than staying only a player therein. Consequentially my PhD thesis was about computational econometrics of exchange rate regimes and monetary policy.

2000-2005: PhD , Exchange Rate Regimes, Computational Econometrics

Economics and econometrics need a new mathematical foundation.

In my PhD thesis I have evaluated empirically groups of European countries for their suitability to form a currency union. The result of this PhD was the software JBendge which defined the state of the art of macroeconometric methods and software used for monetary policy conduct at central banks. The theory underlying the software was published in Econometrica which is among the most sophisticated mathematical journals in economics. Another paper using the underlying numerics for microeconometrics was published in the influential Journal of Econometrics.

However, in spite of these publications the PhD revealed to me that economics has not at its disposal a sufficiently sophisticated economic theory, especially money theory, mathematics and software technology in order to provide a modelling and implementation toolbox for systems as large as exchange rate systems. The research on providing these tools took me on a journey until 2020 where all ingredients are ready for practical industrial applications.

I have also found a first convincing approach to money theory which is a prerequisite for a modern organisation of labour sharing and redistribution of common yields. A suitable organisation of monetary systems is promising to prevent ever repeating financial crises which regularly expose our societies to the risk of wars and disputes on income and wealth distribution. Money is a protocol for sharing the common yields of labor sharing.

We can now implement these structures based on a sound mathematical and software architectural approach. We need accounting systems suitable for consistent records of the exchange of natural and other economic resources, goods and money. Especially, what is missing in current money theory, are bookings which obey to consistency conditions for credit, money and currency creation of commercial, central and banks of central banks. Here we need to go way beyond double entry bookkeeping. Producing IT systems for these monetary structures, with production and exchanges on top of them is what OiCOS was built for.


2005-2010: Reflexive Turn in Economics

The proper foundation for economics is the one of computer  science. Mathematics implements yin and yang in category theory and makes computer science into a social science - any property, object or process is defined by the embedding into its context. It is the key to modularity and process orientation in general. In economics we get the dynamics of institutions and contexts.

My PhD delivered the result that the mathematics in economics is insufficient to provide what economics is meant for, namely providing the institutional setting (also some kind of enterprise resource planing (ERP) IT systems) for large, international and world wide organisations and software protocols for the exchange of money and goods.

Having found this, I dived into the wilderness of interdisciplinary research by asking the most fundamental questions I found. One starting point was Schrödinger's famous "What is Life?" article and the relational and synthetic biology it inspired. My quest was for the suitable foundation for sociology and economics consistent with living systems which to my understanding was developed by Luhmann's reflexive theory of societies.

The crucial link into mathematics and software technology was Paul Grahams "Roots of Lisp". In this article Graham reprograms John McCarthy's meta circular interpreter of Lisp in Lisp. Voodoo, I was thinking! How is this possible and how is it related to my modelling problems in economics? The answer, every computer scientist knows, is reflexivity implemented by the higher order functions of lambda calculus, very much like in my self-referential Excel cells or my first recursive AI in the Atari computer and our later open games theory. Reflexivity has been the most complex pattern that I was not able to resolve during my PhD and which is underlying the Lucas Critique of "theory changing reality" which makes the measurement and forecasts of the performance of a currency union that is only coming into reality into an utterly difficult statistical and modelling problem. Underlying this problem again is the creation of self-maintaining organisations.

Starting from Lisp, I arrived very fast at the suitable foundation of economics: lambda calculus and the modern language of its formulation - category theory which is the foundation of computer science and compiler technology for software that is processing and producing software. No more voodoo but the reflexive math of computability at work which is also suitable to improve the understanding of the organisation of living beings which are carrying a model of themselves and their context in themselves. This insight took me into the next step of putting reflexive economics and game theory on a reflexive foundations.
During this time I came across computational economics at the Argonne institute where I won the best paper award, see also the article about my work in the weekly magazine Wirtschaftswoche. I have met John Nash (a game theorist) and Clive Granger (an econometrician) at the Lindau meeting for Nobel Price Laureates. This period of interdisciplinary research is unlikely to have been possible without the new world wide web full of working papers and informations about any science I needed for inspirations. Studying interdisciplinary fields like I needed to would have been impossible in the library of the University of Mannheim alone.


2010-2015: Reflexive Game Theory and Compositionality

Economics built on a non-reductionistic theory of self-referentiality suitable for living systems.

I have identified programming language semantics, category theory and theoretical computer science as the mathematical foundation for a convincing money theory for exchange rate regimes, aka world financial architecture with sound policies of a bank of central banks. The needs for suitable technologies for such entities are now witnessed by the efforts in crypto currencies, stable coins or decentralised autonomous organisations.

During these years of a search on reflexive structures I have dived into the conferences and networks of theoretical computer science, like CALCO on coalgebras or the University of Nijmegen for a hot spot of all that. I have met Samson Abramsky at a quantum linguistic workshop in Oxford which initiated a paper on game theory with the tools of semantics and category theory. A prequel of this kind of work is the early paper of Dusko Pavlovic on games. He and his laboratory are still a continuous inspiration on topics at the borders of computer science, mathematics and economics. During this time I have met Jules Hedges who wrote his PhD on game theory which is blending some theoretical ideas and directions of my reflexive economics with Jules efforts in category theory. The centre of open games research is now Neil Ghani's research group MSP (mathematical structured programming) in Glasgow.

With open games, see a preprint and the stochastic extension of them into Bayesian Open Games, we can now start building large systems from small ones, which is a prerequisite to arrive where I wanted to: IT systems for large economic entities. During this time there have been also the beginnings of what is now called applied category theory (also here and here). We have set up conferences on interdisciplinary category theoryreflexive economics and also on the logics of decision making.

Reflexive econometrics aims to put the modeller into the model, i.e. the agents we observe and model in economics have to be equipped with the same tools as the observer. AI devices in the internet of things for example are naturally modelled by econometric agents. Or put it more sophisticated: the structures at the (observer's) meta and (observed) object level have to be the same in order to arrive at a compositional theory of economies in order to design suitable institutional setups for macroeconomic systems which are also in line with nature by being built on fractal and by that compositional structures of self-maintained organisations. Taming this kind of fractality is the prerequisite for what is to become the product of OiCOS: operating systems for macroeconomic systems like exchange rate systems with custom currencies, smart cities or energy grids. The reflexivity of dynamic institutions or user generated contexts will play out as an unrivalled adaptivity necessary to succeed in ever faster dynamics of our economies and societies.
Actually, we should not only talk about operating systems for biological and social systems but about consciousness being a most high level form of self-maintaining organisations. I have been working on this utterly fascinating topic with Andrée Ehresmann, who provided, to the best of my knowledge, the most sophisticated mathematical model of consciousness - her Memory Evolutive System.

This gives another angle to look at the OiCOS technology: money forms a consciousness for the world economy. The basic analogy is that a consciousness synchronises the time scales of controlling living systems like the brain, organs, tissues and cells where each part is operating on another, from top to down ever shorter time scale. The same happens in human organisations in societies, think about the CEO and the worker in a company. Money as well is a language and communication protocol in order to synchronise control and decision hierarchies over time. The yield curve in central banking theory is a similar temporal structure fulfilling this need of synchronisation across hierarchical levels and time scales. Money is needed in energy networks as well in order to synchronise decentralised control and cooperation across hierarchical levels and their time scales. Hence, decentralisation is hardly understandable without a theory of organisation which in turn is very centralised concept of identity.


2015 -2017: Sensitivity Analysis

Global model checking.

Before setting up OiCOS I have accepted an offer from the ETH in Zürich to do another two years of research on economic sensitivity analysis with sophisticated tools from engineering. Global sensitivity analysis is a technology for understanding how assumptions of a model feed into its conclusions. We have implemented a global sensitivity analytical approach for the first time in economics in order to answer crucial questions before models can be put in place as a decision algorithm. After this research I was ready to put OiCOS on the ground.


2017-2020: OiCOS

Learning the language of industrial clients in order to translate the concepts of the economic engine of OiCOS. Implementing a company as the ultimate economic reflexivity of a theory that is changing reality.

The first years of OiCOS have been devoted to learn the translations between scientific and industrial concepts of business people. I was occupied by a collaboration with Tim Mennel at the large renewable energy consulting company DNVGL. We have talked endless hours about how to unite economics and physics for the ongoing renewable energy network revolution. The challenges are stunning since the systems are large and hierarchical with many devices which have to be equipped with intelligence for an autonomous decision making. The economic organisations which we need to coordinate the now decentralised energy production with the ever since decentralised energy consumption are rather complex, both in terms of economics and software design. We have now a promising approach to these issues with the economical challenges being identified and the technological tools from research are in place for the applications we need in energy grids.

During the last year in 2019 we have conducted with OiCOS a project with a large German E-vehicle and  a home battery and solar panel producer for a combined E-mobility and home energy production bundle. OiCOS delivered a system configuration simulator and an optimal control AI software for the prosumers starting from measured human behavioural data. This software is the current state of a first product of OiCOS which we are now aiming to extend towards a compositional operating system for energy grids consisting of many prosumers and E-vehicles in hierarchical layers.

The underlying accounting and money exchange systems do also challenge the tools I have developed for money theory. The smart contracts underlying monetary policy record in energy network the way energy and money are shifted in time. This puts an old question of money theory into a new perspective and possibility: what is the ultimate resource on earth on on which we need to denominate our money systems? The answer seems to be rather simple: the sun which is measurably delivering 1367 W of energy per m^2 to the earth and which is the ultimate resource of  economic decision making. The universe as the first investor on earth.

I have developed the money theory of OiCOS with Renee Menendez and married his approach with a national and world accounting system based on the idea of a circular economy together with the computer scientific tools underlying OiCOS. The accounting system is sound and is already implemented as a corecursive proof of concept code witnessing that money can stabilise labor sharing processes during economic production. This is an encouraging and fundamental result regarding the instability of existing monetary systems. The involved composable financial contracts are monetary units within an exchange rate system to compose the energy networks starting with the individual prosumers at the level of individual houses and solar energy productions.

The accounting system of OiCOS is circular, has any possible unit of account, is compositional and the commonality of lambda calculus with synthetic biology makes it suitable for the allocation of natural resources and is programmable. The circular accounting system can serve the purpose of a base to any economic production system at any level from households and companies up to national accounting systems and further up to the world economy in international organisations. The production functions we reformulate now in terms of compositional algebraic geometry actually implement the invisible hand of Adam Smith as a method to parallelise and decentralise the production as well as the computations for an optimal management with hierarchical control across time scales.

However, to be clear, production based on decentralised competitive markets for centralised companies is not enough as an organisational pattern for human economies. We still need democratic and tax based centrally financed public goods which can not be produced at competitive markets. Hence, the ultimate structure in need within an OiCOS operating system are the participative and democratic tools of political institutions. Only then we will complement the economic engine of OiCOS into a consciousness of economies, prioritising goals and synchronising control structures optimally allocated in the hierarchical layers of societies, energy networks or any other complex system of interacting humans, machines and nature.