My dream is to contribute towards the development of a mathematical theory of history and society. I believe the 21st century is ripe for a breakthrough in this regard thanks to (1) theoretical advances in sociology, especially moves towards the resolution of the structure-agency debate, and (2) the emergence of new conceptual and mathematical tools for studying complex systems.
I am inspired by Isaac Asimov's character Hari Seldon, with his psychohistorical equations, and the idea that a theoretical/mathematical understanding of history can lead to more rapid recovery from depressions, dark ages and other sociological setbacks experienced by humanity.
Although history and society are widely believed to be idiosyncratic and not amenable to formal, mathematical description, there are in fact very clear regularities in social phenomena, such as Zipf's law of city size distribution, which cannot be put down to mere chance or coincidence. These phenomena show that societies and social change are constrained to follow certain well-defined paths and patterns, and there are already some convincing and highly developed theories to suggest why such paths and patterns occur.
My specific areas of interest and research are as follows:
Economic diversity, and how it depends on population size, population density, and transport and communications technology. We know that the diversity of goods and services on offer increases with the size of a community, from village, through town to city. Similarly, the diversity of goods and services has also increased over time. This is related to gains of scale, whereby larger markets, due to higher population size and/or increased connectivity and ease of movement, make it possible to sustain ever more elaborate and specialised goods and services. The challenge is to express this mathematically, so that, given a particular population and communications/transport technology, one can state the range of goods and services that will be supported. In fact, the available goods and services amount to the technological level, so that one is really looking at self-consistent solutions. The population and technology determine the level of population and technology that can be supported.
The phoenix principle. This is my term for the principle of creative destruction, or the idea that sociological setbacks, like the Great Depression, are necessary for human progress. When something works and has been working, people become committed to it. They are only inclined to change when it stops working, i.e. when it fails. Since the world keeps changing--not least because those people who failed in the last round are always looking for ways to recover and overtake the more successful people--the eventual failure of any system is inevitable. In this sense, success is the precursor to failure, and failure is the precursor to success. This is why, for example, the losers in World War Two, Japan and Germany, rebounded strongly in the post-war era. They had already lost everything, they had nothing more to lose, and they were able to re-tool and adopt economic and social strategies optimised for the new shape of the world. The victors, by contrast, experienced the winners' curse and were left still clinging to their outmoded but seemingly successful practices. American electronics companies, for example, rejected the transistor and preferred to stick with thermionic valves, because they did not want to redesign their entire factories and product offerings. The transistor patent was purchased instead by a Japanese start-up, the Sony Corporation. The challenge is to model this effect and show how it accounts for the ups and downs of history and the ever-changing fortunes of people, firms, cities, regions and nations.
Quaternions and octonions. I identify four key variables that capture the state of a sociological entity: its scale (a measure of social interactivity--a medieval village has low scale, a modern city has high scale), its political integration (a measure of the extent to which the entity behaves in a unified manner under the direction of a single will), its economic organisation (a measure of the division of labour and the availability of diverse goods and services), and its social cohesion (a measure of the commitment and loyalty of the entity's members, i.e. the extent to which they choose to put the entity's goals and well-being ahead of their own personal goals and well-being). A given sociological entity not only possesses internal scale, integration, organisation and cohesion, with respect to the interactions among its own members, but is also subject to external scale, integration, organisation and cohesion, through its interaction with other entities. For instance, a country has an internal integration deriving from the effectiveness of its government, while also experiencing an external integration to the extent that other countries are able to coerce and impose their own will on it. The eight variables (internal and external scale/integration/organisation/cohesion) can be regarded as the eight components of an octonion. Furthermore, each of the eight variables has both a real and a perceived value (e.g. the real extent to which the government is able to impose order = real integration, and people's perception of how orderly their society is = perceived integration). The state of a society is thus described by a real and a perceived octonion. The challenge is to construct, and solve, octonion equations that indicate how the real and perceived states change as a function of their current values.
I use Mathematica to help me:
Solve equations I come across in my investigations of the above areas
Process and visualise data that bear on these problems
Conduct modelling and simulation to test and evaluate theories
I have very much welcomed Mathematica's addition of tools for handling networks, and of access to curated socio-politico-econo data, both of which are very helpful for exploring social theory. I would like to see future releases provide more historical data, going back to medieval, classical and even prehistoric times, as well as built-in capabilities for handling octonions, to extend the existing quaternion package.