Mathematical Systems Thinking in Human Ecology

A reflection on the places and roles of mathematics in human ecological undertakings would without a doubt be misleading if it were not to include mention of differential equations and their long history of being used for dynamical systems modelling.

Differential equations are a classic tool for dynamical systems modelling, as they offer researchers a starting point from which to model real-world interactions in environments without having to know the exact processes that underly them. In essence, they allow researchers to study the dynamics of a system from derivatives of the ”exact processes” functions that compose it, and depending on whether a system of differential equations is solvable, that is to say, whether one can arrive at the definitive ”exact process” functions that the differential equations of the system are derivatives of, they may allow researchers to model a given dynamic system completely.

Differential equations are the go-to tool for complex dynamical systems modelling and analysis, and are used extensively to model physical, chemical, and purely mathematical processes. From the beginning of the century to the mid-1970s, ecologists, meteorologists, and financiers attempted to apply the same techniques to their fields (see for instance the work of Edward Lorenz or Stephen Smale). Ecologists and social scientists, including human ecologists use these approaches extensively (see An Ecological Introduction, K. E. Boulding, 1950, or Complex ecosystem and sustainable development, Ma Shijun & Wang Rusong, 1990) But while these attempts in some cases revealed new facets of mathematics, they generally failed to do what they set out to achieve. Why?

The answer lies in the features of naturally occurring systems. In a controlled environment, sets of ”exact processes” functions can be enforced as axioms of a closed system, outside functions apply to a system and manage its behaviour. In that case, systems of differential equations may be an appropriate modelling tool. Sadly, there’s a catch: Most naturally occurring systems are adaptive: as their dynamics evolve, the ”exact processes” functions that constitute an adaptive system change, are in unpredictable and constant flux. The cause for these changes is that the ”exact processes” functions are part of the systems themselves, and dependent on material and relational processes that occur within the respective systems, which are described by other ”exact processes” functions. The interactions between these functions can form extensive networks which describe intricate mixes of flows, stocks, and transformations. Differential equations can model Dynamic Systems but not Adaptive Systems. For this latter type of system, differential equations are of very limited help.

Do we need to model everything?

Before I reflect on other mathematical techniques and philosophies, and how I feel they relate to Human Ecology and could find their place in human ecological dialogue, I would like to consider the modes of apprehension of the world that emboldened many researcher throughout history to attempt to model the natural world.

It is not self-evident that the world needs to be modelled. In fact, the world exists already, it’s out there waiting for us, what use is a copy of the process to us if it is only a shadow of the real thing? Why not go out and enjoy it in person? There are so many ways of learning about the world and the systems within it. Often neglected is hands-on learning one can earn from experiencing the world from within: physically and emotionally, to be consumed and surrounded by the experience of being, feeling, seeing, and touching.

For my part, I like to think that while experiential learning in the world is a primordial need which all should get to experience, there is still a place for analysis, deconstruction, and quantitative reason- ing. For one, all of these things are part of the world too, they are aspects of our environment. If human ecology concerns itself with human-environment interactions and the place of human beings in an ecology, then we should see mathematics mathematics as an essentially human ecological venture. Mathematics are a secretive language of nature which rarely shares its secrets. While counting is not an exclusively human trait, the depth and variety of human counting techniques are unique in the animal world.

In the natural sciences, I believe the desire to model and replicate nature as an attempt to under- stand it can be traced back to European materialism, utilitarianism, and determinist philosophy of 17th century Europe. These frameworks travelled extensively through trade, and through the colonial rule and subjugation of nation states by European powers, and spread rapidly worldwide. I bring this up not to brush off the desire to model nature as an outdated cultural relic, but to illustrate that it is a worldview that exists in continuity with the cultures and writing of a very small group of people in a very small part of the world, and at a very specific time in history. We are perhaps needlessly limiting ourselves to a relationship to nature which in some capacity must factor in this lens.

I’ll end on counting as an interesting subject: counting - what counts? Pun intended. Do trees count? Are the processes through which trees grow and multiply, branch, and ache cognisant enough to count as ”counting”? Or are they just material reactions? And does it matter? Can an animate but inorganic process count?

Not-entirely deterministic systems

As a language of nature, mathematics has the enviable property that its descriptions of mathematical phenomena transcend mathematics into all dimensions of the natural world. If a mathematical phenomenon produces a certain type of behaviour, and as observers, we notice a similar phenomenon in nature, we can be quite certain certain that the natural phenomenon is related to the mathematical one. Even though the same process may not be a play in either one, both share a sameness, and this sameness opens a pathway to mathematics ←→ nature exchanges.

We are used to using mathematics to discover Nature, or using mathematics to explain it, but we hardly ever hear of the reverse: using nature to explore mathematics, to explain mathematics, or even as a map to mathematical exploration.

I have a strong hunch that mathematics discovered through this process, especially mathematical processes and algorithms discovered through such an approach would replicate the properties of pro- cesses that occur in nature. These processes would often be not entirely deterministic, part of not entirely deterministic systems, of which we understand very little today. Exploration of mathematics through nature might yield (with some luck and creative thinking) insight into the inner functioning of complex adaptive systems, and help us breakaway from a purely normative or descriptive understand- ing of mathematics which sees mathematics as either dictating nature’s inner workings, or passively reflecting it. Mathematics are a conceptual framework that abstracts relationships that occur in nature, but we only have use for mathematics insofar as they describe the nature that they are a part of. How could nature not describe maths, just as mathematical assertions assert the dynamics, relationships, and laws of nature?

Mathematics as a shimmer through a lens

To close this reflection piece, I’d like to consider one outcome of using nature as an exploratory tool for maths: L-systems.

L-systems are formal systems which use a combination of rules or substitution of symbols of a formal language for others using sometimes deterministic, sometimes stochastic, and sometimes a com- bination of both rules. L-systems were first discovered by Aristid Lindenmayer, a biologist who studied the growth patterns of bacteria such as the Anabaena catenula. Lindenmayer originally intended these systems to provide a formal description of the development of bacteria, and to illustrate the neighbourhood relationships between plant cells. However, they were quickly extended to describe larger organisms with branching structures such as plants.

The following figures showcase some structures generated through L-systems. As you view them, I would like to invite you to think of them not as a flawed mathematical representation of nature, but as a beautiful human enabled nature-informed representation of maths.

Note: L-system models generated using Vlab on macOS, generously provided courtesy of the Algorithmic Botany lab at University of Calgary.