On Infinity and Biology
On Infinity and Biology
By Vera Calenbuhr, Founder, Blue Marble Sustainability Research & Consulting, Switzerland
Fascinating reflections and discussions on infinity are often rooted in philosophy, mathematics and physics. This raises the question, if and how notions of infinity relate to and reach into the realm of biology? We explore this by having a closer look at biological function, self-organization and complexity.
Since the middle of the 20th century, the spontaneous emergence of order and structure has been described and studied in many systems far from thermodynamic equilibrium: fish schools and flocks of birds displaying spontaneously formed structures that can dynamically adapt and reorganize if disturbed e.g., by a predator; foraging ants forming scent trails and being capable of collective decision making and solving optimization problems. These self-organization phenomena are just a few examples from biology. Numerous self-organizing systems are also known from physics (cloud rolls), chemistry (oscillating reactions), biochemistry (biological clocks, enzymes operating in synchronicity), and human behavior (collective decisions) to mention just a few.
However diverse self-organization phenomena may be, they all share some common features:
The subsystems (individual ants, termites, birds, people, molecules) interact with each other and individual behavior is amplified through positive or reinforcing feedback. Generally, a macroscopic pattern emerges in a bottom-up process from the non-linear interactions between the sub-systems at a lower scale.
For self-organization to occur, systems need to be far from equilibrium regarding the exchange of energy and matter with their environment.
Many classes of self-organizing systems are independent of their physical realization and they are found starting from the microscopic level (molecules, cells) to the macroscopic level (animals, people) obeying the same or very similar equations and principles.
Self-organization in biology is closely related to function; and notions of infinity play an important role in it. A conjecture by Konrad Lorenz provides a clue to the concept of 'biological function'. Lorenz suggested that the structure and shape of birds' wings evolved to a degree where they quasi-form a mirror image of the fluid dynamics properties of air.
This 'mirror-image' view of evolution of biological function does not only hold at the level of individuals or organs; it is also nicely illustrated by ants' behavior, i.e., by groups of individuals. Self-organization enables foraging ants to cope with a number of challenges. Prey or other food can be found in many and varying places; in different quantities; in rugged or non-rugged terrain; in changing conditions of temperature and air humidity and so on. And, ants' foraging behavior needs to be fault tolerant and involve fall back options, in case individual ants fail, get lost or suffer similar fates. Groups of ants are able to do all this without a central governing agency or brain. Their 'swarm-intelligence' exceeds the capacities of individual ants by orders of magnitude. Self-organized behavior helps the ants to cope with the complexity and unpredictability of their environment. If the conditions in the environment change, the whole group can adapt collectively and dynamically to the new situation.
In a similar way, self-organized behavior allows birds, fish, termites, people etc. to cope collectively with the complexity and unpredictability of the real world.
In this perspective, the dynamic nature of the emerging self-organized behavior quasi-mirrors the complexity and unpredictability of the environment.
One may argue that a finite material world cannot contain infinite amounts of objects. But what about behavior and complexity? Although foraging ants do certainly encounter limits of dynamic adaptability, there is a continuum in the parameter space within which they act. In the same way that we imagine an infinite number of points on the continuum of a ruler, self-organized systems adapting to their environment may be looked at as exploring an infinite number of points in the multi-dimensional continuous parameter space formed by the complexity of reality! So, one could ask if the ability to navigate in the continuous parameter spaces of complex environments, leading to the function of "adaptive reliable operation" can be considered as the competitive advantage that has emerged in the course of evolution?
Self-organizing behavior emerges, if the system is pushed beyond a critical point as regards e.g., energy or material supply. The science of complex systems - including self-organizing ones - knows many such critical points, bearing notions of infinity and leading to further intriguing questions in biology.
The so-called logistic map describes e.g., insects' populations changing seasonally; it is an example of a simple equation displaying very rich and complex behavior. Depending on the reproduction parameter the populations can be stable from season to season, oscillate or show values changing irregularly displaying chaos. On the way towards chaotic behavior, which is again a class of non-linear systems, some interesting things happen: the periods of the oscillations double when increasing the reproduction parameter. These points, where the period doubling occurs are - again - critical (or bifurcation) points. Finally, the ratio of the parameter intervals to the next between each period doubling converges to a constant value, the so-called Feigenbaum constant (roughly 4.669....) - named after Mitchell Feigenbaum. The period-doubling route to chaos and the Feigenbaum constant are universal properties for a large class of non-linear systems irrespective of their physical nature. Some consider the Feigenbaum constant as important and fundamental as the number pi (π) !
Given that the Feigenbaum constant is the result of the limit of the ratios of increasingly smaller parameter intervals, infinity plays here again a role in a non-linear or complex system.
This would not be so special as there are many processes in nature which are described by limit processes involving scaling to infinitely small or large dimensions. Yet, unlike the case of the infinite number of points on a ruler, there is an intricate structure here. The Feigenbaum constant is closely related to the Mandelbrot-set, which exhibits a further interesting property: the structure found at one scale is repeated infinitely as we go to smaller scales. Mathematical and physical objects displaying such behavior are called fractals.
As we know today, fractals are related in many ways to the complexity emerging from non-linear systems found in nature. So, one may ask if structure in the infinite is a hallmark of complexity in nature, and hence in biology?
And if the emergence of complex self-organizing behaviors beyond critical points is related to biological function - well, this raises the question if biological function has a relation to the structure of infinity?
"Infinite in all directions" is the title of a famous book by physicist Freeman Dyson. "Infinity in all directions" seems also to be an appropriate metaphor for the infinity found in many classes of non-linear systems including the ones displaying self-organization. Infinity seems to be a property straddling many systems. And beyond being mere abstractions, here infinity is related to biological function. We can only speculate what Konrad Lorenz might have said!
Vera Calenbuhr is a scientist, university lecturer, entrepreneur and sailor. She holds a Chemistry Diploma from the University of Cologne and a Ph. D in Natural Sciences from the Free University of Brussels. As the founder of BlueMarble, an international virtual think tank, Dr. Calenbuhr helps to strengthen sustainable development. She is a long-time lecturer at the University of Basel. Dr. Calenbuhr spent most of her career bringing science to policy at the Joint Research Centre of the European Commission. With A.S. Mihailov she co-authored the book "From Cells to Societies - Models of Complex Coherent Actions".
March 14, 2024
