|
Mathematical models are scarcely adequate in (plankton) ecology
It is impossible to mathematically model open systems where interactions between processes occur concurrently. A more extended explanation can be found in Wegner (1997): Why interaction is more powerful than algorithms, Comm. ACM, 40 (5) : 80-91. Wegner’s opinion may be summarized as follows. Mathematical reasoning is of algorithm kind, i.e. sequential and transformational: if [this proposition] then [this consequence], else [a mutually exclusive alternative]. Everything flows logically from a restricted set of initial propositions (axioms). Every step in the reasoning is consistent with the preceding one: in a mathematical demonstration, a proposition cannot be true and false at the same time. Every algorithm can be simulated with a Turing machine. In a Turing machine, all transitions from one state to another one are coded from the start. An input I operating on state A gives state B. During the transition from A to B, no external event can change the state of A. The Turing machine, as well as the mathematical reasoning, applies to a closed universe. This is not true for the external world, where, for example in the biological world, processes occur in parallel. In the real world, A may be modified by another event during the transition from A to B. We deal with open systems, able to have numerous states simultaneously. One extreme case is the human brain, with its 14 billions of neurons; it can present more states than the number of seconds elapsed since the onset of the universe. But simpler systems can also present an astonishing number of state combinations. Gödel’s theorem on the incompleteness of the integers extends to algorithmic reasoning. No formal description of any open system is possible. Mathematics is the royal avenue for a self-consistent description of closed systems. However, this complete safety, so satisfying for the human mind, should be abandoned in the case of open systems. To regain this safety when dealing with open systems, one is often tempted to assume that the system is closed. This may be acceptable for modeling a chemostat, or an area with very few exchanges with the outside. This is what is usually done in biogeochemistry. At a global scale , the closed world hypothesis may hold. But what can we do with that? Predicting that it will be warm in summer, and cold in winter? In ecology the closed world hypothesis is usually not tenable. It is not possible to exclude concurrency of processes. Here mathematical model should be abandoned in favor of the richer description offered by software models.
Last modified 2002-11-19 |