The Reasonable Ineffectiveness of Mathematics in the Biological Sciences

To many the title presents a poor parody of the famous quote "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", made by the 20th century physicist Eugene Wigner. In Wigner's era physics ruled the domain of natural sciences riding on the twin triumph of relativity and quantum physics, and what he really meant was the unique ability of mathematics in advancing the physical understanding of the Universe. Examples abound: Newton invented calculus (years before Leibniz, who was a mathematician) to develop his laws, and Einstein learnt Minkowski and Reimann geometry to develop his relativity theory. Likewise the eventual success of quantum physics depended heavily on Schroedinger equation. In subsequent decades biological sciences have made rapid strides (ignoring Darwin's evolution theory, the biggest stride of them all, which occurred in mid 19th century), with amazing progress in genetics, molecular biology and ecology, and can today rightfully claim to be an equal partner in describing the science of the nature. This has brought up the question, particularly after Wigner made his poignant observation, about the comparative lameness of mathematics when it comes to biology. I have myself mused on it for some time, and finally decided to put my thoughts in writing after reading a recent discussion on NECSI listserv.

Of course, while flipping through a biology textbook today, one would come across quite a few mathematical notations and equations. It is also true for the numerous research papers published in various peer-reviewed biology journals. This is all because of the development of an entire new field of mathematical biology, with the increasing realization that mathematics can make the job of describing many biological phenomena relatively simple. But Wigner did not use the word "effective" to mean mathematics merely as a convenient tool or a language to describe physics; to him discovery of the physical laws would be at best indefinitely delayed, and at worst impossible, without fundamental contribution from mathematics. (I am skating on thin ice here: there are biologists with considerably more authority than me who can point to a number of similar contributions of mathematics to biology, such as Mendelean laws of inheritence, Hardy-Weinberg law of population genetics, Trophic-Dynamic concept of energy transfer, and so on. But the argument against these examples again boils down to whether mathematics was really needed for their discovery, or merely served as a suitable language to present them.)

The reason for this discrepancy clearly lies in the difference in the very structure of the two sciences. Physics is a science of universal phenomena, where each physical law is reproducible anytime anywhere in the Universe given identical conditions (water always boils at 100^oC and freezes at 0^oC under the same ambient temperature and pressure). Biology, by contrast, is a science of here-and-now; that is, each and every biological entity and event is unique and distinct from any other. No two human beings are identical, even when from the same location, race or family. Same holds for individuals belonging to any non-human species. The rules and conditions governing the persistence (or extinction) of a species in two habitats are never same, because the detailed biotic and abiotic environments in the two habitats always differ to some degree. However, at the very basic level all biological systems are composed of electrons, protons, neutrons etc., and as such they must abide by physical laws at this elementary level. Also, universality holds for many biological principles: for instance, Gause's law of competitive exclusion, which states that one species will always outcompete all others when vying for identical resources (the biggest tree always wins against smaller plants when they compete for soil nutrition and sunlight on the same habitat patch). It is easy to see that given identical environmental conditions, this principle should hold everywhere. The catch is in the detail: environmental conditions are never identical in any two places, and therefore the notion of universality, while still valid in biology, is irrelevant in addressing pressing problems of species conservation in as varied places as, for example, Alaska and Hawaii. Thus, biologists often must forego unifying approaches, and study their systems on case-by-case basis. As someone in NECSI forum pointed out, "Simpson doctrine" says it best: Physicists study principles which apply to all phenomena; biologists study phenomena to which all principles apply.

Where does mathematics fit in to this absence of universality in biological sciences? Before we get into that, I should qualify the kind of mathematics that was prevalent in Wigner's time, which was the so called traditional, or continuous, mathematics such as differential and integral calculus with the implicit assumption of continuous (and infinite) time and space. Walter M. Elsasser proposed that it is this property (or limitation) of the traditional mathematics that makes it so effective in physical sciences and the same time ineffective in biological sciences. He delineated sciences into studying two broad classes of objects: the homogeneous (and infinite) class which is characterized by a small number of kinds of objects with a large (almost infinite) number of exact copies of each kind, and the heterogeneous (and finite) class that concerns with objects whose variety far exceeds the number of copies of each kind. Physics, with its requirement of universality and reproducibility of the laws under identical conditions, makes an assumption of homogeneity of space and time as well as the existence of infinite copies, and therefore deals with homogeneous class of objects. Biology, by contrast, because of the sheer variety of the subjects with almost no two identical copies in nature, is said to study objects of the heterogeneous class. It was Elsasser's contention that the continuous mathematics is historically well adapted for the homogeneous class and thus for physical sciences, but not the heterogeneous class for which he claimed that a new kind of logic is needed.

With the advent of computer, many believe such a logic can now be built on computer-based discrete mathematics (including information theory, combinatorics, algorithmics etc.), which has the potential to make fundamental contributions to biological sciences. To give an example, it is possible to develop theoretical models of oceanic plankton dynamics with differential equations representing the population size as a continuous variable, because a typical plankton colony in bloomtime contains upto trillions of individuals. But a population of the bighorn sheep, a threatened species with urgent conservation concern, typically numbers only in 50's and 100's of individuals, and modeling the dynamics of such a population would involve finite mathematics in which each individual needs to be tracked separately. Such an endeavor is possible only in computer-based models similar to the cellular automata type systems, which was developed by Stephen Wolfram and is strongly advocated in his somewhat ostentatiously titled book "A New Kind of Science". (I remember listening to one of his promotional lectures in the University of Michigan few years ago, and was unimpressed by his arrogance.) Another type of mathematics that has become particularly relevant to population biology and ecology is the stochastic mathematics, which is often used to introduce variations into population parameters of the computer models, in order to capture the uniqueness of the individuals making up the population. Thus, it seems we are not far from experiencing a sense of wonder at the effectiveness of the "new mathematics" in biology, similar to what Wigner felt in 1960.