How Mathematics Can Help Conservation Biology

Michael J. Vandeman, Ph.D.

January 7, 2003

     Mathematics enjoys the highest consensus of any science. Conservation biology, on the other hand, is "fuzzy" and "suffers" considerable controversy. They are the perfect couple! Mathematics can lend conservation biologists much-needed rigor and credibility. Conservation biology can supply mathematicians with a way to satisfy their need for ethical engagement in the real world -- to feel useful in a highly important way.

     I would like to suggest four mathematical ideas that offer significant benefits to the study and practice of conservation biology: (1) The Axiomatic Method: create a parsimonious set of basic assumptions (we want to assume as little as possible, in order to minimize objections), from which the principles of conservation can be deduced. E.g., we value genetic diversity. It follows that wildlife (at least in their reproductive years) should not be killed. (2) The existence and significance of the first human. (3) The concepts of Finiteness and Infinitude: when we are young, we feel, and act as if, we will live forever. As we get older, we come to realize just how finite our lifespan and energy are. There is a parallel with how we treat wildlife: some policies implicitly assume that habitat and genetic resources are infinite (e.g. every development is okay, because the wildlife can always find someplace else to live), whereas in fact they are finite. Similarly, most people assume that it is acceptable to kill insects, because their numbers are essentially "infinite". (4) Probability: Explaining evolutionary gradualism vs. saltation.

     Although it has limitations (e.g. as described by Kurt Goedel), the axiomatic method has served mathematics well. The goal is to reduce a field of knowledge to a small set of simple, basic assumptions ("axioms" or "postulates") from which all other knowledge ("theorems") can be deduced by means of logic. It is hoped that everyone will agree that the axioms are true, and therefore that the theorems derived from them are also true. In practice, it was found that it is not necessarily obvious that even such simple axioms are true. For example, Euclid's Parallel Postulate may or may not be true in the "real world", depending on whether space is flat, positively curved (as on a sphere), or negatively curved (as on a hyperbolic surface).

     For example, we could take as axiomatic that genetic diversity is one of our highest values. Hopefully, all humans could be persuaded to accept this postulate. From this we could deduce (the "theorem") that living organisms, at least before and during their reproductive period, should not be killed. I reason as follows: Where does biodiversity come from? From mutations: if there were no mutations, we would all still be bacteria. And where do mutations happen? The first time that a given mutation happens, it most likely happens (probability, another branch of mathematics, enters here) in only a single individual. Therefore, if that individual were to be killed (at least prior to being able to reproduce itself and pass on that mutation), that mutation would disappear from the world, and might not reappear for a very long time, if ever.

     To bring this idea closer to home, look at Homo sapiens. Pick a genetic characteristic -- any characteristic -- that can distinguish humans from other related species. Call it, say, gene H. The set of all humans (i.e., organisms with gene H) that have ever been born is a finite set (more mathematics!). Therefore, it has a first member. In other words, there was a first human. There may have been more than one first human created simultaneously, but, as I explained earlier, most likely, there was only one. Therefore, if that first human were killed prior to being able to reproduce, the human race would never exist, at least not for a very long while (since successful mutations are extremely rare). We humans, at least, would consider that a tragedy.

     From this, we can see the danger of killing reproducing individuals: it reduces genetic diversity, and thus violates what we initially claimed was one of our highest values. Many people have argued the value of preserving species, but we less often hear discussion of the need to preserve diversity within species, and why, therefore, we need to pay attention to individuals. The Endangered Species Act, which is restricted to conserving species, is therefore obsolete.

     The concept of infinity is one of the most interesting and useful ideas ever to arise from mathematics. But it is almost universally abused. Humans act as if natural resources are infinite. The notion that petroleum is finite, and therefore needs to be used only for our most critical needs, is generally ignored. The atmosphere, ocean, and land realms are treated as if they were infinite: we use them to "dump" our wastes, and assume that we will never have to face any negative consequences from doing so. We commandeer and convert more and more land to human uses, and assume that the wildlife we have displaced will either (1) find a new place to live or (2) die and not be missed. Of course, neither are true!

     For the past 15 years, I have been arguing that mountain (off-road) biking should not be allowed, due to the harm it does to people and wildlife. One mountain biking proponent claimed that "trails are infinitesimal in size". In other words, every "small" piece of wildlife habitat is insignificant. This ignores the facts: (1) Wildlife habitat and all living things are finite! If enough are destroyed, species will go extinct and not be replaced. (2) Even the smallest bit of land or water is the entire known universe, for some of the creatures living there. (3) Animals have much more acute senses than we do. Grizzlies can hear us from a mile away, and smell us from five miles away. Thus, the impact of a trail extends for a significant distance on either side. (4) The death of an insect, for example, may not seem very significant to us, but it is certainly significant to that insect!

     (I once spoke to a group of college students in Chennai, India, about the value of wildlife. I explained that as a child, I believed that animals that lived outdoors deserved to live, but anything found indoors deserved to be killed, because it didn't "belong" there. As an adult, I revised my view. Now, when I find a fly indoors, I find a way to shoo it outside, without killing it. A girl in the front row asked a very good question: "What is the value of one fly?" I said that its life is very significant, to it. Later, I came up with a better answer: If any life is significant, isn't all life significant?!)

     Mathematical and ecological fallacies are by no means restricted to young people. As recently as 1979 (Wilkins and Peterson, p. 178), we find statements like “Populations of wild animals can have the annual surplus cropped without harm”. Insect field guides, e.g. Powell and Hogue (1979), also recommend collecting insects as “an exciting and satisfying hobby for anyone” (p. 359). Does that mean that collecting grizzlies or tigers is also an acceptable “hobby”?

     For some reason, there is a controversy about evolutionary gradualism vs. "saltation". Some people think that the theory of evolution implies that changes must be gradual. However, even if a change involves a mutation in only a single nucleotide base pair, the effect needn't be small. The fallacy is assuming that a "small" change in a molecule will result in a small effect on a living thing. That may be true on a macro-scale (where statistical mechanics applies), but it is not always the case on a micro-scale, the domain of quantum mechanics and the Uncertainty Principle. In other words a probability based on the law of large numbers doesn't apply when N is small.

References:

Aczel, Amir D., The Mystery of the Aleph -- Mathematics, the Kabbalah, and the Search for Infinity. New York: Four Walls Eight Windows, 2000.

Ehrlich, Paul R. and Ehrlich, Anne H., Extinction: The Causes and Consequences of the Disappearances of Species. New York: Random House, 1981.

Foreman, Dave, Confessions of an Eco-Warrior. New York: Harmony Books, 1991.

Gould, Stephen Jay, Wonderful Life -- the Burgess Shale and Nature of History. New York: W. W. Norton, 1989.

Heims, Steve J., John Von Neumann and Norbert Wiener: from Mathematics to the Technologies of Life and Death. Cambridge, Mass.: MIT Press, 1980.

Hersh, Reuben, What is Mathematics, Really? New York: Oxford University Press, 1997.

Knight, Richard L. and Kevin J. Gutzwiller, eds. Wildlife and Recreationists. Covelo, California: Island Press, c.1995.

Livingston, John A., Rogue Primate. Toronto, Ontario: Key Porter Books, 1994.

Mayr, Ernst, "Darwin's Influence on Modern Thought" in Timothy Ferris, ed., The Best American Science Writing 2001. New York: Harper Collins,  2001.

Noss, Reed F. and Allen Y. Cooperrider, Saving Nature's Legacy: Protecting and Restoring Biodiversity. Island Press, Covelo, California, 1994.

Powell, Jerry A. and Charles L. Hogue, California Insects. Berkeley: University of California Press,  c. 1979.

Stone, Christopher D., Should Trees Have Standing? Toward Legal Rights for Natural Objects. Los Altos, California: William Kaufmann, Inc., 1973.

Vandeman, Michael J., http://mjvande.info, especially http://mjvande.info/ecocity3.htm, http://mjvande.info/india3.htm, http://mjvande.info/sc8.htm, http://mjvande.info/sustain.htm, and http://mjvande.nfs.host/scb4.htm.

Ward, Peter Douglas, The End of Evolution: On Mass Extinctions and the Preservation of Biodiversity. New York: Bantam Books, 1994.

"The Wildlands Project", Wild Earth. Richmond, Vermont: The Cenozoic Society, 1994.

Wilkins, Bruce J. and Steven R. Peterson, “Nongame Wildlife”, in Wildlife Conservation: Principles and Practices, Richard D. Teague and Eugene Decker, eds. Washington, D. C.: The Wildlife Society, c. 1979.