copyright 1997 MARTIN E. WALTER

The Why? Who? and What? of this Course

I created this class a few years ago to fill a void. As a part of my job, for many years I have taught algebra, trigonometry, calculus and assorted other ``introductory'' (or ``terminal'') mathematics classes. A few of my beginning students have been extremely mathematically gifted, a few have not shown up for class even when they were sitting in front of me. I fear many students have graduated from college without a deep understanding or technical command of humble subjects such as fractions. Far too many students have expressed math anxiety and a fear of failing, based on negative experiences in high school, elementary school, kindergarten and beyond. Most students see little connection between their math class and life as they intend to live it. Of course, math is required, math builds character -- or at least tolerance for adversity. Which brings me to why.

Why did I write this book?

It is quite clear to me (and at least a few others) that today's civilization/economy is like a jet -- a technological marvel, apparently defying the law of gravity, until the fuel tanks hit empty. We clever humans have designed a way of living that apparently violates many laws of Nature, until an irreplaceable resource runs out or there is a design failure that exposes false assumptions. If at least a few understand this human predicament, then an unpleasant crash might be avoided; or at least the seeds of our successors can parachute to safety.

To call this problem to people's attention in a context where it can be calmly, carefully studied and ruminated over -- this is my first reason for writing this book. Now the ``bad news'' is that a certain amount of mathematics seems essential to reach an understanding of the fix most humans are in. The ``good news'' is that most of the mathematics required is quite elementary, adding and multiplying numbers and the like. Thus we have the second reason for writing this book: to teach some mathematics and its connections with life in such a way that the graduating student is fluent enough to use it. Which brings me to who.

Who should take this course?

Of course, I am biased; I think that everyone should. I have had quite a variety of students successfully take this course. At one end of the spectrum have been students very ``bad at math'' who were sufficiently interested and intrigued by the connections with the environment that they studied anyway in the face of utter ``hoplessness.'' Many have buckled down and DONE THE HOMEWORK! Doing the homework, with a tutor if necessary, has raised many an F to a C, even to the occasional B and A. Most importantly, regardless of grade, consciousness has been raised about many issues. Doing the homework, thinking on your own -- these are the times when you make the most progress.

At the other end of the spectrum I have had some students with exceptional talent for mathematics who wanted to see some connection with the environment. These students ran through the material faster than a cheetah after dinner, and they ended up doing ambitious special projects in such areas as chaos, population dynamics, analysis of systems, the future of the internet and so forth.

I have had environmental activists and anti-activists take this course. Most of them came to the same conclusion: it is quite important and in any citizen's best self-interest to have a fluent command of at least elementary mathematics and to see the relationship of mathematics to some of the biggest problems of our time. Which brings me to content, i.e., what.

What is in this course?

For me, mathematics is the search for and study of patterns. Patterns are ideas, idealized observations about Nature that help our minds grasp Her wonder. Patterns in counting end up becoming the mathematics of numbers and algebra. Using these patterns to study populations we discover that the numbers of any species tend to grow in certain ways. The mathematics does not predict the future, it just tells you what you can likely expect to happen. What we learn about population growth has important implications for your life and mine.

Patterns in human thinking become the mathematics of logic. You probably have heard of Aristotelian logic and a deductive system (such as is encountered in Euclidean geometry which captures some of our perceived patterns of space). Such logical systems do reflect human thinking, but not all human thinking. Artistotelian logic is a useful approximation -- simplification -- of reality. It has served us very well, and will continue to do so. However, other patterns of thinking have been captured in the mathematics of fuzzy logic, and it is interesting to note that billions of dollars of business are built on it.

The longer we study Nature and patterns, the more patterns we see; the more mathematics we create. Contrary to some popular opinions, mathematics is alive and growing; and there is beauty and excitement to be experienced. Mathematics provides a special way of looking at the world.

One of the time tested, powerful techniques of mathematics starts by identifying among a body of patterns certain patterns that can be regarded as most basic. These patterns are called axioms (or ``laws''), and different people can come up with different axioms. Our mathematical technique then directs us to build up as many patterns as we can using the axioms as building blocks and certain clearly stated logical rules. The deductive system of Euclidean geometry is one of the most famous examples of this technique. Euclid organized all of the then known facts (``patterns'') of geometrical knowledge into a system built upon a foundation of basic axioms (`self-evident patterns''). Going through this process clarified human understanding of geometry and helped humans communicate their ideas about geometry to one another. We will use this technique to build patterns in numbers, arithmetic and algebra using certain axioms of numbers as building blocks. One of the goals of this exercise is to put your knowledge of numbers on such a solid footing that you ``have a command of fractions.''

Another of my goals, only partially achieved in this course, is to do an ``axiomatic analysis'' of both my culture/civilization/economy and of the Nature in which it is embedded. (One of my more famous colleagues in economics, the late Kenneth Boulding, once said that the laws of the social sciences are just as rigorous as the laws of the natural sciences, it just takes a lot longer to find them.) It turns out that patterns observed in the study of ecology form a natural bridge that leads me to at least attempt to implement this grandiose scheme which includes looking for the axioms of economics, history, sociology, biology, political science and other disciplines. Just attempting to understand the laws (axioms) of Nature helps us understand Nature better and helps us communicate what we think we know about Nature amongst ourselves. Hopefully the same will be true of an attempt to find the laws of human society.

This course is in part a ``snapshot'' of where this ``critical thought'' analysis of Nature and society is at the moment of writing. By attempting to clearly state some of the axioms that determine the design of our civilization/economy, I try to show how this design is flawed in the sense that it violates the axioms of Nature. Nature includes us and all that we do, and I assume that everything in Nature must eventually obey all the laws of Nature, including us! Consider a concrete example. I know of almost no serious decision makers, e.g., legislators, judges, executives of countries or corporations, who do not seek after, promote and applaud continuous (material) growth. The axiom is ``Growth is good. Resources are practically infinite.'' On the contrary, in Nature we observe that growth is an adolescent phase in the life of an organism. Organisms usually do not grow forever without end. As author Edward Abbey said, ``Growth for the sake of growth is the ideology of the cancer cell. Cancer has no purpose but growth; but it does have another result -- the death of the host.'' Populations do not grow forever either. In Nature the axiom appears to be ``There are limits to growth. Earth is finite.''

The purpose of doing this analysis of Nature and society is not to clarify problems and then fall into a state of hopeless despair. My purpose is to see our society's problems clearly and then attempt to formulate clear solutions. It is even conceivable that we might build a society in which most of us are even happier than we are now. According to Carl Gustav Jung, a famous Swiss psychologist, one of the keys to personal happiness is to see yourself as part of something bigger than yourself. I would suggest that seeing ourselves as part of the community of life, as opposed to the center of and ``manager'' of all life, would be a natural, but not uncontroversial, place to start.

It is often possible to do ``pure'' mathematics without generating controversy or disagreement. Such is not possible if we are to attempt to follow the program I have outlined above, for this program includes but is not limited to ``pure'' mathematics. Our subject necessarily will strike close to home and expose each of our values, beliefs, and where we stand on the economic ladder. To be honest and comprehensive, issues and ideas that upset people must be discussed. There is no such thing in reality as a reasoning mind devoid of emotion. Since we have emotions and lots of other things that make us alive besides intellect, I suggest that we not deny it but use it. Use your passions to propel your reason further than it could otherwise go. Just be honest about what is passion and what is reason.