Mathematics for the Environment

Martin Walter

• Publisher: Chapman and Hall/CRC
• Year: 2011
• ISBN:
  9781439884591
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Mathematics for the Environment shows how to employ simple mathematical tools, such as arithmetic, to uncover fundamental conflicts between the logic of human civilization and the logic of Nature. These tools can then be used to understand and effectively deal with economic, environmental, and social issues. With elementary mathematics, the book se

Part I

Mathematics Is Connected
to Everything

Chapter 1

---

Earth’s Climate and Some Basic Principles

1.1 One of the Greatest Crimes of the 20^th Century

What was this Crime? “…In 1949 a federal jury convicted GM (General Motors), Standard Oil of California, Firestone Tire, and others of conspiring to dismantle trolley lines throughout the country (U.S.A.).”¹

My father once told me when I was very young, as we sat in a traffic jam, that there used to be electric-powered trolleys (also called streetcars, or light rail) in Los Angeles, before General Motors came in and bought up the trains, tracks and rights of way and did away with them. That comment sat deep in my unaccessed memory until decades later I went to a showing of the movie, “Taken for a Ride,” by Jim Klein and Martha Olson,² at my public library.

Discussion of this crime is not yet part of the educational experience of most Americans, while the effects of this crime have been global. That is one of the reasons I bring it to your attention now. The fact itself is important for understanding current transportation and energy troubles, as we shall see; but the lesson in media literacy that it begins (more later) is equally important. This crime is also connected to many other important and timely topics to be discussed momentarily, some with strong mathematical content. Thus this section is an illustration of a pattern about which famous conservationist John Muir (1838–1914) said: “When we try to pick out anything by itself, we find that it is bound fast by a thousand invisible cords that cannot be broken to everything in the universe.”

Less poetically I enshrine this pattern in the following assumption:

Connection Axiom: Everything is connected to everything else.

1.2 Feedback

There are two examples of feedback loops associated with the elimination of intraurban electric rail transportation systems in the 20^th century: one with positive growth and one with negative growth.

General Motors established the National Highway Users Conference, consisting of over 3,000 businesses associated with cars, to lobby the federal government for highways. One of their legislative triumphs was to get a federal tax on gasoline dedicated exclusively to road construction, without annual review, [416, p. 12]. Thus gas taxes lead to more roads which lead to more driving (as long as oil is sufficiently cheap) which leads to more gas taxes and repeat. In this next exercise we play with some (made-up to be simple) numbers to see how the mathematics works in a feedback loop with positive growth.

Exercise 1.1 A Feedback Loop with Positive Growth

(i) Let’s simplify the numbers as follows, until we understand the ideas involved. Suppose that for each 1 mile of road $100,000 is generated each year in gas and other user taxes. Suppose that it costs $1,000,000 to build 1 mile of road. In year 0 start with 1 mile of road. After 1 year, this mile has generated $100,000. Our 1 mile of road can then be extended by how many miles using the $100,000 generated that first year? (Hint: the answer is a fraction of a mile.)

(ii) How much gas tax will 1.1 miles of road generate in year 2? (For simplicity, we assume that the 1 mile of road instantly becomes 1.1 miles long at the end of the first year. Make a similar assumption at the end of the second year and so on.)

(iii) How much additional road can be built for $110,000?

(iv) At the end of 10 years how much road will there be and how much gas tax will it generate in the following year?

(v) After 10 years will any money be required to go back and renovate “potholes” or other degradation of the existing roads?

Over $220 billion was spent on road construction by the U.S. federal government in the last three decades of the last century; with state and local governments spending far more than that on roads. From 1956 to the 70s for every dollar the federal government spent on rail transit it spent over $88 on highways. This ratio improved in the direction of mass transit in the 70s, never reaching parity, but in the 80s federal highway spending increased by 85% while mass transit spending lost 50%, a trend continuing into the 90s, cf., [416, p. 13]. Consider the following exercises about “percents.” cf., Exercise 13.2 (viii).

Exercise 1.2 Percent Change

(i) During the 1980s federal highway spending increased by 85%. Now 85% means

, a fraction, which can also be written .85; thus given $1, the increase is $1 ∗ .85, or $.85, i.e., 85 cents. What is the sum of the base amount, $1, and 85% of the base amount? Note that ∗ means multiplication.

(ii) During the 1980s federal spending on mass transit decreased by 50%. So each dollar spent before 1980 is reduced to how much after 1980?

(iii) Suppose the following: in 1980 you spent $1. In 1981 you decreased your spending by 50% over the year 1980. Then in 1982 you increased your spending by 50% over the year 1981. How much did you spend in 1982? Exactly one dollar? More than a dollar? Less than a dollar? How much exactly?

The second feedback loop employed by the GM-led conspiracy was to use a bus company, like National City Lines, which the general public did not know was associated with GM, to go into cities. They would buy up the trolley service, then decrease service thereby diminishing demand, leading to further cuts in service—and then repeat until the trolleys ceased to exist. By 1949 more than 100 electric transit systems in more than 45 cities (90% of the trolley network) had been destroyed, replaced first by buses that were slower and less popular, and then eventually by cars, cf., [416, 646]. Let’s look at how the math works in such a feedback loop with negative growth.

Exercise 1.3 A Feedback Loop with Negative Growth

(i) Suppose you start with 1 unit of demand for an existing 1 unit of trolley service in some city. Suppose National City Lines buys the trolleys and decreases service by 20%. Do you see that now there is .8 units of trolley service?

(ii) The decrease in service makes it more difficult or impossible for some people to use the service so suppose demand drops to .8 units, to meet the amount of available service. Citing the decrease in demand, National City Lines cuts service by 20% again. Do you see that now there are .8 − .8 ∗ .2 = .8 ∗ (1 − .2) = .8 ∗ .8 = .64 units of trolley service?

(iii) Suppose that demand drops to meet available service once more, and that service is once more cut by 20%. How many units of trolley service exist now?

(iv) When service drops to less than ⅓ of a unit, say, the trolleys are discontinued. How many iterations, i.e., repetitions, of the above process does it take for this to happen?

1.3 Edison’s Algorithm: Listening to Nature’s Feedback

An algorithm, when defined generally, is a step-by-step problem-solving procedure. Thomas Alva Edison (1876–1933) was a famous American inventor with more than a thousand patented inventions, including the incandescent light. Edison reportedly replied to those who commented on the enormous number of “failures” he encountered as he tried one material after another in the search for a functioning filament for his “light bulb:” Those were not 100 failures, for I now know 100 things that do not work.

When you find yourself in a situation with insufficient information to solve some problem (and that’s all of us at one time or another), you may be left with no other alternative than to try various things and sec what happens. Of course, “insufficient information” rarely equates to “no information at all,” if for no other reason than other folks have encountered similar problems before us—tried this or that out—and records of what works and what doesn’t were kept.

Humans have been doing lots of things for a long time, and learning from previous mistakes might be considered a hallmark of cultural progress. I make explicit this folk wisdom in the following.

Edison’s Algorithm: Given a problem, research what is known about what might work and what might not work, i.e., what might lead to a solution. Using this information attempt solutions (experiment: observe Nature), and record the results—to the best of your ability.

I have found this algorithm surfacing in my own life in organic farming, house building, teaching, and especially in mathematics! Keep it in mind while tackling the homework problems.

A corollary of Edison’s Algorithm, or a product thereof, is a list of rules. (A corollary of a statement 1 is another statement 2 that logically and easily follows from statement 1.) As mistakes (a mistake by definition leads to undesirable results) are made, investigations are (should be) carried out which isolate the cause(s), and rules are (should be) adopted to avoid making those same mistakes anew.

Corollary: Lists of rules for solving a particular problem and avoiding previously made mistakes arise from successful implementation of Edison’s Algorithm.

Now Edison’s Algorithm and its corollary are so simple and obvious that you might wonder why I take the time to write them down. True, following a set of rules does not guarantee success, but not following “the rules” usually leads to the opposite. Unfortunately, powerful decision-makers all too often ignore the simple wisdom of following rules proven to work. Disasters result. Let’s look at some specific examples.

Exercise 1.4 Edison’s Algorithm and Lists of Rules

(i) On April 5, 2010 there was an explosion at Massey Energy’s Upper Big Branch (Coal) Mine killing 29 miners. According to the Mine Safety and Health Administration (MSHA), see http://www.msha.gov, this mine had 515 citations and orders in 2009, and 124 in 2010 as of May 24, 2010. (See also page 29.) A citation is given for violation of a rule. For example, there are standards for mine ventilation which when followed prevent the buildup of methane gas to dangerous (explosive) levels. Investigate at least two of the violations which were likely proximate causes of the explosion. Discuss the extent to which this is an example of not following Edison’s Algorithm and Corollary—even a pattern of such. How much authority does MSHA have? How big are the fines associated with the violations? Is there too much regulation of coal mines, too little? Why? Are whistleblowers fired, see part (viii) below? List all of the causes of system failure you can find in this case and rank them in order of importance.

(ii) On April 20. 2010. the Macondo oil well of BP-Deepwater Horizon in the Gulf of Mexico blew out. 11 workers killed. It took about 100 days to bring the well under control. Find 2 rules (at least) BP violated. (See, for example, CBS 60 Minutes: http://www.cbsnews.com/stories/2010/05/16/60minutes/main6490197.shtml, “Blowout: The Deepwater Horizon Disaster.” See also “BP’s Deep Secrets,” by Julia Whitty, Mother Jones, Sept/Oct 2010.) Just the year before, the Montera oil rig, operated by PTTEP Australasia in the Timor Sea, blew out on August 21, 2009 and was not capped until November 1, 2009. Previously in the Gulf of Mexico, the Ixtoc 1 (1924’ N, 9212’ W) oil well being drilled by Pemex blew out on June 3, 1979. It was not capped for 297 days. On January 28, 1969, Union Oil’s Platform A off the coast of Santa Barbara, California blew out and went uncontrolled for about 8 to 10 days. Oil disasters are too numerous to list in this book. For example, research the fairly well-known oil pollution in Nigeria and Ecuador, and the not-so-well-known oil disaster in Bolivia, cf., [616, Chapter 2]. Then there was the Exxon-Valdez, cf., page 25, and Exercise 13.12.

Two decades after the Exxon-Valdez serious environmental impacts remain, contrary to some prognostications in 1989–90. (On June ...

Table of contents

• Cover
• Title Page
• Copyright
• Dedication
• Contents
• List of Tables
• List of Figures
• Why Did I Write This Book?
• Reading, Learning and/or Teaching from This Book
• Acknowledgments
• Part I: Mathematics Is Connected to Everything
• Part II: Math and Nature: The Nature of Math
• Part III: One of the Oldest Mathematical Patterns
• Part IV: Counting
• Part V: Box Models: Population, Money, Recycling
• Part VI: Chance: Health, Surveillance, Spies, and Voting
• Part VII: Economics
• Part VIII: Media Literacy
• References
• Index