Mathematics for the Environment: 1st Edition (Hardback) book cover

Mathematics for the Environment

1st Edition

By Martin Walter

Chapman and Hall/CRC

679 pages | 54 B/W Illus.

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pub: 2011-01-18
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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 seeks answers to a host of real-life questions, including:

  • How safe is our food and will it be affordable in the future?
  • What are the simple lessons to be learned from the economic meltdown of 2008–2009?
  • Is global climate change happening?
  • Were some humans really doing serious mathematical thinking 50,000 years ago?
  • What does the second law of thermodynamics have to do with economics?
  • How can identity theft be prevented?
  • What does a mathematical proof prove?

A truly interdisciplinary, concrete study of mathematics, this classroom-tested text discusses the importance of certain mathematical principles and concepts, such as fuzzy logic, feedback, deductive systems, fractions, and logarithms, in various areas other than pure mathematics. It teaches students how to make informed choices using fundamental mathematical tools, encouraging them to find solutions to critical real-world problems.


The book can be recommended to all those readers who are interested in applied mathematics as well as to those who do not think of themselves as mathematicians yet being interested in laws and relationships in which mathematics may be a helpful tool.

—Herbert S. Buscher, Zentralblatt MATH 1211

The book is heavily referenced … there are many detailed exercises designed to highlight how mathematics can be used to explain natural phenomena and human behavior and its consequences. … this book could serve as a text for courses in applied mathematics and a resource for study material in many other subject areas …

MAA Reviews, July 2011

"Recently I purchased Mathematics for the Environment and find it to be one of the most fascinating and comprehensive that I have ever encountered. Next semester I will be teaching a class on mathematical modeling for seniors in our department, and intend to use (with attribution of course) some of the examples and questions. Never have I seen such an eclectic set of topics in a single volume. Basically I am writing to thank you for it, and to say ‘Bravo’!"

—John A. Adam, Ph.D., University Professor and Professor of Mathematics Department of Mathematics & Statistics Engineering & Computational Sciences Building Old Dominion University, Norfolk, Virginia, USA

Table of Contents


Earth’s Climate and Some Basic Principles

One of the Greatest Crimes of the 20th Century


Edison’s Algorithm: Listening to Nature’s Feedback

Fuzzy Logic, Filters, the Bigger Picture Principle

Consequences of the Crime: Suburbia’s Topology

A Toxic Consequence of the Crime

Hubbert’s Peak and the End of Cheap Oil

Resource Wars: Oil and Water

The CO2 Greenhouse Law of Svante Arrhenius

Economic Instability: Ongoing Causes

Necessary Conditions for Economic Success

The Mathematical Structure of Ponzi Schemes

Dishonest Assessment of Risk

One Reason Why Usury Should Again Be Illegal

What Is Mathematics? More Basics

The Definition of Mathematics Used in This Book

The Logic of Nature and the Logic of Civilization

Box-Flow Models

Cycles and Scales in Nature and Mathematics

The Art of Estimating

We All Soak in a Synthetic Chemical Soup

Thomas Latimer’s Unfortunate Experience

What’s in the Synthetic Chemical Soup?

Synthetic Flows and Assumptions

The Flow of Information about Synthetic Flows

You Cannot Do Just One Thing: Two Examples

Mathematics: Food, Soil, Water, Air, Free Speech

The "Hour Glass" Industrial Agriculture Machine

Industrial Agriculture Logic vs. the Logic of Life

Fast Foods, Few Foods, and Fossil Fuels

Genetic Engineering: One Mathematical Perspective

Toxic Sludge Is Good for You!

Media Concentration

Oceans: Rising Acidity and Disappearing Life

Stocks, Flows and Distributions of Food

My Definition of Food

Choices: Central vs. Diverse Decision Making


Mathematics and Energy

How Much Solar Energy Is There?

Solar Energy Is There, Do We Know How to Get It?

Four Falsehoods

Nuclear Power: Is It Too Cheap to Meter?

Net Primary Productivity and Ecological Footprints

NPP, Soil, Biofuels, and the Super Grid

The Brower–Cousteau Model of the Earth

How Heavily Do We Weigh upon the Earth?

Mining and Damming: Massive Rearrangements

Fish, Forests, Deserts, and Soil: Revisited

The Cousteau–Brower Earth Model

Fuzzy Logic, Sharp Logic, Frames, and Bigger Pictures

Sharp (Aristotelian) Logic: A Standard Syllogism

Measuring Truth Values: Fuzzy/Measured Logic

Definitions, Assumptions and the Frame of Debate

Humans in Denial — Nature Cannot Be Fooled — Gravity Exists

The Bigger Picture Principle

The Dunbar Number

The Sustainability Hypothesis: Is It True?

The Dunbar Number

Public Relations, Political Power, and the Organization of Society

Political Uses of Fear

Confronting Fear (and Apathy): Organizing Your Community for Self-Preservation and Sustainability


One Pattern Viewed via Geometry and Numbers: Mathese

The Square Numbers of Pythagoras

The Language of Mathematics: Mathese

A General Expression in Mathese: A Formula for Odd Numbers

An Important Word in Mathese: Σ

Sentences in Mathese: Equations with Σ and a Dummy Variable

Induction, Deduction, Mathematical Research, and Mathematical Proofs

What Is a Mathematical Proof?

What Is a Deductive System?

Originalidad es volver al Origen

Axioms and Atoms

Molecules and Atoms; the Atomic Number and the Atomic Mass Number of an Atom

Scaling and Our First Two Axioms for Numbers

Our First Axiom for Numbers

Number 1: Its Definition, Properties, Uniqueness

The Definition of Multiplicative Inverse

Our Second Axiom for Numbers

If … , Then … . Our First Proofs

Return to the Problem: How Many Protons in One Gram of Protons?

What Is a Mole? Scaling Up from the Atomic to the Human Scale

Five More Axioms for Numbers

Associativity, Identity, and Inverses for +

Commutativity of + and *


What Patterns Can Be Deduced in Our Deductive System?

Playing the Mathematics Game

Rules for Playing the Mathematics Game

The Usual Rules for Fractions Are Part of Our Deductive System

Can You Tell the Difference between True and False Patterns?

More Exercises


A Short Story and Some Numberless Mathematics

Relations Defined as Collections of Ordered Pairs

Symmetric Relations

Transitive and Reflexive Relations

Equivalence Relations

Relations That Are Functions

A Set of Social Rules for the Warlpiri People

The Section Rule

The Mother Relation Rules

The Marriage Rules

The Father Relation Rules

Cultural Contexts in Which Mathematics Is Done


Counting Exactly


Counting Social Security Numbers among Other Things

Permutations: Order Matters

There Are n! Permutations of n Distinct Objects

Counting Connections: Order Does Not Matter

Equivalence Relations and Counting

Using Equivalence Relations to Count

Combinations: Order Does Not Matter

Additional Counting Problems

DNA Computing

More Exercises


Some Population Numbers

Counting People in the World

A Fundamental Axiom of Population Ecology

Counting People in the United States

Basic Mathematical Patterns in Population Growth

Schwartz Charts Are Box-Flow Models

Our First Population Model: Simple Boxes and Flows

Three Basic Operations: Addition, Multiplication, and Exponentiation

Defining Logarithm Functions

Computing Formulas for Doubling Times

Natural Logarithms

Logarithms to Any Base

Further Study: More Complicated Models and Chaos Theory

The World’s Human Population: One Box

Box Models: Money, Recycling, Epidemics

Some Obvious Laws Humans Continue to Ignore

A Linear Multiplier Effect: Some Mathematics of Money

Multiplier Effects Arising from Cycles: The Mathematics of Recycling

A Simple Model of an Influenza Epidemic


Chance: Health and News

If You Test HIV Positive, Are You Infected?

Chance and the "News

Surveillance, Spies, Snitches, Loss of Privacy, and Life

Is Someone Watching You? Why?

Living with a Police Escort?

I’m Not Worried, I’ve Done Nothing Wrong

Identity Theft, Encryption, Torture, Planespotting

Encryption Mathematics and Identity Protection

Extraordinary Rendition = Kidnapping and Torture

Planespotting: A Self-Organizing Countermeasure the CIA Did Not Anticipate

Bigger Pictures and the CIA

Voting in the 21st Century

Stealing Elections Is a Time Honored Tradition

A Simple Solution Exists

Two Modest Proposals


What Exactly Is Economics?

It Takes the Longest Time to Think of the Simplest Things

A Preview of Two Laws of Nature

Three Kinds of Economists

The Human Economy Depends on Nature’s Flows of Energy and Entropy

Nature’s Services and Human Wealth: Important Calculations

How We Treat Each Other: How We Treat Nature — The Tragedy of the Commons

Mathematical Concepts and Economics

Misapplied Mathematics

New Mathematical Patterns: Self-Organizing Systems

Finding a Niche: Habits and Habitats

The Concept of Money

Financial Wealth and Real Wealth

Is Financial Collapse Possible Now?

Follow the Money

Are You Paying More or Less Than Your Fair Share of Taxes?

Financial Growth vs. Fish Growth

Fractional Reserve Banking: An Amazing Mathematical Trick

Distributed vs. Centralized Control and Decision Making

Farms: To Be Run by Few or by Many?

Utilities: MUNI or Investor-Owned?

Linux vs. Microsoft

Medicine for People or for Profit or Both?

A Little History

An Example of the Need for Fuzzy Logic: The Definition of Poverty

Energy and Thermodynamics

Energy and the First Law of Thermodynamics

The First Law of Thermodynamics

Entropy and the Second Law of Thermodynamics

Early Statements of the Second Law of Thermodynamics

Algebraic Statement of the Second Law of Thermodynamics

So What Is Entropy and Can We Measure It?

Some Applications of the Second Law of Thermodynamics: Power Plants and Hurricanes

Hiking up a Mountain

Understanding Entropy with a Little Mathematics

The Financial Mathematics of Loans, Debts, and Compound Interest

Simple and Compound Interest: A Review

How Much Does a Debt Really Cost You? Buying on Time and/or Installment Plans. Amortization. The Four Important Numbers: P, R, r, n

Examples of Individual Debt: Rent-to-Own, Credit Cards, and Loans


Information Flow in the 21st Century

Investigative Journalism Requires Cash

Thesis: The Range of Debate is Too Narrow Now

Time Series Test and Multiple Source Test

Measuring the Range of Debate

Distractions and Illusions

Media Literacy: Censorship and Propaganda

Filters and Censors

Censorship: External and Internal

Conclusion and Epilog: Where Are the Adults?



About the Author

Martin Walter is a professor in the Department of Mathematics at the University of Colorado at Boulder. Dr. Walter is a Sloan, Woodrow Wilson, and National Science Foundation Fellow as well as a member of the American Mathematical Society and Mathematical Association of America. He has lectured or taught in various countries, including Japan, China, Poland, Romania, Australia, Belgium, Norway, Sweden, Denmark, England, Germany, India, Italy, Mexico, Puerto Rico, Canada, and Brazil.

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General