Mathematics for the Environment  book cover
1st Edition

Mathematics for the Environment

ISBN 9781439834725
Published January 18, 2011 by Chapman & Hall
679 Pages 54 B/W Illustrations

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Book Description

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.

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?



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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.


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