Comments on Chapter 1 “Euclid’s Method”

If one holds that mathematics is about the world, the two fundamental questions for elementary mathematics are:

1. How does geometry relate to the world?
2. How do numbers (especially irrational numbers) relate to the world?

Chapter 1 answers the first and Chapter 4, on the basis of Chapter 2 answers the second.

As presented in Euclid’s Elements, straight lines are infinitely thin, continuous and infinitely straight. Lines on earth, at the microscopic level, are none of these. So how can Euclid’s propositions apply to shapes and relationships in the world and how can his arguments reflect and capture relationships in the world? And if everything that we manufacture is shaped and measured according to our understanding of Euclid’s geometry is that to be expected or is it a happy accident?

Answering these questions is the essential burden of Chapter 1. Some highlights:

• A closed figure with three straight edges is a triangle when, considered as a shape there are three relevant sides with no relevant bending or discontinuities.
• Euclid’s postulates are all primitive measurements, either of distance or direction
• A Euclidean argument is a series of abstract measurements. It is a recipe for establishing the asserted indirect measurement by a series of more direct measurements, reducing ultimately to Euclid’s postulates.

Chapter 1 is the first instance of the broader principle that a) we need mathematics to establish quantitative relationships to support indirect measurement b) we establish quantitative relationships by mathematical arguments that embody a series of abstract measurements. In sum, measurement is both the purpose and the method of mathematics.

Suggestions for the reader

This book is for anyone who wants to understand how mathematics relates to the world. But not all chapters are equally relevant to a particular reader.

Chapter 1 “Euclid’s Method” is the most important chapter in the book. If you read only one chapter it should be this one. Chapter 1 identifies indirect measurement as the reason for a science of mathematics and depicts abstract measurement as the essential method of establishing connections in geometry.

Chapter 2 provides a geometric perspective on magnitude, drawing out the relationships among magnitudes that underpin the application of real numbers to measure magnitude. Chapter 2 provides an essential context for chapter 4 on the real number system. If you read only three chapters, read chapters 1, 2, and 4.

Chapter 3 is a continuation of Chapter 1, but more focused on Euclid’s geometry. Its focus is Euclid’s fifth postulate: how it relates to the world and how Euclid uses it to measure area and to establish his theory of geometric proportion (the basis of trigonometry). Chapter 3 is optional in relation to the rest of the book and can be read at any time after Chapter 1.

Chapter 4, a core chapter, offers a reality-oriented perspective on the real number system. Chapter 4 offers a reality-based reformulation of the standard “constructions” while rejecting the idea that numbers are constructed objects. They are, rather, methods to identify and specify relationships in the world. Chapter 4 closes by relating its approach to the work of Dedekind, Cantor, and Heine in the late nineteenth century.

Chapter 5 is a very short chapter identifying a role for geometry (an abstract focus on the objects of measurement) that transcends its restriction to the measurement of three dimensional spatial relationships. As Chapter 5 notices, the application of geometric methods to numerical relationships and to magnitudes such as force dates back at least to Euclid and Archemedes.

The last three chapters, 6-8, though written for a general audience, are intended primarily for college math majors and mathematically advanced high school students. These chapters are included to illustrate how my approach to understanding mathematics applies to advanced mathematics. They develop and motivate key concepts that are often (if not typically) left unmotivated in standard presentations and these chapters are essential reading for math majors who want to understand what they are being taught.

Chapter 6 answers the question: “But what about set theory?” Chapter 6 finds a valid need for set theory in mathematics as a methodological device, but rejects any notion that set theory provides a foundation of mathematics. Closing with a brief discussion of the standard set theory axioms, Chapter 6 points out that, by design, these axioms, building an entire edifice on the empty set, are meaningless. Finally, as an illustration of the value of a proper set theory, Chapter 6 offers a motivation of the key concepts in point set topology.

Chapter 7 motivates and relates key concepts in vector spaces and linear transformations/matrices. It should be read by anyone studying linear algebra.

Abstract mathematical groups, discussed in Chapter 8, play an essential role in advanced mathematics and physics. Chapter 8 develops and motivates key concepts in group theory and group representations from an elementary perspective. To understand group theory one needs to broaden one’s concepts of quantity and measurement and Chapter 8 provides the required perspective. As such, it is the final test of this book’s central thesis. Chapter 8 should be read by anyone who wants to understand how group theory relates to the world.

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Book Description:

What is mathematics about? Is there a mathematical universe glimpsed by a mathematical intuition? Or is mathematics an arbitrary game of symbols, with no inherent meaning, that somehow finds application to life on earth? Robert Knapp holds, on the contrary, that mathematics is about the world. His book develops and applies its alternative viewpoint, first, to elementary geometry and the number system and, then, to more advanced topics, such as topology and group representations. Its theme is that mathematics, however abstract, arises from and is shaped by requirements of indirect measurement. Eratosthenes, in 200 BC, demonstrated the power of indirect measurement when he estimated the circumference of the earth by measuring a shadow at noon, in Alexandria, on the day of the summer solstice. Establishing geometric relationships, solving equations, finding approximations, and, generally, discovering quantitative relationships are tools of indirect measurement: They are the core of mathematics, the drivers of its development, and the heart of its power to enhance our lives.

About the Author:

Robert Knapp earned his Ph.D. in mathematics from Princeton University in 1972. He has published work on differential geometry and partial differential equations, and, after a year at the Insitute for Advanced Studies in Princeton, taught graduate and undergraduate mathematics at Purdue University. His study and appreciation of abstract mathematics began in high school and his conviction that mathematics, including abstract mathematics, is about the world began then, as well. Although he retired from the profession in the late 1970s, his study of the content, history and application of mathematics continues to this day. In recent years he has presented his unique perspectives on geometry and the number system in a series of lectures at Objectivist Summer Conferences organized by the Ayn Rand Institute. He has lived in the Philadelphia area for almost 30 years.