Axiom of Archimedes in Chapter 2 – Summary Comments

The Axiom of Archimedes usually takes the form that, given any positive real number x, there is a positive rational number less than x. In this form, the Axiom of Archimedes is involved in any demonstration that, given any real numbers x < y, there is a rational number r such that x < r < y. And this fact one depends upon whenever one uses decimals to approximate a number.

Yet the Axiom of Archimedes did not begin life as a statement about numbers; it began as a statement about magnitudes. In Aristotle’s Physics it takes the form:

 “…for every finite magnitude is exhausted by means of any determinate quantity however small.”

 I point out in the section that the Axiom of Archimedes does not come from a mathematical derivation, but is a basic observation about the world, specifically, about the nature of magnitudes such as length and weight.

As I delineate in Chapter 2, the central import of the Axiom of Archimedes is that magnitudes are measurable; that any positive magnitude Y can be measured by any other magnitude X of the same type: that for any finite precision requirement, there exists a rational number A, such that Y = AX. That is, within the precision context, Y is indistinguishable from AX.

Taking X and Y as concretely given magnitudes, I point out in the section that A is not unique, that for any prescribed precision standard there are multiple rational numbers A such that Y = AX. But suppose that X and Y arise in an abstract setting. Consider, for example, the relationship between the diagonal X and the edge Y of a square. In such cases, one wants a number A, not necessarily rational for which Y = AX regardless of precision context. In this example, one wants to show that A (in this case the square root of 2) is unique.

I argue in this case, using the Axiom of Archimedes, that A is unique, point out that existence of A requires more argument, and defer that argument to Chapter 4.

Finally, at the end of the section, I present the standard argument that, for any real numbers x < y, there exists a rational number r such that x < r < y.

Chapter 2 – Geometry of Magnitudes

We learn about numbers when we learn to count things and then, a little later, when we begin to measure magnitudes such as length. In our first encounters, numbers are referential; they identify relationships in the world. But theoretical treatments typically treat numbers as, in Dedekind’s words, “free creations of the human mind.”

To rehabilitate numbers one does not start with the concept of number as if it were a given; one starts with the quantities that they are used to relate and measure. As the most interesting case, one starts with magnitudes. Numbers relate to each other the way that they do because of the nature of the quantities-the multitudes and magnitudes-that they measure. Zeroing in on these relationships, on what I call the pre-arithmetic of magnitudes, is the purpose of Chapter 2. As such, Chapter 2 sets the context for chapter 4, most especially for its discussion of irrational numbers.

Chapter 2 addresses questions such as:

  • What does it mean to add magnitudes, as opposed to adding the numbers one might use to measure these magnitudes?
  • What does it mean to multiply magnitudes, when is it meaningful, and what makes it meaningful? Why is length times width a measure of area? What does it mean to multiple a length times a weight?
  • How did the ancient Greeks measure ratios of incommensurable magnitudes?
  • For all their sophistication in comparing ratios of pairs of magnitudes, the Greeks had no mathematical means to express quantities such as speed. What is it that we take for granted that the Greeks were missing?
  • What does the famous Axiom of Archimedes say about magnitudes?

The main conceptual difficulty in Chapter 2 is to focus on quantitative relationships without thinking in terms of feet or pounds. And, where appropriate, to regard algebraic unknowns as standing for magnitudes instead of numbers. And, indeed, to keep in mind which unknowns stand for magnitudes and which ones stand for numbers! My hope, certainly my intention, is that the reader will find it worth the struggle.

However, there is one point in Chapter 2 in which wading through the algebraic expressions may not be worth it to many readers, namely the section on the Axiom of Archimedes. Here, I suggest reading to at least the top of page 123. After that, if one starts getting bogged down, skip to the beginning of the next section on page 129 on “Multitudes, Units, and Ratios.”

I will be adding a follow-up to this post, summarizing the Axiom of Archimedes section. My goal will be to summarize the content for those who skip the section and to provide some orientation for those who persist.

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.