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.

I have been reading the book slowly and been satisfied with my level of understanding up until a got to chapter 7 and Vector Spaces. When I reached p. 380 I had to stop. I cannot get past the statement “To put this point another way,..” This is then followed by the statement: “And, by inspection, the coordinates (a,b,c)…

For my level of understanding or background, which is a BA in Math and Stat, there are too many steps being skipped for me to appreciate this discussion any further. I am planning to skip to the discussion of matrices, and hopefully be able to follow.

If there are any followup, more detailed explications, on Vector Spaces that will be published, I would like to be informed, and, perhaps, pick up where I left off.

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Sorry for the confusion.

The text establishes that, for the vectors v1 and v2, and for any two numbers A and B, Av1 + Bv2 = (A + B, -A + 2B, B).

So suppose (a, b, c) = (A + B, -A + 2B, B). Then, by substitution and then simplifying, one calculates a + b – 3c = (A + B) + (-A + 2B) -3(B) = A + B – A + 2B – 3B = 0. This, a + b – 3c = 0, is the asserted relationship.

The point of establishing this formula relating the coordinates is that there are many vectors (a, b, c) for which the equation a + b – 3c = 0 is false. Such vectors cannot be expressed as linear combinations of v1 and v2. So v1 and v2 do not span the vector space of 3-tuples.

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