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.