Chapter 3 on Context and Implications of the Parallel Postulate

My approach to understanding Euclid, both mathematically and philosophically, is fully presented in Chapter 1. To understand Euclid’s method as abstract measurement is to understand both Euclid’s logical structure and, most critically, the way that Euclid’s geometry relates to the world.

Chapter 3 continues the discussion of Euclid’s fifth postulate. It further examines the context of the parallel postulate and explains its application to Euclid’s theory of geometric area and geometric proportion.

The 19th through the early 20th century, brought the transformation of mathematics from the science of quantity, with a Euclidean geometric base, to a formal system built on ontologically meaningless axioms. This transformation was driven by a philosophical perspective. But it transpired on a stage set by the development of non-Euclidean geometry and the later application of non-Euclidean geometry to Einstein’s relativity.

But these mathematical and scientific developments in no way impact my thesis that mathematics is about the world. It is not only true that the fifth postulate is independent of the other four, but, in light of its measurement implications, this independence should be expected. And since all measurements require physical means, one should not expect to understand geometric relationships established by light rays (general relativity) without reference to the nature of light rays. Yet the perceptual level remains the base and relativistic corrections consist in relating such relationships to the Euclidean geometry of the perceptual level. So the first part of chapter 3 examines the context of the parallel postulate.

The measurement of area and the application of geometric proportion to trigonometric calculation is based, inescapably on the parallel postulate. In modern treatments, this relationship is hidden, even if it’s hidden in plain sight. In Euclid, this dependence is front and center and Euclid’s treatment provides a level of understanding and a perspective on area and proportion that is missing from modern treatments. Understanding area and proportion the way that Euclid did illuminates the geometry and provides an ideal theater for watching Euclid’s method in action. Elucidating these applications completes Chapter 3.

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.