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.