Numbers are used to measure multitudes and, derivatively, to measure or relate magnitudes. Although a comprehensive treatment of number would need to start with multitudes, the intent of Chapter 4 is to focus on the more difficult application of numbers to identify or specify a relationship between two magnitudes.
To grasp a multitude is to have identified or distinguished its unit (any one of the items comprising the multitude). The multitude, as such, has a specific numerical value to be ascertained by counting.
This is not the case when one grasps a magnitude: To measure a magnitude, it remains to specify a unit.
In either case, Chapter 4 holds that a number is an identification or specification of a relationship between two multitudes or two magnitudes of the same kind. This characterization is intended to include the case in which one relates a multitude or magnitude to a unit.
Chapter 4 treats such questions as:
- Why do we need irrational numbers? Given that all precision is finite, how can we meaningfully distinguish irrational numbers from rational numbers?
- Is there a systematic way to specify real numbers (i.e., rational and irrational numbers)?
- How does the principle that all measurements have finite precision apply to the use of numbers to specify quantitative relationships?
- What does it mean to say that a (Cauchy) sequence of numbers converges to a number?
- What does it mean to say that the real number system is complete?
- How should one regard the constructions by Dedekind or, respectively by Cantor, of the real number system?