Chapter 6 on Set Theory Part I

This is a first posting on Chapter 6; there may be others.

Mathematics is not about sets; it is about measurement of the world. But this book needs to assess set theory because:

  • Set theory is common currency in mathematics. It is presupposed and taken for granted in all advanced mathematical writings since the early 20th century
  • It is often alleged to be the foundation of mathematics. Most attempts to find a realist approach to mathematics ultimately connect to set theoretic axioms
  • The validity of set theory in general or, at least, the formal axiomatic approach to set theory, is highly questionable in a reality-based approach.
  • If set theory is valid, or can properly be rehabilitated, a reality-based answer to the question: “What is a set?” is essential.
  • One needs to address the key question: What, properly, would be the measurement function performed by sets?
  • It is essential to assess the status of the proclaimed axioms of set theory
  • If there is a valid concept of mathematical set, it remains to assess its value. The clearest way to offer an affirmative answer is to provide a non-trivial application to something of demonstrable value

These are a lot of bases to cover and even readers sympathetic to my approach are likely to find Chapter 6 the most difficult one in the book. The chapter begins with a positive treatment of mathematical sets and only offers its critical review of formal axiomatic set theory at the end. Chapter 6 covers such topics as:

  1. What is a mathematical set? How does a set differ from a concept? In what sense is a set open-ended, like a concept? In what sense is it not open-ended? What is its proper sphere? What measurement-related function does it serve?
  2. Proper domain of set theory. Set theory as specifically applicable to mathematics. Mathematics conceptualizes the world with respect to measurement of differences. Set theory as presupposing mathematical concepts. An indication of proper hierarchy in mathematics.
  3. Sets as a) primitive measurements b) performing a function of isolation c) a perspective on distinguished objects of measurement. A reality perspective on set theoretic “constructions.” Constructions as recognitions of relationships among quantities.
  4. Point set topology as a non-trivial and important application of sets. What is the purpose of point set topology and why is it important?
  5. Axioms of set theory, context and nature of the ZF axioms and their application to mathematics. Critical comments.
  6. Why has mathematics survived? (I claim that it has.)

Chapter 5 on Geometry and Human Cognition

Geometry, from the start, was about more than plane and solid shapes. When Pythagoras arranged pepples in various shapes, he was, certainly, studying shapes, but he was also studying numbers. Two centuries later, Euclid was drawing lines, in Book VII, to illustrate the rightly celebrated Euclidean Algorithm, used, to this day, to find the greatest common divisor of a pair of numbers.

The Euclidean Algorithm is a cornerstone of number theory. But one can also apply that very algorithm to the edge and diagonal of a square, discovering, rather quickly, that the algorithm will never terminate. It cannot terminate because the two magnitudes have no “common measure.” The magnitudes are incommensurate.

Geometry, for the ancient Greeks, was a perspective and a method. And they applied both the geometric perspective and their geometric method to the entire domain of quantity.

Chapter 5, “Geometry and Human Cognition” closes the Elementary section of the book and is, by far, the shortest chapter in the book. Its purpose is to understand the pervasive role of geometry in mathematics, to indicate its scope, to distinguish and relate the geometric perspective and the measurement perspective, and to apprehend the cognitive contribution of the geometric perspective. Chapter 5 pulls together themes from the first four chapters and sets the stage for the remaining 3.

On a personal note, I have been fascinated by the role of geometry in mathematics since my undergraduate years. This chapter began life as a significantly longer essay, with the same title (never published). It contained, in embryonic form, the key underpinnings of Chapter 1, in turn, the foundational chapter of the entire book. Chapter 2 also began as a short section in that earlier essay. Chapter 3 is a continuation of Chapter 1. And the book, as a whole, grew out of an exploration of themes first introduced in the early essay, “Geometry and Human Cognition.” Yet the core of the original essay, after all the excisions, remains intact, interesting, and important, providing a coda to Part 1 before the more advanced and demanding Part 2.

Chapter 4 on Rational and Irrational Numbers

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?