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:
- 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?
- 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.
- 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.
- 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?
- Axioms of set theory, context and nature of the ZF axioms and their application to mathematics. Critical comments.
- Why has mathematics survived? (I claim that it has.)