Mathematics, Logic, Set Theory, and Objectivism Options

[PhilE.Stein]

Topic Description: A general discussion of Objectivist ideas of mathematics, especially regarding mathematical logic and set theory.

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I'd like to know more about Objectivist views of mathematics, especially regarding mathematical logic and set theory. I have a number of questions. I encourage anyone with insight to address any of these questions or to offer their thoughts irrespective of my questions or to ask additional questions. I hope for a productive discussion that will probe for strengths and weaknesses of classical mathematical logic and set theory, intuitionist mathematics, Objectivist proposals for mathematics, and other approaches to mathematics.

1. What is the state of the art in Objectivist philosophy of mathematics? What Objectivist writers have made significant progress in this area, and what publications may one view?

2. What synopsis can be given of Objectivist views on mathematics? Has a systematic and rigorous statement of an Objectivist philosophy of mathematics been offered?

3. Are there Objectivist mathematicians who have published mathematics that embody particularly Objectivist mathematical concerns, approaches, or philosophy? Are there proposals for new systems of mathematical logic or for new axioms for set theory or mathematics? Or do Objectivist mathematicians find already existing systems and axioms satisfactory so that Objectivist alternatives are not needed?

4. How would mathematicians implement Objectivist ideas about mathematics? If the ideas are not suggested for implementation in mathematical theories, then in what way is it hoped that the ideas will affect mathematical practice?

5 How do Objectivists view mathematical logic and set theory? Is there a preference for classical mathematical logic, for intuitionist mathematical logic, or for other systems, approaches, or philosophies? Most particularly, if there is an Objectivist critique of classical mathematical logic and set theory, what is the critique and what alternatives do Objectivists offer?

6. Probably the most common definition of 'finite' and 'infinite' among mathematicians these days is:

x is finite if and only if x is equinumerous with a natural number.
x is infinite if and only if x is not finite.
(Note: The subject of the empty set had been raised in another thread. The empty set is finite, according the above definitions, since the empty set is a natural number.)

Also, set theories such as Zermelo-Fraenkel set theory, have an axiom that there exists a set w such that 0 is a member of w, and if n is a member of w, then the union of n and the singleton of n is a member of w.

What objections, if any, do Objectivists have to these formulations?

7. While constructivist and/or intuitionist mathematics might not be a perfect fit for Objectivism, these might be closer to Objectivism than mathematics set in classical mathematical logic. Though I know of Heyting arithmetic, I have not been able to find intuitionist axioms for other concerns, especially the real numbers or for set theory. What are some intuitionist systems for real numbers or for set theory and how do Objectivists regard these proposals?

8. Would non-standard analysis be preferred by Objectivists or does non-standard analysis carry assumptions that are more disagreeable to Objectivism than notions of infinity in standard analysis?

9. I've heard a little about mathematics that allows only finite sets (if I'm putting this correctly). How does such mathematics express the real numbers? How does such mathematics present its own meta-theory? Do such efforts offer alternatives for Objectivists?

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It is not required that in this thread we address all of these questions, but these questions do give us plenty to talk about. And I hope that all participants of this thread are allowed to comment or even digress upon anything that falls in the general area of mathematics or other subjects, such as epistemology or philosophy, that contribute to our understanding of mathematics. As well, I hope participants will be allowed to discuss the particulars of certain matters of mathematics as these may arise in the course of the larger subject.

Thanks in advance for whatever light anyone may shed upon this subject.