Positions On The Philosophy Of Math

What does a mathematical proof show? In what sense, if any, do the objects which mathematicians reason about exist? A program of study on the philosophy of mathematics might consider some of the following views:

• Bourbaki - Structuralism
• Brower - Intuitionism
• Frege - Logicism (?)
• Gödel - Platonism
• Hilbert - Formalism
• Lakatos - Proofs and refutations in the tradition of Popper
• J. S. Mill - Math as empirical generalization
• Poincare - Intuitionism (?)
• Russell - Logicism
• Wittgenstein - Constructivism (?)

And one might look at more recent academic interpretations and commentary, such as Putnam or Kripke's interpretation of Wittgenstein.

Philosophers of math often discuss various interesting bits of mathematics. These include the construction of real numbers as equivalence classes of Cauchy-convergent sequences of rationals, of rational numbers as equivalence classes of ordered pairs of integers, and of integers as functions mapping (subsets of) the natural numbers to the natural numbers. Each construction comes with definitions of <, +, and *. The notion of an isomorphism is important in these constructions.

Those who think mathematics is in need of a foundation have often looked for one in terms of logic and set theory. Different axiom systems have been offered for sets. Russell's theory of types contrasts with the Zermelo-Fraenkel (ZF) system. In these set theories, there is an infinity of orders of infinity. I've always like the proof that the power set of a set, that is, the set of all subsets of a set, cannot be put in a one-to-one relationship with the original set. Thinking about applying that theorem recursively to the set of the natural numbers soon exhausts my imagination. Someday I would like to understand Gödel's proofs that if ZF is consistent, then ZP with the axiom of choice (ZFC) is consistent. And if ZFC is consistent, then Cantor's continuum hypothesis is consistent with ZFC. Paul Cohen went further. He proved, in 1963, the continuum hypothesis is independent of the axioms of ZFC. I guess this relates to model theory. I gather the Löwenheim-Skolem theorem is a surprising result.

Philosophers of mathematics often discuss certain important results from comutability theory and the theory of automata. Among these are Gödel's imcompleteness theorem. (Barkley Rosser, Sr., generalized Gödel’s work, from ω-consistency to consistency.) I gather the unsolvability of Diophantine equations, in general, follows from Gödel's theorem. The existence of uncomputable functions is of interest. Every computer programmer should be aware of the halting problem. I find interesting the Church-Turing thesis, the Chomsky hierarchy, and the question of whether the set of problems that can be solved in polynomial time by a deterministic Turing machine is equivalent to the set of problems that can be solved in polynomial time by a nondeterministic Turing machine.