Lectures on the Philosophy of Mathematics, Oxford, Michaelmas 2018

This will be a series of lectures on the philosophy of mathematics, given at Oxford University, Michaelmas term 2018. The lectures are mainly intended for undergraduate students preparing for exam paper 122, although all interested parties are welcome.

My approach to the philosophy of mathematics tends to be grounded in mathematical arguments and ideas, treating philosophical issues as they arise organically. The lectures will accordingly be organized around mathematical themes, in such a way that naturally brings various philosophical issues to light.

Here is a tentative list of topics, which may be updated as the term approaches.

Lecture 1. Numbers. Numbers are perhaps the essential mathematical idea, but what are numbers? Our various number systems — natural numbers, integers, rational numbers, real numbers, complex numbers, hyperreal numbers, surreal numbers, ordinal numbers, and more —  provide a background for classical arguments on incommensurability, the irrationality of $\sqrt{2}$, the infinitude of primes and transcendental numbers, leading to discussions of Platonism, Frege’s number concept, Peano’s numbers, Dedekind’s arguments, and the philosophy of structuralism.

Lecture 2. Rigour. We shall treat the problem of mathematical rigour in the development of the calculus. Informal limit concepts and the use of infinitesimals ultimately led to formal concepts of limit and continuity, as well as a capacity for refined notions such as uniform continuity, accompanied by increasing abstraction in the function concept, which we shall illustrate with the Conway base 13 function and space-filling curves.

Lecture 3. Infinity. Beginning with Zeno’s paradox and classical ideas on potential versus actual infinity, we shall follow the allegory of Hilbert’s hotel to the equinumerosity relation and the notion of countability. Cantor’s diagonal arguments, meanwhile, reveal uncountability and a vast hierarchy of different orders of infinity. Some arguments give rise to the distinction between constructive and non-constructive proof. Time permitting, we shall count into the transfinite ordinals.

Lecture 4. Geometry. We shall discuss classical Euclidean geometry, accompanied by the Euclidean proof concept and the ideal of straight-edge and compass construction, which leads to the concept of constructible numbers. The impossibility of certain constructions, such as duplicating the cube or trisecting the angle, serves as a counterpoint. The rise of non-Euclidean geometry, especially in light of scientific observations and theories suggesting that physical reality may not be Euclidean, challenges previous accounts of what geometry is about. We shall discuss formalizations of geometry, including Russell on Euclid, Russell and Tarski on betweenness, and the significance of Tarski’s decision procedure.

Lecture 5. Proof. What is proof? What is the relation between proof and truth? After clarifying the distinction between syntax and semantics, we shall introduce formal proof systems and highlight the importance of soundness, completeness and effectivity in any such system. We shall discuss the increasing significance of computer-verified proof, presenting the four-color theorem as an illustrative example.

Lecture 6. Computability. What does it mean for a function to be computable? While Gödel’s early approach via primitive recursive functions fell short, Turing’s machine concept proved extremely robust, forming a foundation ultimately for the contemporary computer era. We shall discuss the Church-Turing thesis, in both weak and strong forms, as well as the undecidability of the halting problem. Turing’s oracle concept leads to a mathematical theory of information content. Time and space permitting, we shall carry out a supertask computation, transcending the Turing limit.

Lecture 7. Incompleteness. The Hilbert program, seeking to secure the consistency of higher mathematics by finitary reasoning about the formal system underlying it, was dashed by Gödel’s incompleteness theorems, which show that no consistent formal system can prove even its own consistency, let alone the consistency of a higher system. We shall describe several proofs of the first incompleteness theorem, via the halting problem, via self-reference, and via definability. After this, we’ll discuss the Rosser variation, the second incompleteness theorem, and Tarski on the non-definability of truth. Ultimately, one is led to the inherent hierarchy of consistency strength underlying all mathematical theories.

Lecture 8. Set theory. We shall discuss the emergence of set theory as a foundation of mathematics. An initially naive theory, challenged fundamentally by the Russell paradox, grew into Zermelo’s formal set theory, founded on the idea of a cumulative universe of sets and providing a robust general context in which to undertake mathematics, while also enabling the clarification of fundamentally set-theoretic issues surrounding the axiom of choice, the continuum hypothesis and an increasingly diverse hierarchy of large cardinal concepts. The development of forcing solved many stubborn questions and illuminated a ubiquitous independence phenomenon, feeding into philosophical issues concerning the criteria by which one should add new axioms to mathematics and the question of pluralism in mathematical foundations.