Philosophy of Mathematics, graduate lecture seminar, Oxford, Trinity term 2019

This will be a graduate-level lecture seminar on the Philosophy of Mathematics, run jointly by Professor Timothy Williamson and myself, held during Trinity term 2019 at Oxford University. We shall meet every Tuesday 2-4 pm during term in the Ryle Room at the Radcliffe Humanities building.

We shall discuss a selection of topics in the philosophy of mathematics, based on the readings set for each week, as set out below. Discussion will be led each week either by Professor Williamson or myself.

In the classes led by Williamson, we shall discuss issues concerning the ontology of mathematics and what is involved in its application. In the classes led by me, we shall focus on the philosophy of set theory, covering set theory as a foundation of mathematics; determinateness in set theory; the status of the continuum hypothesis; and set-theoretic pluralism.

Week 1 (30 April)
Discussion led by Williamson. Reading: Robert Brandom, ‘The significance of complex numbers for Frege’s philosophy of mathematics’, Proceedings of the Aristotelian Society (1996): 293-315

Week 2 (7 May)
Discussion led by Hamkins. Reading: Penelope Maddy, Defending the Axioms: On the Philosophical Foundations of Set Theory, OUP (2011), 150 pp.

Week 3 (14 May)
Discussion led by Hamkins. Reading: Donald Martin, ‘Multiple universes of sets and indeterminate truth values’, Topoi (2001): 5-16.

Week 4 (21 May)
Discussion led by Hamkins. Reading: Chris Freiling, ‘Axioms of symmetry: throwing darts at the real number line’, Journal of Symbolic Logic (1986) And: Solomon Feferman, ‘The Continuum Hypothesis is neither a definite mathematical problem nor a definite logical problem’,

Week 5 (28 May)
Discussion led by Hamkins. Reading: his ‘The set-theoretic multiverse’, Review of Symbolic Logic (2012): 416-449 And: Penelope Maddy, ‘Set-theoretic foundations’, Contemporary Mathematics (2017).

Week 6 (4 June)
Discussion led by Williamson. Reading: Cian Dorr, ‘Of numbers and electrons’, Proceedings of the Aristotelian Society (2010): 133-181.

Week 7 (11 June)
Discussion led by Williamson. Reading: Otávio Bueno and Mark Colyvan, ‘An inferential conception of the application of mathematics’, Noûs (2011): 345-374.

Week 8 (18 June)
Discussion led by Williamson. Reading: his ‘Alternative logics and applied mathematics’, Philosophical Issues (2018): 399-424.

Faculty respondent to paper of Ethan Jerzak on Paradoxical Desires, Oxford Graduate Philosophy Conference, November 2018

The Oxford Graduate Philosophy Conference will be held at the Faculty of Philosophy November 10-11, 2018, with graduate students from all over the world speaking on their papers, with responses and commentary by Oxford faculty.

I shall be the faculty respondent to the delightful paper, “Paradoxical Desires,” by Ethan Jerzak of the University of California at Berkeley, offered under the following abstract.

Ethan Jerzak (UC Berkeley): Paradoxical Desires
I present a paradoxical combination of desires. I show why it’s paradoxical, and consider ways of responding to it. The paradox saddles us with an unappealing disjunction: either we reject the possibility of the case by placing surprising restrictions on what we can desire, or we revise some bit of classical logic. I argue that denying the possibility of the case is unmotivated on any reasonable way of thinking about propositional attitudes. So the best response is a non-classical one, according to which certain desires are neither determinately satisfied nor determinately not satisfied. Thus, theorizing about paradoxical propositional attitudes helps constrain the space of possibilities for adequate solutions to semantic paradoxes more generally.

The conference starts with coffee at 9:00 am.  This session runs 11 am to 1:30 pm on Saturday 10 November in the Lecture Room.

Conference Program | Conference web page

Here are the notes I used for my response.


On set-theoretic mereology as a foundation of mathematics, Oxford Phil Math seminar, October 2018

This will be a talk for the Philosophy of Mathematics Seminar in Oxford, October 29, 2018, 4:30-6:30 in the Ryle Room of the Philosopher Centre.

Abstract. In light of the comparative success of membership-based set theory in the foundations of mathematics, since the time of Cantor, Zermelo and Hilbert, it is natural to wonder whether one might find a similar success for set-theoretic mereology, based upon the set-theoretic inclusion relation $\subseteq$ rather than the element-of relation $\in$.  How well does set-theoretic mereological serve as a foundation of mathematics? Can we faithfully interpret the rest of mathematics in terms of the subset relation to the same extent that set theorists have argued (with whatever degree of success) that we may find faithful representations in terms of the membership relation? Basically, can we get by with merely $\subseteq$ in place of $\in$? Ultimately, I shall identify grounds supporting generally negative answers to these questions, concluding that set-theoretic mereology by itself cannot serve adequately as a foundational theory.

This is joint work with Makoto Kikuchi, and the talk is based on our joint articles:

The talk will also mention some related recent work with Ruizhi Yang (Shanghai).


Parallels in universality between the universal algorithm and the universal finite set, Oxford Math Logic Seminar, October 2018

This will be a talk for the Logic Seminar in Oxford at the Mathematics Institute in the Andrew Wiles Building on October 9, 2018, at 4:00 pm, with tea at 3:30.

Abstract. The universal algorithm is a Turing machine program $e$ that can in principle enumerate any finite sequence of numbers, if run in the right model of PA, and furthermore, can always enumerate any desired extension of that sequence in a suitable end-extension of that model. The universal finite set is a set-theoretic analogue, a locally verifiable definition that can in principle define any finite set, in the right model of set theory, and can always define any desired finite extension of that set in a suitable top-extension of that model. Recent work has uncovered a $\Sigma_1$-definable version that works with respect to end-extensions. I shall give an account of all three results, which have a parallel form, and describe applications to the model theory of arithmetic and set theory.


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? We have many kinds of numbers—natural numbers, integers, rational numbers, real numbers, complex numbers, hyperreal numbers, surreal numbers, ordinal numbers, and more—and these number systems provide a fruitful background for classical arguments on incommensurability, the irrationality of $\sqrt{2}$, transcendental numbers, the infinitude of primes, and lead naturally to discussions of platonism, Frege’s number concept, Peano’s numbers, Dedekind’s categoricity arguments, and the philosophy of structuralism.

Lecture 2. Rigour. Let us consider the problem of mathematical rigour in the development of the calculus. Informal continuity concepts and the use of infinitesimals ultimately gave way to formal epsilon-delta limit concepts, which provided a capacity for refined notions, such as uniform continuity, equicontinuity and uniform convergence. Nonstandard analysis resurrected the infinitesimal concept on a more secure foundation, providing a parallel development of the subject, which can be understood from various sweeping perspectives. Meanwhile, increasing abstraction emerged in the function concept, which we shall illustrate with the Devil’s staircase, space-filling curves and the Conway base 13 function.

Lecture 3. Infinity. We shall follow the allegory of Hilbert’s hotel and the paradox of Galileo 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. Zeno’s paradox highlights classical ideas on potential versus actual infinity. Time permitting, we shall count into the transfinite ordinals.

Lecture 4. Geometry. Classical Euclidean geometry, accompanied by its ideal of straightedge and compass construction and the Euclidean concept of proof, is an ageless paragon of deductive mathematical reasoning. Yet, the impossibility of certain constructions, such as doubling the cube, trisecting the angle or squaring the circle, hints at geometric realms beyond Euclid, and leads one to the concept of constructible and non-constructible numbers. 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 and changes our understanding of the nature of geometric and indeed mathematical ontology. New formalizations, such as those of Hilbert and Tarski, replace the old axiomatizations, augmenting and correcting Euclid with axioms on completeness and betweenness. Ultimately, Tarski’s decision procedure hints at the tantalizing possibility of automation in our geometrical reasoning.

Lecture 5. Proof. What is proof? What is the relation between proof and truth? Is every mathematical truth, true for a reason? After clarifying the distinction between syntax and semantics, we shall discuss formal proof systems and highlight the importance of soundness, completeness and verifiability in any such system, outlining the central ideas used in proving the completeness theorem. The compactness theorem distills the finiteness of proofs into an independent purely semantic consequence. Computer-verified proof promises increasing significance; it’s role is well illustrated by the history of the four-color theorem. Nonclassical logics, such as intuitionistic logic, arise naturally from formal systems by weakenings of the logical rules.

Lecture 6. Computability. What is computability? Gödel’s primitive recursive functions were a robust class, yet he gave reasons to despair of a fully satisfactory answer. Nevertheless, Turing’s machine concept, growing out of his careful philosophical analysis of computability, laid a foundation for the contemporary computer era, and the widely accepted Church-Turing thesis asserts that Turing has the right notion. Meanwhile, the distinction between computable decidability and computable enumerability, highlighted by the undecidability of the halting problem, shows that not all mathematical problems can be solved by machine, and a vast hierarchy looms in the Turing degrees, an infinitary information theory. Complexity theory refocuses this on the realm of feasible computation, with the still-unsolved P vs. NP problem standing in the background of nearly every serious issue in theoretical computer science.

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.

Oxford University, Professor of Logic & Sir Peter Strawson Fellow, University College Oxford

In September 2018, I took up a new position in Oxford:

I am looking forward to starting this new chapter in my life and academic career.

Wish me luck!


Modal principles of potentialism, Oxford, January 2018

This was a talk I gave at University College Oxford to the philosophy faculty.

Abstract. One of my favorite situations occurs when philosophical ideas or issues inspire a bit of mathematical analysis, which in turn raises further philosophical questions and ideas, in a fruitful cycle. The topic of potentialism originates, after all, in the classical dispute between actual and potential infinity. Linnebo and Shapiro and others have emphasized the modal nature of potentialism, de-coupling it from infinity: the essence of potentialism is about approximating a larger universe or structure by means of partial structures or universe fragments. In several mathematical projects, my co-authors and I have found the exact modal validities of several natural potentialist concepts arising in the foundations of mathematics, including several kinds of set-theoretic and arithmetic potentialism. Ultimately, the variety of kinds of potentialism suggest a refocusing of potentialism on the issue of convergent inevitability in comparison with radical branching. I defended the theses, first, that convergent potentialism is implicitly actualist, and second, that we should understand ultrafinitism in modal terms as a form of potentialism, one with suprising parallels to the case of arithmetic potentialism.

Here are my lecture notes that I used as a basis for the talk:

For a fuller, more technical account of potentialism, see the three-lecture tutorial series I gave for the Logic Winter School 2018 in Hejnice: Set-theoretic potentialism, and follow the link to the slides.