Please enjoy my conversation with Rahul Sam for his podcast, a sweeping discussion of topics in the philosophy of mathematics—potentialism, pluralism, Gödel incompleteness, philosophy of set theory, large cardinals, and much more.

# Tag Archives: philosophy of mathematics

# Philosophy of Mathematics and Truth, interview with Matthew Geleta on Paradigm Podcast

We had a sweeping discussion touching upon many issues in the philosophy of mathematics, including the nature of mathematical truth, mathematical abstraction, the nature of mathematical existence, the meaning and role of proof in mathematics, the completeness theorem, the incompleteness phenomenon, infinity, and a discussion about the motivations that one might have for studying mathematics.

# Paradox, Infinity, & The Foundations of Mathematics, interview with Robinson Erhardt, January 2023

This was an interview with Robinson Erhardt on Robinson’s Podcast, part of his series of interviews with various philosophers, including many philosophers of mathematics and more.

We had a wonderfully wide-ranging discussion about the philosophy of mathematics, the philosophy of set theory, pluralism, and many other topics. The main focus was the topic of infinity, following selections from my new book, The Book of Infinity, currently being serialized on my substack, joeldavidhamkins.substack.com, with discussion of Zeno’s paradox, the Chocolatier’s Game, Hilbert’s Grand Hotel and more.

Robinson compiled the following outline with links to special parts of the interview:

- 00:00 Introduction
- 2:52 Is Joel a Mathematician or a Philosopher?
- 6:13 The Philosophical Influence of Hugh Woodin
- 10:29 The Intersection of Set Theory and Philosophy of Math
- 16:29 Serializing the Book of the Infinite
- 20:05 Zeno of Elea, Continuity, and Geometric Series
- 39:39 Infinite Games and the Chocolatier
- 53:35 Hilbert’s Hotel
- 1:10:26 Cantor’s Theorem
- 1:31:37 The Continuum Hypothesis
- 1:43:02 The Set-Theoretic Multiverse
- 2:00:25 Berry’s Paradox and Large Numbers
- 2:16:15 Skolem’s Paradox and Indescribable Numbers
- 2:28:41 Pascal’s Wager and Reasoning Around Remote Events
- 2:49:35 MathOverflow
- 3:04:40 Joel’s Impeccable Fashion Sense

Read the book here: joeldavidhamkins.substack.com.

# Frege’s philosophy of mathematics—Interview with Nathan Ormond, December 2021

I was interviewed by Nathan Ormond for a discussion on Frege’s philosophy of mathematics for his YouTube channel, Digital Gnosis, on 10 December 2021 at 4pm.

The interview concludes with a public comment and question & answer session.

# Lectures on the Philosophy of Mathematics, Michaelmas term 2021, Oxford

Philosophy of Mathematics, Exam Paper 122, Oxford University

Wednesdays 12-1 during term, Radcliffe Humanities Lecture Room

Joel David Hamkins, Professor of Logic

This series of self-contained lectures on the philosophy of mathematics is intended for students preparing for Oxford Philosophy exam paper 122. All interested parties from the Oxford University community, however, are welcome to attend, whether or not they intend to sit the exam. The lectures will be organized loosely around mathematical themes, in such a way that brings various philosophical issues naturally to light. Lectures will loosely follow the instructor’s book Lectures on the Philosophy of Mathematics (MIT Press 2021), with supplemental suggested readings each week.

Previously recorded lectures from last year are available on the lecturer’s YouTube channel, below.

In light of the earlier lectures being available online, the plan for the lectures this year will be to feel somewhat more free occasionally to focus on narrower topics, and also to entertain at times a discussion format. Therefore kindly bring questions and well-thought-out opinions to the lecture.

The lectures this term will be held in person. The lecturer requests that students be vaccinated, wear masks, and observe social distancing as practicable. If this proves impossible or unsustainable, we shall regretably revert to online lectures on short notice.

**Lecture 1. Numbers**

Numbers are perhaps the essential mathematical idea, but what are numbers? There are 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 and transcendentality, while setting the stage for discussions of platonism, logicism, the nature of abstraction, the significance of categoricity, and structuralism.

**Lecture 2. Rigour**

Let us consider the problem of mathematical rigor in the development of the calculus. Informal continuity concepts and the use of infinitesimals ultimately gave way to the epsilon-delta limit concept, which secured a more rigorous foundation while also enlarging our conceptual vocabulary, enabling us to express more refined notions, such as uniform continuity, equicontinuity, and uniform convergence. Nonstandard analysis resurrected the infinitesimals on a more secure foundation, providing a parallel development of the subject. 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. Finally, does the indispensability of mathematics for science ground mathematical truth? Fictionalism puts this in question.

**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 nonconstructive proof. Zeno’s paradox highlights classical ideas on potential versus actual infinity. Furthermore, we shall count into the transfinite ordinals.

**Lecture 4. Geometry**

Classical Euclidean geometry is the archetype of a mathematical deductive process. Yet the impossibility of certain constructions by straightedge and compass, such as doubling the cube, trisecting the angle, or squaring the circle, hints at geometric realms beyond Euclid. The rise of non-Euclidean geometry, especially in light of scientific theories and observations suggesting that physical reality is not Euclidean, challenges previous accounts of what geometry is about. New formalizations, such as those of David Hilbert and Alfred Tarski, replace the old axiomatizations, augmenting and correcting Euclid with axioms on completeness and betweenness. Ultimately, Tarski’s decision procedure points to a tantalizing possibility of automation in 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 and discussing various views on the nature of proof, including proof-as-dialogue, we shall consider the nature of formal proof. We shall highlight the importance of soundness, completeness, and verifiability in any formal proof system, outlining the central ideas used in proving the completeness theorem. The compactness property distills the finiteness of proofs into an independent, purely semantic consequence. Computer-verified proof promises increasing significance; its role is well illustrated by the history of the four-color theorem. Nonclassical logics, such as intuitionistic logic, arise naturally from formal systems by weakening the logical rules.

**Lecture 6. Computability**

What is computability? Kurt Gödel defined a robust class of computable functions, the primitive recursive functions, and yet he gave reasons to despair of a fully satisfactory answer. Nevertheless, Alan Turing’s machine concept of computability, growing out of a careful philosophical analysis of the nature of human computability, proved robust and laid a foundation for the contemporary computer era; the widely accepted Church-Turing thesis asserts that Turing had the right notion. 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 the subject on the realm of feasible computation, with the still-unsolved P versus NP problem standing in the background of nearly every serious issue in theoretical computer science.

**Lecture 7. Incompleteness**

David Hilbert sought to secure the consistency of higher mathematics by finitary reasoning about the formalism underlying it, but his program 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, self-reference, and definability, showing senses in which we cannot complete mathematics. After this, we shall discuss the second incompleteness theorem, the Rosser variation, and Tarski’s theorem on the nondefinability of truth. Ultimately, one is led to the inherent hierarchy of consistency strength rising above every foundational mathematical theory.

**Lecture 8. Set Theory**

We shall discuss the emergence of set theory as a foundation of mathematics. Cantor founded the subject with key set-theoretic insights, but Frege’s formal theory was naive, refuted by the Russell paradox. Zermelo’s set theory, in contrast, grew ultimately into the successful contemporary theory, founded upon a cumulative conception of the set-theoretic universe. Set theory was simultaneously a new mathematical subject, with its own motivating questions and tools, but it also was a new foundational theory with a capacity to represent essentially arbitrary abstract mathematical structure. Sophisticated technical developments, including in particular, the forcing method and discoveries in the large cardinal hierarchy, led to a necessary engagement with deep philosophical concerns, such as the criteria by which one adopts new mathematical axioms and set-theoretic pluralism.

# Infinite sets and Foundations—Interviewed on the Daniel Rubin Show

I was interviewed 26 August 2021 by mathematician Daniel Rubin on his show, and we had a lively, wideranging discussion spanning mathematics, infinity, and the philosophy of mathematics. Please enjoy!

**Contents**

0:00 Intro

2:11 Joel’s background. Interaction between math and philosophy

9:04 Joel’s work; infinite chess.

14:45 Infinite ordinals

22:27 The Cantor-Bendixson process

29:41 Uncountable ordinals

32:10 First order vs. second order theories

41:16 Non-standard analysis

46:57 The ZFC axioms and well-ordering of the reals

58:11 Showing independence of statements. Models and forcing.

1:04:38 Sets, classes, and categories

1:19:22 Is there one true set theory? Are projective sets Lebesgue measurable?

1:30:20 What does set theory look like if certain axioms are rejected?

1:36:06 How to judge philosophical positions about math

1:42:01 Concrete math where set theory becomes relevant. Tarski-Seidenberg on positive polynomials.

1:48:48 Goodstein sequences and the use of infinite ordinals

1:58:43 The state of set theory today

2:01:41 Joel’s recent books

Go check out the other episodes on Daniel’s channel!

# Reading and discussion of Lectures on the Philosophy of Mathematics, Amsterdam, March 2021

This will be an event for the $\Phi$-Math Reading Group at the Institute for Logic, Language, and Computation (ILLC) at the University of Amsterdam, 19 March 2021 6pm CET (5pm GMT). Zoom access here.

I shall make a brief presentation of the overall contents of the book, including a discussion of my perspective on the subject, and then get into some of the detailed issues with which the book engages. After this, we shall open up for discussion and comments.

# Graduate Seminar in Philosophy of Logic, Oxford, Hilary Term 2021

This is a graduate seminar in the Philosophy of Logic at the University of Oxford, run jointly by myself and Volker Halbach in Hilary Term 2021.

The theme will be self-reference, truth, and the hierarchy of consistency strength.

A detailed schedule, including the list of topics and readings is available on Volker’s web site.

The seminar will be held Fridays 9-11 am during term, online via Zoom at 812 2300 3837.

The final two sessions of term will be specifically on the hierarchy of consistency strength, based on my current article in progress concerning the possibility of natural instances of incomparability and ill-foundedness in the hierarchy of large cardinal consistency strength.

# Lectures on the Philosophy of Mathematics, Oxford Michaelmas Term 2020

This series of self-contained lectures on the philosophy of mathematics, offered for Oxford Michaelmas Term 2020, is intended for students preparing for philosophy exam paper 122, although all interested parties are welcome to join. The lectures will be organized loosely around mathematical themes, in such a way that brings various philosophical issues naturally to light.

Lectures will follow my new book Lectures on the Philosophy of Mathematics (MIT Press), with supplemental readings suggested each week for further tutorial work. The book is available for pre-order, to be released 2 February 2021.

Lectures will be held online via Zoom every Wednesday 11-12 am during term at the following Zoom coordinates:

https://us02web.zoom.us/j/82822228760?pwd=UHU1cjVxSFpmd3haak5vYU51eFRrUT09

Meeting ID: 828 2222 8760

Passcode: 242081

All lectures will be recorded and made available at a later date.

## Lecture 1. Numbers

Numbers are perhaps the essential mathematical idea, but what are numbers? There are 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 and transcendentality, while setting the stage for discussions of platonism, logicism, the nature of abstraction, the significance of categoricity, and 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 the epsilon-delta limit concept, which secured a more rigourous foundation while also enlarging our conceptual vocabulary, enabling us to express more refined notions, such as uniform continuity, equicontinuity, and uniform convergence. Nonstandard analysis resurrected the infinitesimals on a more secure foundation, providing a parallel development of the subject. 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. Finally, does the indispensability of mathematics for science ground mathematical truth? Fictionalism puts this in question.

## 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 nonconstructive proof. Zeno’s paradox highlights classical ideas on potential versus actual infinity. Furthermore, we shall count into the transfinite ordinals.

## Lecture 4. Geometry

Classical Euclidean geometry is the archetype of a mathematical deductive process. Yet the impossibility of certain constructions by straightedge and compass, such as doubling the cube, trisecting the angle, or squaring the circle, hints at geometric realms beyond Euclid. The rise of non-Euclidean geometry, especially in light of scientific theories and observations suggesting that physical reality is not Euclidean, challenges previous accounts of what geometry is about. New formalizations, such as those of David Hilbert and Alfred Tarski, replace the old axiomatizations, augmenting and correcting Euclid with axioms on completeness and betweenness. Ultimately, Tarski’s decision procedure points to a tantalizing possibility of automation in 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 and discussing various views on the nature of proof, including proof-as-dialogue, we shall consider the nature of formal proof. We shall highlight the importance of soundness, completeness, and verifiability in any formal proof system, outlining the central ideas used in proving the completeness theorem. The compactness property distills the finiteness of proofs into an independent, purely semantic consequence. Computer-verified proof promises increasing significance; its role is well illustrated by the history of the four-color theorem. Nonclassical logics, such as intuitionistic logic, arise naturally from formal systems by weakening the logical rules.

## Lecture 6. Computability

What is computability? Kurt Gödel defined a robust class of computable functions, the primitive recursive functions, and yet he gave reasons to despair of a fully satisfactory answer. Nevertheless, Alan Turing’s machine concept of computability, growing out of a careful philosophical analysis of the nature of human computability, proved robust and laid a foundation for the contemporary computer era; the widely accepted Church-Turing thesis asserts that Turing had the right notion. 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 the subject on the realm of feasible computation, with the still-unsolved P versus NP problem standing in the background of nearly every serious issue in theoretical computer science.

## Lecture 7. Incompleteness

David Hilbert sought to secure the consistency of higher mathematics by finitary reasoning about the formalism underlying it, but his program 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, self-reference, and definability, showing senses in which we cannot complete mathematics. After this, we shall discuss the second incompleteness theorem, the Rosser variation, and Tarski’s theorem on the nondefinability of truth. Ultimately, one is led to the inherent hierarchy of consistency strength rising above every foundational mathematical theory.

## Lecture 8. Set Theory

We shall discuss the emergence of set theory as a foundation of mathematics. Cantor founded the subject with key set-theoretic insights, but Frege’s formal theory was naive, refuted by the Russell paradox. Zermelo’s set theory, in contrast, grew ultimately into the successful contemporary theory, founded upon a cumulative conception of the set-theoretic universe. Set theory was simultaneously a new mathematical subject, with its own motivating questions and tools, but it also was a new foundational theory with a capacity to represent essentially arbitrary abstract mathematical structure. Sophisticated technical developments, including in particular, the forcing method and discoveries in the large cardinal hierarchy, led to a necessary engagement with deep philosophical concerns, such as the criteria by which one adopts new mathematical axioms and set-theoretic pluralism.

# Lectures on the Philosophy of Mathematics

[bibtex key=”Hamkins2021:Lectures-on-the-philosophy-of-mathematics”]

- Book profile at MIT Press, with links to bookstores
- Amazon.com
- Amazon.co.uk
- Order through your local bookstore
- Follow the conversation on Twitter at #PhilMaths
- My book was included on the Bloomberg Businessweek Best Books of 2021

### From the Preface:

Philosophical conundrums pervade mathematics, from fundamental questions of mathematical ontology—What is a number? What is infinity?—to questions about the relations among truth, proof, and meaning. What is the role of figures in geometric argument? Do mathematical objects exist that we cannot construct? Can every mathematical question be solved in principle by computation? Is every truth of mathematics true for a reason? Can every mathematical truth be proved?

This book is an introduction to the philosophy of mathematics, in which we shall consider all these questions and more. I come to the subject from mathematics, and I have strived in this book for what I hope will be a fresh approach to the philosophy of mathematics—one grounded in mathematics, motivated by mathematical inquiry or mathematical practice. I have strived to treat philosophical issues as they arise organically in mathematics. Therefore, I have organized the book by mathematical themes, such as number, infinity, geometry, and computability, and I have included some mathematical arguments and elementary proofs when they bring philosophical issues to light.

# Philosophical Trials interview: Joel David Hamkins on Infinity, Gödel’s Theorems and Set Theory

I was interviewed by Theodor Nenu as the first installment of his Philosophical Trials interview series with philosophers, mathematicians and physicists.

# Lectures on the philosophy of mathematics, Oxford, Michaelmas term 2019

This will be a series of self-contained lectures on the philosophy of mathematics, given at Oxford University in Michaelmas term 2019. We will be meeting in the Radcliffe Humanities Lecture Room at the Faculty of Philosophy every Friday 12-1 during term.

All interested parties are welcome. The lectures are intended principally for students preparing for philosophy exam paper 122 at the University of Oxford.

The lectures will be organized loosely around mathematical themes, in such a way I hope that brings various philosophical issues naturally to light. The lectures will be based on my new book, forthcoming with MIT Press.

There are tentative plans to make the lectures available by video. I shall post further details concerning this later.

**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 and transcendentality, while setting the stage for discussions of platonism, logicism, the nature of abstraction, the significance of categoricity, and 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. Whether the indispensibility of mathematics for science grounds mathematical truth is put in question on the view known as fictionalism.

**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 new views on the dialogical nature of proof. With formal proof systems, we shall 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 defined the primitive recursive functions, a robust class of computable functions, yet he gave reasons to despair of a fully satisfactory answer. Nevertheless, Turing’s machine concept, growing out of a careful philosophical analysis of computability, laid a foundation for the contemporary computer era; the widely accepted Church-Turing thesis asserts that Turing has the right notion. 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 the subject 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 second incompleteness theorem, the Rosser variation, 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. Cantor founded the subject with key set-theoretic insights, but Frege’s formal theory was naive, refuted by the Russell paradox. Zermelo’s set theory, in contrast, grew ultimately into the successful contemporary theory, founded upon the cumulative conception. Set theory was simultaneously a new mathematical subject, with its own motivating questions and tools, but also a new foundational theory, with a capacity to represent essentially arbitrary abstract mathematical structure. Sophisticated technical developments, including especially the forcing method and discoveries in the large cardinal hierarchy, led to a necessary engagement with deep philosophical concerns, such as the criteria by which one adopts new mathematical axioms and set-theoretic pluralism.

# 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 https://www.jstor.org/stable/pdf/4545241.pdf

**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. https://link.springer.com/article/10.1023%2FA%3A1010600724850

**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) https://www.jstor.org/stable/pdf/2273955.pdf. And: Solomon Feferman, ‘The Continuum Hypothesis is neither a definite mathematical problem nor a definite logical problem’, https://math.stanford.edu/~feferman/papers/CH_is_Indefinite.pdf

**Week 5** (28 May)

Discussion led by Hamkins. Reading: his ‘The set-theoretic multiverse’, *Review of Symbolic Logic* (2012): 416-449

https://doi.org/10.1017/S1755020311000359. And: Penelope Maddy, ‘Set-theoretic foundations’, *Contemporary Mathematics* (2017). http://philsci-archive.pitt.edu/13027/1/MaddyFoundations.pdf

**Week 6** (4 June)

Discussion led by Williamson. Reading: Cian Dorr, ‘Of numbers and electrons’, *Proceedings of the Aristotelian Society* (2010): 133-181. https://onlinelibrary.wiley.com/doi/epdf/10.1111/j.1467-9264.2010.00282.x

**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. https://onlinelibrary.wiley.com/doi/epdf/10.1111/j.1468-0068.2010.00772.x

**Week 8** (18 June)

Discussion led by Williamson. Reading: his ‘Alternative logics and applied mathematics’, *Philosophical Issues* (2018): 399-424. https://onlinelibrary.wiley.com/doi/epdf/10.1111/phis.12131

# 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.