PH34215 Philosophy of Mathematics
Introductory description
Do mathematical objects such as numbers and sets exist or are they merely useful fictions? What is the nature of mathematical knowledge and how is it distinct from our knowledge of the physical world? What, if any, is the connection between the two? What role does mathematics play in the empirical sciences? What is the correct logic for reasoning about mathematics? Are formally undecidable statements (e.g. the Parallel Postulate, the Gödel sentence, the Continuum Hypothesis) objectively true or false? This module will explore different ways in which philosophy might be of help in answering these questions, both from the contemporary perspective and that of the major foundational schools of the late 19th and early 20th centuries: logicism, intuitionism, and formalism.
Module aims
This module has two goals: 1) to familiarise students with major developments in the foundations of mathematics from the late 19th century onward; and 2) to illustrate how these developments inform contemporary debates in philosophy of mathematics.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
Week 1: Introduction to the module. The infinite in ancient Greek thought.
 Core reading: introduction and chapters 1 and 2 of A. W. Moore, The Infinite (Routledge, 2001).
Week 2: Infinity in mathematics. Cantor’s theory of sets.
 Core reading: part I (pages 1–34) of M. Giaquinto, The Search for Certainty (OUP, 2002).
Week 3: The classtheoretic paradoxes. Type theory and limitation of size.
– Core reading: chapter 10 of Moore (2001), part II (pages 35–65) of Giaquinto (2002).
Week 4: The axiomatic method. Hilbertian finitism.
– Core reading: Chapters IV.3–4 of Giaquinto (2002), D. Hilbert, On the infinite (1926).
Week 5: Constructivism in Brouwer and Heyting.
– Core reading: chapter 7 of S. Shapiro, Thinking About Mathematics (OUP, 2000), introduction to Brouwer's Cambridge lectures on intuitionism (CUP, 1981).
Week 7: The Löwenheim–Skolem theorem. Skolem’s paradox.
– Core reading: Chapters IV.1 and IV.2 of Giaquinto (2002), P. Benacerraf, Skolem and the skeptic (1985).
Week 8: The continuum problem. Realism and indeterminacy in set theory.
– Core reading: Chapter VI.1 of Giaquinto (2002), K. Gödel, What is Cantor's continuum problem? (1947).
Week 9: Categoricity and determinacy. Structuralism.
– Core reading: Chapter 10 of Shapiro (2000), P. Benacerraf, What numbers could not be (1965).
Week 10: Potential infinity revisited. Modality and potentiality.
– Core reading: Linnebo and Shapiro, Actual and potential infinity (2019).
Learning outcomes
By the end of the module, students should be able to:
 Demonstrate knowledge of some of the central topics in the philosophy of mathematics, and of the historical development of key approaches to the philosophy of mathematics (Subject knowledge and understanding)
 Understand the significance that questions in the philosophy of mathematics have to wider issues in philosophy and the foundations of mathematics (cognitive skills)
 Articulate their own view of the relative merits of different theories and engage critically with the arguments put forward in support of them (key skills)
 Show an understanding of methodological issues in the philosophy of mathematics, and of questions of demarcation between philosophy and mathematics (subjectspecific skills)
Indicative reading list
Much of the background and historical reading will be drawn from two books:
The Infinite (2nd ed.) by A. W. Moore (Routledge, 2001).
The Search for Certainty by M. Giaquinto (Oxford University Press, 2002).
There will also be a substantial use of original sources and recent scholarship. Many important papers can be found in the following collections.
Philosophy of Mathematics: Selected Readings (2nd ed.), edited by P. Benacerraf and H. Putnam (Cambridge University Press, 1983).
From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, edited by J. van Heijenoort.
An excellent handbook in philosophy of mathematics is:
The Oxford Handbook of Philosophy of Mathematics and Logic, edited by S. Shapiro (Oxford University Press, 2005).
A survey of topics in mathematical logic that are relevant for this course is:
Mathematical Logic by J. R. Shoenfield (AddisonWesley, 1967).
Subject specific skills
Show an understanding of methodological issues in the philosophy of mathematics, and of questions of demarcation between philosophy and mathematics
Transferable skills
Understand how major debates in the philosophy of mathematics—e.g. between logicism, formalism, and intuitionism—are related to topics in the history of philosophy, metaphysics, and epistemology. Appreciate how developments in mathematical logic—e.g. axiomatic set theory, proof theory—grew out of concern for foundational issues in the 19th and early 20th century.
Study time
Type  Required 

Lectures  9 sessions of 2 hours (12%) 
Seminars  8 sessions of 1 hour (5%) 
Private study  124 hours (83%) 
Total  150 hours 
Private study description
No private study requirements defined for this module.
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Students can register for this module without taking any assessment.
Assessment group D8
Weighting  Study time  

1000 word essay  20%  
Online Examination  80%  
2 hour exam ~Platforms  AEP

Assessment group D9
Weighting  Study time  

1000 word essay  20%  
Online Examination  80%  

Feedback on assessment
Written feedback on essays and exams.
Prerequisites
PH136 (Logic 1) is recommended as a prerequisite. Otherwise, the module is designed to be as selfcontained as possible. However, you should be aware that several of the topics we will discuss are related to developments in mathematical logic (as treated in modules like PH210 Logic II, PH340 Logic III and MA3H3 Set Theory), and also build on philosophical themes which are covered in modules like PH251 Metaphysics, PH252 Epistemology, and PH144 Mind and Reality. Background in these subjects will therefore be helpful for fully engaging with the module content.
To take this module, you must have passed:
Courses
This module is Core for:
 Year 4 of UMAAGV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations
This module is Optional for:

UHIAV1V8 Undergraduate History and Philosophy (with Year Abroad and a term in Venice)
 Year 3 of V1V8 History and Philosophy (with Year Abroad and a term in Venice)
 Year 4 of V1V8 History and Philosophy (with Year Abroad and a term in Venice)
 Year 3 of UHIAV1V7 Undergraduate History and Philosophy (with a term in Venice)

UPHAV700 Undergraduate Philosophy
 Year 2 of V700 Philosophy
 Year 2 of V700 Philosophy
 Year 3 of V700 Philosophy
 Year 3 of V700 Philosophy
 Year 4 of UPHAV701 Undergraduate Philosophy (wiith Intercalated year)
 Year 4 of UPHAV702 Undergraduate Philosophy (with Work Placement)
 Year 4 of UPHAV7MM Undergraduate Philosophy, Politics and Economics (with Intercalated year)
This module is Core option list A for:

UMAAGV17 Undergraduate Mathematics and Philosophy
 Year 3 of GV17 Mathematics and Philosophy
 Year 3 of GV17 Mathematics and Philosophy
 Year 3 of GV17 Mathematics and Philosophy
 Year 3 of UMAAGV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations
This module is Core option list B for:

UMAAGV17 Undergraduate Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy
 Year 2 of UMAAGV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations
This module is Core option list C for:
 Year 4 of UMAAGV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations
This module is Core option list F for:

UMAAGV18 Undergraduate Mathematics and Philosophy with Intercalated Year
 Year 4 of GV18 Mathematics and Philosophy with Intercalated Year
 Year 4 of GV18 Mathematics and Philosophy with Intercalated Year
This module is Option list A for:

UPHAVL78 BA in Philosophy with Psychology
 Year 2 of VL78 Philosophy with Psychology
 Year 3 of VL78 Philosophy with Psychology
 Year 4 of UPHAVL79 BA in Philosophy with Psychology (with Intercalated year)
This module is Option list B for:

UMAAG105 Undergraduate Master of Mathematics (with Intercalated Year)
 Year 2 of G105 Mathematics (MMath) with Intercalated Year
 Year 4 of G105 Mathematics (MMath) with Intercalated Year
 Year 5 of G105 Mathematics (MMath) with Intercalated Year

UMAAG100 Undergraduate Mathematics (BSc)
 Year 2 of G100 Mathematics
 Year 2 of G100 Mathematics
 Year 2 of G100 Mathematics
 Year 3 of G100 Mathematics
 Year 3 of G100 Mathematics
 Year 3 of G100 Mathematics

UMAAG103 Undergraduate Mathematics (MMath)
 Year 2 of G100 Mathematics
 Year 2 of G103 Mathematics (MMath)
 Year 2 of G103 Mathematics (MMath)
 Year 3 of G100 Mathematics
 Year 3 of G103 Mathematics (MMath)
 Year 3 of G103 Mathematics (MMath)
 Year 4 of G103 Mathematics (MMath)
 Year 4 of G103 Mathematics (MMath)

UMAAG106 Undergraduate Mathematics (MMath) with Study in Europe
 Year 2 of G106 Mathematics (MMath) with Study in Europe
 Year 4 of G106 Mathematics (MMath) with Study in Europe
 Year 2 of UMAAG1NC Undergraduate Mathematics and Business Studies
 Year 2 of UMAAG1N2 Undergraduate Mathematics and Business Studies (with Intercalated Year)
 Year 2 of UMAAGL11 Undergraduate Mathematics and Economics
 Year 2 of UECAGL12 Undergraduate Mathematics and Economics (with Intercalated Year)

UMAAG101 Undergraduate Mathematics with Intercalated Year
 Year 2 of G101 Mathematics with Intercalated Year
 Year 4 of G101 Mathematics with Intercalated Year

UPHAVQ72 Undergraduate Philosophy and Literature
 Year 2 of VQ72 Philosophy and Literature
 Year 3 of VQ72 Philosophy and Literature
 Year 4 of UPHAVQ73 Undergraduate Philosophy and Literature with Intercalated Year
This module is Option list C for:
 Year 3 of UHIAV1V5 Undergraduate History and Philosophy
 Year 4 of UHIAV1V6 Undergraduate History and Philosophy (with Year Abroad)
This module is Option list D for:

UHIAV1V5 Undergraduate History and Philosophy
 Year 2 of V1V5 History and Philosophy
 Year 3 of V1V5 History and Philosophy
 Year 4 of UHIAV1V8 Undergraduate History and Philosophy (with Year Abroad and a term in Venice)
 Year 4 of UHIAV1V6 Undergraduate History and Philosophy (with Year Abroad)

UHIAV1V7 Undergraduate History and Philosophy (with a term in Venice)
 Year 2 of V1V7 History and Philosophy (with a term in Venice)
 Year 3 of V1V7 History and Philosophy (with a term in Venice)