PH34215 Philosophy of Mathematics
 Department
 Philosophy
 Level
 Undergraduate Level 3
 Credit value
 15
 Module duration
 10 weeks
 Assessment
 Multiple
 Study location
 University of Warwick main campus, Coventry
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 nineteen and early twentieth centuries  i.e. logicism, intuitionism, and formalism.
Module aims
This module has two goals:
 to familiarise students with major developments in the foundations of mathematics from the late 19th century onward;
 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.
Indicative schedule and summary of readings
Lecture 1: Introduction, philosophical terminology
 Shapiro chapters 1, 2
 Benacerraf “Mathematical truth”
Lecture 2 : Classical perspectives: Plato & Aristotle
 Shapiro chapter 3
 Plato selections from the Republic and the Meno
 Aristotle selections from Metaphysics M
Lecture 3: Rationalism and empiricism: Kant & Mill
 Shapiro chapter 4
 Kant selections from Prolegomena
 Mill selections from A system of logic
Lecture 4: Logic and set theory interlude/review
 George and Velleman chapter 3
 Set Theory (SEP)
 Secondorder and Higherorder Logic (SEP)
 Some formal theories of arithmetic (SEP)
Lecture 5: Logicism: Frege & Russell
 Shapiro chapter 5
 Frege selections from The foundations of arithmetic
 Heck “Frege’s Theorem: An Introduction”
 “Frege’s logic, theorem, and foundations for arithmetic” [SEP]
Lecture 6: Intuitionism: Brouwer & Heyting
 Shapiro chapter 6
 Heyting “Disputation” [BP]
 George and Velleman chapters 4 and 5
 Optional: Dummett “The philosophical basis of intuitionistic logic”
Lecture 7: Formalism and the Hilbert Programme
 Shapiro chapter 7
 George and Velleman chapter 6
 Hilbert “On the infinite” [BP]
 Zach “Hilbert’s program” [SEP]
Lecture 8: Paradoxes, G¨odel’s Theorems, and set theoretic independence
 Shapiro chapter 8
 Russell’s Paradox [SEP]
 G¨odel’s Incompleteness Theorems [SEP]
 The Continuum Hypothesis [SEP]
 Optional: G¨odel “What is Cantor’s Continuum Hypothesis?”, Dummett “The philosophical significance of G¨odel’s Theorem”
Lecture 9: Structuralism
 Shapiro chapters 9,10
 Structuralism and Nominalism [SEP]
 Benacerraf “What numbers could not be” [BP]
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 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
Our primary texts will be
 Thinking about mathematics, Stewart Shapiro, Oxford University Press, 2000.
 Philosophies of mathematics, Alexander George and David Velleman, WileyBlackwell, 2001.
Many of the other reading are available in  Philosophy of Mathematics: Selected Readings edited by Paul Benacerraf and Hilary Putnam,
Cambridge University Press, 1983. [BP]  The Stanford Encyclopedia of Philosophy [SEP].
Other sources are available through the Moodle page.
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 D5
Weighting  Study time  

1000 word essay  20%  
Online Examination  80%  
2 hour exam

Assessment group D6
Weighting  Study time  

1000 word essay  20%  
Online Examination  80%  

Assessment group D7
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. But you should be aware that several of the topics we will discuss are
related to developments in mathematical logic (as treated in modules like Logic II/III and Set
Theory) and also build on philosophical themes which are covered in modules like Metaphysics
and Epistemology. So background in these subjects will 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:

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 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 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:
 Year 2 of UHIAV1V5 Undergraduate History and Philosophy
 Year 2 of UMAAG105 Undergraduate Master of Mathematics (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 2 of UMAAG106 Undergraduate 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

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
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)