Michael Potter | University of Cambridge (original) (raw)

Books by Michael Potter

Research paper thumbnail of Set theory and its philosophy: a critical introduction

Research paper thumbnail of The Rise of Analytic Philosophy 1879-1930: From Frege to Ramsey

In this book Michael Potter offers a fresh and compelling portrait of the birth of modern analyti... more In this book Michael Potter offers a fresh and compelling portrait of the birth of modern analytic philosophy, viewed through the lens of a detailed study of the work of the four philosophers who contributed most to shaping it: Gottlob Frege, Bertrand Russell, Ludwig Wittgenstein, and Frank Ramsey. It covers the remarkable period of discovery that began with the publication of Frege's Begriffsschrift in 1879 and ended with Ramsey's death in 1930. Potter—one of the most influential scholars of this period in philosophy—presents a deep but accessible account of the break with absolute idealism and neo-Kantianism, and the emergence of approaches that exploited the newly discovered methods in logic. Like his subjects, Potter focusses principally on philosophical logic, philosophy of mathematics, and metaphysics, but he also discusses epistemology, meta-ethics, and the philosophy of language. The book is an essential starting point for any student attempting to understand the work of Frege, Russell, Wittgenstein, and Ramsey, as well as their interactions and their larger intellectual milieux. It will also be of interest to anyone who wants to cast light on current philosophical problems through a better understanding of their origins.

Research paper thumbnail of Wittgenstein's Notes on Logic

Wittgenstein's philosophical career began in 1911 when he went to Cambridge to work with Russell.... more Wittgenstein's philosophical career began in 1911 when he went to Cambridge to work with Russell. He compiled the Notes on Logic two years later as a kind of summary of the work he had done so far. Russell thought that they were 'as good as anything that has ever been done in logic', but he had Wittgenstein himself to explain them to him. Without the benefit of Wittgenstein's explanations, most later scholars have preferred to treat the Notes solely as an interpretative aid in understanding the Tractatus (which draws on them for material), rather than as a philosophical work in their own right.

Michael Potter unequivocally demonstrates the philosophical and historical importance of the Notes for the first time. By teasing out the meaning of key passages, he shows how many of the most important insights in the Tractatus they contain. He discusses in detail how Wittgenstein arrived at these insights by thinking through ideas he obtained from Russell and Frege. And he uses a challenging blend of biography and philosophy to illuminate the methods Wittgenstein used in his work.

The book features the complete text of the Notes in a critical edition, with a detailed discussion of the circumstances in which they were compiled, leading to a new understanding of how they should be read.

Contents

* Introduction
* 1. Finding a Problem
* 2. First Steps
* 3. Matter
* 4. Analysis
* 5. The Fundamental Thought
* 6. The Symbolic Turn
* 7. Simplicity
* 8. Unity
* 9. Fregean Propositions
* 10. Assertion
* 11. Complex and Fact
* 12. Forms
* 13. Russell's Theory of Judgment
* 14. Meaning
* 15. Metaphysics
* 16. Sense
* 17. Truth-Functions
* 18. Truth-Operations
* 19. Molecular Propositions
* 20. Generality
* 21. Resolving the Paradoxes
* 22. Typical Ambiguity
* 23. Identity
* 24. Sign and Symbol
* 25. Wittgenstein's Theory of Judgment
* 26. The Picture Theory
* 27. Tractarian Objects
* 28. Philosophy
* 29. Themes

Appendices

* A. History of the text
* B. The Notes on Logic

Research paper thumbnail of The Cambridge Companion to Frege

Gottlob Frege (1848–1925) was unquestionably one of the most important philosophers of all time. ... more Gottlob Frege (1848–1925) was unquestionably one of the most important philosophers of all time. He trained as a mathematician, and his work in philosophy started as an attempt to provide an explanation of the truths of arithmetic, but in the course of this attempt he not only founded modern logic but also had to address fundamental questions in the philosophy of language and philosophical logic. Frege is generally seen (along with Russell and Wittgenstein) as one of the fathers of the analytic method, which dominated philosophy in English-speaking countries for most of the twentieth century. His work is studied today not just for its historical importance but also because many of his ideas are still seen as relevant to current debates in the philosophies of logic, language, mathematics and the mind. The Cambridge Companion to Frege provides a route into this lively area of research.

Contents

Preface
Note on translations
Chronology
1. Introduction Michael Potter
2. Understanding Frege's project: Joan Weiner
3. Frege's conception of logic Warren Goldfarb
4. Dummett's Frege Peter Sullivan
5. What is a predicate? Alex Oliver
6. Concepts, objects, and the context principle Thomas Ricketts
7. Sense and reference Michael Kremer
8. On sense and reference: a critical reception William Taschek
9. Frege and semantics Richard Heck
10. Frege's mathematical setting Mark Wilson
11. Frege and Hilbert Michael Hallett
12. Frege's folly Peter Milne
13. Frege and Russell Peter Hylton
14. Inheriting from Frege: the work of reception, as Wittgenstein did it Cora Diamond.

Research paper thumbnail of Set Theory and its Philosophy

A comprehensive philosophical introduction to set theory. Anyone wishing to work on the logical f... more A comprehensive philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. I offer a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. I discuss in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. I offer a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unusual is that it interweaves a careful presentation of the technical material with a detailed philosophical critique: I do not merely expound the theory dogmatically but at every stage discuss in detail the reasons that can be offered for believing it to be true.

Contents

Part I: Sets

1. Logic
2. Collections
3. The hierarchy
4. The theory of sets

Part II: Numbers

5. Arithmetic
6. Counting
7. Lines
8. Real numbers

Part III: Cardinals and Ordinals

9. Cardinals
10. Basic cardinal arithmetic
11. Ordinals
12. Ordinal arithmetic

Part IV: Further Axioms

13. Orders of infinity
14. The axiom of choice
15. Further cardinal arithmetic

Appendices

1. Traditional axiomatizations
2. Classes
3. Sets and classes

Research paper thumbnail of Mathematical Knowledge

What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it s... more What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions. Written by some of the world's leading philosophers of mathematics, psychologists, and mathematicians, Mathematical Knowledge gives a lively sense of the current state of debate in this fascinating field.

Contents

1. Mary Leng: Introduction
2. Michael Potter: What is the problem of mathematical knowledge?
3. Tim Gowers: Mathematics, memory, and mental arithmetic
4. Alan Baker: Is there a problem of induction for mathematics?
5. Marinella Cappelletti and Valeria Giardino: The cognitive basis of mathematical knowledge
6. Mary Leng: What's there to know? A fictionalist account of mathematical knowledge
7. Mark Colyvan: Mathematical recreation versus mathematical knowledge
8. Alexander Paseau: Scientific platonism
9. Crispin Wright: On quantifying into predicate position: Steps towards a (new)tralist position

Research paper thumbnail of Reason's Nearest Kin: Philosophies of arithmetic from Kant to Carnap

How do we account for the truth of arithmetic? And if it does not depend for its truth on the way... more How do we account for the truth of arithmetic? And if it does not depend for its truth on the way the world is, what constrains the world to conform to arithmetic? Reason's Nearest Kin is a critical examination of the astonishing progress made towards answering these questions from the late nineteenth to the mid-twentieth century. In the space of fifty years Frege, Dedekind, Russell, Wittgenstein, Ramsey, Hilbert, and Carnap developed accounts of the content of arithmetic that were brilliantly original both techically and philosophically. Michael Potter's innovative study presents them all as finding that content in various aspects of the complex linkage between experience, language, thought, and the world. Potter's reading places them all in Kant's shadow, since it was his attempt to ground arithmetic in the spatio-temporal structure of reality that they were reacting against; but it places us in Godel's shadow too, since his incompleteness theorems supply us with a measure of the richness of the content they were trying to explain. This stimulating reassessment of some of the classic texts in the philosophy of mathematics reveals many unexpected connections and illuminating comparisons, and offers a wealth of ideas for future work in the subject.

Research paper thumbnail of Sets: An Introduction

Research paper thumbnail of Set Theory and its Philosophy

A comprehensive philosophical introduction to set theory. Anyone wishing to work on the logical f... more A comprehensive philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. I offer a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. I discuss in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. I offer a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unusual is that it interweaves a careful presentation of the technical material with a detailed philosophical critique: I do not merely expound the theory dogmatically but at every stage discuss in detail the reasons that can be offered for believing it to be true.

Contents

Part I: Sets

1. Logic
2. Collections
3. The hierarchy
4. The theory of sets

Part II: Numbers

5. Arithmetic
6. Counting
7. Lines
8. Real numbers

Part III: Cardinals and Ordinals

9. Cardinals
10. Basic cardinal arithmetic
11. Ordinals
12. Ordinal arithmetic

Part IV: Further Axioms

13. Orders of infinity
14. The axiom of choice
15. Further cardinal arithmetic

Appendices

1. Traditional axiomatizations
2. Classes
3. Sets and classes

Papers by Michael Potter

Research paper thumbnail of Wittgenstein's Tractatus

Research paper thumbnail of Abstraction by Recarving

Proceedings of the Aristotelian Society, 2001

Research paper thumbnail of Foundations without Foundationalism: A Case for Second-Order Logic

The Philosophical Review, 1994

Research paper thumbnail of Proof and Knowledge in Mathematics

Philosophical Books, 1993

Routeledge edited by Michael DETLEFSEN 65-109 Logicism (by Steven J. WAGNER) 92 mathématiques = c... more Routeledge edited by Michael DETLEFSEN 65-109 Logicism (by Steven J. WAGNER) 92 mathématiques = clôture apriorique des intuitions primitives Imagine an ideal reasoner, beginning with a priori intuitions and proceeding by explanatory inference, deduction, and further intuition, without relying on science. Let us say (tendentiously!) that she relies on strictly mathematical reasons. (Her mathematics could be called the a priori closure of the original intuitions.

Research paper thumbnail of Proof and knowledge in mathematics

First published 1992 by Routledge 11 New Fetter Lane, London EC4P 4EE This edition published in t... more First published 1992 by Routledge 11 New Fetter Lane, London EC4P 4EE This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge's collection of thousands of eBooks please go to www. eBookstore. ...

Research paper thumbnail of Wittgenstein's philosophy of mathematics

Oxford Handbook of Wittgenstein, 2011

Research paper thumbnail of Frege, Russell and Wittgenstein

Routledge Companion to the Philosophy of Language, 2012

Research paper thumbnail of Abstractionist class theory: Is there any such thing?

A discussion of the philosophical prospects for basing a neo-Fregean theory of classes on a princ... more A discussion of the philosophical prospects for basing a neo-Fregean theory of classes on a principle that attempts to articulate the limitation-of-size conception.

Research paper thumbnail of The logic of the Tractatus

Handbook of the History of Logic, 2009

Describes some of the main features of the logic and metaphysics of Wittgenstein's Tractatus.

Research paper thumbnail of The birth of analytic philosophy

Routledge Companion to Twentieth-Century Philosophy, 2008

Tries to identify some strands in the birth of analytic philosophy and to identify in consequence... more Tries to identify some strands in the birth of analytic philosophy and to identify in consequence some of its distinctive features.

Research paper thumbnail of What is the problem of mathematical knowledge?

Mathematical Knowledge, 2007

Suggests that the recent emphasis on Benacerraf's access problem locates the peculiarity of mathe... more Suggests that the recent emphasis on Benacerraf's access problem locates the peculiarity of mathematical knowledge in the wrong place. Instead we should focus on the sense in which mathematical concepts are or might be "armchair concepts" — concepts about which non-trivial knowledge is obtainable a priori.

Research paper thumbnail of Set theory and its philosophy: a critical introduction

Research paper thumbnail of The Rise of Analytic Philosophy 1879-1930: From Frege to Ramsey

In this book Michael Potter offers a fresh and compelling portrait of the birth of modern analyti... more In this book Michael Potter offers a fresh and compelling portrait of the birth of modern analytic philosophy, viewed through the lens of a detailed study of the work of the four philosophers who contributed most to shaping it: Gottlob Frege, Bertrand Russell, Ludwig Wittgenstein, and Frank Ramsey. It covers the remarkable period of discovery that began with the publication of Frege's Begriffsschrift in 1879 and ended with Ramsey's death in 1930. Potter—one of the most influential scholars of this period in philosophy—presents a deep but accessible account of the break with absolute idealism and neo-Kantianism, and the emergence of approaches that exploited the newly discovered methods in logic. Like his subjects, Potter focusses principally on philosophical logic, philosophy of mathematics, and metaphysics, but he also discusses epistemology, meta-ethics, and the philosophy of language. The book is an essential starting point for any student attempting to understand the work of Frege, Russell, Wittgenstein, and Ramsey, as well as their interactions and their larger intellectual milieux. It will also be of interest to anyone who wants to cast light on current philosophical problems through a better understanding of their origins.

Research paper thumbnail of Wittgenstein's Notes on Logic

Wittgenstein's philosophical career began in 1911 when he went to Cambridge to work with Russell.... more Wittgenstein's philosophical career began in 1911 when he went to Cambridge to work with Russell. He compiled the Notes on Logic two years later as a kind of summary of the work he had done so far. Russell thought that they were 'as good as anything that has ever been done in logic', but he had Wittgenstein himself to explain them to him. Without the benefit of Wittgenstein's explanations, most later scholars have preferred to treat the Notes solely as an interpretative aid in understanding the Tractatus (which draws on them for material), rather than as a philosophical work in their own right.

Michael Potter unequivocally demonstrates the philosophical and historical importance of the Notes for the first time. By teasing out the meaning of key passages, he shows how many of the most important insights in the Tractatus they contain. He discusses in detail how Wittgenstein arrived at these insights by thinking through ideas he obtained from Russell and Frege. And he uses a challenging blend of biography and philosophy to illuminate the methods Wittgenstein used in his work.

The book features the complete text of the Notes in a critical edition, with a detailed discussion of the circumstances in which they were compiled, leading to a new understanding of how they should be read.

Contents

* Introduction
* 1. Finding a Problem
* 2. First Steps
* 3. Matter
* 4. Analysis
* 5. The Fundamental Thought
* 6. The Symbolic Turn
* 7. Simplicity
* 8. Unity
* 9. Fregean Propositions
* 10. Assertion
* 11. Complex and Fact
* 12. Forms
* 13. Russell's Theory of Judgment
* 14. Meaning
* 15. Metaphysics
* 16. Sense
* 17. Truth-Functions
* 18. Truth-Operations
* 19. Molecular Propositions
* 20. Generality
* 21. Resolving the Paradoxes
* 22. Typical Ambiguity
* 23. Identity
* 24. Sign and Symbol
* 25. Wittgenstein's Theory of Judgment
* 26. The Picture Theory
* 27. Tractarian Objects
* 28. Philosophy
* 29. Themes

Appendices

* A. History of the text
* B. The Notes on Logic

Research paper thumbnail of The Cambridge Companion to Frege

Gottlob Frege (1848–1925) was unquestionably one of the most important philosophers of all time. ... more Gottlob Frege (1848–1925) was unquestionably one of the most important philosophers of all time. He trained as a mathematician, and his work in philosophy started as an attempt to provide an explanation of the truths of arithmetic, but in the course of this attempt he not only founded modern logic but also had to address fundamental questions in the philosophy of language and philosophical logic. Frege is generally seen (along with Russell and Wittgenstein) as one of the fathers of the analytic method, which dominated philosophy in English-speaking countries for most of the twentieth century. His work is studied today not just for its historical importance but also because many of his ideas are still seen as relevant to current debates in the philosophies of logic, language, mathematics and the mind. The Cambridge Companion to Frege provides a route into this lively area of research.

Contents

Preface
Note on translations
Chronology
1. Introduction Michael Potter
2. Understanding Frege's project: Joan Weiner
3. Frege's conception of logic Warren Goldfarb
4. Dummett's Frege Peter Sullivan
5. What is a predicate? Alex Oliver
6. Concepts, objects, and the context principle Thomas Ricketts
7. Sense and reference Michael Kremer
8. On sense and reference: a critical reception William Taschek
9. Frege and semantics Richard Heck
10. Frege's mathematical setting Mark Wilson
11. Frege and Hilbert Michael Hallett
12. Frege's folly Peter Milne
13. Frege and Russell Peter Hylton
14. Inheriting from Frege: the work of reception, as Wittgenstein did it Cora Diamond.

Research paper thumbnail of Set Theory and its Philosophy

A comprehensive philosophical introduction to set theory. Anyone wishing to work on the logical f... more A comprehensive philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. I offer a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. I discuss in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. I offer a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unusual is that it interweaves a careful presentation of the technical material with a detailed philosophical critique: I do not merely expound the theory dogmatically but at every stage discuss in detail the reasons that can be offered for believing it to be true.

Contents

Part I: Sets

1. Logic
2. Collections
3. The hierarchy
4. The theory of sets

Part II: Numbers

5. Arithmetic
6. Counting
7. Lines
8. Real numbers

Part III: Cardinals and Ordinals

9. Cardinals
10. Basic cardinal arithmetic
11. Ordinals
12. Ordinal arithmetic

Part IV: Further Axioms

13. Orders of infinity
14. The axiom of choice
15. Further cardinal arithmetic

Appendices

1. Traditional axiomatizations
2. Classes
3. Sets and classes

Research paper thumbnail of Mathematical Knowledge

What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it s... more What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions. Written by some of the world's leading philosophers of mathematics, psychologists, and mathematicians, Mathematical Knowledge gives a lively sense of the current state of debate in this fascinating field.

Contents

1. Mary Leng: Introduction
2. Michael Potter: What is the problem of mathematical knowledge?
3. Tim Gowers: Mathematics, memory, and mental arithmetic
4. Alan Baker: Is there a problem of induction for mathematics?
5. Marinella Cappelletti and Valeria Giardino: The cognitive basis of mathematical knowledge
6. Mary Leng: What's there to know? A fictionalist account of mathematical knowledge
7. Mark Colyvan: Mathematical recreation versus mathematical knowledge
8. Alexander Paseau: Scientific platonism
9. Crispin Wright: On quantifying into predicate position: Steps towards a (new)tralist position

Research paper thumbnail of Reason's Nearest Kin: Philosophies of arithmetic from Kant to Carnap

How do we account for the truth of arithmetic? And if it does not depend for its truth on the way... more How do we account for the truth of arithmetic? And if it does not depend for its truth on the way the world is, what constrains the world to conform to arithmetic? Reason's Nearest Kin is a critical examination of the astonishing progress made towards answering these questions from the late nineteenth to the mid-twentieth century. In the space of fifty years Frege, Dedekind, Russell, Wittgenstein, Ramsey, Hilbert, and Carnap developed accounts of the content of arithmetic that were brilliantly original both techically and philosophically. Michael Potter's innovative study presents them all as finding that content in various aspects of the complex linkage between experience, language, thought, and the world. Potter's reading places them all in Kant's shadow, since it was his attempt to ground arithmetic in the spatio-temporal structure of reality that they were reacting against; but it places us in Godel's shadow too, since his incompleteness theorems supply us with a measure of the richness of the content they were trying to explain. This stimulating reassessment of some of the classic texts in the philosophy of mathematics reveals many unexpected connections and illuminating comparisons, and offers a wealth of ideas for future work in the subject.

Research paper thumbnail of Sets: An Introduction

Research paper thumbnail of Set Theory and its Philosophy

A comprehensive philosophical introduction to set theory. Anyone wishing to work on the logical f... more A comprehensive philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. I offer a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. I discuss in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. I offer a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unusual is that it interweaves a careful presentation of the technical material with a detailed philosophical critique: I do not merely expound the theory dogmatically but at every stage discuss in detail the reasons that can be offered for believing it to be true.

Contents

Part I: Sets

1. Logic
2. Collections
3. The hierarchy
4. The theory of sets

Part II: Numbers

5. Arithmetic
6. Counting
7. Lines
8. Real numbers

Part III: Cardinals and Ordinals

9. Cardinals
10. Basic cardinal arithmetic
11. Ordinals
12. Ordinal arithmetic

Part IV: Further Axioms

13. Orders of infinity
14. The axiom of choice
15. Further cardinal arithmetic

Appendices

1. Traditional axiomatizations
2. Classes
3. Sets and classes

Research paper thumbnail of Wittgenstein's Tractatus

Research paper thumbnail of Abstraction by Recarving

Proceedings of the Aristotelian Society, 2001

Research paper thumbnail of Foundations without Foundationalism: A Case for Second-Order Logic

The Philosophical Review, 1994

Research paper thumbnail of Proof and Knowledge in Mathematics

Philosophical Books, 1993

Routeledge edited by Michael DETLEFSEN 65-109 Logicism (by Steven J. WAGNER) 92 mathématiques = c... more Routeledge edited by Michael DETLEFSEN 65-109 Logicism (by Steven J. WAGNER) 92 mathématiques = clôture apriorique des intuitions primitives Imagine an ideal reasoner, beginning with a priori intuitions and proceeding by explanatory inference, deduction, and further intuition, without relying on science. Let us say (tendentiously!) that she relies on strictly mathematical reasons. (Her mathematics could be called the a priori closure of the original intuitions.

Research paper thumbnail of Proof and knowledge in mathematics

First published 1992 by Routledge 11 New Fetter Lane, London EC4P 4EE This edition published in t... more First published 1992 by Routledge 11 New Fetter Lane, London EC4P 4EE This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge's collection of thousands of eBooks please go to www. eBookstore. ...

Research paper thumbnail of Wittgenstein's philosophy of mathematics

Oxford Handbook of Wittgenstein, 2011

Research paper thumbnail of Frege, Russell and Wittgenstein

Routledge Companion to the Philosophy of Language, 2012

Research paper thumbnail of Abstractionist class theory: Is there any such thing?

A discussion of the philosophical prospects for basing a neo-Fregean theory of classes on a princ... more A discussion of the philosophical prospects for basing a neo-Fregean theory of classes on a principle that attempts to articulate the limitation-of-size conception.

Research paper thumbnail of The logic of the Tractatus

Handbook of the History of Logic, 2009

Describes some of the main features of the logic and metaphysics of Wittgenstein's Tractatus.

Research paper thumbnail of The birth of analytic philosophy

Routledge Companion to Twentieth-Century Philosophy, 2008

Tries to identify some strands in the birth of analytic philosophy and to identify in consequence... more Tries to identify some strands in the birth of analytic philosophy and to identify in consequence some of its distinctive features.

Research paper thumbnail of What is the problem of mathematical knowledge?

Mathematical Knowledge, 2007

Suggests that the recent emphasis on Benacerraf's access problem locates the peculiarity of mathe... more Suggests that the recent emphasis on Benacerraf's access problem locates the peculiarity of mathematical knowledge in the wrong place. Instead we should focus on the sense in which mathematical concepts are or might be "armchair concepts" — concepts about which non-trivial knowledge is obtainable a priori.

Research paper thumbnail of Ramsey's transcendental argument

Ramsey's Legacy, 2005

Explores the historical and philosophical background to a curious argument of Ramsey's that in th... more Explores the historical and philosophical background to a curious argument of Ramsey's that in the Tractatus the possibility of the infinite proves its actuality.

Research paper thumbnail of What is wrong with abstraction?

Philosophia Mathematica, 2005

We correct a misunderstanding by Hale and Wright of an objection we raised in 'Hale on Caesar' to... more We correct a misunderstanding by Hale and Wright of an objection we raised in 'Hale on Caesar' to their abstractionist programme for rehabilitating logicism in the foundations of mathematics.

Research paper thumbnail of Recarving content: Hale's final proposal

Proceedings of the Aristotelian Society, 2002

A follow-up, showing why Bob Hale's revision of his notion of weak sense is still inadequate.

Research paper thumbnail of Was Gödel a Gödelian platonist?

Philosophia Mathematica, 2001

Gödel's appeal to mathematical intuition to ground our grasp of the axioms of set theory is notor... more Gödel's appeal to mathematical intuition to ground our grasp of the axioms of set theory is notorious. I extract from his writings an account of this form of intuition which distinguishes it from the metaphorical platonism of which Gödel is sometimes accused and brings out the similarities between Gödel's views and Dummett's.

Research paper thumbnail of Intuition and reflection in arithmetic

Aristotelian Society Supplementary Volume, 1999

Classifies accounts of arithmetic into four sorts according to the resources they appeal to in co... more Classifies accounts of arithmetic into four sorts according to the resources they appeal to in constructing its subject matter.

Research paper thumbnail of Classical arithmetic as part of intuitionistic arithmetic

Grazer Philosophische Studien, 1998

Argues that classical arithmetic can be viewed as a proper part of intuitionistic arithmetic. Sug... more Argues that classical arithmetic can be viewed as a proper part of intuitionistic arithmetic. Suggests that this largely neutralizes Dummett's argument for intuitionism in the case of arithmetic.

Research paper thumbnail of Hale on Caesar

Philosophia Mathematica, 1997

Presents a battery of difficulties for the notion that arithmetic can be based on Hume's principle.

Research paper thumbnail of Taming the infinite

BJPS, 1996

A critique of Shaughan Lavine's attempt in Understanding the Infinite to reduce talk about the in... more A critique of Shaughan Lavine's attempt in Understanding the Infinite to reduce talk about the infinite to finitely comprehensible terms.

![Research paper thumbnail of Critical notice of `Parts of classes' by David Lewis](https://a.academia-assets.com/images/blank-paper.jpg)

The Philosophical Quarterly, 1993

Argues that Lewis is not as ontologically innocent as he pretends.

Research paper thumbnail of More thoughts on replacement

Research paper thumbnail of Is the Tractatus really nonsense?

Research paper thumbnail of Privacy and acquaintance in Frege, Russell and Wittgenstein

Research paper thumbnail of Frege's influence on Wittgenstein

Research paper thumbnail of Two kinds of amiguity?

Research paper thumbnail of The role of classes in Principia

The axiom of reducibility in Principia can be thought of as a class existence axiom. Yet class te... more The axiom of reducibility in Principia can be thought of as a class existence axiom. Yet class terms are incomplete symbols which disappear on analysis. I shall discuss the interaction between these two features of the system and relate them to later critcisims of Principia by Wittgenstein and Ramsey.

Research paper thumbnail of Why ZF is not a foundation for mathematics (but nor is ZU)

Research paper thumbnail of Abstractionist set theory

Research paper thumbnail of More on replacement

Research paper thumbnail of More on replacement

Research paper thumbnail of Wittgenstein on solipsism

Research paper thumbnail of Why study philosophy?

Research paper thumbnail of Wittgenstein 1916

Research paper thumbnail of Where does arithmetic fit into transcendental philosophy?

Research paper thumbnail of Wittgenstein and the club but not the poker

Research paper thumbnail of TBC

Research paper thumbnail of Wittgenstein in conflict: How the Tractatus was born

Research paper thumbnail of How abstract is topology?

Research paper thumbnail of Wittgenstein's debt to Frege

Research paper thumbnail of The genesis of the Notes on Logic

Research paper thumbnail of Propositions in Wittgenstein and Ramsey

Proceedings of the 41st International Ludwig Wittgenstein Symposium, 1999

In Begriffsschrift Frege proposed to ignore the part of content that is irrelevant to logic; what... more In Begriffsschrift Frege proposed to ignore the part of content that is irrelevant to logic; what remains he called ‘conceptual content’. In ‘On sense and reference’ he renamed this ‘sense’ but failed to stress that it is a notion belonging to the philosophy of logic, not of language. Russell seems to have seen the importance of the notion only briefly. Wittgenstein did not make use of the notion until he was in Norway, and only introduced the terminology of ‘sign’ and ‘symbol’ to mark the distinction while composing the Tractatus. Ramsey proposed to treat sign and symbol as merely two different ways of typing token inscriptions, but this unduly brushes over the difficulties the notion of a symbol involves. The most striking feature of Wittgenstein’s thinking on this is the way that he generalized Frege’s argument for the notion of sense so as to bypass his incorrect particularization to the case of identity.