Philosophy of Mathematics

Below are readings and essay questions for tutorials in Philosophy of Mathematics. Many of the readings are available online, and all are easily obtained from the college or other libraries in Oxford, but if you are struggling to get hold of anything, email me, as I have PDF copies of nearly everything.

I have divided the reading for each topic into CORE READING and FURTHER READING with more introductory texts marked with a star (*). Focus on the Core Reading suggestions in writing tutorial essays, and use the Further Reading suggestions as starting points for exploring topics in more depth during vacations. You will find more suggestions in the Faculty Reading List.

The default plan is to cover the TUTORIAL TOPICS. There are also (or will eventually be) some OTHER TOPICS, which I’ll provide reading lists for as and when I get some time to do so. In setting the essay questions and readings for the TUTORIAL TOPICS, I’ve taken as a point of departure James Studd’s reading list, which you can find on Talis.

TUTORIAL TOPICS

1. Frege’s and Russell’s Logicisms
2. Intuitionism
3. Hilbert’s Formalism
4. Set Theory
5. Realism: Benacerraf’s Dilemma
6. Nominalism: Indispensability
7. Structuralism
8. Neo-Logicism

OTHER TOPICS

• Kant
• Empiricism
• Predicativism

ANTHOLOGIES and TEXTBOOKS

The following anthologies and collections are particularly useful, and contain many of the key readings. I will refer to them below as Benacerraf and Putnam, van Heijenoort, and Shapiro respectively:

It will be to your advantage to have done some vacation reading beforehand. I recommend the following:

I have set chapters from Bostock (2009) and Shapiro (2000) as introductory Core Reading, while Hamkins (2021) is the text on which the lectures (and, in recent years, many of the exam questions) are based. Any would serve well as vacation reading, ideally alongside Alexander George and Daniel Velleman (2002) Philosophies of Mathematics (Blackwell), which treats the first three topics in a little more depth. You’ll also find good overviews of the topics we’ll be looking at in the Stanford Encyclopedia of Philosophy. The following is a good place to start—you’ll find links to other relevant entries at the bottom of the article:

TUTORIAL TOPICS

1. FREGE’S and RUSSELL’S LOGICISMS

TOP

ESSAY QUESTION

Explain and assess Frege’s and Russell’s attempts to reduce mathematics to logic.

CORE READING

FURTHER READING

In theory, questions on Frege’s and Russell’s views in the philosophy of mathematics, which used to be set for the old Frege, Russell, Wittgenstein paper, are now set for this one, but in practice this doesn’t seem to happen, so it’s perhaps best to regard this topic as preparatory for future weeks. In pursuing it in more depth, you might want to start with Ch. 2 and 3 of George and Velleman (2002), which cover much the same material as Bostock in the Core Reading, but in a little more detail; Ch. 3, among other things, explains how to derive Peano’s axioms of arithmetic in ZF via the von Neumann ordinals. See also Burgess (2005) and Demopoulos and Clark’s contribution to Shapiro. For further discussion of Frege, see Dummett (1991) and various papers in Demopoulos, ed. (1995), starting perhaps with those by Benacerraf and Parsons. We’ll look at attempts to rescuscitate something like Frege’s program when we look at NEO-LOGICISM in week 8. For further discussion of Russell’s philosophy of mathematics, see, in the first instance, Sainsbury (1979). You can focus for now on Russell’s so-called simple theory of types—see Copi (1971) and Urquart’s paper in Griffin, ed. (2003). For the ramified theory, see the readings for the PREDICATIVISM. You might also want to think about the question whether second-order logic is really logic. For more on this, see Shapiro’s own contribution to Shapiro, as well as the reading for the SECOND-ORDER LOGIC topic in the PHILOSOPHICAL LOGIC paper.

PAST PAPER QUESTIONS

EITHER
(a) Explain Frege’s concept of number and whether it follows the philosophy of structuralism.
OR
(b) ‘Neo-logicists have failed to justify the conceptual necessity of the existence of infinitely many mathematical objects.’ Discuss. (2019)

EITHER
(a) How should logicists respond to the ‘Julius Caesar’ problem?
OR
(b) ‘Hume’s Principle is ontologically committing, and so cannot be analytic.’ Discuss. (2018)

Assess the advantages and disadvantages of using second-order logic in mathematics? (2017)

Is second-order logic really logic, or is it a form of set theory? (2016)

2. INTUITIONISM

TOP

ESSAY QUESTION

How does intuitionist logic and mathematics differ from it classical counterpart? Should philosophical considerations lead us to renounce classical mathematics?

CORE READING

FURTHER READING

Benacerraf and Putnam contains various pieces by Brouwer and Heyting. For overviews, see Posy in Shapiro and van Stigt’s introduction to Part I of Mancosu, ed. (1998). Your time’s probably best spent focusing on Dummett, however. His (1978) reprints ‘The Philosophical Significance of Intuitionistic Logic’ alongside various other important papers; see especially ‘Wang’s Paradox’ and (looking ahead to next week) ‘The Philosophical Significance of Gödel’s Theorem’. But perhaps the most important thing to read is his (2000) book, updating the first edition, published in 1977; see especially the introduction, Ch. 1 and Ch. 7, but also Ch. 2 and 3, on intuitionistic mathematics. George and Velleman (2002) is, again, an invaluable discussion. Ch. 4 focuses on the differences between intuitionism and realism; Ch. 5 on intuitionistic mathematics. For some additional critical discussion, you might try Hellman (1989) and Rumfitt (2000), as well as Cook’s piece in Shapiro. For a nice overview of intuitionistic logic, try Burgess (2009), while, for more on intuitionism in contemporary mathematics, try McCarty’s contribution to Shapiro.

PAST PAPER QUESTIONS

Outline the main differences between the classical and intuitionistic forms of logic, arithmetic, and analysis. What reason, if any, is there to prefer the latter? (2019)

EITHER
(a) Is it a matter of choice whether one endorses classical or intuitionistic mathematics?
OR
(b) Is it possible to perform an infinite number of calculations in a finite period of time? What is the significance of your answer? (2018)

EITHER
(a) Was Brouwer right that classical mathematics breeds inconsistency?
OR
(b) Is intuitionistic mathematics adequate for science? (2017)

Outline the main differences between classical analysis and intuitionistic analysis. Which is correct? (2016)

3. HILBERT’S FORMALISM

TOP

ESSAY QUESTION

Can Hilbert’s finitism be defended in the wake of the Incompleteness Theorems?

CORE READING

FURTHER READING

As ever, George and Velleman (2002) is well worth a look; Ch. 6 and 7 cover much the same ground as Bostock in the Core Reading, but in more depth. See also the excellent discussion in Potter (2000), and the historical overview of formalisms of all sorts in Detlefsen’s paper in Shapiro. Of the primary literature, perhaps the most useful thing to look at is Hilbert (1928). See also Part III of Mancosu, ed. (1998), including the useful introduction by Mancosu himself, placing Hilbert’s program within the context of debates about intuitionism. Blanchette (1996) discusses Hilbert’s early views, and his debate with Frege on the significance of consistency. Detlefsen (1986) attempts to defend Hilbert’s program in the face of Gödel’s results. Despite this, the program is generally regarded a failure—Giaquinto, in the Core Reading, is the best discussion I know of. But less ambitious programs remain. Simpson (2009) is an introduction to one of these: reverse mathematics, pioneered by Harvey Friedman.

PAST PAPER QUESTIONS

Can there be a consistent extension of Peano Arithmetic that proves its own inconsistency? If so, provide one explicitly and prove that it has this property, or argue that there can be no such theory. (Assume that Peano Arithmetic itself is consistent.) What are the philosophical implications of your answer with regard to Hilbert’s Programme? (2019)

EITHER
(a) Why did Hilbert classify the statement ‘x + 2 = 2 + x’ as finitary?
OR
(b) ‘If mathematics were merely the manipulation of meaningless symbols, its applicability would be inexplicable.’ Do you agree? (2018)

EITHER
(a) Are Hilbert’s philosophical ideas still relevant today?
OR
(b) How do we know that arithmetic is consistent? (2017)

Is there a well-motivated distinction between finitary and infinitary mathematics? (2016)

4. SET THEORY

TOP

ESSAY QUESTION

What justification do we have for ZFC?

CORE READING

FURTHER READING

In thinking more about this topic, start with debate concerning the iterative conception. How is it to be understood? Why think it’s correct? Which axioms of ZFC are justified on it? In particular, does it sanction Replacement or Choice? In approaching these questions, you could do worse than work through Incurvati (2020), a recent and accessible discussion of the iterative and other conceptions of set. The iterative conception itself goes back to Zermelo and von Neumann, but was first made explicit by Gödel; see his ‘What is Cantor’s Continuum Problem?’ in Benacerraf and Putnam. See the epilogue to Ferreirós (2007) for more on the history, and, for some other classic discussions, Gödel’s ‘Russell’s Mathematical Logic’ and the pieces by Parsons and Wang in Benacerraf and Putnam. Other, more recent discussions include Maddy (1988), Potter (1993), and Linnebo (2013). Further questions emerge in connection with the Continuum Hypothesis (CH). Owing to work by Gödel and Cohen, we know that both CH and ¬CH are consistent with ZFC. So how, if at all, is the question of whether or not CH is true to be settled? How might additional axioms that settle it be motivated? Should we think of there being such a thing as a determinate set-theoretic reality in which CH holds or doesn’t in the first place? Is there rather a plurality of set-theoretic universes, in some of which CH holds and in others of which it doesn’t? The issues here are technical, but rewarding. Ch. 12 of Linnebo (2017), listed in the Core Reading, provides a short introduction, though Ch. 8 of Hamkins (2021), conveying some of the richness and depth of the issues, is perhaps a better place to start. See also Maddy (2017), examining how set theory provides a foundation for mathematics.

PAST PAPER QUESTIONS

Explain how to interpret and derive the axioms of Peano Arithmetic in ZF. Does the derivation show that numbers are sets? (2019)

‘The fact that much of mathematics can be modelled in set theory is of epistemological but no ontological significance.’ Discuss. (2018)

What features would a theory need to have in order to replace set theory as a framework for contemporary mathematics? (2017)

EITHER
(a) How are the axioms of ZFC set theory justified?
OR
(b) Does V=L? (2016)

5. REALISM: BENACERRAF’S DILEMMA

TOP

ESSAY QUESTION

How should the mathematical realist respond to Benacerraf’s dilemma?

CORE READING

FURTHER READING

The background to Benacerraf’s dilemma is the realist position developed by Gödel. See his papers ‘Russell’s Mathematical Logic’ and (especially) ‘What is Cantor’s Continuum Problem?’ in Benacerraf and Putnam. For discussion and criticism of Gödel, see Chihara (1973) and (1990) (the latter targeting Maddy too). Useful discussions of the dilemma itself include Field (1988), Clarke-Doane (2016), and Nutting (2016). Maddy develops her response to the dilemma in her (1990) book. Balaguer (1998) offers a different response on behalf of the realist, that of the plenitudinous platonist. For Quine’s holistic realism, see, at least in the first instance, Resnik’s piece in Shapiro; beyond that, see the various suggestions given for EMPIRICISM.

PAST PAPER QUESTIONS

‘[I]t is a crime against the intellect to try to mask the problem of naturalizing the epistemology of mathematics with philosophical razzle-dazzle. Superficial worries about the intellectual hygiene of causal theories of knowledge are irrelevant to and misleading from this problem, for the problem is not so much about causality as about the very possibility of natural knowledge of abstract objects.’ (W.D. HART) Discuss. (2019)

When we see an object, do we also see the singleton of it? (2018)

EITHER
(a) How can we explain the fact that the mathematical claims arising from the work and thought of the mathematical community turn out, on the whole, to be true?
OR
(b) ‘The most plausible version of the causal theory of knowledge admits platonism, and the version most antagonistic to platonism is implausible.’ (STEINER) Discuss. (2018)

Is there a sound epistemological objection to Platonism? (2017)

6. NOMINALISM: INDISPENSABILITY

TOP

ESSAY QUESTION

How successful is Field’s response to the Putnam-Quine indispensability argument?

CORE READING

FURTHER READING

Field (1989) develops and defends his fictionalist response. It includes, among other things, his response to an interesting objection raised in Shapiro (1983), who argues that Field’s program faces problems analogous to those posed by Gödel for Hilbert’s program. More recent critical discussions of Field include Macbride (1999) and Burgess and Rosen (1997), which is essential reading for anyone interested in nominalism more generally—though you might want to start with their contribution to Shapiro, a short paper in which they set out what they take to be the central challenges to nominalism. See also Chihara’s contribution, which defends his own variety of nominalism from their objections. Shapiro (2000), in the Core Reading, discusses and provides references for some more recent nominalist developments. More recently still, you might look at Leng (2010). Other important criticisms of the Quine-Putnam indispensability argument include those of Maddy (1992) and Sober (1993).

PAST PAPER QUESTIONS

EITHER
(a) ‘Empiricism fails because no experience could rationally compel us to give up our most fundamental mathematical beliefs’. Discuss.
OR
(b) Is the existence of abstract mathematical objects indispensable to physics? (2019)

Could mathematics be a body of falsehoods? (2018)

Are nominalist reformulations of scientific theories superior to the originals? Does it matter? (2017)

‘Even if nominalism about mathematics is tenable, nominalism about logic is not.’ Discuss. (2016)

7. STRUCTURALISM

TOP

ESSAY QUESTION

Is there a tenable account of mathematics as the study of mathematical structures?

CORE READING

FURTHER READING

Structuralist ideas go back to Dedekind (1888), though they first emerged in English-speaking philosophy of mathematics in the 1960s with Benacerraf’s ‘What Numbers Could Not Be’ and Putnam’s ‘Mathematics without Foundations’. If you’re pursuing this topic in more depth, you’ll want to read all three. (Putnam’s paper, like Benacerraf’s, is in Benacerraf and Putnam.) Benacerraf and Putnam both attacked set-theoretic reductionism—something which, by now, you will no doubt have a good handle on, but see Ch. 3 of George and Velleman, listed as Further Reading for FREGE’S and RUSSELL’S LOGICISMS, for a refresher. The 1980s saw further developments, with Hellman, drawing on Putnam, developing a form of eliminative structuralism and Resnik and Shapiro, drawing more on Benacerraf, developing forms of non-eliminative structuralism. You’ll want to work through the differences between these; see, in addition to the pieces by Hellman and Shapiro in the Core Reading, Resnik (1997). Parsons (1990), who introduced the terminology of eliminative and non-eliminative structuralism, is another must-read from around this time, developing a form of non-eliminative structuralism owing more to mathematical practice. See also the set-theoretic structuralism of Reck and Price (2000). For discussion of various problems and challenges for structuralism, see also Reck (2003), Shapiro (2006), and Macbride’s contribution to Shapiro, and, for useful introductory discussion, Hamkins (2021).

PAST PAPER QUESTIONS

Explain Dedekind’s categoricity argument for the arithmetic of natural numbers and explain how it is relevant for the philosophy of structuralism. (2019)

In the slogan ‘mathematics is the study of structure’, is there any way of understanding ‘structure’ which is both epistemologically and mathematically acceptable? (2018)

EITHER
(a) ‘Analysis is about all the models that satisfy its axioms.’ Discuss.
OR
(b) Could the ordered pair ⟨a, b⟩ be the set {{a}, {a, b}}? (2017)

Does structuralism provide a way of understanding mathematics that does not require the existence of mathematical objects? (2016)

8. NEO-LOGICISM

TOP

ESSAY QUESTION

Does the derivation of Peano Arithmetic from Hume’s Principle have any philosophical significance?

CORE READING

FURTHER READING

There are various forms of neo-logicism, but the main approach to think about is the neo-Fregean approach of Bob Hale and Crispin Wright, which takes as its point of departure the fact that Peano Arithmetic can be derived from Hume’s Principle (HP) in the context of (standard) second-order logic. Hale and Wright developed their approach in various books and papers; besides the papers in their (2001) book, see their contribution to Shapiro and Wright’s (1983) book. In thinking about the approach more, there are various more specific issues to think about. One is the Julius Caesar problem. Neo-Fregeans suggest we can take HP as some sort of definition of the concept of number, but Frege himself thought that it could not be, as it doesn’t settle such questions as whether Julius Caesar is the number 0. Hale and Wright set out their official solution in ‘To Bury Caesar…’ in the (2001) book, but see also pp. 107-17 of Wright (1983), where Wright sets out a rather different (and on some views, better) solution. A different set of issues concern HP itself. It’s important to the neo-Fregean position that it is an analytic truth of some sort. But is it? Boolos (1997) is a classic discussion of some of the main problems here. One is the so-called bad company objection: what distinguishes HP from abstraction principles which are either, like Frege’s Basic Law V, inconsistent, or at least not consistent with HP? For discussion this and other issues concerning HP, see Rayo’s contribution to Shapiro, Burgess (2005), Hale and Wright (2009), Linnebo, ed. (2009), Studd (2016), and various papers in Ebert and Rossberg, eds. (2016).

PAST PAPER QUESTIONS

EITHER
(a) Explain Frege’s concept of number and whether it follows the philosophy of structuralism.
OR
(b) ‘Neo-logicists have failed to justify the conceptual necessity of the existence of infinitely many mathematical objects.’ Discuss. (2019)

EITHER
(a) How should logicists respond to the ‘Julius Caesar’ problem?
OR
(b) ‘Hume’s Principle is ontologically committing, and so cannot be analytic.’ Discuss. (2018)

EITHER
(a) Can neo-Fregeans account for ordinary mathematical knowledge?
OR
(b) ‘Hume’s Principle is analytic but not logically true.’ Discuss. (2017)

EITHER
(a) Can neo-logicism successfully account for other branches of mathematics than arithmetic?
OR
(b) What is the Bad Company objection to neo-logicism? Is it successful? (2016)

• Home

Reading Lists

• Early Modern Philosophy
• Knowledge and Reality
• Ethics
• Mind
• Logic and Language
• Mathematics
• Cognitive Science
• Philosophical Logic

Lectures

• Frege and Tarski, HT 2018
• Philosophy of Logic, HT 2017
• General Philosophy, HT 2016