PHIL 30067: Logicism and Neo-Logicism


Practical matters

Lecturer: Dr Øystein Linnebo,

Office: B41 in the basement of 11 Woodland Rd (next to the Philosophy Department)



Class: Lectures Tuesday 12-1pm in the Seminar Room in the basement of the Philosophy Department

Seminars Wednesday 10-11am and 11-12am (you will be assigned one of these times) in my office (see above) 

Attendance in lectures and seminars is mandatory. Students are expected to participate actively in the seminar.


Office hours: My office hours are Tuesday and Thursday 2-3pm in my office (see above).


Readings: Core and optional readings are listed below. You must have read the core readings before class; otherwise you won’t be able to follow the discussion. You should also expect to have to read the core readings several times.


Books: Students are encouraged to buy copies of the following books:

- Michael Beaney (ed.), The Frege Reader (Blackwell, 1997) [Available from Amazon for £20 by clicking here.]

- William Demopoulos (ed.), Frege’s Philosophy of Mathematics (Harvard UP, 1995) [Appears to be unavailable.]

Also useful are

- Stewart Shapiro, Thinking about Mathematics (Oxford UP, 2000) [Available from Amazon for £15 by clicking here or from Blackwells.]

- Paul Benacerraf and Hilary Putnam, Philosophy of Mathematics: Selected Readings 2nd ed. (Cambridge UP, 1983) [Available from Amazon for £31 by clicking here or (apparently faster) from Blackwells.]


Essay: An essay of 2500 to 3500 words must be handed in to the Department Office by (but preferably before) 4.00pm on 28 April. A list of suggested essay questions will be distributed in due course.


Seminar presentation: All students will be asked to make seminar presentations.


Exam: There will be a 3-hour exam in the exam period.


Course overview

Logicism is the view that pure mathematics is ultimately based on logic and definitions and thus is analytic a priori. More precisely, the view is that the basic concepts of pure mathematics can be defined in purely logical terms, and that, given these definitions, the axioms of pure mathematics are capable of purely logical proofs.


We begin by examining in the philosophical and logical work of the father of modern logicism, Gottlob Frege. Unfortunately, Frege’s logicism foundered on the rock of Russell’s paradox. But fortunately, recent work has shown how the bulk of Frege’s technical work can be salvaged. The trick is to start with Hume’s Principle, which says that the number of Fs is identical with the number of Gs iff the Fs and the Gs can be one-to-one correlated. Hume’s Principle is consistent. And a technical result known as Frege’s Theorem says that all of ordinary (Peano) arithmetic follows (in second-order logic) from the Principle and some very natural definitions. This has motivated the so-called “neo-logicism” of Bob Hale and Crispin Wright, which will be our concern in the second half of the course.


Central issues will include the following:

-         What is “pure logic”? In particular, what is the status of second-order logic?

-         How did Frege (and how should we) understand the distinctions between analytic/synthetic and a priori/a posteriori?

-         How should ascriptions of number (e.g. “There are five apples on the table”) be analyzed?

-         Is Frege right that the numbers are objects?

-         The philosophical status of Hume’s Principle

-         The “Caesar Problem”: If arithmetic is based on Hume’s Principle, how can the number 3 be distinguished from the Roman emperor Julius Caesar?

-         The “Bad Company Problem”: Although Hume’s Principle is consistent, some closely related principles are not.

-         How faithful is Hume’s Principle to our ordinary understanding of arithmetical vocabulary?



Introductory readings

For some excellent introductions to our topic, see

-         R. Heck, “An Introduction to Frege’s Theorem,” Harvard Review of Philosophy 7 (1999), pp. 56-73

-         G. Boolos, “Gottlob Frege and the Foundations of Arithmetic,” in his Logic, Logic, and Logic (Harvard UP, 1998), pp. 143-54 [Master copy]

-         S. Shapiro, Thinking about Mathematics (Oxford UP, 2000), ch. 5 [Master copy]


Work by Frege

-         Begriffsschrift, transl. and repr. in J. van Heijenoort (ed.), From Frege to Gödel (Harvard University Press, 1967)

-         Foundations of Arithmetic, transl. by J.L. Austin (Blackwell, 1953); partially repr. in Beaney (ed.), The Frege Reader (Blackwell, 1997) and in P. Benacerraf and H. Putnam (eds.), Philosophy of Mathematics: Selected Readings, 2nd ed. (Cambridge UP, 1983)

-         Basic Laws of Arithmetic, transl. by M. Furth (University of California Press, 1964); partially repr. in Beaney


Other monographs and anthologies

-         G. Boolos, Logic, Logic, and Logic (Harvard UP, 1998) (esp. essays 9, 10, 12, 19)

-     W. Demopoulos (ed.), Frege’s Philosophy of Mathematics (Harvard UP, 1995) (esp. essays 1, 2, 6, 7, 8)

-     M. Dummett, Frege: Philosophy of Mathematics (Harvard UP, 1991)

-     B. Hale and C. Wright, Reason’s Proper Study (Oxford UP, 2001) (esp. essays 4, 5, 12, 14)


Week 1. What is logicism? [handout]

Class: Overview. What is logicism? What might its philosophical payoff be?

- Kant’s Critique of Pure Reason, B-Edition Introduction, sections I–IV

- Begriffsschrift, Preface, in Beaney

- Foundations, Introduction, §§1-4, 14, 87-91, 105; (mostly) in Beaney


Seminar: Frege’s epistemological views

- Philip Kitcher, “Frege’s Epistemology,” Philosophical Review 88 (1979), 235-62


Week 2. The “arithmetization” of analysis [handout]

Class: Kant’s philosophy of mathematics; formalization; elimination of appeals to intuition

- Kant’s Critique of Pure Reason, B-Edition Introduction, section V; “The Discipline of Pure Reason in Its Dogmatic Use” (Part II, Ch. 1, Section 1; esp. A712/B740-A724/B752)

- Shapiro, Thinking about Mathematics, Ch. 4, Sections 1 and 2 [Master copy]

- Alberto Coffa, “Kant, Bolzano, and the Emergence of Logicism,” in Demopoulos


Seminar: Frege’s rationalism

- Paul Benacerraf, “Frege: the Last Logicist,” in Demopoulos [Master copy]

- Tyler Burge, “Frege on Knowing the Third Realm,” Mind 101 (1992), 633-50


Week 3. What is “pure logic”? [handout]

Class: Frege’s logic in Begriffsschrift, second-order logic, developing mathematics in second-order logic

- (most important) George Boolos, “Reading the Begriffsschrift,” in his Logic, Logic, and Logic and in Demopoulos

- Frege, Begriffsschrift, ch. 1, in Beaney

- Enderton, A Mathematical Introduction to Logic, ch. 4, especially pp. 268-71 and 277-81 [Master copy]


Seminar: Some views on second-order logic

- W.V. Quine, Philosophy of Logic 2nd ed. (Harvard UP, 1986), ch. 5 [Master copy]

- G. Boolos, “To Be Is to Be a Value of a Variable (Or to Be Some Values of Some Variables),” in his Logic, Logic, and Logic

- (optional) A. Rayo and S. Yablo, “Nominalism through De-Nominalization,” Nous 35 (2001), pp. 74-79


Week 4. Ascriptions of number [handout]

Class: Frege’s analysis of ascriptions of number, cardinality quantifiers

- Foundations, §§29-54 (ca. 28pp) (esp. §§45-54); mostly in Beaney

- Dummett, Frege: Philosophy of Mathematics, ch. 8 [Master copy]


Seminar: Can numbers be ascribed to other things than concepts?

- B. Yi, “Is Two a Property?Journal of Philosophy 96 (1999), pp. 163-90 (esp. pp. 163-177)

- I. Rumfitt, “Concepts and Counting,” Proceedings of the Aristotelian Society 101 (2001), pp. 41-68


Week 5. Numbers as objects [handout]

Class: Hume’s Principle, the Julius Caesar Problem; “the linguistic turn”

- Foundations, §§55-69 and 106-109, in Beaney

- Richard Heck, “An Introduction to Frege’s Theorem,” Harvard Review of Philosophy 7 (1999), pp. 56-73 (Sections 1-4)

- Dummett, Frege: Philosophy of Mathematics, ch. 10 (don’t worry about pp. 119-124) [Master copy]


Seminar: Crispin Wright on numbers as objects

- Crispin Wright, Frege’s Conception of Numbers as Objects, Sections i, ii, iii, vii, viii [Master copy]


Week 6. Russell’s Logicism

Class: Cancelled because of strike.



- Shapiro, Thinking about Mathematics, pp. 115-124 [Master copy]

- Russell, Introduction to Mathematical Philosophy (1919), pp. 1-19 and 181-206 [Except for pp. 181-193, these pages are reprinted in Benacerraf & Putnam. However, these missing pages are very important; so please use the master copy in the Dept Office.]



Week 7. The Paradoxes and Responses to Them [handout]

- Marcus Giaquinto, The Search for Certainty, pp. 37-41, 49-56, 58-116 (66 pp of text)



- Frank Ramsey, “The Foundations of Mathematics,” in his The Foundations of Mathematical Logic and Other Logical Essays (Routledge & Kegan Paul, 1925); read pp. 20-21, 24-9 (8 pp) [Master Copy]

- Priest, “The Structure of the Paradoxes of Self-Reference,” Mind 103(1994), pp. 26-34


Week 8. The logicism of the logical positivists [handout]


- Ayer, “The A Priori,” in B&P

- Quine, “Two Dogmas of Empiricism”, in his From a Logical Point of View (Harvard UP, 1953)



- Carnap, “Empiricism, Semantics, and Ontology,” in B&P

- Quine, “Carnap and Logical Truth,” in B&P




Week 9. Frege’s Theorem, ordinals vs cardinals [handout]

Class: Frege Arithmetic vs. ordinary arithmetic

- Richard Heck, “An Introduction to Frege’s Theorem,” Harvard Review of Philosophy 7 (1999), pp. 56-73

- Charles Parsons, “Intuition and Number,” (esp. Section III) [master copy]



- R. Heck, “Cardinality, Counting, and Equinumerosity,” Notre Dame Journal of Formal Logic 42.3 (2000), pp. 187-209 (esp. Sections 3-5)

- O. Linnebo, “Frege’s Context Principle and Reference to Natural Numbers,” Section 3


ESSAY DUE 28 APRIL 4PM [Suggested essay topics]


Week 10. Implicit definitions [handout]

- Hartry Field, “Platonism for Cheap?,” Canadian Journal of Philosophy 14 (1984) [master copy]

- Bob Hale and Crispin Wright, “Implicit Definitions and the Apriori,” [master copy]



- John MacFarlane, “Double Vision


Week 11. Is Hume’s Principle analytic? [handout]

- George Boolos, “Is Hume’s Principle Analytic?” [master copy]

- Crispin Wright, “Is Hume’s Principle Analytic?” (you may skip the Appendix)



- Oystein Linnebo, “Frege’s Context Principle and Reference to Natural Numbers,” forthcoming in Logicism, Intuitionism, and Formalism—What has become of them?, eds. S. Lindström et al., Springer


Week 12. The Julius Caesar Problem [handout]

- Foundations §§66-69, in Beaney

- B. Hale and C. Wright, “Caesar Interred,” forthcoming in Identity and Modality: New Essays on Philosophy of Mathematics and Metaphysics edited by F. MacBride (OUP) [Master copy]

- Richard Heck, “The Julius Caesar Objection,” esp. Sections III-V, in his (ed.), Language, Thought, and Logic (Oxford UP, 1997)



- Oystein Linnebo, “To Be Is to Be an F”, Dialectica 59 (2005), pp. 201-222