PHIL 30067: Logicism and NeoLogicism
Practical matters
Lecturer: Dr Øystein
Linnebo,
Office: B41
in the basement of 11 Woodland Rd (next to the Philosophy Department)
Email: Oystein.Linnebo@bristol.ac.uk
Class: Lectures Tuesday 121pm in the Seminar Room in the basement of the Philosophy Department
Seminars Wednesday 1011am and 1112am (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 23pm 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 2^{nd} 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 3hour 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 onetoone correlated. Hume’s Principle is consistent. And a technical result known as Frege’s Theorem says that all of ordinary (Peano) arithmetic follows (in secondorder logic) from the Principle and some very natural definitions. This has motivated the socalled “neologicism” 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 secondorder 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. 5673
 G. Boolos, “Gottlob Frege and the Foundations of Arithmetic,” in his Logic, Logic, and Logic (Harvard UP, 1998), pp. 14354 [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, 2^{nd} 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, BEdition Introduction, sections I–IV
 Begriffsschrift, Preface, in Beaney
 Foundations, Introduction, §§14, 14, 8791, 105; (mostly)
in Beaney
Seminar: Frege’s epistemological views
 Philip
Kitcher, “Frege’s
Epistemology,” Philosophical Review 88
(1979), 23562
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, BEdition Introduction, section V; “The Discipline of
Pure Reason in Its Dogmatic Use” (Part II, Ch. 1, Section 1; esp.
A712/B740A724/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), 63350
Class: Frege’s logic in Begriffsschrift, secondorder logic, developing mathematics in secondorder 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. 26871 and 27781 [Master copy]
Seminar: Some views on secondorder logic
 W.V. Quine, Philosophy
of Logic 2^{nd} 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 DeNominalization,” Nous
35 (2001), pp. 7479
Week 4. Ascriptions
of number [handout]
Class: Frege’s analysis of ascriptions of number, cardinality quantifiers
 Foundations, §§2954 (ca. 28pp) (esp. §§4554); 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. 16390 (esp. pp. 163177)
 I.
Rumfitt, “Concepts
and Counting,” Proceedings of the
Aristotelian Society 101 (2001), pp. 4168
Week 5. Numbers as
objects [handout]
Class: Hume’s Principle, the Julius Caesar
Problem; “the linguistic turn”
 Foundations, §§5569 and 106109, in Beaney
 Richard Heck, “An Introduction to Frege’s Theorem,” Harvard Review of Philosophy 7 (1999), pp. 5673 (Sections 14)
 Dummett, Frege: Philosophy of Mathematics, ch. 10
(don’t worry about pp. 119124) [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]
Class: Cancelled because of strike.
 Shapiro, Thinking about Mathematics, pp. 115124 [Master copy]
 Russell, Introduction to Mathematical Philosophy (1919), pp. 119 and 181206 [Except for pp. 181193, 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. 3741, 4956, 58116 (66 pp of text)
Seminar:
 Frank Ramsey, “The Foundations of Mathematics,” in his The Foundations of Mathematical Logic and Other Logical Essays (Routledge & Kegan Paul, 1925); read pp. 2021, 249 (8 pp) [Master Copy]
 Priest, “The
Structure of the Paradoxes of SelfReference,” Mind 103(1994), pp. 2634
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)
Seminar:
 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. 5673
 Charles Parsons, “Intuition and Number,” (esp. Section III) [master copy]
Seminar:
 R. Heck, “Cardinality, Counting, and Equinumerosity,” Notre Dame Journal of Formal Logic 42.3 (2000), pp. 187209 (esp. Sections 35)
 O. Linnebo, “Frege’s Context Principle and Reference
to Natural Numbers,”
Section 3
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]
Seminar:
 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 §§6669, 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 IIIV, in his (ed.), Language, Thought, and Logic (Oxford UP, 1997)
Optional
 Oystein Linnebo, “To
Be Is to Be an F”, Dialectica 59 (2005), pp. 201222