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Frege's Logicism and the Neologicists

Practical information

Wednesdays, from 12pm to 2pm
Practice session: Wednesdays, from 2pm to 4pm, every other week
Office hours by appointment, in room 234, Ludwigstrasse 31, second floor
Readings for next week are marked in red on the content list.


Course description

    Logicism is the foundational doctrine in philosophy of mathematics that originated in Frege, according to which mathematics can be reduced to logic. Frege proved that a good deal of mathematics could be reduced to a logical system of his. This system turn out to be inconsistent, as Russell’s paradox shows. Attempts to repair Frege’s logical system and the logicist project were abandoned due to Gödel’s Incompleteness Theorems and the raise of set theory. Towards the end of the XX century the core ideas of logicism were revived by the so-called neologicists. The first half of the course will be focused on the original logicist doctrine, Frege’s results, Russell’s paradox, and Principia Mathematica. In the second half the philosophical and formal aspects of the neologicist project will be introduced, and its achievements and the difficulties it faces will be assessed. 


Preparing for the course

Contents (work in progress)

1. An Introduction to Frege's Logicism
        The epistemic problem of mathematics, the analytic/synthetic distinction, 
  • Kant, Critique of Pure Reason, Introduction, and Part I of the Transcendental Doctrine of Elements, second edition.
  • Frege, Foundations of Arithmetic, Introduction and §§1-69, especially §§3, 5-7, 13, 14, 21, 22, 24, 26, 28, 29. 39, 45, 46, 48, 51, and 53-69. 
  • Cook, "New Waves on an Old Beach: Fregean Philosophy of Mathematics Today", §§1-2.

2. Frege's Logic
        Second-order logic, Frege's Logic, Hume's Principle, Frege's Theorem, Russell's Paradox
    • Boolos, "Gottlob Frege and the Foundations of Arithmetic".
    
      Second-Order Logic
    • Shapiro, Foundations without Foundationalism: A Case for Second-order Logic, chap. 3 and 4. 

3. Neologicism
        Abstractionism and its difficulties: the Julius Caesar problem, the ontological problem, the problem of analyticity, the Bad Company problem, the impredicativity problem

    Abstractionism
    • Ebert & Rossberg, "Introduction to Abstractionism"

    The Julius Caesar Problem
    • Hale & Wright, "To Bury Caesar...".
    
    The Ontological Problem
    • Heck, "The Existence (and Non-existence) of Abstract Objects". 
    • MacBride, "Neo-Fregean Meta-Ontology: Just don't Ask Too Many Questions".

    • Moltmann, F. (2017) "The Number of Planets, a Number-Referring Term?".

    The Problem of Analyticity
    • Wright, "Is Hume's Principle Analytic?".
    • Boolos, "Is Hume's Principle Analytic?".
    • Ebert, “A Framework for Implicit Definitions and the A Priori”.
    • Wright, "Abstraction and Epistemic Entitlement: On the Epistemological Status of Hume’s Principle".

    The Bad Company Problem
    • Cook, “Conservativeness, Cardinality, and Bad Company”.

    The Impredicativity Problem
    • Linnebo, "Impredicativity in the Neo-Fregean Program".


Bibliography

  • Benacerraf, P. & Putnam, H. (eds.) Philosophy of Mathematics: Selected Readings, CUP, second edition, 1984.
  • Boolos, G. (1987) “The Consistency of Frege's Foundations of Arithmetic”, in Thompson (ed.), On Being and Saying: Essays in Honor of Richard Cartwright, MIT Press, pp. 3-20.
  • --- (1990) “The Standard of Equality of Numbers;” reprinted in Burgess and Jeffrey (eds.), Logic, Logic, and Logic, Harvard UP, pp. 202-219.
  • --- (1997) "Is Hume's Principle Analytic?", in Burgess and Jeffrey (eds.), Logic, Logic, and LogicHarvard UP, 1998, pp. 301-314.
  • --- (1998) "Gottlob Frege and the Foundations of Arithmetic", in Burgess and Jeffrey (eds.), Logic, Logic, and Logic, Harvard UP, pp. 143-154.
  • Bueno, O. & Linnebo, Ø. (eds.), New Waves in Philosophy of Mathematics, Palgrave Macmillan, 2009.
  • Burgess, J. P. (2003) Fixing Frege, Princeton UP.
  • Cook, R. (ed.) The Arché Papers on the Mathematics of Abstraction, Springer, 2007. 
  • --- (2009) "New Waves on an Old Beach: Fregean Philosophy of Mathematics Today," in Bueno & Linnebo (eds.), New Waves in Philosophy of Mathematics, Palgrave Macmillan, pp 13-34.
  • --- (2017) “Conservativeness, Cardinality, and Bad Company”, in Ebert & Rossberg (eds.), Abstractionism. Essays in Philosophy of Mathematics, OUP, pp. 223-246.

  • Dedekind, R. (1888) Was sind und was sollen die Zahlen?, Vieweg.
  • Dummett, M. (1991) Frege: Philosophy of Mathematics, Duckworth.
  • Ebert, “A Framework for Implicit Definitions and the A Priori,” in Ebert & Rossberg (eds.), Abstractionism. Essays in Philosophy of Mathematics, OUP, pp. 133-160.
  • Ebert, P. & Rossberg, M. (2007) "What is the Purpose of Neo-Logicism?", Traveaux de Logique 18: 33-61.
  • --- (ed.) Abstractionism. Essays in Philosophy of Mathematics, OUP, 2017.
  • --- (2017) "Introduction to Abstractionism", in Ebert & Rossberg (eds.), Abstractionism. Essays in Philosophy of Mathematics, OUP, pp. 3-33.
  • Frege, G. (1879) Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Louis Nebert; translation by Bauer Mengelberg as Concept Notation: A formula language of pure thought, modelled upon that of arithmetic, in van Heijenoort (ed.), From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879-1931, Harvard UP.
  • --- (1884) Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl, w. Koebner; translated by Austin as The Foundations of Arithmetic: A Logic-Mathematical Enquiry into the Concept of Number, Blackwell, second revised edition, 1974.
  • --- (1893/1903) Grundgesetze der Arithmetik, Band I/II, Verlag Herman Pohle; translation by Ebert & Rossberg (with Wright) as Basic Laws of Arithmetic: Derived using concept-script, OUP, 2013.
  • Hale, B. & Wright, C. (2001) "To Bury Caesar..." in Hale & Wright, The Reason’s Proper StudyOUP, 2001. 
  • Heck, Jr., R. (1993) “The Development of Arithmetic in Frege’s Grundgesetze der Arithmetik”, Journal of Symbolic Logic 58: 579-600.
  • --- (1997) "The Julius Caesar Problem," in his (ed.), Language, Thought, and Logic, OUP.
  • --- (1999) "Frege’s theorem, an introduction", Harvard Review of Philosophy 7, pp. 56-73.
  • --- (2012) Reading Frege’s Grundgesetze, Clarendon Press.
  • --- (2011) Frege’s Theorem, Clarendon Press.
  • --- (2017) "The Existence (and Non-existence) of Abstract Objects", in Ebert & Rossberg (eds.), Abstractionism. Essays in Philosophy of Mathematics, OUP, pp. 50-78.
  • Kant, I., Critique of Pure Reason; translation by Guyer & Wood, CUP, 1999.
  • Linnebo, Ø. (2017) "Impredicativity in the Neo-Fregean Program", in Ebert & Rossberg (eds.), Abstractionism. Essays in Philosophy of Mathematics, OUP, pp. 247-268.
  • MacBride, F. (2003) “Speaking with Shadows: A Study of Neo-Logicism”, British Journal for the Philosophy of Science 54: 103-163.
  • --- (2017) "Neo-Fregean Meta-Ontology: Just don't Ask Too Many Questions", in Ebert & Rossberg (eds.), Abstractionism. Essays in Philosophy of Mathematics, OUP, pp. 94-112.
  • MacFarlane, J. (2002) "Frege, Kant, and the Logic in Logicism", Philosophical Review 3:25-65.
  • Moltmann, F. (2017) "The Number of Planets, a Number-Referring Term?," in Ebert & Rossberg (eds.), Abstractionism. Essays in Philosophy of Mathematics, OUP, pp. 113-129.
  • Russell, B., Letter to Frege and the latter’s reply; translation in van Heijenoort (ed.), From Frege to Gödel, a source book in Mathematical Logic 1879-1931, Harvard UP, pp. 124-8.
  • Shapiro, S. (1991) Foundations without Foundationalism: A Case for Second-order Logic, Clarendon Press.
  • Tennant, N. (1987) Anti-Realism and Logic: Truth as Eternal, Clarendon Library of Logic and Philosophy, OUP.
  • --- "Logicism and Neologicism", The Stanford Encyclopedia of Philosophy (Fall 2014 Edition), Zalta (ed.), URL = <https://plato.stanford.edu/archives/fall2014/entries/logicism/>.
  • Wright, C. (1983) Frege's Conception of Numbers as Objects, Aberdeen UP.
  • --- (1999) "Is Hume's Principle Analytic?", Notre Dame Journal of Formal Logic 40: 6-30.
  • --- "On the philosophical significance of Frege’s theorem", in Hale & Wright, The Reason’s Proper Study, OUP, 2001.

  • --- (2017) "Abstraction and Epistemic Entitlement: On the Epistemological Status of Hume’s Principle," in Ebert & Rossberg (eds.), Abstractionism. Essays in Philosophy of Mathematics, OUP, pp. 161-187.
  • Zalta, E. N., "Frege's Theorem and Foundations for Arithmetic", The Stanford Encyclopedia of Philosophy (Summer 2017 Edition), Zalta (ed.), URL = <https://plato.stanford.edu/archives/sum2017/entries/frege-theorem/>.
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