The Dark Heritage of Logicism
DOI:
https://doi.org/10.48160/18532330me10.208Keywords:
foundations of mathematics, mathematical logic, philosophy of logic, model theory, second-order logic, history of logic and mathematicsAbstract
Logicism finds a prominent place in textbooks as one of the main alternatives in the foundations of mathematics, even though it lost much of its attraction from about 1950. Of course the neologicist trend has revitalized the movement on the basis of Hume’s Principle and Frege’s Theorem, but even so neologicism restricts itself to arithmetic and does not aim to account for all of mathematics. The present contribution does not focus on the classical logicism of Frege and Dedekind, nor on the Russell-Carnap period, and also not on recent neologicism; its aim is to call attention to some forms of heritage from logicism that normally go quite unnoticed. In the 1920s, 1930s and 1940s, the logicist thesis became a stimulus for some deep innovations in the field of mathematical logic. One can argue, in particular, that two key ideas linked with formal semantics had their origins in the conception of logic associated with the logicist trend – the expansion of metamathematics brought about by Tarski, opening the way to model theory, and the insistence on the “full” set-theoretic semantics as “standard” for second-order logic. The paper proposes an analysis of those inheritances and argues that that logical theory ought to avoid some of their implications.
References
Boolos, G. (1986), “Saving Frege From Contradiction”, Proceedings of the Aristotelian Society87: 137-151.
Boolos, G. (1995), “Frege's Theorem and the Peano Postulates”, Bulletin of Symbolic Logic 1: 317-326.
Chang, C. C. y H. J. Keisler (1973), Model Theory, Amsterdam: North-Holland.
Church, A. (1956), Introduction to Mathematical Logic, Princeton, NJ: Princeton University Press.
Dedekind, R. (1888),¿Quéson y para qué sirven los números?, Madrid:Alianza, 1997.Enderton, H.B.(2015), “Second-order and Higher-order Logic”, enZalta,E. N. (ed.), The Stanford Encyclopedia of Philosophy (Fall 2015 Edition), https://plato.stanford.edu/archives/fall2015/entries/logic-higher-order/.
Feferman, A. y S. Feferman (2004), Alfred Tarski: Life and Logic, Cambridge: Cambridge University Press.
Ferreirós, J. (1997), “Notes on Types, Sets, and Logicism, 1930-1950”, Theoria12: 91-124.
Ferreirós, J.(2001), “The Road to Modern Logic –An interpretation”, The Bulletin of Symbolic Logic7: 441-484.
Ferreirós, J. (2009), “Hilbert, Logicism, and Mathematical Existence”, Synthese170(1): 33-70.
Ferreirós, J. (2011), “On Arbitrary Sets and ZFC”, The Bulletin of Symbolic Logic17(3): 361-393.
Ferreirós, J. (2015), “Far from Modelisation: The Emergence of Model Theory”, Oberwolfach Reports12: 2851-2855.
Frege, G. (1884), Die Grundlagen der Arithmetik, Breslau: Max und Hermann Marcus. (Versión castellana de C. Ulises Moulines: “Los fundamentos de la aritmética”, en Frege, G., Escritos filosóficos, Barcelona: Crítica, 1996, pp. 31–144.)
Frege, G. ([1893] 1962), Grundgesetze der Arithmetik,Hildesheim: Georg Olms. (Versión castellana de C. Ulises Moulines de la “Introducción a Las leyes fundamentales de la aritmética” en: Frege, G., Estudios sobre semántica, Barcelona: Ariel, 1973, pp. 157-162; reimpreso en: Frege, G., Escritos filosóficos, Barcelona: Crítica, 1996, pp. 248-252.)
Giaquinto, M. (2002), The Search for Certainty: A Philosophical Account of Foundations of Mathematics,Oxford: Oxford University Press.
Hilbert, D. (1899), Fundamentos de Geometría, Leipzig: Teubner. (Versión castellana de la7aed.: Fundamentos de Geometría, Madrid: CSIC, 1991. Ver en cambio la edition critique: Les fondements de la Géométrie, Paris: Dunod, 1971.)
Hilbert, D. (1918), “Axiomatisches Denken”, Mathematische Annalen78: 405-415. (Versión castellana: “Pensamiento axiomático”, en Hilbert, D., Fundamentos de las matemáticas, México: UNAM, 1993, pp. 23-35.)
Hilbert, D. y W. Ackermann (1928), Grundzüge der theoretischen Logik, Berlin: J. Springer. (Versión castellana de una edición posterior, con grandes cambios: Elementos de lógica teórica, Madrid: Tecnos, 1962.)
Kreisel, G. (1967), “Informal Rigour and Completeness Proofs”, en Lakatos, I. (ed.), Problems in the Philosophy of Mathematics, Amsterdam: North-Holland, pp. 138-171.
Quine, W. V. (1970), Philosophy of Logic, Cambridge, MA: Harvard University Press.
Shapiro, S. (1991), Foundations without Foundationalism: A Case for Second-Order Logic, Oxford: Oxford UniversityPress.
Tarski, A. ([1933] 1935), “Der Wahrheitsbegriff in den formalisierten Sprachen”, Studia Philosophica1 (1935): 261-405. (Original polaco: Pojęcie prawdy w językach nauk dedukcyjnych, Warszowa: Towarzystwo Naukowe Warszawskie 1933. Referencias a la versión inglesa de Joseph Henry Woodger: “The Concept of Truth in Formalized Languages”,en Tarski, A., Logic, Semantics, Metamathematics, Oxford: Oxford University Press, 1956, pp. 152-278.)
Tarski, A. (1940), Introduction to Logicand the Methodology of Deductive Sciences, Cambridge, MA: Harvard University Press. (Versión castellana: Introducción a la lógica y a la metodología de las ciencias deductivas, Madrid: Espasa-Calpe, 1968.)
Tennant, N. (2014), “Logicism and Neologicism”, en Zalta, E. N. (ed.), The Stanford Encyclopedia of Philosophy (Fall 2014 Edition), http://plato.stanford.edu/archives/fall2014/entries/logicism/.
Whitehead, A. N. y B. Russell (1910-1913), Principia Mathematica, Cambridge: Cambridge University Press, 2aed.1925-1927, reimpresión 1978.
Zermelo, E. (1930), “Über Grenzzahlen und Mengenbereiche”, Fundamenta Mathematicae16: 29-47.
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