El Tractatus al rescate de Principia Mathematica: Ramsey y los fundamentos logicistas de las matemáticas

• Emilio Méndez Pinto ^[1]
  1. [1] Universidad Nacional Autónoma de México

     Universidad Nacional Autónoma de México

     México

     
• Localización: Crítica: revista hispanoamericana de filosofía, ISSN 0011-1503, Vol. 54, Nº. 161, 2022, págs. 43-69
• Idioma: español
• DOI: 10.22201/iifs.18704905e.2022.1346
• Enlaces
  • Texto completo
• Resumen
  • español

    Mi objetivo es discutir las principales dificultades que Frank P. Ramsey encontró en Principia Mathematica y la solución que, vía el Tractatus Logico-Philosophicus, propuso al respecto. Sostengo que las principales dificultades que Ramsey encontró en Principia Mathematica están, todas, relacionadas con que Russell y Whitehead desatendieron la forma lógica de las proposiciones matemáticas, las cuales, según Ramsey, deben ser tautológicas.

  • English

    My aim is to expose the main difficulties that Frank P. Ramsey encountered in Principia Mathematica and the solution that, via the Tractatus Logico- Philosophicus , Ramsey proposed in this regard. I contend that the main difficulties that Ramsey encountered in Principia Mathematica were all related to Russell and Whitehead neglecting the logical form of mathematical propositions, which according to Ramsey must be tautological.

• Referencias bibliográficas
  • Anscombe, E., 1989, “The Simplicity of the Tractatus”, Crítica, vol. 21, no. 63, pp. 3–16.
  • Anscombe, E., 1963, An Introduction to Wittgenstein’s Tractatus, St. Augustine’s Press, Indiana.
  • Barceló, A. (manuscrito), Fundacionismo y Anti-Fundacionismo en Filosofía de las Matemáticas. http://www.filosoficas.unam.mx/docs/37/files/Fundacionismo%20y%20Anti-Fundacionismo.pdf)
  • Barceló, A. (manuscrito), Wittgenstein contra la Metamatemática. (http://www.filosoficas.unam.mx/docs/37/files/BarceloWittgensteinMetamatematicas.pdf.)
  • Black, M., 1964, A Companion to Wittgenstein’s Tractatus, Cambridge University Press, Cambridge.
  • Boole, G., 1847, The Mathematical Analysis of Logic, George Bell and Sons, Londres.
  • Bradley, R., 1987, “Tractatus 2.022-2.023”, Canadian Journal of Philosophy, vol. 17, no. 2, pp. 349–359.
  • Burgess, J., 2018, “Putnam on Foundations: Models, Modals, Muddles”, en G. Hellman y R. Cook (eds.), Hilary Putnam on Logic and Mathematics,...
  • Carnap, R., 1963, “Intellectual Autobiography”, en P. Schilpp (ed.), The Philosophy of Rudolf Carnap, Open Court, Illinois, pp. 3–84.
  • Carnap, R., 1937, The Logical Syntax of Language, Kegan Paul, Londres.
  • Carnap, R., 1931, “Die logizistische Grundlegung der Mathematik”, Erkenntnis, vol. 2, no. 1, pp. 91–105.
  • Chihara, C., 1980, “Ramsey’s Theory of Types: Suggestions for a Return to Fregean Sources”, en H. Mellor (ed.), Prospects for Pragmatism:...
  • Church, A., 1996, Introduction to Mathematical Logic, Princeton University Press, Nueva Jersey.
  • Clark, P. y W. Demopoulos, 2005, “The Logicism of Frege, Dedekind, and Russell”, en S. Shapiro (ed.), The Oxford Handbook of Philosophy of...
  • Colyvan, M., 2009, “Applying Inconsistent Mathematics”, en O. Bueno y Ø. Linnebo (eds.), New Waves in Philosophy of Mathematics, Palgrave...
  • Consuegra, F., 1991, “El logicismo russelliano”, Crítica, vol. 23, no. 67, pp. 15–39.
  • Curry, H., 1977, Foundations of Mathematical Logic, Dover, Nueva York.
  • Curry, H., 1951, Outlines of a Formalist Philosophy of Mathematics, North-Holland Publishing Company, Ámsterdam.
  • Detlefsen, M., 2005, “Formalism”, en S. Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic, Oxford University Press,...
  • Dummett, M., 1975, “The Philosophical Basis of Intuitionistic Logic”, Studies in Logic and the Foundations of Mathematics, vol. 80, pp. 5–40.
  • Earman, J. y J. Norton, 1996, “Infinite Pains: The Trouble with Supertasks”, en A. Morton y S. Stich (eds.), Benacerraf and his Critics, Blackwell,...
  • Feferman, S., 2005, “Predicativity”, en S. Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic, Oxford University Press,...
  • Floyd, J., 2005, “Wittgenstein on Philosophy of Logic and Mathematics”, en S. Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics...
  • Frank Plumpton Ramsey Papers, 1920–1930, ASP.1983.01, Archives of Scientific Philosophy, Archives and Special Collections, University of Pittsburgh...
  • Frege, G., 1980, The Foundations of Arithmetic, Northwestern University Press, Evanston.
  • Gandy, R., 1977, “The Simple Theory of Types”, en R. Gandy y J. Hyland, Logic Colloquium 76: Studies in Logic and the Foundations of Mathematics,...
  • George, A. y D. Velleman, 2002, Philosophies of Mathematics, Blackwell, Malden.
  • Gödel, K., 1998, “Russell’s Mathematical Logic”, en P. Benacerraf y H. Putnam (eds.), Philosophy of Mathematics: Selected Readings, Cambridge...
  • Gödel, K., 1995, “The Present Situation in the Foundations of Mathematics”, en S. Feferman (ed.), Kurt Gödel: Collected Works. Volume III:...
  • Goldfarb, W., 2010, “Frege’s Conception of Logic”, en T. Ricketts y M. Potter (eds.), The Cambridge Companion to Frege, Cambridge University...
  • Goldfarb, W., 2002, “Wittgenstein’s Understanding of Frege: The Pre-Tractarian Evidence”, en E. Reck (ed.), From Frege to Wittgenstein: Perspectives...
  • Hahn, H., 1980, Empiricism, Logic, and Mathematics: Philosophical Papers, Reidel, Dordrecht.
  • Hahn, H., O. Neurath y R. Carnap, 1973, “Wissenschaftliche Weltauffassung: Der Wiener Kreis”, en R. Cohen y M. Neurath (eds.), Otto Neurath:...
  • Hamkins, J., 2020, Lectures on the Philosophy of Mathematics, The MIT Press, Cambridge.
  • Hintikka, J., 1998, “El lugar de C.S. Peirce en la historia de la teoría lógica”, en J. Hintikka, El viaje filosófico más largo: De Aristóteles...
  • Jeffrey, R., 1996, “Logicism 2000: A Mini-manifesto”, en A. Morton y S. Stich (eds.), Benacerraf and his Critics, Blackwell, Cornwall, pp....
  • Jourdain, P., 1915, “Notes”, en G. Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, Open Court, Chicago, pp. 202–208.
  • Katz, J., 1996, “Skepticism about Numbers and Indeterminacy Arguments”, en A. Morton y S. Stich (eds.), Benacerraf and his Critics, Blackwell,...
  • Kitcher, P., 1984, The Nature of Mathematical Knowledge, Oxford University Press, Nueva York.
  • Kleene, S., 2009, Introduction to Meta-Mathematics, Ishi Press, Nueva York.
  • Klement, K., 2019, “Russell’s Logicism”, en W. Russell (ed.), The Bloomsbury Companion to Bertrand Russell, Bloomsbury, Londres, pp. 151–178.
  • Köhler, E., 2006, “Ramsey and the Vienna Circle on Logicism”, en M. Galavotti (ed.), Cambridge and Vienna: Frank P. Ramsey and the Vienna...
  • Kreisel, G., 1972, “Hilary Putnam. Mathematics without Foundations, The Journal of Philosophy, vol. 64, pp. 5–22”, Journal of Symbolic Logic,...
  • Kremer, M., 2015, “The Whole Meaning of a Book of Nonsense: Reading Wittgenstein’s Tractatus”, en M. Beaney (ed.), The Oxford Handbook of...
  • Kripke, S., 2006, Wittgenstein a propósito de reglas y lenguaje privado, Tecnos, Madrid.
  • Lakatos, I., 2015, Proofs and Refutations: The Logic of Mathematical Discovery, Cambridge University Press, Cambridge.
  • Linnebo, Ø., 2017, Philosophy of Mathematics, Princeton University Press, Nueva Jersey.
  • Linsky, B., 2011, The Evolution of Principia Mathematica: Bertrand Russell’s Manuscripts and Notes for the Second Edition, Cambridge University...
  • MacBride, F. et al., 2019, “Frank Ramsey”, en E. Zalta (ed.), Stanford Encyclopedia of Philosophy. (https://plato.stanford.edu/entries/ramsey/)
  • MacFarlane, J., 2002, “Frege, Kant, and the Logic in Logicism”, The Philosophical Review, vol. 111, no. 1, pp. 25–65.
  • Maddy, P., 2013, Defending the Axioms: On the Philosophical Foundations of Set Theory, Oxford University Press, Oxford.
  • Mancosu, P., 2021, “Three Letters on the Foundations of Mathematics by Frank Plumpton Ramsey”, Philosophia Mathematica, vol. 29, no. 1, pp....
  • Marion, M., 2012, “Wittgenstein, Ramsey and British Pragmatism”, European Journal of Pragmatism and American Philosophy, vol. IV, no. 2, pp....
  • Marion, M., 2008, Wittgenstein, Finitism, and the Foundations of Mathematics, Oxford University Press, Oxford.
  • Marion, M., 1995, “Wittgenstein and Ramsey on Identity”, en J. Hintikka (ed.), From Dedekind to Gödel: Essays on the Development of the Foundations...
  • Mays, W., 1967, “Recollections of Wittgenstein”, en K. Fann (ed.), Ludwig Wittgenstein: The Man and his Philosophy, Humanities Press, Nueva...
  • McCarty, D., 1993, “On the Failure of Mathematics’ Philosophy”, Synthese, vol. 96, no. 2, pp. 255–291.
  • McGuinness, B., 2006, “Wittgenstein and Ramsey”, en M. Galavotti (ed.), Cambridge and Vienna: Frank P. Ramsey and the Vienna Circle, Springer,...
  • McGuinness, B. y G.H. von Wright, 1995 (eds.), Ludwig Wittgenstein: Cambridge Letters, Wiley-Blackwell, Nueva Jersey.
  • Methven, S., 2015, Frank Ramsey and the Realistic Spirit, Palgrave, Londres.
  • Misak, C., 2020, Frank Ramsey: A Sheer Excess of Powers, Oxford University Press, Oxford.
  • Misak, C., 2019, “Ramsey, Pragmatism, and the Vienna Circle”, European Journal of Pragmatism and American Philosophy, vol. XI, no. 1, pp....
  • Mosterín, J. y R. Torretti, 2010, Diccionario de lógica y filosofía de la ciencia, Alianza, Madrid.
  • Nakano, A., 2021, “On Ramsey’s Reason to Amend Principia Mathematica’s Logicism and Wittgenstein’s Reaction”, Synthese, no. 199, pp. 2629–2646.
  • Potter, M., 2005, “Ramsey’s Transcendental Argument”, en H. Lillehammer y D.H. Mellor (eds.), Ramsey’s Legacy, Oxford University Press, Oxford,...
  • Priest, G., 2017, Logic: A Very Short Introduction, Oxford University Press, Oxford.
  • Putnam, H., 1995, Mathematics, Matter and Method, Cambridge University Press, Cambridge.
  • Putnam, H., 1967, “Mathematics without Foundations”, The Journal of Philosophy, vol. 64, no. 1, pp. 5–22.
  • Quine, W.V., 1982, Mathematical Logic, Harvard University Press, Cambridge.
  • Quine, W.V., 1965, Elementary Logic, Harper, Nueva York.
  • Ramsey, F., 1991, Notes on Philosophy, Probability and Mathematics, Bibliopolis, Pittsburgh.
  • Ramsey, F., 1926, Mathematical Logic, Kegan Paul, Londres.
  • Ramsey, F., 1925, The Foundations of Mathematics, Kegan Paul, Londres.
  • Ramsey, F., 1923, “Critical Notice of L. Wittgenstein’s Tractatus”, Mind, vol. XXXII, no. 128, pp. 465–478.
  • Rayo, A., 2005, “Logicism Reconsidered”, en S. Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic, Oxford University...
  • Reichenbach, H., 1996, Objetivos y métodos del conocimiento físico, FCE, México.
  • Resnik, M., 2005, “Quine and the Web of Belief”, en S. Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic, Oxford University...
  • Richardson, A. y T. Uebel, 2007, The Cambridge Companion to Logical Empiricism, Cambridge University Press, Nueva York.
  • Rodych, V., 2018, “Wittgenstein’s Philosophy of Mathematics”, en E. Zalta (ed.), Stanford Encyclopedia of Philosophy. (https://plato.stanford.edu/entries/wittgenstein-mathematics/)
  • Rodych, V., 1995, “Pasquale Frascolla’s Wittgenstein’s Philosophy of Mathematics”, Philosophia Mathematica, vol. 3, no. 3, pp. 271–288.
  • Russell, B., 1999, “¿Es la matemática puramente lingüística?”, en B. Russell, Análisis filosófico, Paidós, Barcelona, pp. 113–127.
  • Russell, B., 1996, The Principles of Mathematics, Norton, Nueva York.
  • Russell, B., 1920, Introduction to Mathematical Philosophy, George Allen y Unwin, Londres.
  • Russell, B., 1908, “Mathematical Logic as Based on the Theory of Types”, American Journal of Mathematics, vol. 30, no. 3, pp. 222–262.
  • Russell, B. y A. Whitehead, 1962, Principia Mathematica, Cambridge University Press, Cambridge.
  • Soames, S., 2016, “Propositions, the Tractatus, and ‘The Single Great Problem of Philosophy’ ”, Crítica, vol. 48, no. 143, pp. 3–19.
  • Stalnaker, R., 2003, “On what Possible Worlds Could Not Be”, en R. Stalnaker, Ways a World Might Be: Metaphysical and Anti-Metaphysical Essays,...
  • Steiner, M., 1998, The Applicability of Mathematics as a Philosophical Problem, Harvard University Press, Cambridge.
  • Stewart, I., 2017, Infinity: A Very Short Introduction, Oxford University Press, Oxford.
  • Tomasini, A., 2012, Los atomismos lógicos de Russell y Wittgenstein, Instituto de Investigaciones Filosóficas-UNAM, México.
  • Wang, H., 2016, From Mathematics to Philosophy, Routledge, Nueva York.
  • Wang, H., 1967, “Russell and his Logic”, Ratio, vol. VII, no. 1, pp. 1–34.
  • Weaver, N., 2005, “Predicativity beyond Γ0”, Mathematics ArXiv e-prints. (https://arxiv.org/abs/math/0509244)
  • Weyl, H., 2009, Philosophy of Mathematics and Natural Science, Princeton University Press, Princeton.
  • Wittgenstein, L., 2009, Tractatus logico-philosophicus, Alianza, Madrid.
  • Wittgenstein, L., 2007a, Observaciones filosóficas, Instituto de Investigaciones Filosóficas-UNAM, México.
  • Wittgenstein, L., 2007b, Zettel, Instituto de Investigaciones Filosóficas-UNAM, México.
  • Wittgenstein, L., 2003, Investigaciones filosóficas, Instituto de Investigaciones Filosóficas-UNAM, México.
  • Wittgenstein, L., 1983, Remarks on the Foundations of Mathematics, The MIT Press, Cambridge.
  • Wittgenstein, L., 1982, Diario filosófico (1914–1916), Ariel, Barcelona.
  • Wrigley, M., 1998, “A Note on Arithmetic and Logic in the Tractatus”, Acta Analytica, vol. 21, pp. 129–131.
  • Zalta, E., 2008, “A Defense of Logicism”, Internationales Wittgenstein Symposium: Reduktion und Elimination in Philosophie und den Wissenschaften,...