Sobre el concepto de recursión y sus usos
Recursión regla gramatical estructura auto-inclusión auto-referencia
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ene. 15, 2015
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La noción de recursión se emplea en diferentes sentidos, adquiriendo una variedad de significados. Así, en unos casos se usa para caracterizar una regla esencial que constituye un modo de definición en un sistema. Este sentido tiene su origen en la Lógica Matemática y la Teoría de la Computabilidad. En otros casos se aplica para indicar la organización interna de una estructura, tal como sucede en la Ciencia Cognitiva y la Ciencia de la Computación, al tiempo que también se emplea en el sentido anterior en estas mismas disciplinas. El objetivo de este trabajo es mostrar en cada caso las diferencias en el uso de ambos sentidos.
Mota, S. (2015). Sobre el concepto de recursión y sus usos. Praxis Filosófica, (40), 153–181. https://doi.org/10.25100/pfilosofica.v0i40.3016
Abelson, H. & Sussman, G. J. with Sussman, J. (1996). Structure and interpretation of computer programs. Cambridge, MA: MIT Press.
Arsenijevic, B. & Hinzen, W. (2010). Recursion as a human universal and as a primitive. Biolinguistics, 4(2-3), 165-173.
Arsenijević, B. & Hinzen, W. (2012). On the abscence of X-within-X recursion in human grammar, Linguistic Inquiry, 43, 423-440.
Boolos, G. & Jeffrey, R. (1974). Computability and logic. Cambridge: Cambridge University Press.
Boolos, G. (1971). The iterative conception of set, The Journal of Philosophy, 68, 215-231.
Chomsky, N. (1959). On certain formal properties of grammars. Information and Control, 2, 137-167.
Chomsky, N. (1965). Aspects of theory of syntax. Cambridge. MA: MIT Press
Chomsky, N. (1995). The minimalist program. Cambridge. MA: MIT Press.
Chomsky, N. (2006). Language and mind. Cambridge: Cambridge University Press.
Chomsky, N. (2007a). Of minds and language, Biolinguistics, 1, 9-27.
Chomsky, N. (2007b). Approaching UG from below. In U. Sauerland & H. M. Gärtner (Eds.), Interfaces + Recursion = Language? (pp. 1-30). Berlin: Mouton.
Chomsky, N. (2008). On phases. In R. Freidin, C. Otero, & M. L. Zubizarreta (Eds.), Foundational issues in linguistic theory (pp. 133-166). Cambridge. MA: MIT Press.
Chomsky, N. (2010). Some simple evo devo theses: How true might they be for language? In R. Larson, V. Déprez & H. Yamakido (Eds.), The evolution of human language (pp. 45-62). Cambridge: Cambridge University Press.
Chomsky, N. (2011). Language and other cognitive systems. What is special about language?. Language Learning and Development, 7, 263-278.
Chomsky, N. (2012). Some core contested concepts. Proceedings of the CUNY 2012, 1-18.
Church, A. (1932). A set of postulates for the foundation of logic, The Annals of Mathematics, 33, 346-366.
Church, A. (1936). An unsolvable problem of elementary number theory. In M. Davis (Ed.), The undecidable (pp. 88-107). New York: Raven Press.
Corballis, M. C. (2007). Recursion, language and starlings, Cognitive Science, 31, 69-704.
Corballis, M. C. (2011). The recursive mind: The origins of human language, thought, and civilization. Princeton, NJ: Princeton University Press.
Cutland, N. (1980). Computability: an introduction to recursive function theory. Cambridge: Cambridge University Press.
Epstein, R. & Carnielli, W. (1989). Computability: computable functions, logic, and the foundations of mathematics. Pacific Grove, CA: Wadsworth & Brooks/Cole.
Everett, D. (2005). Cultural constraints on grammar and cognition in Pirahã, Current Anthropology, 46(4), 621-646.
Everett, D. (2009). Pirahã culture and grammar: a response to some criticisms, Language, 85(2), 405-442.
Fitch, T. (2010). Three meanings of recursion: key distinctions for biolinguistics. In R. Larson, V. Déprez & H. Yamakido (Eds.), The evolution of human language(pp. 73-90). Cambridge: Cambridge University Press.
Frascolla, P. (1994). Wittgenstein’s philosophy of mathematics. London: Routledge.
Gödel, K. (1931). On formally undecidable propositions of the Principia Mathematica and related systems. I. In M. Davis (Ed.), The undecidable (pp. 4-38). New York: Raven Press.
Gödel, K. (1934). On undecidable propositions of formal mathematical systems. In M. Davis (Ed.), The undecidable (pp. 39-74). New York: Raven Press.
Gödel, K. (1964). Postscriptum to Gödel 1931. In M. Davis (Ed.), The undecidable (pp. 71-73). New York: Raven Press.
Hauser, M., Chomsky, N. & Fitch, T. (2002). The faculty of language: What is, who has it, and how did it evolve?, Science, 298, 1569-1579.
Hrbacek, K. & Jech, T. (1999). Introduction to Set Theory, New York: Marcel Dekker, Inc.
Jackendoff, R. & Pinker, S. (2005). The nature of the language faculty and its implications for evolution of language (reply to Fitch, Hauser, and Chomsky), Cognition, 97, 211-225.
Jackendoff, R. (2011). What is the human language faculty? Two views, Language, 87, 586-624.
Karlsson, F. (2010). Syntactic recursion and iteration. In H. van der Hulst (Ed.), Recursion and human language (pp. 43-67).
Kinsella, A. (2010). Was recursion the key step in the evolution of the human language faculty? In H. van der Hulst (Ed.), Recursion and human language(pp. 179-191).
Kleene, S. C. (1938). On notation for ordinal numbers, The Journal of Symbolic Logic, 3, 150-5.
Kleene, S. C. (1943). Recursive predicates and quantifiers, Transactions of the American Mathematical Society, 53, 41-73.
Kleene, S. C. (1952). Introduction to metamathematics. Amsterdam: North-Holland Publishing.
Lobina, D. J. (2011). Recursion and the competence/performance distinction in AGL tasks, Language and Cognitive Processes, 26, 1563-1586.
Lobina, D. J. (2014a). Probing recursion. Cognitive Processing, DOI 10.1007/s10339-014-0619-z
Lobina, D. J. (2014b). What linguists are talking about when talking about..., Language Sciences, 45, 56-70.
Luuk, E. & Luuk, H. (2011). The redundancy of recursion and infinity for natural language, Cognitive Processing, 12, 1-11.
Marion, M. (1995). Wittgenstein and finitism, Synthese, 105, 141-176.
Marion, M. (1998). Wittgenstein, finitism, and the foundations of mathematics. Oxford: Oxford University Press.
Marion, M. (2009). Radical anti-realism, Wittgenstein and the length of proofs, Synthese, 171, 419-432.
Moro, A. (2008). The boundaries of Babel. Cambridge, MA: MIT Press.
Mota, S. (2013). La propiedad de la recursión en el “Tractatus Logico-Philosophicus” de Wittgenstein y su relación con la Teoría de la Computabilidad y la Lógica Matemática, 17, Observaciones Filosóficas. http://www.obervacionesfilosoficas.net/lapropiedaddelarecursion.htm
Mota, S. (2014). La historia y la gramática de la recursión: una precisión desde la obra de Wittgenstein, Pensamiento y Cultura, 17, 20-48.
Odifreddi, P. (2001). Recursive functions: an archeological look. In C.S. Claude, M.J. Dinneen & S. Sburlan (Eds.), Combinatorics, Computability and Logic(pp. 13-31). London: Springer-Verlag.
Pinker, S. & Jackendoff, R. (2005). The faculty of language: What’s special about it?, Cognition, 95, 201-236.
Post, E. (1921). Introduction to a general theory of elementary propositions, American Journal of Mathematics, 43, 163-185.
Post, E. (1943). Formal reductions of the general combinatorial decision problem, American Journal of Mathematics, 65, 197-215.
Post, E. (1944). Recursively enumerable sets of positive integers and their decision problems. In M. Davis (Ed.), The undecidable (pp. 305-337). New York: Raven Press.
Roberts, E. (2006). Thinking Recursively with Java. Hoboken, NJ: John Wiley and Sons, Inc.
Roddych, V. (1999a). Wittgenstein on irrationals and algorithmic decidability, Synthese, 118, 279-304.
Rodgers, P., Black, P.E. (2004). Recursive data structure. In: Pieterse, V., Black, P.E. (Eds.), Dictionary of Algorithms and Data Structures. Online at: http://www.nist.gov/dads/HTML/recursivstrc.html.
Rodych, V. (1997). Wittgenstein on mathematical meaningfulness, decidability, and application, Notre Dame Journal of Formal Logic, 38, 195-225.
Rodych, V. (1999b). Wittgenstein’s inversion of Gödel’s Theorem, Erkenntnis, 51, 173, 206.
Rodych, V. (2002). Wittgenstein on Gödel: the newly published remarks, Erkenntnis, 56, 379-397.
Rodych, V. (2003). Misunderstanding Gödel: new arguments about Wittgenstein and new remarks by Wittgenstein, Dialectica, 57, 279-313.
Shanker, S.G. (1987). Wittgenstein versus Turing on nature of Church’s thesis, Notre Dame Journal of Formal Logic, 28, 615-649.
Sieg, W. (1997). Step by recursive step: Church ́s analysis of effective calculability, The Bulletin of Symbolic Logic, 3, 154-180.
Skolem, T. (1923). The foundations of elementary arithmetic established by means of the recursive mode of thought, without the use of apparent variables ranging over infinite domains. In J. Van Heijenoort (Ed.), From Frege to Gödel. A source book in mathematical logic, 1879-1931 (pp. 302-333). Cambridge, MA: Harvard University Press.
Soare, R. (1996). Computability and recursion, The Bulletin of Symbolic Logic, 2, 284-321.
Soare, R. (2009). Turing oracles machines, online computing, and three displacements in computability theory, Annals of Pure and Applied Logic, 160, 368-399.
Tomalin, M. (2006). Linguistics and the Formal Sciences. Cambridge: Cambridge University Press.
Tomalin, M. (2007). Reconsidering recursion in syntactic theory, Lingua, 117, 1784-1800.
Tomalin, M. (2011). Syntactic structures and recursive devices: A legacy of imprecision, Journal of Logic, Language and Information, 20, 297-315.
Turing, A. (1937). On computable numbers, with an application to the Entscheidungsproblem. In M. Davis (Ed.), The undecidable (pp. 116-151). New York: Raven Press.
Wirth, N. (1986). Algorithms and data structures. Hemel Hempstead, UK: Prentice Hall. © N. Wirth 1985 (Oberon version: August 2004).
Wittgenstein, L. (1922). Tractatus logico-philosophicus. London: Routledge.
Wittgenstein, L. (1974). Philosophical grammar. Traducción de Luis F. Segura, ed. 1992. México, D.F.: UNAM.
Wittgenstein, L. (1975). Philosophical remarks. Traducción de Alejandro Tomasini Bassols, ed. 1997. México, D.F.: UNAM.
Wittgenstein, L. (1978). Remarks on the foundations of mathematics. Oxford: Blackwell.
Wrigley, M. (1977). Wittgenstein’s philosophy of mathematics, Philosophical Quarterly, 27, 50-59.
Zwart, J.W. (2011). Recursion in language: A layered-derivation approach, Biolinguistics, 5, 43-56.
Arsenijevic, B. & Hinzen, W. (2010). Recursion as a human universal and as a primitive. Biolinguistics, 4(2-3), 165-173.
Arsenijević, B. & Hinzen, W. (2012). On the abscence of X-within-X recursion in human grammar, Linguistic Inquiry, 43, 423-440.
Boolos, G. & Jeffrey, R. (1974). Computability and logic. Cambridge: Cambridge University Press.
Boolos, G. (1971). The iterative conception of set, The Journal of Philosophy, 68, 215-231.
Chomsky, N. (1959). On certain formal properties of grammars. Information and Control, 2, 137-167.
Chomsky, N. (1965). Aspects of theory of syntax. Cambridge. MA: MIT Press
Chomsky, N. (1995). The minimalist program. Cambridge. MA: MIT Press.
Chomsky, N. (2006). Language and mind. Cambridge: Cambridge University Press.
Chomsky, N. (2007a). Of minds and language, Biolinguistics, 1, 9-27.
Chomsky, N. (2007b). Approaching UG from below. In U. Sauerland & H. M. Gärtner (Eds.), Interfaces + Recursion = Language? (pp. 1-30). Berlin: Mouton.
Chomsky, N. (2008). On phases. In R. Freidin, C. Otero, & M. L. Zubizarreta (Eds.), Foundational issues in linguistic theory (pp. 133-166). Cambridge. MA: MIT Press.
Chomsky, N. (2010). Some simple evo devo theses: How true might they be for language? In R. Larson, V. Déprez & H. Yamakido (Eds.), The evolution of human language (pp. 45-62). Cambridge: Cambridge University Press.
Chomsky, N. (2011). Language and other cognitive systems. What is special about language?. Language Learning and Development, 7, 263-278.
Chomsky, N. (2012). Some core contested concepts. Proceedings of the CUNY 2012, 1-18.
Church, A. (1932). A set of postulates for the foundation of logic, The Annals of Mathematics, 33, 346-366.
Church, A. (1936). An unsolvable problem of elementary number theory. In M. Davis (Ed.), The undecidable (pp. 88-107). New York: Raven Press.
Corballis, M. C. (2007). Recursion, language and starlings, Cognitive Science, 31, 69-704.
Corballis, M. C. (2011). The recursive mind: The origins of human language, thought, and civilization. Princeton, NJ: Princeton University Press.
Cutland, N. (1980). Computability: an introduction to recursive function theory. Cambridge: Cambridge University Press.
Epstein, R. & Carnielli, W. (1989). Computability: computable functions, logic, and the foundations of mathematics. Pacific Grove, CA: Wadsworth & Brooks/Cole.
Everett, D. (2005). Cultural constraints on grammar and cognition in Pirahã, Current Anthropology, 46(4), 621-646.
Everett, D. (2009). Pirahã culture and grammar: a response to some criticisms, Language, 85(2), 405-442.
Fitch, T. (2010). Three meanings of recursion: key distinctions for biolinguistics. In R. Larson, V. Déprez & H. Yamakido (Eds.), The evolution of human language(pp. 73-90). Cambridge: Cambridge University Press.
Frascolla, P. (1994). Wittgenstein’s philosophy of mathematics. London: Routledge.
Gödel, K. (1931). On formally undecidable propositions of the Principia Mathematica and related systems. I. In M. Davis (Ed.), The undecidable (pp. 4-38). New York: Raven Press.
Gödel, K. (1934). On undecidable propositions of formal mathematical systems. In M. Davis (Ed.), The undecidable (pp. 39-74). New York: Raven Press.
Gödel, K. (1964). Postscriptum to Gödel 1931. In M. Davis (Ed.), The undecidable (pp. 71-73). New York: Raven Press.
Hauser, M., Chomsky, N. & Fitch, T. (2002). The faculty of language: What is, who has it, and how did it evolve?, Science, 298, 1569-1579.
Hrbacek, K. & Jech, T. (1999). Introduction to Set Theory, New York: Marcel Dekker, Inc.
Jackendoff, R. & Pinker, S. (2005). The nature of the language faculty and its implications for evolution of language (reply to Fitch, Hauser, and Chomsky), Cognition, 97, 211-225.
Jackendoff, R. (2011). What is the human language faculty? Two views, Language, 87, 586-624.
Karlsson, F. (2010). Syntactic recursion and iteration. In H. van der Hulst (Ed.), Recursion and human language (pp. 43-67).
Kinsella, A. (2010). Was recursion the key step in the evolution of the human language faculty? In H. van der Hulst (Ed.), Recursion and human language(pp. 179-191).
Kleene, S. C. (1938). On notation for ordinal numbers, The Journal of Symbolic Logic, 3, 150-5.
Kleene, S. C. (1943). Recursive predicates and quantifiers, Transactions of the American Mathematical Society, 53, 41-73.
Kleene, S. C. (1952). Introduction to metamathematics. Amsterdam: North-Holland Publishing.
Lobina, D. J. (2011). Recursion and the competence/performance distinction in AGL tasks, Language and Cognitive Processes, 26, 1563-1586.
Lobina, D. J. (2014a). Probing recursion. Cognitive Processing, DOI 10.1007/s10339-014-0619-z
Lobina, D. J. (2014b). What linguists are talking about when talking about..., Language Sciences, 45, 56-70.
Luuk, E. & Luuk, H. (2011). The redundancy of recursion and infinity for natural language, Cognitive Processing, 12, 1-11.
Marion, M. (1995). Wittgenstein and finitism, Synthese, 105, 141-176.
Marion, M. (1998). Wittgenstein, finitism, and the foundations of mathematics. Oxford: Oxford University Press.
Marion, M. (2009). Radical anti-realism, Wittgenstein and the length of proofs, Synthese, 171, 419-432.
Moro, A. (2008). The boundaries of Babel. Cambridge, MA: MIT Press.
Mota, S. (2013). La propiedad de la recursión en el “Tractatus Logico-Philosophicus” de Wittgenstein y su relación con la Teoría de la Computabilidad y la Lógica Matemática, 17, Observaciones Filosóficas. http://www.obervacionesfilosoficas.net/lapropiedaddelarecursion.htm
Mota, S. (2014). La historia y la gramática de la recursión: una precisión desde la obra de Wittgenstein, Pensamiento y Cultura, 17, 20-48.
Odifreddi, P. (2001). Recursive functions: an archeological look. In C.S. Claude, M.J. Dinneen & S. Sburlan (Eds.), Combinatorics, Computability and Logic(pp. 13-31). London: Springer-Verlag.
Pinker, S. & Jackendoff, R. (2005). The faculty of language: What’s special about it?, Cognition, 95, 201-236.
Post, E. (1921). Introduction to a general theory of elementary propositions, American Journal of Mathematics, 43, 163-185.
Post, E. (1943). Formal reductions of the general combinatorial decision problem, American Journal of Mathematics, 65, 197-215.
Post, E. (1944). Recursively enumerable sets of positive integers and their decision problems. In M. Davis (Ed.), The undecidable (pp. 305-337). New York: Raven Press.
Roberts, E. (2006). Thinking Recursively with Java. Hoboken, NJ: John Wiley and Sons, Inc.
Roddych, V. (1999a). Wittgenstein on irrationals and algorithmic decidability, Synthese, 118, 279-304.
Rodgers, P., Black, P.E. (2004). Recursive data structure. In: Pieterse, V., Black, P.E. (Eds.), Dictionary of Algorithms and Data Structures. Online at: http://www.nist.gov/dads/HTML/recursivstrc.html.
Rodych, V. (1997). Wittgenstein on mathematical meaningfulness, decidability, and application, Notre Dame Journal of Formal Logic, 38, 195-225.
Rodych, V. (1999b). Wittgenstein’s inversion of Gödel’s Theorem, Erkenntnis, 51, 173, 206.
Rodych, V. (2002). Wittgenstein on Gödel: the newly published remarks, Erkenntnis, 56, 379-397.
Rodych, V. (2003). Misunderstanding Gödel: new arguments about Wittgenstein and new remarks by Wittgenstein, Dialectica, 57, 279-313.
Shanker, S.G. (1987). Wittgenstein versus Turing on nature of Church’s thesis, Notre Dame Journal of Formal Logic, 28, 615-649.
Sieg, W. (1997). Step by recursive step: Church ́s analysis of effective calculability, The Bulletin of Symbolic Logic, 3, 154-180.
Skolem, T. (1923). The foundations of elementary arithmetic established by means of the recursive mode of thought, without the use of apparent variables ranging over infinite domains. In J. Van Heijenoort (Ed.), From Frege to Gödel. A source book in mathematical logic, 1879-1931 (pp. 302-333). Cambridge, MA: Harvard University Press.
Soare, R. (1996). Computability and recursion, The Bulletin of Symbolic Logic, 2, 284-321.
Soare, R. (2009). Turing oracles machines, online computing, and three displacements in computability theory, Annals of Pure and Applied Logic, 160, 368-399.
Tomalin, M. (2006). Linguistics and the Formal Sciences. Cambridge: Cambridge University Press.
Tomalin, M. (2007). Reconsidering recursion in syntactic theory, Lingua, 117, 1784-1800.
Tomalin, M. (2011). Syntactic structures and recursive devices: A legacy of imprecision, Journal of Logic, Language and Information, 20, 297-315.
Turing, A. (1937). On computable numbers, with an application to the Entscheidungsproblem. In M. Davis (Ed.), The undecidable (pp. 116-151). New York: Raven Press.
Wirth, N. (1986). Algorithms and data structures. Hemel Hempstead, UK: Prentice Hall. © N. Wirth 1985 (Oberon version: August 2004).
Wittgenstein, L. (1922). Tractatus logico-philosophicus. London: Routledge.
Wittgenstein, L. (1974). Philosophical grammar. Traducción de Luis F. Segura, ed. 1992. México, D.F.: UNAM.
Wittgenstein, L. (1975). Philosophical remarks. Traducción de Alejandro Tomasini Bassols, ed. 1997. México, D.F.: UNAM.
Wittgenstein, L. (1978). Remarks on the foundations of mathematics. Oxford: Blackwell.
Wrigley, M. (1977). Wittgenstein’s philosophy of mathematics, Philosophical Quarterly, 27, 50-59.
Zwart, J.W. (2011). Recursion in language: A layered-derivation approach, Biolinguistics, 5, 43-56.
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