On the impredicativity and circularity of Frege’s Ancestral

Authors

DOI:

https://doi.org/10.51359/2357-9986.2020.248897

Keywords:

Gottlob Frege, ancestral relation, impredicativity, logicism

Abstract

Among Frege’s major contributions to the philosophy of mathematics lies his fa-mous definition for the Ancestral relation. Presented first in the 1879 Begriffsschrift, it has an important role in Frege’s logicism, given that it provides the conditions for transforming the predecessor relation into a linear-ordered series. One of the objec-tions for Frege’s definition comes from its impredicativity and the circularity thus yielded. Such objection was pointed by Benno Kerry in 1887, and more recently by Ignacio Angelelli in 2012. In this article, I argue from a Fregean perspective that Frege’s Ancestral is not circular, although its inevitably impredicative character.

Author Biography

João Vitor Schmidt, Universidade Estadual de Campinas

Master's degree and PhD candidate in Philosophy at the State University of Campinas - UNICAMP. CNPq Scholar.

References

ANGELELLI, Ignacio. Frege’s Ancestral and its Circularities. Logica Univer-salis, 6(3):477–483, 2012.

BOOLOS, George and HECK, Richard Kimberly. Die Grundlagen der Arith-metik §§82-83. In: HECK, Richard Kimberly, Frege’s Theorem. ClarendonPress: Oxford, 2011.pp.69-89.

CARNAP, Rudolf. Logical Syntax of Language. Routledge University Press:London, 1937.

CARNAP, Rudolf. The Logicist Foundations of Mathematics. In: BENA-CERRAF, Paul; PUTNAM, Hilary (eds). Philosophy of Mathematics: Selec-ted readings. 2ed. Cambridge: Cambridge University Press, 1964.

FEFERMAN, Solomon. Predicativity. In: SHAPIRO, Stewart (ed.) The Ox-ford Handbook of Philosophy of Mathematics and Logic. Oxford UniversityPress: New York, 2005.pp.590-624.

FREGE, Gottlob. The Foundations of Arithmetic. A logico-mathematical en-quiry into the concept of number. Blackwell: Oxford, 1953.

FREGE, Gottlob. Begriffsschrift, a Formula Language, modeled upon that ofArithmetic, for Pure Thought. In: HEIJENOORT, Jean Van (ed.) From Fregeto Gödel: A Source Book in Mathematical Logic, 1879-1931. Harvard Uni-versity Press: Cambridge, 1967. pp.1-82.

FREGE, Gottlob. Collected Papers on Mathematics, Logic and Philosophy. Ba-sil Blackwell: New York, 1984.

FREGE, Gottlob. Basic Laws of Arithmetic. Oxford University Press: Oxford,2013.

FULUGONIO, Maria. Benno Kerry: una crı́tica temprana al logicismo de fre-ge y un antecedente en la historia del problema de la impredicatividad. CLE e-Prints, 8(6), 2008.

GÖDEL, Kurt. Russell’s Mathematical Logic. In: FEFERMAN, Solomon(ed.) Kurt Gödel Collected Works, volume II. Oxford University Press: Ox-ford, 1990.pp.119-141.

HECK, Richard Kimberly. Is Frege’s definition of the Ancestral adequate?Philosophia Mathematica, 24(1):91–116, 2016.

KANT, Immanuel. Critique of Pure Reason. Cambridge University Press:Cambridge, 1998.

KERRY, Benno. Über Anschauung und ihre Psychische Verarbeitung. Viertel-jahrsschrift für Wissenschaftliche Philosophie, III(11):249–307, 1887.

PARSONS, Charles. Realism and the Debate on Impredicativity, 1917–1944. In: PARSONS, Charles. Philosophy of Mathematics in the TwentiethCentury: Selected Essays. Cambridge: Harvard University Press, 2014. pp.40-66.

POINCARÉ, Henri. Mathematics and Science: Last Essays. Dover Publicati-ons, Inc: New York, 1963.

RAMSEY, Frank. The Foundations of Mathematics. In: BRAITHWAITER,Richard (ed.).The Foundations of Mathematics and other Logical Essays.Routledge & Kegan Paul Ltd: London, 1931.pp.1-61.

RUSSELL, Bertrand. On some difficulties in the Theory of Transfinite Num-bers and Order Types. Proceedings of the London Mathematical Society, s2-4(1):29–53, 1907.

RUSSELL, Bertrand. The Principles of Mathematics. W. W. Norton & Com-pany, New York, 1996.

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Published

2020-11-27

Issue

Vol. 47 No. 2 (2020): Filosofia da Lógica

Section

Número Especial sobre Filosofia da Lógica

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