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How to Cite Complete Issue More Information About This Article THEORIA. Revista de Teoría, Historia y Fundamentos de la Ciencia ISSN: 0495-4548 ISSN: 2171-679X [email protected] Universidad del País Vasco/Euskal Herriko Unibertsitatea España Miguel Sagüillo, José Hilary Putnam on the philosophy of logic and mathematics* [1] THEORIA. Revista de Teoría, Historia y Fundamentos de la Ciencia, vol. 33, no. 2, 2018, pp. 183-200 Universidad del País Vasco/Euskal Herriko Unibertsitatea España DOI: https://doi.org/10.1387/theoria.17626 Available in: https://www.redalyc.org/articulo.oa?id=339755528003 How to cite Complete issue Scientific Information System Redalyc More information about this article Network of Scientific Journals from Latin America and the Caribbean, Spain and Journal's webpage in redalyc.org Portugal Project academic non-profit, developed under the open access initiative THEORIA ESTABLISH E D IN 1952 BY MIGU E L SÁNCH E Z -MAZAS Vol. 33/2 • May 2018 Second Series An International Journal for Theory, History and Foundations of Science CALIJ Centro de Análisis, Lógica e Informática Jurídica (CALIJ) http://www.ehu.eus/theoria T H E O R I A REVISTA DE TEORÍA, HISTORIA Y FUNDAMENTOS DE LA CIENCIA AN INTERNATIONAL JOURNAL FOR THEORY, HISTORY AND FOUNDATIONS OF SCIENCE ESTABLISH E D in 1952 by MIGUEL SÁNCHEZ-MAZAS Second Series EDITORIAL BOARD Editor-in-chief: Andoni IBARRA (University of the Basque Country, UPV/EHU) Editors: Cristina CORREDOR (Universidad de Valladolid), Antonio DIÉGUEZ (Universidad de Málaga) Logic and philosophy of logic and mathematics: José Luis ZALABARDO (University College London) Philosophy of language: Genoveva MARTÍ (ICREA & Universitat de Barcelona) Philosophy of mind and philosophy of cognitive science: Agustín VICENTE (Ikerbasque & UPV/EHU) Philosophy of science: José DÍEZ (Universitat de Barcelona) History of science: María Jesús SANTESMASES (CSIC) Science and Tecnhology Studies: Miguel GARCÍA-SANCHO (University of Edinburgh) Managing Editor: Mario SANTOS-SOUSA (University College London) ADVISORY BOARD Juan José Acero (Universidad de Granada), Ignazio Angelelli (University of Texas at Austin), Manuel Atienza (Universitat d’Alacant), Salvador Barberà (Universitat Autònoma de Barcelona), Gustavo Bueno † (Universidad de Oviedo), Nancy Cartwright (Durham University), Newton C.A. da Costa (Universidade de São Paulo), Joseph Dauben (City University of New York), Elías Díaz (Universidad Autónoma de Madrid), Javier Echeverria (Ikerbasque & UPV/EHU), Josep Maria Font (Universitat de Barcelona), Manuel García- Carpintero (Universitat de Barcelona), Manuel Garrido † (Universidad Complutense de Madrid), Jaakko Hintikka (Boston University), Philip Kitcher (Columbia University), Bruno Latour (Sciences Po, Paris), Larry Laudan (UNAM), Javier de Lorenzo (Universidad de Valladolid), Thomas Mormann (UPV/EHU), C. Ulises Moulines (Ludwig-Maxilians-Universität München), Carlos Moya (Universitat de València), Javier Muguerza (UNED), León Olive (UNAM), Carlos París † (Universidad Autónoma de Madrid), Carlos Solís (UNED), Roberto Torretti (Universidad de Puerto Rico), Enric Trillas (Universidad Politécnica de Madrid), Nicanor Ursua (UPV/EHU), Bas C. van Fraassen (San Francisco State University) EDITORIAL OFFICE Editorial manager: María Luisa CUTANDA (CALIJ, Donostia-San Sebastián, Spain) Editorial assistant: Garazi MONTUSCHI (UPV/EHU) Mailing address: CALIJ-THEORIA, Elhuyar plaza 2, 20018, Donostia-San Sebastián, Spain Tel.: (+34) 943 017 447. Fax: (+34) 943 015 990. E-mail: [email protected] THEORIA is a four-monthly journal (issues in January, May and September). The contents of this journal are covered, among others, by the following abstracting and indexing services: Arts & Humanities Citation In- dex®, ISI Alerting Services, Current Contents® / Arts & Humanities, Bulletin Signalétique 519, DICE, ICYT, ISOC and RESH of the Spanish research center CSIC, Mathematical Reviews, Current Mathematical Publica- tions, MathSci, Philosopher’s Index, Repertoire bibliographique de la Philosophie, and SCOPUS. THEORIA’s website http://www.ehu.eus/theoria THEORIA REVISTA DE TEORÍA, HISTORIA Y FUNDAMENTOS DE LA CIENCIA CONTENTS Volume 33/2, May 2018, pp. 157-373 ISSN 0495-4548 Monographic Section I Updating indispensabilities: Putnam in memoriam Guest Editor: Mary Leng 159 Mary Leng Guest Editor’s Introduction 161 Concha Martínez Vidal Putnam and contemporary fictionalism 165 José Miguel Sagüillo Hilary Putnam on the philosophy of logic and mathematics 183 Otávio Bueno Putnam’s indispensability argument revisited, reassessed, revived 201 Sorin Bangu Indispensability, causation and explanation 219 Susan Vineberg Mathematical explanation and indispensability 233 Matteo Plebani The indispensability argument and the nature of mathematical objects 249 Monographic Section II From basic cognition to mathematical practice Guest Editors: María de Paz and José Ferreirós 265 María de Paz and José Ferreirós Guest Editors’ Introduction 267 Rafael Núñez Praxis matemática: reflexiones sobre la cognición que la hace posible 271 Markus Pantsar Early numerical cognition and mathematical processes 285 Roy Wagner Cognitive stories and the image of mathematics 305 José Ferreirós y Manuel J. García-Pérez ¿«Natural» y «euclidiana»? Reflexiones sobre la geométrica práctica y sus raíces cognitivas 325 Valeria Giardino Manipulative imagination: how to move things around in mathematics 345 Sorin Costreie The geometrical basis of arithmetical knowledge: Frege and Dehaene 361 Summary 371 Hilary Putnam on the philosophy of logic and mathematics* 1 José Miguel Sagüillo Received: 04/02/2017 Final version: 17/04/2018 BIBLID 0495-4548(2018)33:2p.183-200 DOI: 10.1387/theoria.17626 ABSTRACT: This paper focuses on Putnam’s conception of logical truth as grounded in his picture of mathematical practice and ontology. Putnam’s 1971 book Philosophy of Logic came one year later than Quine’s homonymous volume. In the first section, I compare these two Philosophies of Logic which exemplify realist-nominalist view- points in a most conspicuous way. The next section examines Putnam’s views on modality, moving from the modal qualification of his intuitive conception to his official generalized non-modal second-order set-theoretic concept of logical truth. In the third section, I emphasize how Putnam´s “mathematics as modal logic” departs from Quine’s “reluctant Platonism”. I also suggest a complementary view of Platonism and modalism showing them perhaps interchangeable but underlying different stages of research processes that make up a rich and dy- namic mathematical practice. The final, more speculative section, argues for the pervasive platonistic concep- tion enhancing the aims of inquiry in the practice of the working mathematician. Keywords: Putnam, Quine, Logic, Indispensability, Ontology, Scientific Language, Mathematical Practice. RESUMEN: Este artículo estudia la concepción de Putnam de verdad lógica que emana de su visión de la práctica de la matemática y de su ontología. Philosophy of Logic, el libro de 1971 de Putnam surge un año más tarde que el homónimo de Quine. En la primera sección, se comparan estas dos Filosofías de la Lógica que ejemplifican los puntos de vista del realismo y del nominalismo de modo conspicuo. La siguiente sección examina el enfoque de la modalidad de Putnam, que va desde la cualificación modal de su caracterización intuitiva de validez ló- gica a su concepción oficial generalizada no-modal conjuntista de segundo orden. La tercera sección subraya el modo en que «la matemática como lógica modal» de Putnam se distancia del «Platonism a regañadientes» de Quine. Aquí se sugiere una visión complementaria del Platonism y del modalismo, los cuales, aunque quizás intercambiables, se muestran subyaciendo a los diferentes estadios del proceso de investigación de una práctica de la matemática rica y dinámica. La sección final, más especulativa, conjetura algunas razones de la persistente concepción platónica implícita en la práctica del matemático. Palabras clave: Putnam, Quine, Lógica, Indispensabilidad, Ontología, Lenguaje Científico, Práctica de la Matemática. * I am grateful to the participants in the workshop “Updating indispensabilities: Hilary Putnam in Me- moriam” held in Santiago de Compostela at the end of November 2016 for discussion that followed my presentation. I am grateful to Concha Martínez, Matteo Plebani, Otávio Bueno, and John Cor co- ran for comments on a previous version. My special thanks also go to two anonymous referees of Theo- ria for their insightful suggestions that significantly improved the present version. One report was es- pecially constructive, dialogical and detailed. 1 The research for this paper was supported by the Spanish Ministry of Economy and Competitivity and FEDER via the research projects FFI 2013-41415-P and FFI2017-82534-P. THEORIA 33/2 (2018): 183-200 184 José Miguel Sagüillo Introduction This paper focuses on Putnam’s philosophy of logic and mathematics. Specifically, I ex- amine his conception of logical truth, which is based on his view of mathematical prac- tice and ontology. His conception can be seen as an endorsement of the mathematical paradigm of contemporary model theory framing a general picture of mathematics and the natural sciences. Even though his 1971 presentation is philosophical, in it Putnam emphasizes what can be thought of as an official view of logic as a mature scientific dis- cipline. However, by considering Putnam’s overall view of logic and science as expressed in his different writings, we can see that he never loses sight
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