Boston Studies in the Philosophy and History of Science
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Boston Studies in the Philosophy and History of Science Volume 339 Series Editors Alisa Bokulich, Boston University Jürgen Renn, Max Planck Institute for the History of Science Managing Editor Lindy Divarci, Max Planck Institute for the History of Science Founding Editor Robert S. Cohen Editorial Board Members Theodore Arabatzis, University of Athens Heather E. Douglas, University of Waterloo Kostas Gavroglu, University of Athens Thomas F. Glick, Boston University Hubert Goenner, University of Goettingen John Heilbron, University of California, Berkeley Diana Kormos-Buchwald, California Institute of Technology Christoph Lehner, Max Planck Institute for the History of Science Peter McLaughlin, Universität Heidelberg Agustí Nieto-Galan, Universitat Autònoma de Barcelona Nuccio Ordine, Universitá della Calabria Ana Simões, Universidade de Lisboa John J. Stachel, Boston University Baichun Zhang, Chinese Academy of Science
The series Boston Studies in the Philosophy and History of Science was conceived in the broadest framework of interdisciplinary and international concerns. Natural scientists, mathematicians, social scientists and philosophers have contributed to the series, as have historians and sociologists of science, linguists, psychologists, physicians, and literary critics. The series has been able to include works by authors from many other countries around the world. The editors believe that the history and philosophy of science should itself be scientific, self-consciously critical, humane as well as rational, sceptical and undogmatic while also receptive to discussion of first principles. One of the aims of Boston Studies, therefore, is to develop collaboration among scientists, historians and philosophers. Boston Studies in the Philosophy and History of Science looks into and reflects on interactions between epistemological and historical dimensions in an effort to understand the scientific enterprise from every viewpoint. More information about this series at https://link.springer.com/bookseries/5710
Gianluigi Oliveri • Claudio Ternullo Stefano Boscolo Editors Objects, Structures, and Logics FilMat Studies in the Philosophy of Mathematics
Editors Gianluigi Oliveri Claudio Ternullo Accademia Nazionale delle Scienze, Lettere Departament de Matemàtiques i Informàtica ed Arti di Palermo Universitat de Barcelona Università di Palermo Barcelona, Spain Palermo, Italy Stefano Boscolo Dipartimento di Filosofia e Beni Culturali Università di Venezia Ca’ Foscari Trevio, Italy ISSN 0068-0346 ISSN 2214-7942 (electronic) Boston Studies in the Philosophy and History of Science ISBN 978-3-030-84705-0 ISBN 978-3-030-84706-7 (eBook) https://doi.org/10.1007/978-3-030-84706-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Gianluigi Oliveri, Claudio Ternullo, and Stefano Boscolo
Part I Mathematical Objects
2 Aristotle’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Luca Zanetti
3 Hofweber’s Nominalist Naturalism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Eric Snyder, Richard Samuels, and Stewart Shapiro
4 Exploring Mathematical Objects from Custom-Tailored
Mathematical Universes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Ingo Blechschmidt
5 Rescuing Implicit Definition from Abstractionism . . . . . . . . . . . . . . . . . . . . . 97
Daniel Waxman
Part II Structures and Structuralisms
6 Structural Relativity and Informal Rigour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Neil Barton
7 Ontological Dependence and Grounding for a Weak
Mathematical Structuralism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Silvia Bianchi
8 The Structuralist Mathematical Style: Bourbaki as a Case Study . . . . 199
Jean-Pierre Marquis
9 Grothendieck Toposes as Unifying ‘Bridges’: A
Mathematical Morphogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Olivia Caramello
vvi Contents
Part III Logics and Proofs
10 Game of Grounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Davide Catta and Antonio Piccolomini d’Aragona
11 Predicativity and Constructive Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Laura Crosilla
12 Truth and the Philosophy of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Andrea Cantini
13 On Lakatos’s Decomposition of the Notion of Proof. . . . . . . . . . . . . . . . . . . . 331
Enrico Moriconi
14 A Categorical Reading of the Numerical Existence Property
in Constructive Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
Samuele MaschioAbout the Editors
Claudio Ternullo (PhD Liverpool, 2012) is currently Beatriu de Pinós (Marie-
Skłodowska Curie Actions COFUND) Postdoctoral Fellow at the University of
Barcelona. Previously, he has held positions as post-doc at the Kurt Gödel Research
Center for Mathematical Logic at the University of Vienna and at the University
of Tartu. His research interests lie in logic and the philosophy of mathematics, in
particular in the philosophy of set theory. His work focuses on the set-theoretic
multiverse, new axioms (and their justification), and mathematical platonism (in
particular, Gödel’s Platonism). He has also done work on issues in ancient and
medieval philosophy.
Stefano Boscolo (PhD Palermo, 2016) is an IT Solution Architect at Volkswagen
Group. His current areas of expertise are advanced analytics, cloud computing, and
machine learning. After receiving his PhD in Logic and Philosophy of Science,
he worked at Ca’ Foscari University of Venice on truth pluralism. Throughout his
academic career, he worked on the philosophy of mathematics, in particular on the
debate between platonism and anti-platonism. His current research interests range
from natural-language processing to deep neural networks.
Gianluigi Oliveri obtained a laurea (BA) in philosophy from the University of Bari
(Italy). After that, he received a DPhil in philosophy from the University of Oxford
(GB) and a PhD in cognitive science from the University of Messina (Italy).
His research interests range from the philosophy of mathematics, artificial intelli-
gence, and the philosophy of science to metaphysics and the history of philosophy.
He has held teaching and research posts at the universities of Reading, Leeds,
Keele, Oxford, and Palermo, and has been a visiting scholar at the Sydney Centre
for the Foundations of Science, University of Sydney, Australia. He is, currently,
Associate Professor of Logic and the Philosophy of Science at the University of
viiviii About the Editors Palermo (Italy); corresponding member of the Accademia Nazionale di Scienze, Lettere ed Arti di Palermo (National Academy of Sciences, Letters and Arts of Palermo); and chairman of the Centro Interdipartimentale per le Tecnologie della Conoscenza (Interdepartmental Centre for the Technologies of Knowledge) at the University of Palermo.
Contributors
N. Barton Fachbereich Philosophie, University of Konstanz, Konstanz, Germany
S. Bianchi University School for Advanced Studies (IUSS), Pavia, Italy
I. Blechschmidt Institut für Mathematik, Universität Augsburg, Augsburg,
Germany
A. Cantini DILEF, Università di Firenze, Firenze, Italy
O. Caramello Dipartimento di Scienza e Alta Tecnologia, Università degli Studi
dell’Insubria, Como, Italy
Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France
D. Catta LIRMM – Montpellier University, Montpellier, France
L. Crosilla Department of Philosophy, IFIKK, University of Oslo, Blindern,
Norway
A. P. d’Aragona Centre Gilles Gaston Granger, Aix-Marseille Univ, CNRS,
Aix-en-Provence, France
J.-P. Marquis Département de Philosophie, Université de Montréal, Montréal, QC,
Canada
S. Maschio Dipartimento di Matematica “Tullio Levi-Civita”, Padova, Italy
E. Moriconi Department of Civilisations and Forms of Knowledge, via P. Paoli 15,
Pisa, Italia
R. Samuels Ohio State University, Columbus, OH, USA
S. Shapiro Ohio State University, Columbus, OH, USA
E. Snyder LMU, Munich, Germany
ixx Contributors D. Waxman Department of Philosophy, National University of Singapore, Singapore L. Zanetti NEtS Center, Department of Humanities and Life Sciences, Scuola Universitaria Superiore IUSS Pavia, Pavia, Italy
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