Models and Idealizations in Science: Artifactual and Fictional Approaches (Logic, Epistemology, and the Unity of Science, 50) 3030658015, 9783030658014

Table of contents :
Preface
About This Book
Contents
Contributors
1 Introduction: Theories, Models, and Scientific Representations
1.1 The Spell of Theories
1.2 Theories and Models
1.3 Two Approaches to the Study Scientific Models
1.4 Models as Representations
1.5 Structuralism and Structural Representation
1.6 Informal Similarity
1.7 Inferentialism
1.8 Idealization
1.9 Models as Fictions
1.10 Models as Epistemic Artifacts
1.11 The Place of Models in Science
References
2 An Artifactual Perspective on Idealization: Constant Capacitance and the Hodgkin and Huxley Model
2.1 Introduction
2.2 Galilean and Minimalist Idealization
2.3 The Artifactual Perspective on Idealization
2.4 The Hodgkin and Huxley Model
2.5 Discussion of Idealizations in the Hodgkin and Huxley Model
2.6 Conclusions
References
3 Informative Models: Idealization and Abstraction
3.1 Introduction
3.2 The Mathematical Theory of Communication (MTC)
3.3 An Analogy: Models as Communication Channels
3.4 Modeling and the Transmission of Information: Some Examples
3.5 Conclusion
References
4 Deidealized Models
4.1 Introduction
4.2 What Should We Understand By Idealization?
4.3 How Idealized Models Are Built
4.4 Costs and Benefits of Idealization
4.5 Deidealizing Models
4.6 Cost and Benefits of Deidealization
4.7 Can All Models Be Deidealized?
4.8 Conclusion
4.9 Afterword: Idealization without Representation
References
5 Scientific Representation as Ensemble-Plus-Standing-For: A Moderate Fictionalist Account
5.1 Introduction
5.2 Settling the Analysis
5.3 The EPS Account: A Simple, Non-Fictionalist Version
5.4 EPS: The Complete, Moderate Fictionalist Version
5.5 Concluding Remarks
References
6 Seven Myths About the Fiction View of Models
6.1 Introduction
6.2 The Fiction View of Models
6.3 First Myth: Fictions Are Falsehoods
6.4 Second Myth: Fictional Models Are Data-Free
6.5 Third Myth: The Fiction View Is Antithetical to Representation
6.6 Fourth Myth: Fiction Trivializes Epistemology
6.7 Fifth Myth: The Fiction View Is Antithetical to Mathematisation
6.8 Sixth Myth: Fiction Misconstrues the Function of Models
6.9 Seventh Myth: The Fiction View Stands on the Wrong Side of Politics
6.10 Conclusion
References
7 Bridging the Gap: The Artifactual View Meets the Fiction View of Models
7.1 Introduction
7.2 The Fiction View of Models
7.3 The Artifactual View of Models
7.4 The Integrated Fiction View of Models
References
8 Models as Hypostatizations: The Case of Supervaluationism in Semantics
8.1 Introduction
8.2 Realism and Irrealism About Fictional Characters
8.3 Models as Fictions: An Illustration from Semantics
8.4 Conclusion: Models and Fictions
References
9 Structural Representation and the Ontology of Models
9.1 Introduction
9.2 Structural Representation: Central Features
9.2.1 Partial Structures
9.2.2 Partial Structures and Structural Representation: A Challenge
9.3 Representation Without Set-Theoretic Structures: The Ontology of Models Set Free
9.3.1 Modalizing Set-Theoretical Structures: A Modal-Structural Interpretation
9.3.2 Partial Structures and Second-Order Logic
9.3.3 Set Theory and Metaphysical Interpretations
9.3.4 Ontologically Neutral Quantification
9.4 Conclusion
References
10 Representation and Surrogate Reasoning: A Proposal from Dialogical Pragmatism
10.1 Introduction
10.2 Representation and Surrogate Reasoning: General Insights
10.3 What is to Represent?
10.4 A Ludic-Dialogical Approach to Representation: Interaction and Applicability of Logic
10.5 Interaction and Language Games in Logic: Origins of Another Paradigm
10.5.1 Dialogical Logic
10.6 The Dialogical take of Representation
10.6.1 Representational Force of the Model
10.7 Final Remarks
Appendix: Standard Dialogical Logic
SR1 (Classical Game-Playing Rule)
References
11 Prediction and Explanation by Theoretical Models: An Instrumentalist Stance
11.1 Introduction
11.2 Causal Explanation and Theoretical Explanation
11.3 Inter-Theoretical Explanation by Theoretical Models
11.4 Predicting New Elements: From Mendeleev’s Table to the Atomic Shell Model
11.5 Conclusion
References
12 Commented Bibliography on Models and Idealizations
References
Name Index
Subject Index

Citation preview

Logic, Epistemology, and the Unity of Science 50

Alejandro Cassini Juan Redmond   Editors

Models and Idealizations in Science Artifactual and Fictional Approaches

Logic, Epistemology, and the Unity of Science Founding Editor John Symons

Volume 50

Series Editor Shahid Rahman, Domaine Universitaire du Pont du Bois, University of Lille III, Villeneuve d’Ascq, France Managing Editor Nicolas Clerbout, Universidad de Valparaíso, Valparaíso, Chile Editorial Board Jean Paul van Bendegem, Gent, Belgium Hourya Benis Sinaceur, Techniques, CNRS, Institut d’Histoire et Philosophie des Sci, Paris, France Johan van Benthem, Institute for Logic Language & Computation, University of Amsterdam, Amsterdam, Noord-Holland, The Netherlands Karine Chemla, CNRS, Université Paris Diderot, Paris, France Jacques Dubucs, Dourdan, France Anne Fagot-Largeault, Philosophy of Life Science, College de France, Paris, France Bas C Van Fraassen, Department of Philosophy, Princeton University, Princeton, NJ, USA Dov M. Gabbay, King’s College, Interest Group, London, UK Paul McNamara, Philosophy Department, University of New Hampshire, Durham, NH, USA Graham Priest, Department of Philosophy, Graduate Center, City University of New York, New York, NY, USA Gabriel Sandu, Department of Philosophy, University of Helsinki, Helsinki, Finland Sonja Smets, Institute of Logic, Language and Computation, University of Amsterdam, Amsterdam, Noord-Holland, The Netherlands Tony Street, Divinity College, Cambridge, UK

Göran Sundholm, Philosophy, Leiden University, Leiden, Zuid-Holland, The Netherlands Heinrich Wansing, Department of Philosophy II, Ruhr University Bochum, Bochum, Nordrhein-Westfalen, Germany Timothy Williamson, Department of Philosophy, University of Oxford, New College, Oxford, UK

Logic, Epistemology, and the Unity of Science aims to reconsider the question of the unity of science in light of recent developments in logic. At present, no single logical, semantical or methodological framework dominates the philosophy of science. However, the editors of this series believe that formal frameworks, for example, constructive type theory, deontic logics, dialogical logics, epistemic logics, modal logics, and proof-theoretical semantics, have the potential to cast new light on basic issues in the discussion of the unity of science. This series provides a venue where philosophers and logicians can apply specific systematic and historic insights to fundamental philosophical problems. While the series is open to a wide variety of perspectives, including the study and analysis of argumentation and the critical discussion of the relationship between logic and philosophy of science, the aim is to provide an integrated picture of the scientific enterprise in all its diversity. This book series is indexed in SCOPUS. For inquiries and submissions of proposals, authors can contact Christi Lue at [email protected]

More information about this series at http://www.springer.com/series/6936

Alejandro Cassini · Juan Redmond Editors

Models and Idealizations in Science Artifactual and Fictional Approaches

Editors Alejandro Cassini Department of Philosophy Universidad de Buenos Aires Ciudad de Buenos Aires, Argentina

Juan Redmond Institute of Philosophy Universidad de Valparaíso Valparaíso, Chile

ISSN 2214-9775 ISSN 2214-9783 (electronic) Logic, Epistemology, and the Unity of Science ISBN 978-3-030-65801-4 ISBN 978-3-030-65802-1 (eBook) https://doi.org/10.1007/978-3-030-65802-1 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved 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

Preface

The first ideas concerning this book originated when the editors jointly organized a Workshop on Models and Idealizations in Science that took place in Valparaíso, Chile, on 29th–30th August 2016. Although the Workshop was not specifically devoted to the artifactual or the fictional approaches to scientific models, we noticed that these were two significant topics that recurred once and again in the questions and discussions that followed the different presentations. We also realized that in the growing philosophical literature on scientific models there was no book or collection of articles in which those two approaches were compared and assessed. The present book is intended as a contribution to the development and epistemological assessment of the artifactual and fictional approaches to models. The book is not based on the presentations to the 2016 workshop. Tarja Knuuttila and Mauricio Suárez participated in that event but only Suárez and Bolinska’s contribution to this book reflects in part Suárez 2016 presentation. The rest of the chapters have been written especially for this book and have not been previously published. Each contribution was written independently from the others and we have in this way avoided cross-references between the different chapters. The aforementioned workshop showed us that the philosophy of scientific modeling had undergone quick developments in recent years. The pervasive use of models and idealizations in all sciences was slowly, and lately, acknowledged by philosophers of science since the past two decades of the twentieth century. By contrast, during the first two decades of the twenty-first century, there was an explosion of different kinds of philosophical studies regarding scientific modeling. The ontology of models and their different functions in scientific practices were submitted to close scrutiny. Many paradigmatic historical examples of models—from Bohr’s atomic model and the liquid drop model of the atomic nucleus to the Lotka–Volterra predator–prey model, to mention just three of the most popular—were revisited from different philosophical and epistemological points of view. The significance of computer simulation models, in turn, was widely recognized, to the point of conceiving of the present situation of science as the era of computer simulations. In recent years, the literature on scientific models was enriched with the publication of several monographic books and collections of articles devoted to different aspects of scientific modeling, as can be seen in the commented bibliography included at v

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the end of this book. However, philosophers of science have not yet reached an agreement concerning the precise terminology to be used in dealing with scientific models and, for that reason, the vagueness and ambiguity of some key terms, such as “model system”, “target”, “abstraction”, and “idealization”, among many others, are still a hindrance for the communication between philosophers of different persuasions. Moreover, how artifactualism and fictionalism concerning models are to be understood is dependent on the meaning assigned to the concepts of artifact and fiction, which do not come from the philosophy of science but from disciplines such as metaphysics, aesthetics, and the philosophy of technology. Last, but not least, the ubiquitous concept of representation, one of the most elusive in Modern philosophy, has defied the many attempts at elucidation in the many philosophical disciplines in which were employed, such as the philosophies of language, mind, and art. This book also aims at contributing to the clarification of these and other concepts that belong to the toolkit with which philosophers of science address such questions as what models are, what they are used for, and how they represent—if they do it—the phenomena we encounter in the real world. The different chapters of the book have been ordered according to the following criteria. The first chapter is a general introduction to the many topics concerning scientific modeling addressed in the remaining chapters. The next three chapters (2, 3, and 4) deal mainly with the concept of idealization and the artifactual approach to models. The next four chapters (5, 6, 7, and 8) deal with the fictional view of models. The remaining chapters (9, 10, and 11) are about other issues concerning modeling, such as structural representation, surrogative reasoning, and the explanatory and predictive use of models. The last chapter is a commented bibliography on the philosophy of models, idealizations, and related topics. The content of each chapter is the following. In the first chapter of this book, Alejandro Cassini and Juan Redmond offer a rather elementary but fairly complete and extensive introduction to the present state of the philosophy of scientific models. It was written with the specific purpose of providing the readers with an accessible account of the main topics that have been discussed and elaborated by the most distinguished philosophers of science in the past two decades. It also provides a brief historical narrative of the rise and the early development of the philosophy of scientific models since the middle of the twentieth century. The commented bibliography at end of the book complements this narrative by offering a classified list of the main relevant books on models and idealizations in science preceded by short commentaries intended to guide the search for further readings on the different topics. Professional philosophers of science or readers acquainted with the philosophical literature on scientific models may skip those materials and proceed directly to the chapters of their interest. All chapters are self-contained and mutually independent, although some of them are more closely related than others. In Chap. 2, Natalia Carrillo and Tarja Knuuttila assert that there are two traditions of thinking about idealization offering almost opposite views on their functioning and epistemic status. While one tradition views idealizations as epistemic deficiencies, the other one highlights the epistemic benefits of idealization. Both of them treat idealizations as deliberate misrepresentations, however. They then argue for an

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artifactual account of idealization, comparing it to the traditional accounts of idealization, and exemplifying it through the Hodgkin and Huxley model of the nerve impulse. From the artifactual perspective, the epistemic benefits and deficiencies introduced by idealization frequently come in a package due to the way idealization draws together different resources in model construction. Accordingly, idealization tends to be holistic in that it is not often easily attributable to some specific parts of the model. They conclude that the artifactual approach lends a unifying view into idealization in that it is able to recover several basic philosophical insights motivating both the deficiency and epistemic benefit accounts, while being simultaneously detached from the idea of distortion by misrepresentation. In Chap. 3, Mauricio Suárez and Agnes Bolinska apply the tools of communication theory to scientific modeling in order to characterize the informational content of a scientific model. They argue that when represented as a communication channel, a model source conveys information about its target, and that such representations are therefore appropriate whenever modeling is employed for informational gain. They then extract two consequences. First, the introduction of idealizations is akin in informational terms to the introduction of noise in a signal; for in an idealization we introduce ‘extraneous’ elements into the model that have no correlate in the target. Second, abstraction in a model is informationally equivalent to equivocation in the signal; for in an abstraction we ‘neglect’ in the model certain features that are obtained in the target. They conclude that it becomes possible in principle to quantify idealization and abstraction in informative models, although precise absolute quantification will be difficult to achieve in practice. In Chap. 4, Alejandro Cassini analyses how highly idealized theoretical models can be deidealized. He argues that idealized models are built with a definite purpose and for that reason, the advantages and disadvantages of idealizing depend essentially on the specific purpose for which a given model is designed. As a consequence, even when deidealization may be feasible, a cost–benefit analysis may suggest avoiding it. He exemplifies those circumstances with a study of deidealized models of the Solar System and physical pendula. He concludes that deidealization has not to be conceived of as an end in itself, or as aiming at a veridical representation of the phenomena, but rather as a means to other ends, such as obtaining better explanations or predictions, or more generally, improving the expediency of our models to solve the problems that originated their construction. In Chap. 5, José A. Díez examines the reasons for claiming that models are fictions. He affirms that it has been argued that, in order to account for some key features of the practice of modeling in science, such as the existence of unsuccessful representations and also of successful yet inaccurate or idealized ones, it is necessary to accept fictional entities. In opposing such a view, he sketches an account of scientific modeling and argues that according to such account there is no need for strong factionalism but only a weak, unproblematic fictionalist component is needed. In Chap. 6, Roman Frigg and James Nguyen present a detailed statement and defense of the fiction view of scientific models, according to which they are akin to the characters and places of literary fiction. They argue that this variety of fictionalism does not regard the products of science as falsehoods; allows that data can be part of

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models; properly understood (as an account of the ontology of models, rather than their function) is perfectly compatible with the fact that they represent; allows that we can learn important truths about the world from models; allows for this to involve mathematics; does not misconstrue the function of scientific models in practice; and finally is no more politically problematic than many other philosophical and scientific ideas. In Chap. 7, Fiora Salis compares the artifactual and the fictional views of models. She argues that both accounts contain several deep insights concerning the nature of scientific models but they also face some difficult challenges. She then puts forward an account of the ontology of models intended to incorporate the benefits of both views avoiding their main difficulties. Her key idea is that models are human-made artifacts that are akin to literary works of fiction. In this view, models are complex objects that are constituted by a model description and the model content generated within a game of make-believe. As per the artifactual view, model descriptions are construed as concrete representational tools that enable and constrain a scientists’s cognitive processes and provide intersubjective epistemic access to their imaginings. As per the fiction view, model descriptions are construed as props in a game of make-believe, where props are concrete objects prescribe certain imaginings. In Chap. 8, Manuel García Carpintero defends a form of antirealism for the explicit talk and thought both about fictional entities and scientific models: a version of Stephen Yablo’s figuralist brand of fictionalism. He argues that in contrast with pretense-theoretic fictionalist proposals, on his view, utterances in those discourses are straightforward assertions with straightforward truth-conditions, involving a particular kind of metaphors or figurative manner. But given that the relevant metaphors are all but ‘dead’, this might suggest that the view is after all realist, committed to referents of some sort for singular terms in the relevant discourses. He revisits these issues from the perspective of the more recent work on them and applies his view to recent debates in semantics on the role and adequacy of supervaluationist models of indeterminacy. In Chap. 9, Otávio Bueno introduces the structural account of representation through partial structures and examines its ontological commitments. He points out that structural approaches to scientific representation emphasize the crucial role played by structures in representing salient features of the world. It is common to present and, in some cases, even to reify such structures as abstract entities, in particular as set-theoretic constructs. Against this view, he argues that no such reification is called for and that several strategies can be articulated to avoid commitment to set-theoretic structures in the defense of structural representation. He concludes that one can favour the latter without being platonist about the former. In Chap. 10, Juan Redmond presents an inferential conception of scientific representation based on the ludic perspective of dialogical pragmatism. He conceives of his proposal as an answer to the question “How models are used to represent the world?” Consequently, he lines up with the host of pragmatic approaches that stress the importance of the notion of use and users for representing and modeling. The main point of his proposal is that users or agents use models to represent their targets

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doing inferences concerning those targets on the basis of reasoning dialogically about the models. In Chap. 11, Andrés Rivadulla argues for an instrumentalist approach to the use of theoretical models in the physical sciences, which, on his view, have not to be conceived of as intended representations of the phenomena, but just as useful tools for explaining and predicting those phenomena. He analyses two examples of theoretical models employed for those aims. The first one is the supernova model, intended mainly as explanatory. The second one is the atomic central field shell model, where the postulated internal structure for atoms—their electronic configuration—made the prediction and discovery of new noble gases possible. He concludes that theoretical models can be regarded as artifacts designed with the specific purpose of facilitating calculations. In Chap. 12, Alejandro Cassini provides a classified and commented bibliography of printed books on the philosophy of scientific modeling and related issues such as representation, idealization, computer simulation, and others. The book’s editors have normalized the spelling of several key terms that admits of a double spelling—such as modeling, artifact, idealization, and practice—to simplify the index of subjects and facilitate the reading of the different chapters. For reasons of coherence, many other terms were normalized to the American spelling, except when they appeared in literal citations or quotations. Any mistake or inconsistency in the spelling of all those terms is entirely our responsibility. Ciudad de Buenos Aires, Argentina Valparaíso, Chile

Alejandro Cassini Juan Redmond

About This Book

This book is intended both as an introduction to the philosophy of scientific modeling and as a contribution to the discussion and clarification of two recent philosophical conceptions of models: the artifactual and the fictional views. The first chapter provides a rather elementary but fairly complete and extensive introduction to the present state of the philosophy of scientific models. It also offers a brief historical narrative of the rise and the early development of the philosophy of scientific models since the middle of the twentieth century. The commented bibliography at end of the book complements this narrative by offering a classified list of the main relevant books on models and idealizations in science preceded by short commentaries intended to guide the search for further readings on the different topics. The rest of the book is a collection of previously unpublished articles by different philosophers of science, who deal with a wealth of topics concerning models and idealizations in science. Among the many issues they address, it can be mentioned the artifactual view of idealization, the use of information theory to elucidate the concepts of abstraction and idealization, the deidealization of models, the nature of scientific fictions, the fiction view of models defended from its critics, the structural account of representation and the ontological status of structures, the role of surrogative reasoning with models, and the use of models for predicting and explaining physical phenomena.

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Contents

1

Introduction: Theories, Models, and Scientific Representations . . . . Alejandro Cassini and Juan Redmond

2

An Artifactual Perspective on Idealization: Constant Capacitance and the Hodgkin and Huxley Model . . . . . . . . . . . . . . . . . Natalia Carrillo and Tarja Knuuttila

1

51

3

Informative Models: Idealization and Abstraction . . . . . . . . . . . . . . . . Mauricio Suárez and Agnes Bolinska

71

4

Deidealized Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alejandro Cassini

87

5

Scientific Representation as Ensemble-Plus-Standing-For: A Moderate Fictionalist Account . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 José A. Díez

6

Seven Myths About the Fiction View of Models . . . . . . . . . . . . . . . . . . 133 Roman Frigg and James Nguyen

7

Bridging the Gap: The Artifactual View Meets the Fiction View of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Fiora Salis

8

Models as Hypostatizations: The Case of Supervaluationism in Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Manuel García-Carpintero

9

Structural Representation and the Ontology of Models . . . . . . . . . . . 199 Otávio Bueno

10 Representation and Surrogate Reasoning: A Proposal from Dialogical Pragmatism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Juan Redmond

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Contents

11 Prediction and Explanation by Theoretical Models: An Instrumentalist Stance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Andrés Rivadulla 12 Commented Bibliography on Models and Idealizations . . . . . . . . . . . 249 Alejandro Cassini Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

Contributors

Agnes Bolinska Department of Philosophy, University of South Carolina, Columbia, SC, USA Otávio Bueno Department of Philosophy, University of Miami, Coral Gables, FL, USA Natalia Carrillo University of Vienna, Vienna, Austria Alejandro Cassini Department of Philosophy, University of Buenos Aires, Buenos Aires, Argentina José A. Díez Department of Philosophy, University of Barcelona, Barcelona, Spain Roman Frigg Department of Philosophy, Centre for Philosophy of Natural and Social Science, London School of Economics and Political Science, London, UK Manuel García-Carpintero University of Barcelona, Barcelona, Spain Tarja Knuuttila University of Vienna, Vienna, Austria James Nguyen Centre for Philosophy of Natural and Social Science, London School of Economics and Political Science, London, UK; Institute of Philosophy, School of Advanced Study, University of London, London, UK; Department of Philosophy, University College London, London, UK Juan Redmond Universidad de Valparaíso, Valparaíso, Chile Andrés Rivadulla Complutense University of Madrid, Madrid, Spain Fiora Salis Department of Philosophy, University of York, Heslington, York, UK Mauricio Suárez Faculty of Philosophy, Department of Logic and Theoretical Philosophy, Complutense University of Madrid, Madrid, Spain

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Chapter 1

Introduction: Theories, Models, and Scientific Representations Alejandro Cassini and Juan Redmond

Abstract Alejandro Cassini and Juan Redmond offer an elementary but fairly complete and extensive introduction to the present state of the philosophy of scientific models. It was written with the purpose of providing the readers an accessible account of the main topics that have been discussed and elaborated on by the most distinguished philosophers of science in the last two decades. It also provides a brief historical narrative of the rise and the early development of the philosophy on scientific models since the middle of the twentieth century. Keywords Theories · Models · Representation · Idealization This introductory chapter aims at providing the reader with the minimal background knowledge required to understand the fictional and the artifactual conceptions of models, which are the leitmotif of the remaining chapters of this book. The chapter also offers a sketch of the philosophical context from which those positions have emerged. Besides those conceptions of models, it deals with several related topics such as the concept of idealization and the different accounts of scientific representation. However, it is not intended as a general overview of the philosophy of scientific models or as a survey of all the different positions concerning representation and idealization. That would be the subject-matter of a whole book. This introduction is almost exclusively focused on the developments of the philosophy of models in the twenty-first century. Nonetheless, it also includes a brief historical sketch of the history of the subject in the second half of the twentieth century that provides the context from which the present philosophical stances on scientific models have arisen. The first section of the chapter is not about models but theories because theories were almost the exclusive issue with which classical philosophers of science, A. Cassini (B) University of Buenos Aires, Buenos Aires, Argentina e-mail: [email protected] J. Redmond Universidad de Valparaíso, Valparaíso, Chile © Springer Nature Switzerland AG 2021 A. Cassini and J. Redmond (eds.), Models and Idealizations in Science, Logic, Epistemology, and the Unity of Science 50, https://doi.org/10.1007/978-3-030-65802-1_1

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A. Cassini and J. Redmond

since the 1930s were concerned with. The classical conception of empirical theories, conceived of as deductive calculi interpreted by means of correspondence rules, was the main outcome of this initial interest. The second section briefly explains how a growing concern with models arose from an alternative conception of theories, the so-called semantic view, according to which theories are identified with collections of related models. The third section presents two approaches to the study of models, the ontological and the pragmatic, which are still nowadays influent through the works of many philosophers of science. The fourth section introduces the representational conception of models and explains some of its difficulties and challenges. The next sections are devoted to the three of the main influential conceptions of scientific representation. The fifth section deals with structuralism and the structural account of representation. The sixth section is about the informal similarity view of representation. The seventh section deals with the inferential conception of representation. The eighth section is an account of the concept of idealization, whose study has been largely independent of that of representation. The ninth section is devoted to the different varieties of the fictional conception of models, which are all based on the analogy between scientific models and literary works of fiction. The tenth section offers a short account of the artifactual view of models. The eleventh and last section contains some general reflections on the place of models in science and the prospects for their further study. We have made our best efforts to offer a clear and concise presentation of the many topics included in this chapter although we know that each one would have deserved more detailed treatment. The references cited at the end of this chapter provide information concerning the main sources of the different approaches discussed. The commented bibliography placed as the last chapter of this book offers a more extended guide for further reading on most issues related to the philosophy of scientific models. We have not intended to write an entirely neutral account of those issues, rather, we have provided a reconstruction of several tenets from our standpoint and, sometimes, we have pointed out to difficulties and unsolved problems the different views have to face. We hope that this rather long introduction be useful to initiate the reader in the complex domain of the philosophy of scientific models, one of the liveliest ones in the present philosophy of science.

1.1 The Spell of Theories Philosophy of science as a professional discipline slowly arose since the beginning of the twentieth century and was consolidated by the 1930s, mainly through the collective work of the members of the Vienna Circle and other European philosophers. Around the 1940s the discipline was firmly established in the United States, where many emigrate from Europe, both scientists and philosophers, managed to obtain academic positions. The first wave of European philosophers of science in the early decades of the century was not integrated by philosophers, but by working scientists, such as Ernst Mach, Henri Poincaré, Pierre Duhem, Norman Campbell,

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and Percy Bridgman, among many others. The second wave brought to the scene the first generation of professional philosophers of science, such as Rudolf Carnap, Hans Reichenbach, Karl Popper, and Carl Gustav Hempel, to mention just a few of the most distinguished. All of them started by writing their first works in the German language, but they emigrated to English-speaking countries, the United Kingdom and the United States, by the middle of the 1930s, soon after the establishment of the Nazi regime in Germany. For that reason, they produced most of their fundamental works in the English language, a language to which their first works were lately translated. As a consequence of the massive emigration of the German-speaking philosophers of science in the 1930s, the technical terminology of the newly born discipline of the philosophy of science, which started to be coined in German, was established in the English language, where it has remained entrenched until present days. This terminology did not include the term “model” as one of its keywords. The first generation of professional philosophers of science was concerned mainly, if not exclusively, with theories. The logical analysis of the structure of physical theories captured almost all their attention. The reasons for this absorbent interest are clear enough in retrospect. As the history of science shows, the emergence of wide-ranging and fundamental theories, such as Copernican astronomy, Newtonian mechanics, or Darwinian evolutionary biology, are rare and rather sparse episodes in the development of science. Nonetheless, the first three decades of the twentieth century witnessed the unexpected irruption of two (or three) of the boldest and most revolutionary theories in the entire history of science: Einstein’s special and general theories of relativity, still classical theories, and the much more disruptive theory of quantum mechanics, whose meaning was unclear from its very inception. The theories of relativity and the new quantum mechanics exerted a deep influence on the philosophers of the Vienna Circle and other European philosophers of science, who followed with enormous interest the rapid development of the physical science. It is then understandable that both theories were the main subject of the logical and philosophical analysis promoted by the logical empiricist philosophers. Each of those theories posed different kinds of problems. The special theory of relativity was presented by Einstein in 1905, in a clean and almost complete formulation, which rested on two simple postulates: the principle of relativity and the principle of the constancy of the speed of light. Nonetheless, it was still an open question whether those postulates were enough to deduce all the new physical results of the theory. Einstein’s formulation of his theory was not strictly axiomatic, at least, it did not satisfy the strictures of the new formal methods developed by David Hilbert and Bertrand Russell, which were taken as paradigmatic examples of logical clarity and rigor by the logical empiricists. Thus, a logical reconstruction of the special theory of relativity, to which Reichenbach devoted his first books, was regarded as a compelling task.1 The general theory of relativity, whose confirmation by Eddington in 1919 launched Einstein’s worldwide legend, was less clear in its foundations. Einstein did not present the theory based on principles when he published its definitive 1 For

instance, Reichenbach (1924), a work that did not make a very good impression on Einstein.

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formulation in 1916. Two years later, he claimed that the theory was based on three foundations: the principle of relativity, the principle of equivalence (between inertia and gravitation), and the Mach principle, according to which the gravitational field “is completely determined by the masses of the bodies” (Einstein 1918: 241). But those did not seem to be the postulates from which Einstein’s field equations were derived. The task of securing the foundations of the general theory of relativity by providing a sound axiomatization of it turned out to be a very difficult enterprise, which was never accomplished. In any event, it looked by the 1920s as a promising venture. On the other hand, quantum mechanics posed very different problems. In the first place, its development showed the kind of collaborative enterprise that the logical empiricists promoted for the construction of a “scientific philosophy”. However, the theory, if it was a theory at all, was not presented in a clear a straightforward formulation comparable with that of the special theory of relativity. In the beginning, the theory did not possess even a unified mathematical language. Dirac’s notation and Von Neumann’s mathematical formulation of the theory in terms of the Hilbert space formalism provided a common language to all physicists by the first years of the 1930s.2 This accomplishment, however, did not solve the problem of interpreting the theory, which remained open until our days. The deep disagreements among the physicists concerning the physical meaning of the new quantum theory, whose paradigmatic example was the famous Einstein-Bohr debate, exerted a great fascination on the philosophers of science at the very moment in which the philosophy of science was starting to be established as an academic discipline. The spell of quantum mechanics extended far beyond the logical empiricists or the philosophers of science in general. Heidegger (1954: 4), for instance, claimed that “science does not think” (although “it has to do with thinking”), but exempted from the reach of this infamous dictum those scientists who debated the meaning of the quantum mechanics. He explicitly acknowledged that “the present leaders of atomic physics, Niels Bohr and [Werner] Heisenberg, think in a thoroughly philosophical way” (Heidegger 1961: 51).3 The great influence exerted by the theories of relativity and the quantum mechanics on the first generation of professional philosophers of science explains, at least to a great extent, that they regarded the logical analysis of the structure of physical theories as one of the fundamental tasks of the new discipline of the philosophy of science. The guidelines for that analysis were provided by Hilbert’s rigorous re-axiomatization of Euclidean geometry (Hilbert 1899). The ideal consisted in presenting all physical theories employing a small set of independent axioms written in a well-regimented formal language. There was no doubt that the ideal was achieved in Hilbert-style mathematics, but it was not obvious at all that it could be achieved in physics or empirical sciences generally. After all, physical theories were not written in formal languages and, with some noticeable exceptions, such as Newton’s mechanics 2 We

refer to the classic books by Dirac (1930) and Von Neumann (1932). remarks were stated in the reverse order: the first (Heidegger 1954), belongs to a course of the years 1951–1952; the second (Heidegger 1961), to a course of the years 1934–1935.

3 Those

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or Einstein’s special relativity, they were not presented by means of axioms or postulates. That is the reason why, according to Carnap and the logical empiricists, physical theories required a “rational reconstruction”, in which their logical foundations were clearly established. The logical reconstruction of theories had to be guided by such epistemic virtues as simplicity, clarity, and logical rigor. It had to start with the conceptual analysis of the language of science, and then proceed step by step in reconstructing all theories as axiomatic systems within a logically regimented formal language. The outcome of the logical empiricist analysis of physical theories was what we will call the classical conception of theories. In different moments, it was called the “standard conception”, the “received view”, the “statement view” and “the syntactic view” of theories. Those tags are still in use but they cannot be regarded nowadays as adequate descriptions of this view of theories. In the first place, because the classical view is not anymore the standard conception of theories and, for that reason, it is not the received view for present philosophers of science. In the second place, because it is not a syntactic conception of theories, as we will see later. Carnap’s work could summarize the development of the classical conception: its first complete formulation can be found in his Foundations of logic and mathematics (Carnap 1939), a short monography written for the Encyclopedia of unified science, and its last reformulation can be found in his long article “The methodological character of theoretical terms” (Carnap 1956). By the end of the 1950s, the classical conception of empirical theories was at bay, because of the persistent criticisms by different analytical philosophers and by historically oriented philosophers of science. The paradigm for the classical conception of theories was the axiomatic method of the mathematical sciences. According to it, a theory is the set of all the logical consequences of a collection of axioms written in a specified language. More simply, a theory is the set of all the theorems that can be derived from the axioms. In symbols: T = Cn (A). When a theory is not axiomatized, it can be defined as the logical closure of a given set of formulae (or sentences) from its language. This is the so-called logical conception of a theory, which seems to be perfectly adequate for mathematical theories. For instance, we can say that Euclidean geometry is the set of all the theorems that follow from Hilbert’s (1899) axioms. However, it is not obvious that this conception of theories could be appropriate for the formulation of empirical theories. According to the classical conception, what distinguishes a purely formal theory from an empirical theory is a new component of the latter that was called rules of correspondence. An empirical theory is then the set of the logical consequences of the union of two different sets of sentences: the theoretical postulates (P) and the rules of correspondence (C). In symbols: T = Cn (P ∪ C). Some specifications are required to properly spell out this last symbolic expression. According to the classical conception of theories, all empirical theories had to be formulated in a formal regimented language L, whose vocabulary included two different kinds of symbols. On the one hand, the logical and mathematical symbols, which were endowed with a fixed logical meaning and, for that reason, were no susceptible to interpretation. On the other hand, the descriptive symbols (essentially predicate symbols) were divided into two exclusive classes: observational terms and

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theoretical terms. In the light of the empiricist view of semantics, within which the distinction was conceived, observational terms were conceived of as having a direct meaning provided by the experience of our senses because those were the terms that referred to perceptible objects, events, and properties. Theoretical terms, by contrast, were conceived of as having no direct meaning because they were the terms that intended to refer to objects, events, or properties not perceivable by unaided human senses. Terms such as “red” or “hot” were the standard examples of observational terms, whereas “electron” or “spin” (in its quantum–mechanical use) were the standard examples of theoretical terms. According to the classical conception, the axiomatic basis of an empirical theory is composed of the theoretical postulates, that is, sentences which contain, besides logical terms, solely theoretical terms as descriptive terms, and the rules of correspondence, which are mixed sentences, that is, sentences that contain at least one theoretical term and at least one observational term as descriptive terms. Purely observational sentences, with which the theory is to be tested, do not need to be included as part of the axiomatic basis, because they logically follow from the theoretical postulates and the rules of correspondence. If so, what distinguishes a purely formal theory from an empirical theory are the rules of correspondence. These are essentially semantic rules, whose main function is to give meaning to the theoretical terms that appear in the postulates of the theory. That is why the classical conception of theories is not syntactic at all, but a straightforward semantic conception of empirical theories. From the standpoint of the classical conception, the logical closure of a set of the theoretical postulates is a syntactic axiomatic system, but it is not an empirical theory; it is a purely formal theory. It lacks empirical content because it does not imply any observational sentence and, consequently, it cannot be tested by experience. The classical conception of theories is nothing but a sophisticated version of the old hypothetico-deductive conception of theories developed in the nineteenth century. In one of his popular writings on scientific methodology, Einstein had expressed this conception in the following terms: “basic laws (axioms) and conclusion together form what is called a ‘theory’” (Einstein 1919: 219). Carnap, Reichenbach, and the rest of the logical empiricists were also in this respect Einstein’s followers.4 Although the logical empiricist philosophers of science promoted the logical reconstruction of physical theories by formalizing them in a first-order regimented language, the classical conception of empirical theories is not necessarily tied to those requirements. A theory can either be formalized in a second-order formal language or the language of naive set theory or can be presented in a natural language, enriched with technical terms, without any resource to formalization, as is usual in most sciences. What is essential to the classical conception is that theories should be presented as axiomatic calculi interpreted by correspondence rules. According to the classical

4 For

a more detailed historical account of the development of the classical conception of theories see the extensive introduction to Suppe (1977), which remains useful, although outdated in some respects.

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view, presenting a theory is just presenting its axiomatic basis, that is, its theoretical postulates and its rules of correspondence. Many philosophers of science have misunderstood the classical conception of empirical theories by conceiving of them as uninterpreted axiomatic calculi. In the late 1930s, Carnap himself had made this point clear: “any physical theory [….] can in this way be presented in the form of an interpreted system, consisting of a specific calculus (axiom system) and a system of semantical rules for its interpretation” (Carnap 1939: 60). Or, more explicitly: “a theory must not be a ‘mere calculus’ but possess an interpretation, on the basis of which it can be applied to facts of nature” (Carnap 1939: 68).

1.2 Theories and Models There was a scarce place for models within the classical conception of theories. Of course, working scientists had been building models at least since the middle of the nineteenth century. Maxwell’s mechanical models of the electromagnetic ether are well-known examples of that practice. Some scientists, such as Boltzmann (1902), explicitly acknowledged the importance of models in the mathematical and physical sciences. Moreover, some early philosophers of science, notably Duhem (1906), discussed the use of models in physical sciences and its costs and benefits. Nonetheless, there were very few analytical treatments of the concept of the scientific model by philosophers of science before the 1950s.5 Those philosophers that elaborated on the classical view of theories acknowledged the existence of models, but they assigned secondary importance to them. Rudolf Carnap, for instance, did not conceive of models as interpretations of a formal calculus because any physical theory must be already interpreted. According to him, models are just a means to make intuitive the abstract content of a theory by resourcing to analogies with familiar objects or processes. This is the case of the well-known mechanical models for Maxwell’s equations, which, according to Carnap (1939: 67), are just “a way of representing electromagnetic micro-processes by an analogy to known macro-processes, e.g., movements of visible things”. Models do not provide new interpretations for a given theory and for that reason, they are “not at all essential for a successful application of the physical theory” (1939: 68). Carnap then concluded that “the discovery of a model has no more than an aesthetic or didactic or at best a heuristic value” (1939: 69). This attitude towards models was characteristic of most philosophers of science that endorsed the classical view of the structure of empirical theories. For example, in the early 1950s, Richard Braithwaite discussed at length the distinction between models and theories but just in order to point out the disadvantages and dangers of using models. According to him, the main danger consists in attributing to the 5 Bailer-Jones

(2009) provides an extended account of the early discussions of models by scientists and philosophers of science before 1950. It includes a useful year-by-year bibliography, which shows how the number of publications on the topic increased from the 1960s.

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things in themselves the false aspects of the models. He made the (retrospectively) interesting remark that using models is engaging in a sort of fictionalist attitude because “thinking of theories by means of models is always as-if thinking; hydrogen atoms behave (in certain respects) as if they were solar systems each with an electronic planet revolving round a protonic sun. But hydrogen atoms are not solar systems; it is only useful to think of them as if they were such systems if one remembers all the time that they are not” (Braithwaite 1953: 93). Ernest Nagel, who in the early 1960s published one of the last comprehensive treatises in the general philosophy of science framed within the classical view of theories, acknowledged to models a more significant role: models are one of the “three major components of theories” (Nagel 1961: 90), besides the uninterpreted syntactic calculus and the semantic rules of correspondence. Nagel conceived of models as interpretations of the abstract calculi in familiar terms, whose function is to “supply some flesh for the skeletal structure in terms of more or less familiar conceptual or visualizable materials” (1961: 90). He admitted that physical theories are usually not presented by means of axiomatic calculi interpreted by rules of correspondence but rather, as was the case with Bohr’s atomic theory, they are “embedded in a model or interpretation” (1961: 95). Bohr’s atomic model is just an interpretation of the abstract postulates of his theory. Those kinds of models, such as the billiard-ball model of the kinetic theory of the gases, provide useful “substantive analogies” in terms of familiar concepts that may help us to understand the theory itself. Moreover, they can be useful heuristic devices for the extension and application of theories. Nagel thought that, despite its advantages, the use of models may be dangerous. The dangers Nagel pointed out were the same stressed by Braithwaite: attributing “some inessential feature of the model” to “the theory embedded in it”, and “confusing the model with the theory itself” (1961: 115). As a consequence, he concluded that “a model may be a potential intellectual trap as well as an invaluable intellectual tool” (1961: 115). The treatment of models by Carnap, Braithwaite, and Nagel shows that the classical view of theories had neither an easy way of accommodating the use of models in science nor a clear conception of how theories and models should be related. As Nagel (1961: 95) remarked, “models are not substitutes for rules of correspondence”, which means that the presentation of a theory is complete when the theoretical postulates and the rules of correspondence are presented. If this is so, how models could be the third component of theories? In the last resource, models are as inessential for Nagel as they were for Carnap or Braithwaite. The main function of models is, for all the endorsers of the classical view, merely auxiliary: they are useful heuristic or didactical devices for understanding, extending, or applying theories. For that reason, they cannot be autonomous entities in any way, they are always models of some specific theories and are, so to speak, dependent on them. In the meantime, the concept of a model had a life on its own in the domain of formal logic and formal sciences generally. It became well-established in the field due to the work of Alfred Tarski and other mathematicians since the 1930s. The expression “model theory” to refer to a definite branch of mathematical logic was in use since the 1950s, although a systematic treatment of the topic was not

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accomplished until the early 1970s.6 In present days, model theory is an extremely complex and technical branch of pure mathematics, which has firmly established foundations. The concept of a model in the formal sciences is clear and well-defined so that there is not very much philosophical discussion on the meaning of that concept. By contrast, within the natural and the social sciences, as well as philosophy, this concept is employed with a great diversity of meanings and has a startling variety of uses. We will be concerned here with the different concepts of the model in the field of the empirical sciences, but, because some philosophers of science have claimed that the mathematical concept of a model can be applied to all sciences, we will provide o brief account of the latter concept. From the mathematical point of view, a model is just a set-theoretical structure. A model is then an ordered set that includes one or more domains, which are non-empty sets of objects of any kind, and one or more relations or functions defined on those domains. The structure may include also some distinguished elements of the domain. In symbols, a model is a structure of the form S = D1 , …, Dn , R1 , …, Rm , f 1 , …, f i , a1 , …, ak . A structure does not need to contain all these elements. One domain together with one relation or function or distinguished element is enough to form a structure (so that D, R, D, f , and D, a are three different structures). Notice that any structure, and consequently any model, is a set, whose elements (D, and, R, or f ) are also sets. Moreover, if D is a set of sets, the distinguished elements of D are also sets. From the ontological point of view, a model is always an abstract entity, like any set. A model M of a formal theory T is a structure in which all the theorems of T are true. For instance, the models of group theory are all the structures in which the theorems of group theory are true. Theories that have models are called satisfiable. If a theory is satisfiable, that is, if it has a least one model, it is logically consistent. Inconsistent theories have no models because no structure could verify contradictory sentences. The converse of this relationship does hold for first-order theories, but it does not hold generally; there are consistent second-order theories that have no models.7 The different models of a given mathematical theory may have different formal relations between them. For instance, they can be isomorphic, which means that they share a structure of the same type. If all the models of a satisfiable theory are isomorphic between them, the theory is said to be categorical. Besides isomorphism, there are many formal relations two models may have between them, such as homomorphism, partial isomorphism, embedding, and others.8 It should be noticed that all those formal relations are defined exclusively between set-theoretical structures.

6 Although there were several books on model theory written in the 1960s, Chang and Keisler (1973)

was possibly the first comprehensive treatise on model theory and remained for many years as a work of reference in the field. 7 See Shapiro (1991) for an extensive discussion of second-order theories and their properties. 8 More information on formal relations between structures is given in Sect. 1.5 of this chapter, which deals with structural representation.

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Thus, if we say that two entities, say A and B, are homomorphic (isomorphic, etc.) between them, A and B necessarily have to be set-theoretical structures. The introduction of the concept of a model into the mainstream philosophy of empirical sciences came from the so-called semantic view of theories, which started to be developed in the 1950s with the work of Patrick Suppes. According to Suppes, the mathematical concept of a model, the one employed by mathematical logicians, can also be applied to all empirical sciences. There is just one concept of a model adequate for all sciences.9 Empirical theories are then better identified through a collection of related models than by means of a set of axioms written in a given language. This is the basic tenet of the semantic conception of empirical theories. It was developed in different directions by many philosophers, but it is not necessary to account for all the varieties of this view here.10 What is common to all the different semantic views of theories is the following insight: empirical theories are collections of models related by some specific relation. The different varieties of semanticist philosophers differ between them in two fundamental respects: what models are and which is the relation between the many models of the same theory. For instance, according to the structural conception of theories, developed by Sneed (1971) and Stegmüller (1973), among others, all models of empirical theories are set-theoretical structures, and the relation between them is a formal relation such as isomorphism, homomorphism, and others of that kind.11 In the other extreme of the spectrum, according to Giere (1988, 2006), models are different kinds of entities (such as maps, icons, equations, etc.) used to represent the phenomena and the relation between them is that of informal similarity (in some respects and to a specified degree). These two approaches to theories have different consequences for the identity of empirical theories. In the case of the structural account, theories are well-defined entities because formal relations (such as isomorphism, which is an equivalence relation) demarcate without any vagueness the models that belong to a given theory from the models that do not belong to it. By contrast, in the case of Giere’s account, the informal relation of similarity that unifies the models of one theory admits of degrees and it is not generally transitive so that some models are more or less similar to other models of the theory in question, and some models of the same theory may be not similar to each other. Consequently, as Giere himself pointed out, empirical theories are vague entities, whose limits are not well-defined. This vagueness does not affect mathematical theories, because the set of all the theorems that follow from a given axiomatic basis is, like any set, determined by its members, and, for that reason, it is always well-defined. Notice that, for the same reason, semanticists cannot define empirical theories as consisting of sets (or classes) of models because in that case, 9 The

last chapter of Suppes (1957) a textbook in first-order logic, contains the first comprehensive formulation of the semantic view of empirical theories. Suppes (1969, 1993, 2002) include most of the content of his previous articles on models and theories. 10 One of those varieties of semanticism, not described here, is the state-space account of theories, developed by Van Fraassen (1980, 1989) and Suppe (1989). 11 The most detailed structuralist account of theories is still that of Balzer et al. (1987).

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they would become rigid (and abstract) entities, like any set. They are forced to characterize those theories, in contrast to mathematical theories, as collections of models, in some informal meaning of the term. It is still usual to oppose the semantic view of empirical theories to the supposedly “syntactic view” received from the logical empiricist philosophers. As we have seen, the classical view is also semantic, so that the syntactic-semantic distinction does no capture any difference between the two accounts of theories. A better name for the semantic view would be the model-theoretic view of theories. This name points to a substantial difference from the classical conception, in which models play no role in order to identify a given theory. It is also generally assumed that the classical view is a statement view of theories, that is, that theories are conceived of as linguistic entities, whereas the semantic view is a non-statement view of theories. This, again, is not a useful distinction. In the first place, if, according to the classical view, theories are defined as sets of sentences, they are not linguistic, but abstract entities. Of course, theories are presented using axioms expressed as sentences written in a given language, but this is also true of the models of theories in the semantic view (how could be otherwise?). In the second place, according to the classical view, theories can be defined as sets of propositions, not of sentences or statements. A given formulation of a theory is certainly presented by means of sentences of some language, but one and the same theory can be formulated in different languages using different axiomatic bases. From this point of view, a theory can be defined as the equivalence class of all its formulations, or as the set of all the propositions expressed by anyone of its formulations. In any event, a theory is always an abstract entity. On the other hand, each semantic view of theories includes a linguistic (or propositional) component, that component that the structuralist philosophers call “empirical assertions” and Giere calls “theoretical hypotheses”. These are sentences that relate the models of a given theory with the phenomena to which they are applied. A simple collection of interrelated models is not yet a theory, because such collection by itself does not bear any relationship to the phenomena of our experience. After all, the semantic conception contains some linguistic elements, and for that reason is not a purely non-statement view of theories. Endorsers of the model-theoretic view of empirical theories claim that they conceive of theories as flexible entities, in opposition to the classical view in which theories are rigid entities. According to the classical view, theories cannot be modified without losing their identity, in particular, the addition of one rule of correspondence to the axiomatic basis of a given theory implies unavoidably a change of theory. By contrast, the collection of models associated with a given theory allows for changes that preserve the identity of the theory: some models can be added or removed and this does not imply a change of theory, but just an extension or a restriction of the domain of application of that theory. This fact, according to semanticist philosophers, fits with the practice and the history of science, where we can see that theories are corrected or modified. Changes have to be limited in some way or another because we surely do not admit the replacement of all the models of a theory. Thus, a distinction between nuclear and peripherical models is introduced, so that a theory preserves its

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identity to the extent that its nuclear models are not changed. Theories then are not entirely flexible because any theory must contain a rigid collection of models. Among the costs of the model-theoretic view of empirical theories, we can mention the following. First, it is assumed that models are presented by means of a set of axioms, but the theory is not identified with those specific axioms (and their logical consequences), but with the class of models of such axioms. This presupposes that the set of axioms is consistent otherwise the class of its models would be empty. According to the semantic view, there cannot be inconsistent theories. All inconsistent theories would collapse to the same theory (the empty collection of models) if it could be called a theory at all. Second, some metatheoretical well-established properties of formal and mathematical theories, such as completeness or saturation, cannot be applied to empirical theories. The completeness of a collection of models is simply not defined within the semantic view. By contrast, according to the classical view of theories, there can be inconsistent theories, and the concepts of completeness or saturation are well defined. Third, the semantic view has to be limited to empirical theories because it cannot be applied to mathematical theories. We can perfectly conceive of formal theories without models, and even of consistent (second-order) theories without models. In opposition to this, the classical conception of theories can be applied both to empirical and mathematical or formal theories. The structure of both kinds of theories is essentially identical, being the main difference in the fact that empirical theories have to be formulated using interpreted languages. The semantic view of theories represented the introduction of the concept of a model in the mainstream of the philosophy of empirical sciences. Nonetheless, there are several limitations imposed by this approach to the study of models. Above all, the fundamental concern of semanticist philosophers is still focused on theories. Indeed, they are not only interested in the structure of theories (like logical empiricists were), but also in the dynamics of theories, a topic that includes the intertheoretic relations as well as the change and replacement of theories through the history of science. Nonetheless, theories are still conceived as the fundamental products of science and as the main vehicles of scientific knowledge, which persist as the central unity of epistemological analysis for the philosophy of science. Within the semantic approach, models are, so to speak, subordinated to theories. Models count just as models of a given theory. Models are not autonomous entities or genuine vehicles of knowledge by themselves. They are conceived merely as the ingredients that compose scientific theories and cannot be understood without them. Many pioneering books and articles on scientific models were published before the 1990s, but it was this last decade of the twentieth century that showed the rise of systematic studies concerning the use of models and idealizations in empirical sciences.12 After 1990 models started to be studied by themselves, independently of scientific theories. The book edited by Morgan and Morrison (1999) can be regarded as the starting point of the contemporary approach to scientific models. This work 12 See

Bailer-Jones (2009) for detailed references to the philosophical literature on models from 1950 to 1990. In that period, Hesse (1963) was the only monographic book devoted to scientific models.

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had the effect of freeing the study of models from that of the scientific theories. According to Morgan and Morrison, most scientific models are not models of theories, but new entities that are built out of many heterogeneous components. Principled wide-ranging theories, such as special relativity or quantum field theory, do provide guidance for building some specific models, but this is not always the case. Models are used in many domains of science in which there are no such comprehensive theories or even no theories at all. In the view of Morgan and Morrison, models are often very complex scientific constructs, which make use of several hypotheses belonging to different theories, as well as empirical knowledge (such as statistical regularities and different kinds of data) coming from a variety of sources. Although theories are used to build models, these are often independent of any specific theory. Morgan and Morrison called models autonomous agents and conceived of them as a sort of mediators between the theories and the world. They also pointed out, borrowing the title of Hacking (1983) well-known book, that models are instruments employed both to represent the phenomena and to intervene on them. Cartwright (1999) simultaneously stressed the role of models as mediators between abstract theories and concrete phenomena. According to her, theories do not represent (at least directly) the real world, only models, in which general principles are exemplified, have such representative function.13 By the beginning of the twenty-first century, most philosophers of science already accepted that scientific models are highly independent of theories and are employed to fulfill a great diversity of functions. They also accepted that there exist many distinct types of models. There are purely predictive models, explanatory models, exploratory models, and many others. A given model can be used for many different purposes in many different contexts of application. Model building is pervasive in science, both in domains where there are well-established theories as well as in domains where there are almost no theories at all. This tenet is still the starting point for most philosophical studies on modeling in the sciences. As a consequence of the above-mentioned developments, the relationship between theories and models has become blurred. Scientists and philosophers of science alike do not agree on this question. It can be easily showed that scientists use the term “model” with a startling diversity of meanings. On many occasions, they use “theory” and “model” as synonymous or as interchangeable terms. On other occasions, they reserve the use of “model” to refer to as yet incomplete theories, or to theories of restricted domains of application, or even to provisional or exploratory hypotheses. Some semanticist philosophers of science persist in regarding all models as constitutive components of theories, although they acknowledge that this view cannot account for the many uses of the term “model” within the sciences. By contrast, the majority of the philosophers of science grant that many scientific models are not models of a specific theory. For the moment, there is no general agreement concerning the question of how models and theories are or should be related. Nonetheless, there is a growing consensus on the fact that the scientific practice (or at least that of what 13 It should be noticed that the view of models as mediators between theories and phenomena was pioneered by Bunge (1973).

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we could call “normal science”) does not consist primarily in inventing theories, but rather in building models. The use of models has been widely acknowledged as pervasive both in the theoretical and the experimental sciences, whether the natural or the social sciences.

1.3 Two Approaches to the Study Scientific Models If we were to ask working scientists what models are and how are they used in practice, we surely will obtain a surprising variety of different and conflicting answers. BailerJones (2009) conducted a series of qualitative interviews, which showed precisely this result. Some years later, Gerlee and Lundh (2016) conducted a more varied sample of interviews with scientists of many scientific disciplines with similar results. They reported that “models play a central role in the work of [all] these scientists” and, at the same time, that “it is evident that the concept of a model differs between the disciplines” (2016: 66). Moreover, they noticed that scientists frequently do not distinguish between models and theories and, when they do it, they hold different conceptions concerning how they are related. Over the years, we have inquired many scientists from different disciplines and specializations with, again, the same outcome. On many occasions, scientists call models to those entities the philosophers of science traditionally called theories or hypotheses; even a single conjecture or mathematical equation is sometimes called a model. At least one working scientist told us that models are one of the following things: mathematical equations, diagrams, drawings, graphics, prototypes, layouts, scale artifacts, icons, maps, computer-generated images, computer programs, and computer simulations. Sometimes, we were told that a model was built by combining different elements, such as equations and images. When we inquired what models were used for, the answers varied from the general idea of “solving a problem” to a diversity of functions such as “discovering, exploring or investigating new phenomena”, “describing objects, facts, phenomena or processes”, “explaining facts or phenomena”, “predicting facts, events or phenomena”, “understanding things, facts, phenomena or processes”, “representing objects, facts, phenomena or processes”, “applying a theory or a hypothesis”, “solving the equations of a theory”, “constructing or building a theory”, “testing a theory or a hypothesis”, and many others. It is far from clear how to introduce at least a bit of order into this heterogeneous diversity. Taken at face value, the scientists’ answers show several characteristics of scientific models as they are used in science: first, there is a broad variety of uses of the term “model” to name different kinds of both concrete and abstract objects; second, there is no focal or fundamental meaning of the word that refers to some specific kind of objects; third, models are employed to perform a variety of functions, or, to put it in other terms, they are built to solve very different types of problems; and fourth, there seems to be no fundamental function to which all models can be said to contribute. Perhaps, what could be said about all models is that they permit

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us to have epistemic access to a complex variety of phenomena. Gerlee and Lundh (2016: 66) concluded in the same way that “despite the fact that all the scientists mean different things by the word ‘model’ there is something essential that binds all its uses together. The models provide access to a reality that in almost all cases is of such complexity that it cannot be described or controlled without some prior simplification and abstraction”. What then seems to be common to every scientific model is the fact of being idealized in some way or another and to a certain degree. This property of models is the key to gaining access to phenomena that are too complex to be described or understood in all their details. Facing this heterogeneous class of entities called models and the diversity of the functions they can fulfill, philosophers of science have developed two general approaches to the study of scientific models. The first one is the ontology of models, that is, the attempt at answering the question of what models are and how could be classified according to which kind of entities they are. The general and most obvious answer to this question is that some scientific models are concrete material objects whereas other models are abstract objects. This by itself is not very enlightening but at least it should warn us that it would be very unlikely to reduce the diversity of objects which are called models in the sciences to one kind of entity, such as, for instance, mathematical structures. The second approach is the pragmatics of models, that is, the attempt at determining how models are used in practice by working scientists and which their functions are in different domains of research. Again, the most general and obvious answer is that models are employed to reach most of the aims that philosophers have traditionally attributed to scientific theories or hypotheses: discovering, exploring, describing, explaining, and predicting the phenomena. As we have said, it seems that this diversity of functions can be accounted for by saying that models are used to get epistemic access to the phenomena or to obtain information on the phenomena or, in more general terms, that models are the vehicles of scientific knowledge. This seems acceptable with two provisos. The first is that we should acknowledge that models are not the only vehicle of scientific knowledge; theories, among other things, are also such vehicles. The second is that we should not intend to obtain a complete account of all the functions that models fulfill within the different sciences. This is an excessively ambitious venture, which will be frustrated by the very development of the sciences. Scientists almost surely will build new kinds of models designed to solve new types of problems. For that reason, we could hardly expect to classify all types of models and all the functions they can fulfill. We simply will not have at our disposal a sufficient knowledge of the domain of the objects to be classified. A persistent hindrance for both approaches is the fact that there is no agreement among the philosophers of science concerning how to demarcate the domain of scientific models from other scientific constructs, such as hypotheses, theories, experiments, and computer simulations. The usual strategy was and still is, to characterize the class of the entities to be called models by identifying several paradigmatic examples of successful scientific models. The philosophers’ favorite examples have been contemporary physical models, such as Rutherford’s, Thomson’s, and Bohr’s atomic models, the liquid drop and the shell models of the atomic nucleus, the quark

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model, and the standard model of elementary particles. Many earlier models have been included in the paradigmatic class, such as, for instance, Newton’s gravitational models of the Solar System, Maxwell’s mechanical models of the ether, and the ideal gas model. The Lotka-Volterra model of the prey-predator dynamics and the WatsonCrick double-helix model of the DNA structure have been two of the most frequently mentioned paradigmatic models in the biological sciences. Computational models of climate dynamics are more recent examples. There are many other examples but the aforementioned are enough to determine what philosophers of science have understood by scientific models. Nonetheless, the demarcation problem remains unsolved and we frequently encounter in the literature that different philosophers disagree on the meaning they assign to the terms model and theory. The ontology and the pragmatic of models, of course, are not independent issues. Moreover, they both have a bearing on the question of how models represent the phenomena. The idea one may have concerning how models represent and how they are used in practice can be strongly determined by one’s assumptions on the ontology of models. For instance, if all models are conceived as set-theoretical structures, this fact constraints the relations they can have with the phenomena they are supposed to represent; representation has then to be understood as some kind of formal relationship between structures. However, there is no general one-to-one correspondence between a determinate position concerning the ontology of models and a determinate conception of scientific representation. A specific ontology of models may be compatible with different conceptions of representation, and vice versa. On the other hand, the same kind of object may perform different representative functions and may be used to reach very different aims. Ideally, a complete account of scientific models should provide definite answers to the ontological and pragmatic questions but in practice, philosophers do not proceed that way. Many accounts of scientific representation (as we will see in the later sections) do not rest on a well-defined ontology for scientific models. Conversely, some conceptions of the ontological status of models are not committed to a definite functional approach or a definite account of representation.

1.4 Models as Representations Given the diversity of models that are built in the sciences and the variety of functions for which they are used, a pressing question is the one concerning what they have in common, besides the indisputable fact that they are used to obtain epistemic access to the phenomena. Is there any concept that permits us to unify this plurality? Many philosophers of science have found the answer appealing to the concept of representation. Accordingly, what most scientific models would have in common is the fact that they represent the phenomena. The representational conception of scientific models has become the standard view, or at least the mainstream view, among philosophers of science. It should be noticed, however, that representationalist philosophers do not claim that models are the only vehicles of scientific representation. Which the other

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vehicles are is a disputed issue that depends on the concept of a model employed in each case. Theories, when conceived of as different from models, are sometimes regarded as representational. Drawings, graphs, diagrams, and maps, if they are not regarded as models, are also possible vehicles of representation. On the other hand, representationalists do not claim that all scientific models are representational; they acknowledge that some models are not built or used to represent the phenomena. This is the case, for instance, of purely predictive models, such as statistical climate models or numerical computer simulations. The question of how to distinguish representational from non-representational models has not a clear answer yet, that is, we do not have a general criterion to be applied in all cases. The distinction has to be made case by case taking into account, among other things, the context in which such models are used and the purposes of the agents that build them. In any event, representationalist philosophers do claim that the vast majority of the paradigmatic scientific models -such as all the examples given in the previous section- are intended to represent some phenomena in the real world. There is no agreement on a general definition of the concept of a representational model but this characterization can be taken as fairly comprehensive: a model is an idealized representation of some phenomenon or domain of phenomena. This account employs three key-concepts, which require elucidation: representation, idealization, and phenomena. Let us start with the last one. The concept of a phenomenon is part and parcel of the language of the philosophical tradition, whose origins can be traced back to the ancient Greek philosophy. This is not the place to do a detailed historical analysis, but some remarks will be in order. The original meaning of this term refers to “what appears” to human senses, say, this piece of glass, as opposed to what it is not apparent or it is hidden to our senses, for instance, the atoms of which, supposedly, the glass is made of. Modern scientists use the term in a very broad sense to refer to everything that happens, whether observable or not. The fall of a rock from a mountain and the collision of two electrons are both called physical phenomena by most physicists. Many philosophers, following Kant, have opposed the phenomena, which are the objects of our perception, to the things in themselves, which cannot be perceived or known at all. This is not the standard use of the term in science or the present philosophy of science. We will use it, like most philosophers who are concerned with scientific models, in a wide sense that includes physical systems in general, without discriminating between observable and unobservable entities. This is certainly not a very precise philosophical elucidation of the concept, but it will suffice for our purposes here. Representation, in turn, is one of the fundamental concepts of modern philosophy, which since the seventeenth century has conceived of human knowledge as a representation (Vorstellung) of the real world. In recent times, the concept has been extensively employed and discussed in such fields as the philosophy of language and the philosophy of mind, among other disciplines. After many decades of effort, the ideas of linguistic representation and mental representation have been refractory to precise analysis. The idea of scientific representation was introduced later in the domain of the philosophy of science, around the early 1990s, mainly in connection

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with the view of models as representations. Again, it has been an elusive concept, on the meaning of which no substantial agreement has been reached among philosophers. There is a whole spectrum of positions concerning what a representation is and concerning the properties it has, which cannot be accounted for in a limited space. We will not intend to survey here this issue.14 We just will address some significant questions concerning scientific representations before accounting for specific conceptions of representation. The first question we want to address is whether “to represent” is a binary relation, a ternary relation, or a relation of more than three places. Most philosophers of science have assumed that representation is a two-place relation between a source, that is the object used to represent something, and a target, that is the thing that is represented. In the specific case of scientific models, the model itself (which can be a concrete or abstract object) is the source of the representation and the phenomena that the model represents is the target of the representation. Targets have been characterized, without very much precision, as “selected parts or aspects of the world” (Frigg and Nguyen 2017: 51), or as “abstractions over phenomena” (Weisberg 2013: 90). In any case, targets cannot be conceived of as the things in themselves, nor the realworld phenomena, but as human constructions that do not exist independently of the agents that build the models. It seems pretty clear that selected aspects of the world cannot exist without the agents that select them and that abstractions over phenomena cannot exist without the agents that abstract them. Nonetheless, many philosophers of science still use to speak about the target of a model as if it were a part of the world or a phenomenon that exists by itself. In opposition to that, we believe that both models and targets are human constructs and, for that reason, they have to be regarded as artificial objects. Targets are not parts of nature that are there waiting to be modeled by scientists. Every target is constructed from the phenomena through a complex process that involves redescription and abstraction.15 For the sake of simplicity, we will refer to targets as the modeled phenomena but we should warn that by phenomena we do not mean something that exists by itself in the real world. A typical example of representing is a map that is used to represent a certain territory. Giere (1997, 1999, 2006) has stressed the analogy between scientific models and maps, an analogy according to which a model represents its target as a map represents a territory. Here, representation is a two-place relationship endowed with well-defined logical properties. It seems intuitively clear that maps do not represent themselves, and that territories do not represent maps. As a consequence, if we follow the analogy, the relation of representation between models and their targets should be irreflexive and asymmetric. It is not very common to find a map that represents another map, which in turn represents a territory. If this were the case, we surely will say that the first map does not represent the territory represented by the 14 There is an extensive literature on scientific representation. The main works on the topic are listed

and commented on in the bibliography at the end of this book. 15 This is not the place to elaborate on the details of that process. For a thorough constructivist account of targets, according to which there are no targetless models, see Cassini (2018). We do not assume the details of this specific account here.

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second map. As a consequence, the relation of representation should be intransitive. If we were to define or elucidate the concept of representation using other relational concepts, these must have the right logical properties. This will exclude, as Suárez (2003) has argued, some candidates such as isomorphism (or morphisms in general) and similarity, which are all reflexive relations. Isomorphism, for instance, being an equivalence relation, has none of the three logical properties we intuitively ascribe to representation. The second question we want to address concerns the role of the agents that use models to represent the phenomena. At first sight, we surely think that some representations are correct or adequate, others are misrepresentations, and some things are not representations at all. Nonetheless, it is hard to make sense of these distinctions without appealing to the agents that build the models with some definite purposes in mind. As Callender and Cohen (2006) have pointed out in a very muchdiscussed article, in principle, anything can be taken as the vehicle to represent any target by mere stipulation. For instance, we can use two plastic balls of the same size and different colors to represent two molecules, say of iron and copper, or two planets, say Venus and Mars, or two persons sitting at the table right now. For the same purpose, we can use very different pairs of distinguishable objects, such as two pencils of different colors, or two watches of different shapes, and so on. Can we say that one definite pair of objects provides a better representation of the targets than the other pairs? There is no possible answer to this question without taking into account the purposes for which the models were built. If the model builders wanted to represent the relative spatial distances between the two objects (be they molecules, planets, or persons) anyone of those pairs are suitable sources for the representation. If they wanted to represent the relative sizes of the two objects, two balls of the same volume are not suitable vehicles, but two pencils of different longitude could be. If they wanted to represent, say, the shape and the relative size of Mars and Jupiter, neither the balls nor the pencils are adequate vehicles to do the job; we will need, for instance, two balls of different volumes that have the adequate proportions. It is widely accepted that the representational nature of scientific models is determined both by the intentions of the agents and the properties of the model. By itself, no object represents any other object, regardless of the properties they may have or the relations that they may hold between them. Moreover, one and the same object can be used to represent many different things, depending on the intentions of the agents that use it as a representational vehicle. However, it is not the case that any object can be used to represent adequately any target; sometimes, an object simply lacks the required properties, that is, those that are regarded as relevant to solve a posed problem. This implies accepting that representation is at least a three-place relationship between an agent, a model, and its target. According to Giere’s agentbased view of models, we should include the agents’ purposes as the fourth place of the representational relationship (Giere 2004, 2006, 2010). We can then say that an agent A uses a model M to represent a target T for some specific purposes P. Accordingly, representation is characterized as a four-place relationship. This has not been the only way of conceiving of the notion of representation among the mainstream representational view of scientific models; nonetheless, it is a sufficiently

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refined view to capture some fundamental features of the practice of representing in science. Besides, this conception is not committed to Giere’s specific conception of representation as informal similarity; on the contrary, it is broad enough to be used by any representational view of models. There is a fairly extended consensus among representationalist philosophers of science on the fact that representation is a pragmatic relation established by an agent (the model builder or user) who intentionally uses some object as a vehicle to represent a given target. Without the agent’s intentions, there is no representation at all. We can take the condition (i) “an agent A uses a model M to represent a target T for some specific purposes P” as a necessary condition for any account of how models represent. Condition (i) is by itself a sufficient condition for representation in general: if an agent A intends to represent a target T by means of model M, then, M represents T for A; if a different agent A’ intends to represent another target T ’ by means of the same model M, then, M represents T ’ for A’; if no agent uses M to represent any target, then, M does not represent anything. This condition, which conceives of representation as a three-place relation, is sufficient to solve the problems of the directionality of all representations and the possible mistargeting of them. The relationship of representation between a model M and a definite target T is irreflexive, asymmetric, and intransitive because of the decision of a given agent A. Likewise, M represents T, and not a different target T ’ because agent A has decided to represent that specific target. The direction of representation and the target to which it is addressed are entirely dependent on the agent’s intentions. The purposes of the agent, in turn, are necessary to characterize the concept of adequate representation. We will say that a model M provides an adequate representation of its target T if it satisfies the specific purposes P of the agents that built M. The adequacy of a given representation, which can be a matter of degree, depends in this way on the agent´s purposes. Given a different purpose, any previously adequate representation of a definite target may become inadequate. The different philosophical theories of scientific representation can be characterized as attempts at providing the necessary and/or sufficient conditions for adequate representation. It is not our aim to survey here all conceptions of representation employing models.16 We will offer just a brief account of some traditional theories, which form the contrast class, so to speak, for the fictional and artifactual accounts of models, with which this book deals. One family of theories is the one that appeals to the notion of similarity in order to specify a necessary condition for adequate representation; its fundamental tenet is that models represent their targets in virtue of being similar to them. We will distinguish two species of similarity theories. The first one is the formal similarity account, according to which a model represents its target in virtue of having a structure that is similar to the structure of the target. We will refer to it as 16 Frigg

and Nguyen (2017) is a detailed review article on the main conceptions of scientific representation by means of models; it contains extensive bibliographical references. We have benefited from this article in writing the following three sections. Suárez (2010, 2016) and Teller (2014) provide shorter general accounts of the same topic, from different points of view. Frigg and Nguyen (2020) is now the most extended account of scientific representation. The book was still in the press when this chapter was written and for that reason we could not take it into account here.

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the structural conception of representation. The second one is the informal similarity account, according to which a model represents its target in virtue of being similar to it in some respects and to some degrees to be specified. Another theory is the inferential account of representation, which states that a model represents its target if it permits the informed agents to draw specific inferences regarding the target. We will offer here just a sketch of these theories in the following sections.

1.5 Structuralism and Structural Representation The structuralist conception of scientific representation is a direct byproduct of the semantic or model-theoretic view of scientific theories. Structuralist philosophers of science have taken as their standpoint the idea put forward by Suppes (1960) according to which models in the empirical sciences are just mathematical models, , more precisely, set-theoretical structures. Given that functions can be expressed as relations, most philosophers of science have adopted a narrow concept of structure for the sake of discussing the notion of representation: structures are ordered sets of the form S = D, R1 , …, Ri , where D, the domain of the structure, is a non-empty set and R1 , …, Ri is a non-empty family of relations defined (extensionally) on the elements of D. We will follow this convention here.17 Mathematical structures may have different formal relations between them. A broad class of such relations is that of morphisms, which includes several different species. A morphism is a structure-preserving mapping from one structure to another (not necessarily different) structure of the same type. Two structures S 1 = D1 , R1  and S 2 = D2 , R2  are said to be of the same type (sometimes, similar) if and only if they have the same number of domains and relations, and the corresponding relations are of the same degree. The more general morphism is that of homomorphism, which in set theory can be used with the same extension as a morphism in general. A homomorphism between two structures of the same type is a mapping that preserves the structure. An isomorphism is just a bijective homomorphism. Two structures are isomorphic if and only if there is a structure-preserving bijection between them. Many structuralist philosophers have privileged the relation of isomorphism between structures but this is not necessary for the structuralist view of representation. Homomorphism is a more general concept that seems more adequate. Structuralists can also appeal to other weaker formal relations between structures, such as embedding, an injective mapping that preserves the structure.18 In what follows we will use the concept of homomorphism to characterize structural representation.

17 A

structure, as we have said in Sect. 1.2, may contain any number of domains, distinguished elements of those domains, and functions and operations defined on those domains. For the sake of simplicity, we will ignore these complications. 18 Rigorous definitions of these concepts can be found in any textbook on model theory, such as Tent and Ziegler (2012), chapter 1.

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In a broad sense, the standard structuralist view of representation maintains that a model M represents a target T iff M is homomorphic to T. The general conception of structuralist representation is then that a model M represents a target T iff (i) an agent A intends to represent T by means of M, and (ii) M is homomorphic to T. The first condition is the general condition for any representation; the second condition is the specific structuralist criterion for distinguishing adequate representations from misrepresentations. This characterization avoids the traditional objection according to which morphisms are not suitable to define representation because they do not have the required logical properties, as Suárez (2003) has argued. Actually, homomorphism between structures is a reflexive and transitive relation, whereas isomorphism is an equivalence relation (that is, reflexive, symmetric, and transitive). By contrast, as we have said in Sect. 1.4, we intuitively regard representation as an irreflexive, asymmetrical, and intransitive (or at least, non-transitive) relation. That objection would be efficient if we were to take condition (ii) alone as defining the structural view of representation, as it has been usual among structuralist philosophers. However, it is precisely condition (i) the one that establishes the correct directionality of the relation of representation. If an agent selects an object as a vehicle to represent another object, this pragmatically established relation between the two objects can be taken as irreflexive, asymmetric, and intransitive by decision. A model M is selected to represent a specific target T but not to represent other possible targets that may have similar structures. This strategy excludes by decision unintended targets and precludes that models represent themselves or that targets represent models. However, there is a remaining difficulty with the second condition. Morphisms are defined exclusively between set-theoretical structures, whereas in the factual sciences many models and targets do not seem to be that kind of abstract entities. Models are often concrete material objects, such as prototypes, icons, or scale models. In a literal sense, they cannot be homomorphic (isomorphic, etc.) to their targets because they are not structures. Moreover, targets are sometimes supposed to be physical phenomena or physical entities. Even in the case of purely mathematical models, of which we can assume they are set-theoretical structures, it would be a categorical mistake to say that they represent physical phenomena because ¿how could an abstract structure be homomorphic to a concrete object? A plausible structuralist answer to this problem is that although set-theoretical structures are abstract entities, they are instantiated by concrete objects and aggregates of objects. In this way, we can say that a map is homomorphic to a given territory because they instantiate homomorphic structures, or that a scale model of a warship and the ship itself are isomorphic because they instantiate isomorphic structures. This answer certainly poses deep metaphysical questions because instantiation was originally conceived of as a metaphysical relation between universals and particulars, where a particular is said to instantiate a universal property, such as being yellow, a property that in turn can be instantiated by many other particulars. Surely, any object or aggregate of concrete objects can simultaneously instantiate many different structures, among them some that are homomorphic to one of the many structures instantiated by a model. This last fact could jeopardize the whole

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structuralist enterprise because any object or aggregate of objects always instantiates some structure that is homomorphic to some of the many structures instantiated by a given model. As a consequence, any model might adequately represent any target. It is by no means easy to account for misrepresentation from the structuralist point of view. We certainly cannot say that a model M misrepresents a target T when M and T do not instantiate two homomorphic structures. It is manifest that any object instantiates many structures that are homomorphic to many structures instantiated by any different object. For instance, let us take two material objects and divide them into two parts that are unequal in some respect; then construct a structure whose domain is the set that contains as elements the two parts of the first object ordered by the relation of being unequal in that respect (for example, “being heavier than”) and construct a similar structure with the two parts of the second object and the same relation. By this procedure, we have obtained two homomorphic (actually, isomorphic) structures. There are less trivial ways of constructing such structures but the example shows that those pairs of objects instantiate at least two isomorphic structures. More generally, any object composed of parts can instantiate several different structures and some of them will be homomorphic to some of the structures instantiated by other objects composed of parts. In this way, structural representation seems to be trivial because any object can provide an adequate representation of any other object. ¿How misrepresentation could be accounted for within the structuralist framework? To solve this problem, structuralists can appeal to the purposes of the agents to select one definite structure S M among the many structures instantiated by a model M and one definite target T that instantiates a homomorphic structure S T . In this way, the agents are who determine which the representative structures are. Suppose that the structure S M is selected as the representative structure of model M and that several aggregates of concrete objects instantiate structures that are homomorphic to S M . In that case, the intentions of the agents determine that M represents the actual target T and does not represent other possible targets, despite that all of them are structurally similar to the model. The target of any structural representation has to be conceived of as an intended target. This is also true of any kind of representation. Following the above considerations, structural adequate representation can be defined this way: a model M adequately represents a target T iff (i) an agent A intends to represent T by means of M, and (ii) M and T instantiate two selected homomorphic structures S M and S T . Given that morphisms, and formal relations generally, do not admit of degrees, a model cannot be a more adequate representation of its target than another model. As a consequence, a model misrepresents its target if condition (i) is satisfied and condition (ii) is not, that is, M and T do not instantiate the two homomorphic selected structures S M and S T . Finally, a model does not represent a target T if condition (i) is not fulfilled, independently of whether condition (ii) is true or not. In this way, the structuralist is capable of distinguishing representation from misrepresentation and this latter one from non-representation. What cannot be distinguished on this account are different degrees of adequate representation concerning one selected structure S. For that reason, every representation is either an adequate representation or a misrepresentation of its intended target.

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It is possible to make sense of degrees of adequate representation by different models but regarding different selected structures. We can say that a model M 2 provides a more adequate representation of the intended target T than a model M 1 if (i) M 2 instantiates a structure S M2 that is homomorphic to a structure ST 2 instantiated by T, and (ii) M 1 does not instantiate S M2 but just a structure S M1 that is a substructure of S M2 . Given that the target T instantiates the two structures S T 1 and S T 2 , both M 1 and M 2 count by themselves as adequate representations of T. However, we can say that M 2 is a better representation of T than M 1 because it instantiates a more specific and rich structure. In that way, M 2 provides a more complete and more informative representation of T than M 1 .19 By contrast, when two models instantiate two structures, none of which is a substructure of the other, they cannot be compared concerning the adequacy of the representations they provide of the same intended target. If they are both adequate representations of the same target, no one can be regarded as a better representation than the other. On many occasions, structuralist philosophers seem to think that the relation of isomorphism between structures is the one that provides the most adequate representation of a given target. From this point of view, it could be said that a model M 1 provides a more adequate representation of an intended target T than another model M 2 if M 1 and T instantiate two isomorphic structures S M1 and S T , whereas M 2 instantiates a structure S M2 that is just homomorphic to S T . This comparison, however, is relative to some selected structures, such as S M1 and S M2 , because M 2 may also instantiate many other structures isomorphic to S T . This shows again that the structuralist definition of adequate representation has to be relativized to some selected structures instantiated by the model and the target. Another brand of structuralism is the one that appeals to representation by partial structures, as advocated by Da Costa and French (2003) and by Bueno and French (2018). A partial structure S is a set of the form D, R, where D is a non-empty set and R is a partial relation defined over the elements of D.20 An n-place partial relation R is, in turn, a set of the form R1 , R2 , R3 , where R1 , R2 , and R3 are mutually disjoint sets such that R1 ∪ R2 ∪ R3 = Dn . Here R1 is the set of n-tuples that are known to belong to R, R2 is the set of n-tuples that are known not to belong to R, and R3 is the set of n-tuples for which it is not known whether they belong or not to R. When R3 is empty, R is a standard n-place relation that reduces to R3 . This formalism permits to define the relations of partial homomorphism and partial isomorphism between partial structures.21 In general terms, two partial structures S 1 and S 2 are partially homomorphic (or isomorphic) when they contain two partial substructures S’1 ⊂ S 1 and S’2 ⊂ S 2 , which are homomorphic (or isomorphic) between them. Those partial relations have the usual logical properties, for instance, partial isomorphism is an 19 To take an example from pure mathematics, we can say that group theory and field theory provide adequate structural representations of the set of the real numbers with usual arithmetic operations, but field theory is a more adequate representation, according to our criterion. 20 For the sake of simplicity, we will use the simplest partial structure. More complex structures of this kind may include several different relations defined on their domains. 21 Further technical details can be found in Da Costa and French (2003: 48–52) and Bueno and French (2018: 41–44).

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equivalence relation between partial structures. Moreover, they are more general than (and, consequently weaker than) the standard relations of homomorphism and isomorphism between full structures because they contain them as special cases (when R3 is empty). Within this framework, it can be said that a model M adequately represents a target T iff (i) an agent A intends to represent T by means of M, and (ii) M and T instantiate two selected partial homomorphic structures S M and S T . In principle, the resource to partial structures allows accounting for the incompleteness of our knowledge of the modeled phenomena. It also permits to accommodate Hesse’s (1963) notion of neutral analogies between models and targets, that is, those properties of which we do not know whether are shared or not be the model and the target.22 A definite answer to the ontological question seems to follow immediately from the structuralist conception of model: models are structures, and structures are sets and, consequently, abstract entities. More precisely, models are mathematical objects of the same kind as numbers and functions (when they are defined, as is customary, within the set theory). How should we conceive of mathematical entities, and abstract entities generally, is a metaphysical question that cannot be addressed here. In any event, structuralism is not committed to holding that all scientific models are settheoretical structures. Given that morphisms are defined exclusively between structures, this claim implies that targets must be also structures. This was Suppes’ position in the 1960s but it is not anymore feasible. Structuralists are not forced to conceive of all scientific models as set-theoretical structures, as Bueno and French (2018: 70) have pointed out. As we have seen, in order to represent a given target a model must instantiate a structure that is homomorphic to a structure instantiated by the target. If we admit that physical objects and aggregates of objects can instantiate structures, models and targets can be both physical objects. More generally, the structuralist answer to the ontological question must be that a model is whatever object that has (or instantiates) a structure. A model can be a set-theoretical structure, like all mathematical models, but it also can be one of the many different kinds of entities that are called models in empirical sciences, among them, single physical objects.23

1.6 Informal Similarity The informal similarity theory of scientific representation is a descendant of Mary Hesse’s (1963) account of models as analogies (which, by the way, has a long history in nineteenth century physics). According to Hesse, models are based on positive and negative analogies with their targets. Positive analogies are just the properties shared by the source (the model) and the target (the modeled phenomena), whereas negative analogies are the properties of the source that the target does not possess (or, to be more cautious, that the modelers believe the target does not possess). Besides, 22 See 23 See

the beginning of Sect. 1.6 for an account of Hesse’s conception of analogy. also Bueno’s contribution to this book.

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neutral analogies are those properties that have not been explored yet and for that reason, we cannot ascertain whether they are shared or not by the model and the target. Models are then said to be similar to their targets to the extent that there are some positive relevant analogies between them. The specific properties that are regarded as relevant and consequently selected as positive analogies are dependent on the purposes for which the models are built. Roughly speaking, there are two different kinds of analogies a model may have with its target: it may be analogous to the structural features of the target or its behavior. Let us consider one example of each kind. The plastic balls and rods model of molecules, widely used in teaching chemistry, is intended to represent the three-dimensional structure of the different kinds of molecules.24 Each plastic ball represents an atom and each rod represents the chemical bond between different atoms. Atoms of different elements are represented by balls of different colors. It is not supposed that real molecules are composed of spheres linked by rigid rods but it is assumed that they have a spatial structure analogous to the disposition of the balls and the rods in the model. On the other hand, the model is not intended to represent the behavior of the different kinds of molecules, say, their motions or their collisions. By contrast, the Newlyn-Phillips hydraulic machine is a model intended to represent the macroeconomic behavior of a national economy.25 It models the flux of money inside the economic system as a flux of water inside tubes, which connect several tanks and reservoirs, forming a closed circuit. The hydraulic machine shows how the economic system works but not how it looks like. There is no structural similarity between water and money, or between the national bank and a tank of water. The machine does not resemble its target as a prototype or scale model resembles the modeled phenomena (say, a bridge). In any event, despite their differences, it can be said that both models are similar in some respects to their targets. According to the informal similarity account of representation, models represent their targets by being similar to them in some respects and to some degree. This is not tenable without further restrictions. If two objects are said to be similar when they have (or instantiate) some shared properties, it follows that anything is similar to anything else because all objects have at least one property in common. However, we cannot, at the risk of trivializing the whole issue, accept that any object provides a representation or even an adequate representation of any other object. Ronald Giere, the main endorser of this account, has acknowledged that the model builders have to specify the similarity by selecting the features of the model that are claimed to be similar to some features of the target and determining the required degree of similarity in each case. As Giere (2006: 64) has put it, “it is the possibility of specifying such similarities that makes possible to use the model to represent the real system in this way”.

24 This

model, and its historical development, is analyzed in detail by Toon (2012), chapters 4 and

5. 25 The

model, and its historical precedents, is described by Morgan (2012), chapter 5.

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Even so, there could be too many similarities at stake. Which ones are regarded as relevant similarities and which are not is something that depends upon the purposes for which a given model was designed in the first place. Here a pragmatic and contextual element seems unavoidable. Each model is designed to perform a specific task, that is, to solve a well-defined problem. As a consequence, the properties of the model which are regarded as relevant are determined by the kind of solution that the model users expect to find for the problem that originated the construction of the model. Even the required degree of similarity between the properties of the model and the properties of the target is dependent upon that expected solution to the problem. And this seems hardly quantifiable. Giere (2006: 64) has claimed that representing “does not require the existence of a general measure of similarity between models and real system”. In each particular case, the model builders have to specify the required degree of similarity in a rather informal way. As he states it, what counts as “similar enough” in each case “would depend on the purposes for which the model is being applied”, and this “is a function of the context and not merely a relationship between the model and the system to which is applied” (Giere 2006: 64). This seems a fairly reasonable pragmatic solution to the problem of the vagueness of the informal similarity relationship. Sometimes, the informal similarity account of representation is introduced through the following definition: a model M represents the target T iff M is similar to T in certain (specified) respects and to certain (specified) degrees. When defined in this way, the relation of representation does not have the required logical properties because informal similarity is a reflexive, symmetric, and non-transitive (and also non-intransitive) relation. As a consequence, representation lacks the required directionality, which has to be established by the agent’s intentions, as Giere pointed out. For that reason, it cannot be regarded as an acceptable definition. On the basis of Giere’s account of representation as a four-place relationship, we can say that a model M represents a target T only if (i) M is used by some agent A to represent T with some specific purposes P. A model M adequately represents a target T iff (i) an agent A intends to represent T by means of M with some specific purposes P, and (ii) M is sufficiently similar to T as to satisfy the agent’s purposes P. On the other hand, M misrepresents T iff (i) an agent A intends to represent T by means of M with some specific purposes P, and (iii) M is not sufficiently similar to T as to satisfy the agent’s purposes P. Finally, a model M does not represent a given target T if (iv) M is not intended by any agent A to represent T. These are not the standard definitions in the literature. However, they provide a reasonable reconstruction of the informal similarity theory of representation. Notice that on the informal similarity account the concept of representation is not defined, rather, is taken as a primitive notion. Condition (i) provides just a necessary condition for representation in general. If we were to regard it as a definition, it would be circular because the very concept of representation is used both in the definiendum and the definiens. This condition provides the required directionality of representation. Conditions (i) and (ii) are individually necessary and jointly sufficient for adequate representation; likewise, conditions (i) and (iii) are jointly necessary and sufficient conditions for misrepresentation. They provide explicit definitions for those

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concepts. Notice, however, that in our reconstruction adequate representation and misrepresentation are relative to the purposes of the agents, say, the model builders or users. As a consequence, no model by itself provides an adequate representation or misrepresentation of its target; the same model can be regarded as an adequate representation for an agent A with purposes P and as a misrepresentation for another agent A’ with different purposes P’. Condition (iv), in turn, provides a sufficient condition for non-representation. In principle at least, the agent-based similarity account of representation, as we have formulated it, provides an answer to the questions of when a model represents, adequately represents, misrepresents, and does not represent a target. Notice that, because informal similarity admits of degrees, adequate representation and misrepresentation are matters of degree, so that we can compare different models of the same target and conclude, say, that one of them provides a better -or worse- representation of the target. A model can be similar to its target to different degrees and, as a consequence, it can be more or less satisfactory regarding the agent’s purposes. Giere’s favorite example of representation -a map and its territory- provides a good illustration of the above conditions. We assume that all maps are produced with the intention of representing some aspects of a given territory. If this condition is not satisfied, they are not representations at all (and perhaps we will not call them maps, but mere drawings). Given some specific purposes by the users, for instance, exploring every branch of a river that forms a delta, a map may provide a more or less adequate representation of the territory to be explored. We can say that, for the purpose in question, the map misrepresents the territory if one or more of the branches of the river are not depicted in the map. If all branches are depicted, the map is an adequate representation of the territory. Nonetheless, a different map may provide a more adequate representation, for instance, if it shows with higher accuracy the intricacies of the course of each branch than the other map, facilitating in this way the navigation of the river. The informal similarity account of models does not imply any specific answer to the ontological question. No definite ontology of models follows from this view. Giere and other endorsers of this account acknowledge that there exists a diversity of physical and theoretical models and that a model can be either a concrete physical object or an abstract entity. The more general ontological problem is then how to define similarity in terms so general as to allow us to say that any two objects are or not similar. The most intuitive answer seems to be that they instantiate the same properties. This answer, however, is deceptively simple. We can think that objects instantiate properties and relations, and consequently, that two objects are similar when they have some specified properties in common. But if theoretical models are conceived of as abstract entities that represent physical phenomena, it is far from clear how they could have properties in common; abstract entities certainly do not have physical properties. Because informal similarity comes in degrees, one model can be more similar to a given target than other models of the same target. Nonetheless, it is conceivable that we were to face situations in which we find ourselves unable to compare different models; for instance, given several maps of the same territory ¿how could we decide

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which one is more similar to the territory? The problem lies in the fact that informal similarity is a purely qualitative concept. In order to compare two or more models regarding their similarity to the target, we would need to introduce a comparative concept of similarity, such as, for example, the comparative concept of hardness for rigid solids. The problem will disappear if we were to have at our disposal a measure for the concept, but this presupposes a metrization of the similarity notion, which is something we have not achieved yet. If we were to accept that similarity consists in sharing some properties, we would say that an object O1 is more similar to another object O2 than to a third object O3 if O1 has more properties in common with O2 than with O3 . However, it is not easy to elaborate this intuition, given that any object, in principle, has an indefinite number of properties. A workable comparison is possible only if we specify a finite set of well-defined properties that belong to the objects to be compared. Weisberg (2013, chapter 8) has made an attempt at providing a similarity measure for models, which is not free of difficulties. For instance, on Weisberg’s account similarity is an asymmetric relation, as representation should be, but it is nonetheless reflexive and not always intransitive. On the other hand, the similarity measure is not absolute but relative to a set of selected properties, a set whose elements are dependent on relevance considerations, which are surely different in diverse contexts. Finally, it does not provide an answer to the problem of how could we compare an abstract object with a concrete object. Weisberg’s similarity measure is, like probability, a real number between 0 and 1. But even assuming that we can measure the degree of similarity between a model and a target, this will not solve the problem of deciding when the model is similar enough to the target, given the purposes of the agents . In any event, the decision has to be made in each context and probably numbers will not help very much. It then seems plausible to accept, as Giere has pointed out, that the similarity account of representation does not require a measure of the similarity between models and targets.

1.7 Inferentialism The inferential conception of representation does not appeal to any kind of formal or informal similarity between models and targets. According to this view, models do not represent their targets in virtue of being similar to them. Mauricio Suárez, who originated this account, called it a deflationary or minimalist theory of representation. Inferentialism does not attempt at defining representation, that is, at finding necessary and sufficient conditions for it; it merely intends to provide some necessary conditions.26 26 Suárez

(2004) is the fundamental article. It started from Suárez’s (2003) criticism of both the formal and the informal similarity conceptions of scientific representation. Suárez (2015) elaborates further details of his account. For two short general accounts of inferentialism see also Suárez (2010, 2016).

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The key idea of the inferential conception of representation is that scientific models are epistemic representations of their targets because they permit us to perform a surrogative reasoning about them. This notion was introduced into the studies of scientific representation by Swoyer (1991), who endorsed a variety of the structuralist view of representation. We perform this kind of reasoning when studying the structure or the behavior of a model we draw valid or sound inferences concerning the structure or the behavior of its target. For instance, from an analysis of the properties of a given model, we can infer a prediction concerning the future behavior of its target under certain specific conditions. The inferences in question do not have to be necessarily deductive, they can be inductive in a broad sense of the word (which includes every kind of non-deductively valid argument, such as those employed in abductive and analogical reasoning). Because surrogative reasoning can make use of non-deductive inferences, it would be better to speak here of acceptable inferences, that is, inferences we regard as sound despite being deductively invalid. The acceptable inferences extracted from a model do not need to be true statements concerning the target. Surrogative reasoning has to be conceived as a means to form or generate hypotheses concerning the target. And of course, some of the inferences may drive us to entertain false hypotheses. Suárez (2004: 773) provides the following characterization of the inferential view of representation: “A scientific model M represents a target T only if (i) the representational force of M points towards T, and (ii) M allows competent and informed agents to draw specific inferences regarding T.”27 This cannot be taken as a definition of the concept of representation because it does not state necessary and sufficient conditions to apply the term but just necessary conditions.28 Consequently, it cannot be used as a criterion to determine that a given model is effectively a representation of its target. It merely provides a criterion to determine when a model does not represent a target. If at least one of the two conditions do not obtain, M is not a representation of T. In principle, we could be able to specify additional necessary conditions, which together with (i) and (ii) be jointly sufficient for scientific representation. Because inferentialism does not attempt at providing such additional conditions, many philosophers probably will not regard it as a candidate for a complete theory of representation. Suárez (2004, 2015) himself admits that he calls “minimalist” and “deflationary” his inferential account of representation because, among other things, it does not intend to provide a full theory of representation. In his own words, inferentialism “refuses to lay down necessary and sufficient conditions on any instances of representation. Representational force and inferential capacity are taken to be only general features (and therefore at best necessary conditions) on representation, but they are neither jointly nor individually sufficient for representation” (Suárez 2015: 45). There is a sense, after all, in which inferentialism is not a theory of scientific representation 27 We

have replaced the letters A and B in the original by M and T, respectively. and Solé (2006: 41) employ a biconditional to introduce conditions (i) and (ii), but this seems to be just an occasional slip. 28 Suárez

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and for that reason, it cannot be regarded as a rival of other theories, such as the structuralist and the informal similarity accounts. The first condition in Suárez’s account of inferentialism establishes the directionality of the representation. In this way, it solves the problem of the logical properties of representation, avoiding at the same time the problem of unintended targeting. What is called “representational force” is surely the outcome of the agent’s decisions and, for that reason, this condition can be taken as equivalent to what we called condition (i) in the formulation of the similarity view of representation. As we said, it seems reasonable to think that this is a necessary condition for any kind of representation. The second condition is just a statement of the surrogative reasoning condition. This condition permits to distinguish epistemic representation from mere representation by arbitrary stipulation or denotation. Inferences about the target should be specific in the sense that they could not be obtained from an arbitrary object employed as a model of that target, or, equivalently, from “the mere fact that the source represents its target” (Suárez 2016: 456). Undoubtedly, it is reasonable to think that scientific models should allow surrogative reasoning and, more generally, that this is a necessary condition of epistemic representation. In particular, all partisans of some similarity account of representation will grant that scientific models, in virtue of being structurally or informally similar to their targets, permit informed agents to draw specific inferences concerning those targets. Suárez’s inferentialism is compatible with this stance. In many cases, acceptable inferences about a target are based on similarities with the models that represent it. As minimal constraints on representing by using models, Suárez’s conditions are hardly disputable. Suárez (2015: 45) admits that “normally some other conditions -such as isomorphism or similaritywould need to obtain in each concrete case of representation. This seems to suggest that inferentialism is not only compatible with other conceptions of scientific representation, such as the structuralist or the informal similarity accounts, but also that those accounts may supplement the inferential view with the required sufficient conditions for representing. Because inferentialism does not provide necessary and sufficient conditions for representation, a fortiori it cannot provide definitions of adequate representation or misrepresentation. Nonetheless, it can provide sufficient conditions. We can say that a model M adequately represents a target T if all the hypotheses inferred from M are true of T and that a model M misrepresents a target T if all the hypotheses inferred from M are false of T. The converse of these conditions is obviously unacceptable. We surely will grant that an adequate representation of a given target sometimes will permit us to infer some false hypotheses about it, and, at the same time, that even the worse misrepresentation of a target sometimes will lead us to infer a true hypothesis about it. The inferential conception does not provide any account of degrees of adequate representation or misrepresentation either. We could think of weakening the sufficient conditions: for instance, instead of “all hypotheses” we could say “most hypotheses”, or “the majority of the hypotheses”, or “a certain defined ratio of hypotheses”. However, given that the number of inferences we can draw from a model is potentially infinite, it is not clear what these weaker conditions could mean. Likewise, we could weaken the concepts of true and false hypotheses by

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qualifying them as “approximate”. But again it is far from clear what approximately true or false hypotheses are in themselves. There are many other possible weaker conditions, we will not explore here. In any event, it seems clear that scientific models, being always idealized, will license in most cases approximately true -but nonetheless false- hypotheses concerning their targets. Regarding the ontological question, nothing definite follows from the inferential conception of representation. In principle, we can perform surrogative reasoning with every kind of model, either concrete or abstract. We can use any kind of object to represent another object if the first permits us to generate useful hypotheses about the second. The only requirement that Suárez (2010: 98) demands is that the model must be an object endowed with a sufficiently rich internal structure in terms of its parts and the relations between them. The inferential view of representation allows for different models of the same target and even for incompatible models of such a target. It does not place any additional constraint on the ontological nature of scientific models. This fact can be viewed as a virtue to the extent that it permits to accommodate all the different uses of the concept of a model in scientific practice. Inferentialism is sufficiently flexible to admit that a given target can be represented by set-theoretical structures or sets of mathematical equations, as well as by a variety of concrete physical objects. Inferentialism seems to be an excessively weak account of scientific representation. It just captures the fact that representative models must allow for surrogative reasoning, but it does not explain why models allow for that kind of reasoning. This has to be taken either as a brute fact or as something to be explained by another account of scientific representation (as Suárez himself seems to acknowledge). By contrast, the similarity view can explain that fact by appealing, precisely, to the relationship of similarity between models and targets. All successful models permit surrogative reasoning because they are similar to their targets in some selected respects. And this goes beyond mere stipulation by the agents. Any object can be used to represent any other object by mere stipulation but this is not by itself an epistemic representation and does not permit surrogative reasoning without further assumptions, at least, it will not enable us to draw specific inferences about the target, in Suárez’s sense. We can use a steel pen to represent the Eiffel Tower and truly infer from that model that the French tower is made of steel. Or we can slightly tilt that pen to use it as a model of the Leaning Tower of Pisa and truly infer that the inclination angle of the Italian tower is around four degrees. Nonetheless, in both cases, we are making those inferences on the basis of assuming some similarities between some selected properties of the pen (being made of steel or being tilted) and some selected properties of the two towers. It is clear from these examples that a pen can be used as an adequate representation of the two towers, depending on our purposes. If we want to represent the material of which the Eiffel Tower is made, a steel pen is adequate; if we want to represent the shape of that tower, it is not. The inferential account does not have the required tools to account for those kinds of examples nor to distinguish adequate from inadequate representations. For those reasons, it cannot be regarded as a rival theory of the different similarity accounts of representation.

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1.8 Idealization It is generally acknowledged that all scientific models are idealized to some degree, and, at the same time, that some models can be deidealized in order to get a more complete or realistic representation of the phenomena to which they apply. At first sight, all this sounds quite reasonable but it is not easy to make these claims welldefined. Actually, despite being accepted as one of the essential features of scientific models, the concept of idealization has received less attention from the philosophers of science than that of representation. As a consequence, we still lack a standard or mainstream account of idealization and even a non-ambiguous vocabulary to refer to this feature of models. Scientists, in turn, make frequent use of the term but mostly in a broad and rather loose sense. Given the present situation of the philosophical studies on idealization, we cannot expect very much terminological precision or agreement in distinguishing different meanings of the term. This unavoidably leads to the necessity of stipulating a more or less well-defined meaning for some concepts. The concept of idealization has been related in the philosophical literature with those of abstraction, approximation, distortion, incompleteness, and simplification, among others. In turn, those concepts are not well-defined either and have been characterized in different ways. Scientists who build or use models often agree in describing them by using those terms but they generally do not provide a precise elucidation of them. A model is sometimes described as a simplified representation, description, or account of a complex phenomenon. It is said that a simplified model is obtained by abstracting some properties of the modeled phenomena, that is, by selecting some reduced set properties regarded as relevant or significant for some purposes and neglecting all its remaining properties. For the same reason, it is often said that models are abstract representations of the phenomena. On many occasions, scientists attribute to models certain properties that they believe the phenomena do not possess, or at least that they do not possess to some degree. For this reason, it is said that the model is (to some degree) a distorted representation of the real-world phenomena. As a consequence of being abstract and simplified, models are said to be incomplete. Because they are distorted, models often provide only descriptions of the structure or predictions of the behavior of the modeled phenomena that are not entirely exact or precise, and, for that reason, they are said to be approximations to such phenomena. We will not intend to survey here the different accounts of idealization available in the philosophical literature. The topic has been addressed from different points of view because, besides modeling, it is relevant to different issues in the philosophy of science, such as realism, explanation, laws of nature, and the aims of scientific inquiry.29 There are many different reasons for building highly idealized models. One of the most pervasive is the mathematical or computational intractability of complex models. Many abstractions, simplifications, and distortions are embedded into a 29 For references, see the section on idealization in the commented bibliography at the end of this book.

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model in order to make it tractable. For instance, the gravitational interaction of three or more bodies is mathematically intractable and for that reason, physicists build simplified two-body models, in which bodies are considered as perfect homogeneous spheres and the influence of other bodies is neglected, as well as that of other non-gravitational forces. According to a fundamental theorem of Newton’s celestial mechanics, homogeneous spheres gravitate as if all their masses were concentrated in their centers of gravitation; for that reason, they can be treated as point-like particles. We know that stars and planets are not perfect homogeneous spheres but given that they are approximately spherical and not highly inhomogeneous, they can be approximated without much distortion this way. We also know they do experience the gravitational influence of multiple bodies; nonetheless, in certain conditions, those facts can be neglected without affecting very much the explanatory or predictive power of the model. For instance, when the distance between the bodies is very large compared to their dimensions, they can be considered as point-like masses and when the masses of the other bodies are not very large, or they are very far apart of a two-body system, their gravitational influence can be neglected. In principle, any pair of bodies, regardless of their shape and composition, can be treated as two particles that interact gravitationally if they are sufficiently separated between them and relatively isolated, because of long distances, from other bodies. We also know that the two-body system is subject to non-gravitational forces but they are neglected because their influence is too feeble compared to the one exerted by gravitation. In this way, the Newtonian celestial mechanics makes use of idealized models of two-body subsystems of the Solar System, such as the Sun-Earth system or the SunMars system. Those simplified models are useful for certain purposes but they cannot be applied in all circumstances. If we were interested in calculating the orbit of Mars to high accuracy, we could not use a Sun-Mars two-body model; the gravitational influence of Jupiter, which has a large mass and it is relatively near to Mars, could not be neglected. The less idealized Sun-Mars-Jupiter three-body model is mathematically intractable but an approximate and quite complicated calculation can be performed by introducing other idealizations into the model. If we were to build a model of the gravitational interaction between all the planets of the Solar System, the resulting many-body model would be too complicated and intractable to be useful for calculating the planetary orbits. And surely, we cannot expect to build a model of the gravitational interaction between all the bodies that compose the Solar System -including every dwarf planet, satellite, asteroid, and minor body- because we do not even know how many bodies we should include into the model. Finally, a model that intended to take into account the gravitational influence of all the stars in the Milky Way on planetary motions is presently beyond our imagination. As the planetary models example shows, all idealizations are consciously introduced into a model to fulfill a given purpose, in this case, that of calculating the planetary orbits to a previously determined level of accuracy. The expediency of the models is determined by the established purposes. If we want to predict the positions of Mars with a low level of accuracy, the two-body model may be useful; if we want a higher precision, we must appeal to the three-body model. There is always a trade-off between different desiderata we want to obtain from a model because

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those desiderata usually cannot be maximized at the same time. We cannot have a highly accurate and mathematically simple (or even tractable) model of planetary motions; we must resign some accuracy in order to get tractability. This is a typical situation in scientific modeling. The modelers must make a decision concerning which virtues are to be privileged in each case. The example also shows that abstracting and distorting are the central procedures that lead to most scientific idealizations. In building a two-body model of the gravitational interaction between the Sun and a determinate planet, we make an act of abstraction when we neglect the gravitational influence of all other bodies of the Solar System (or the whole Universe) and we make a distortion when we regard the Sun and the planets as homogeneous spheres or point-like masses. The fact that the resulting model constitutes a simplification of the phenomena and a mere approximation to its structure or behavior is just a consequence of being abstract and distorted. Finally, the example can be employed to show that models can be deidealized, at least to some degree. There is a clear intuitive sense in which the Sun-Mars model is more idealized than the Sun-Mars-Jupiter model. Models are deidealized by removing some of the abstractions and/or distortions they include. The three-body planetary model is not less distorted than the two-body model because it considers the three bodies that compose it as point-like masses; nonetheless, it is less abstract than the two-body model because it does not neglect the gravitational influence of at least one of the planets of the Solar System. We can easily imagine a sequence of less and less idealized models obtained by the successive deidealization of a basic simple model. The question of the identity of the resulting deidealized models -and all models generally- is rather a matter of convention. We can think of the SunMars-Jupiter model either as a modified Sun-Mars model or as a different model. For practical reasons, it is convenient to assign an identity to each model, so that we should regard every model of the sequence of deidealized models as a different model from their predecessors. In the case of theoretical models, idealization is obtained by a double process of abstraction and distortion: first, some parameters that could represent properties of the modeled phenomena are not included into the model, and second, the values of the parameters that are included into the model are set to values we think are not the actual values of the properties belonging to the phenomena (frequently, those parameters are set to infinite, 1 or 0, often as limiting values). As a consequence, the deidealization of a given model can be accomplished in two different ways: either by adding new parameters, which results in a less abstract model, and/or resetting the values of some model’s parameters to new and more adequate values, which results in a less distorted model. Deidealized models are often called more “realistic” models, that is, models that provide a better and more complete representation of some complex phenomena. A realist construal of scientific theories and models is often an implicit assumption or starting point of the very formulation of the question of idealization. According to this assumption, the fundamental aim of modeling is to provide a more complete and realistic description of the targeted phenomena as possible or, in other words,

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a veridical representation of the world. As Weisberg (2013: 98) puts it, “targetdirected modeling aims to give a realistic picture of a single target using a single model”; that is why “we can think of idealization as a departure from a complete, veridical representation of real-world phenomena”. According to this realist stance, scientific models attempt at providing a sort of approximately true description of some aspects of the world. Deidealized models then should be understood as truer than more idealized models, or at least as less unrealistic representations.30 Every realist conception of scientific models must address the problem of idealization, given that idealized models cannot be faithful representations of their targets. Scientific models cannot aim at providing literally true and exact descriptions of the world. We cannot conceive of a model without any kind of idealization; such a model would be as complicated and useless as a one-to-one scale map of a territory. Moreover, the idea of providing a complete representation of a domain of phenomena is not tenable. As Paul Teller has pointed out, all human representations are “ubiquitously inexact” (either imprecise or inaccurate, or both) due to “the limits of human representational powers relative to -at least in practice- unlimited complexity of the world” (Teller 2014: 492). Those limitations show up in the fact that scientists often build and use different incompatible models of the same phenomena, such as the many models of the atomic nucleus.31 From a realist standpoint, such models have to be regarded as incomplete or partial representations of the nuclear phenomena. Sometimes, scientific models cannot be deidealized without compromising some of the purposes for which they were built in the first place, for instance, without making the mathematical equations of the model intractable. Instead of regarding deidealized models as more realistic or truthlike representations of phenomena, it would be more convenient to speak of them as richer representations of their targets, that is, representations that add more details to the model without being necessarily more realistic descriptions of the phenomena. We should not forget that a model -no matter how much deidealized- always contains some idealizations. For a non-realist construal of the aims of science, idealization is less problematic. Highly idealized models are just useful devices designed to solve some well-defined problems and have to be regarded as successful to the extent that they permit to obtain acceptable solutions to such problems. From this point of view, deidealized models are not necessarily better representations or descriptions of the phenomena. They can be regarded as more refined instruments used to solve more specific problems.

30 Realist views of models and idealizations include the pioneering articles by Mc Mullin (1985) and

Laymon (1985), as well as the works of Giere (2006) and Morrison (2015), among others. For more references see Cassini’s contribution to this book, which provides further examples of deidealized models. 31 See a clear account of nuclear models in Cook (2010) and a philosophical discussion of incompatible models of the atomic nucleus in Morrison (2015), chapter 5.

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1.9 Models as Fictions The main idea of the fictional conception of models is that scientific models must be conceived of as the same kind of entities as literary fictional characters and places, such as Cervantes’ Don Quixote or Tolkien’s Middle Earth. From this point of view, fictionalism is essentially an answer to the ontological question concerning the nature of models. What kind of entities fictions are is above all a metaphysical problem, to which the philosophy of science cannot provide a specific solution.32 There are different philosophical accounts of fiction and for that reason “fictionalism” has become an ambiguous term. The very term “fiction” has different senses in natural languages. Fictionalist philosophers of science have adopted one of two different accounts of fiction. The first account conceives of fictions as non-existent beings and, accordingly, of fictional terms as non-denoting ones. This is an extended common usage in non-scientific contexts: on many occasions, when we say that something is fictitious, we usually mean that it is not real. In this sense, fiction opposes to reality. The second account conceives of fictions as the products of imagination, which does not imply that they do not exist or that fictional terms do not denote anything in the real world. Imaginary entities are not always unreal or non-existent. The two senses of fiction are not mutually incompatible, given that some products of our imagination may not exist in the real world. We can imagine a two-horned rhinoceros as well as a four-horned rhinoceros, although the first one is real and the other is fictitious in the two senses of the word. Nonetheless, these two different senses of the term fiction have originated two distinguishable varieties of fictionalism in the philosophy of science. One traditional brand of fictionalism has adopted the first concept of fiction. A thorough fictional account of science based on the conception of fictions as nonexistent beings was developed by the end of the nineteenth century by Hans Vaihinger, whose fundamental work, Die Philosophie der Als Ob, was written around 1880, although it was not published until 1911. According to Vaihinger (1927), most scientific concepts, such as those of atom or force, are entirely fictional and, as a consequence, do not refer to anything in the real world. Vaihinger distinguished between full fictions, which he conceives of as self-inconsistent concepts, and semi-fictions, which he conceives of as consistent but non-referring concepts. Accordingly, those statements containing full fictional terms are false for purely logical reasons, whereas those statements containing semi-fictional terms are false for factual reasons. This brand of fictionalism is a full antirealist conception of science. If realism is defined by the conditions of reference and truth, according to which (i) at least some typical theoretical terms are genuinely referential, and (ii) at least some scientific hypotheses and theories are true or approximately true, then, Vaihinger’s fictionalism implies the rejection of both conditions. Field’s (2016) mathematical fictionalism, according to which numbers and other mathematical entities do not exist and mathematical 32 Thomasson

(1999) is a classic study in the metaphysics of fiction. For more references see the section on fictionalism in the commented bibliography at the end of this book.

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statements are thus false, is also a form of antirealism based on that concept of fiction. This variety of fictionalism has to face two serious problems. The first one concerns non-denoting terms or empty descriptions. If fictional terms do not denote, statements containing such terms, say, “Protons are made of quarks” or “4−6 = −2”, cannot be true, rather they have to be regarded either as false or as lacking truth-value. The two possibilities clash with classical bivalent logic. This is obvious if we say that no fictional statement has a definite truth-value. On the other hand, we cannot maintain consistently that all fictional statements are false: if one of them is false, its negation must be true. The kind of non-classical logic adequate to the fictional discourse has been a very much-discussed topic, on which we do not dwell here.33 The second problem is that inexistence cannot be generally proved. How could we know that a given kind of entity does not exist? Can we prove in any way that quarks or numbers do not exist? Have we proven that the luminiferous ether does not exist? Certainly, we cannot do those things. We can only assert that impossible objects do not exist, that is, those objects whose existence cannot be postulated without contradiction. We have disposed of the hypothesis that postulated the existence of the ether because we have invented a theory -special relativity- that solves (or dissolves) all the conceptual and empirical problems of the older electromagnetic theories without assuming the existence of the ether. However, we cannot prove in any way that the ether does not exist, if the very concept of the luminiferous ether is not self-inconsistent. The same remark holds for mathematical objects and any other logically possible entity. One of the main virtues of Vaihinger’s fictionalism is that it can evade the two problems. The “as if” clause, which Vaihinger borrows from Kant, provides the clue to the solution of the existential question. The philosophy of the as if essentially consists in treating fictional statements as if they were true factual hypotheses. For instance, mathematical fictionalism proceeds as if numbers exist and for that reason, it can apply classical logic and semantics to all mathematical statements. The same strategy holds for the hypotheses about unobservable physical entities or properties. No change of logic is then required. Assuming that the concept of the electromagnetic field is not inconsistent, we can proceed to develop a physical theory of that field as if it were a real entity. We do not need to venture into the impossible task of proving that fields do not exist. The main problem with this variety of fictionalism is that it cannot distinguish a useful fictional statement from a successful empirical hypothesis, at least in the case of semi-fictions. Vaihinger (1927, chapter 24) mentions some criteria to distinguish between fictions and hypotheses, but they hardly apply to scientific hypotheses as we presently conceive of them. He considers fictions as provisional and convenient means to a definite end, which are justified in light of their utility to solve a problem. In Vahinger’s words, fictions are expedient (Zweckmäßig), another term borrowed from Kant’s philosophy, on which Vaihinger was a leading expert. Moreover, all fictions are introduced into science with the explicit awareness that they do not claim 33 Woods

(2007) is a detailed technical survey of the topic. Woods (2018) provides an extended philosophical discussion. See also Redmond’s contribution to this book.

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to be actual or true. Any antirealist philosopher of science will acknowledge that these are precisely some of the main characteristics of a good scientific hypothesis. Even from a realist standpoint, the difference between semi-fictions and hypotheses is not always clear, as Vaihinger himself often points out. Vaihinger’s fictionalism had a second life after Fine (1993) vindicated it. At the end of his article, Fine made a connection between fictionalism and the practice of building idealized models in science. According to Fine (1993: 16), “the industry devoted to modeling natural phenomena, in every area of science, involves fictions in Vaihinger’s sense”. In particular, computer simulations make extensive use of fictionalization techniques, such as using discrete space and time grids, multiple spatial dimensions, and many others. These brief remarks inaugurated one of the present-day strands of fictionalism concerning models, a brand of a definite antirealist variety. This variety of fictionalism is not easily compatible with a representational view of models. At least, it does not seem compatible with the theories of representation (such as those by Hughes 1997 and Frigg and Nguyen 2016) that include denotation as a necessary condition of representation. If “quark” is regarded as a fictional term without denotation, the quark model cannot represent any phenomena. Thus, within this brand of fictionalism, it seems more natural to deny that models represent the phenomena. Concerning the pragmatic of models, this fictional account of models is akin to pragmatism and scientific instrumentalism, as Fine (1998) has pointed out. It emphasizes that the value of models does not consist in providing truthlike descriptions of the phenomena in the real world; they are just useful devices to solve specific theoretical and practical problems. In principle, this task could be accomplished without any appeal to the representational capacity of the employed models. Successful models are simply those models that satisfactorily solve the problems that originated their construction. Should we call fictitious every model that includes just one fiction? Vaihinger’s and Fine’s strategies seem to imply a positive answer to this question. They do not even distinguish between idealizations and fictions. Some realist philosophers of science, such as Giere (2009) and Teller (2009, 2012), have replied that this would be a misuse, or at least an abuse, of the term fiction. They have pointed out that even if we were to call fiction every idealization contained in a model, it does not follow from this that the model as a whole is a fictional entity. A successful model can provide an (approximately) veridical representation of the world despite including fictional elements. Giere and Teller have a point here. If we pay attention to the usual attitude of scientists towards a model that contains some elements that are recognized as fictions, we will usually find that the model is taken at least as a partial or approximate representation of some aspects of its target. Let us take the example of the planetary models of the internal structure of atoms. According to the standard quantum theory, elementary particles, such as electrons, do not have definite trajectories in spacetime. For that reason, the electronic orbitals cannot be taken as real. We could say that they are useful fictions (certainly, inherited from pre-quantum models). Most textbooks on quantum mechanics or atomic physics make use of those planetary models -as well as visual representations of them- with the explicit proviso

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that the orbitals should not be taken as depictions of real trajectories. However, no textbook or treatise in physics, to the best of our knowledge, includes such proviso concerning the very existence of the electrons or the atomic nuclei. Electrons are by no means regarded as fictions by working physicists. Nobody thinks that electrons are similar to tiny spheres or to the plastic balls with which they are often represented, but (almost) everybody accepts that those elements of the atomic model represent real entities. That is why atomic planetary models as a whole are never called fictions in the scientific literature, although it is admitted that they contain both fictions and idealizations. The second brand of fictionalism concerning models is the one that conceives of them as products of the imagination, which are not essentially different from literary works of fiction. Scientific models are similar to literary works because they describe imaginary systems, such as frictionless pendula or infinite populations. This stance, which could be called the imagination view of models, was put forward by GodfreySmith (2006, 2009), Frigg (2010a, b, c), and Frigg and Nguyen (2016), under the heading of the fiction view of models.34 This view is based on some positive analogies between scientific models and literary works of fiction. However, those analogies can be employed to solve different kinds of problems concerning models. The analogy between models and literary fictions can be used to provide an answer to the ontological problem. In this respect, the answer would be that scientific models are the same kind of entities as literary fictions. This, by itself, is not very enlightening because the ontological status of fictional characters, and fictions generally, is far from being a clear issue. There are many different and conflicting conceptions of fictional entities in the vast philosophical literature on this topic and it is by no means obvious which one is the most adequate for scientific models. One traditional option is to conceive of fictions as abstract entities that are in many respects similar to sets, numbers, and other mathematical entities. Those realist or Platonist positions have to face very well-known problems concerning our epistemic access to nonspatiotemporal entities. Moreover, this account seems more adequate to theoretical models, say, Bohr’s atomic model, than to material models. A different option is to conceive of models as abstract artifacts, that is, as “cultural creations similar in kind to stories, theories, and laws” (Thomasson 2020: 67). Again, this fits better with theoretical models than with material models, such as the Newlyn-Phillips hydraulic machine designed to represent the flux of money in a country. Other positions include those that conceive of models as possible entities, or as non-existing concreta, instead of as existing abstracta. In each of these realist options, we will have to face hard metaphysical problems. There are also a variety of antirealist positions, according to which there are no fictional entities.35 As a consequence, one may be a realist or 34 Frigg

and Nguyen (2020) contains a systematic presentation of this view of models. Levy and Godfried-Smith (2020) includes several chapters that discuss different accounts of models as fictions. 35 See Friend (2007) for a general account of realist and antirealist conceptions of literary fictions; and Levy and Godfrey-Smith (2020) for different conceptions of fictionalism about scientific models. For more references concerning the metaphysics of fictions see the section on fictionalism in the commented bibliography at the end of this book.

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antirealist concerning models, depending on the position one has adopted about the ontology of fictions. In conclusion, it is not manifest that the analogy with fictions can provide clarification to the problems associated with the ontology of scientific models. The metaphysical problems posed by the existence of fictional entities may be a very interesting issue in itself but its relevance to the philosophy of models is still an open question. A different possible use of the analogy between models and literary fictions consists in elucidating how models represent the phenomena. According to this use of the analogy, scientific models represent their targets as literary fictions represent reality. This assumes that a fictional narrative aims at representing reality, a hypothesis that could be questioned. In any event, it can be granted that at least some subgenre of fictional narrative, exemplified by the realist novel from the nineteenth century, does aim at representing reality. More generally, it can be acknowledged that some visual arts, those which are called figurative, also aim at representing reality. What exactly follows from this analogy? Those fictionalist philosophers that appeal to this analogy think that it has some heuristic value to clarify the concept of scientific representation. However, if fiction is characterized by departure from reality, nonreferentiality and falsity, it will have a scarce bearing on the idea of representation. That is why this brand of fictionalism is based on the view according to which the defining feature of literary fiction is not falsity. As Frigg and Nguyen have put it, “what makes a text fictional is the attitude that the reader is expected to adopt towards it”, for instance, “when reading a novel we are not meant to take the sentences we read as reports of fact; rather we are supposed to imagine the events described” (Frigg and Nguyen 2017: 85). This notion of fiction as imagination, nonetheless, does not imply by itself any definite account of representation; in principle, it is compatible with different theories of scientific representation. It simply provides a heuristic standpoint to explore the issue. Walton’s (1990) theory of fictions in representational arts as make-believe games has been highly influential on several philosophers who endorsed the fiction view of models. Frigg (2010a, b, c), Toon (2012), and Frigg and Nguyen (2016, 2017, 2020), among others, have developed the analogy according to which building scientific models is like engaging in make-believe games by using some props as vehicles to represent some phenomena. In Walton’s account, props and principles of generation prescribe us to imagine some fictional propositions, which are then true in the game of make-believe, that is, true in the fiction. Something analogous happens with models: they prompt us to imagine some model systems of which they provide veridical descriptions. Sentences that are true in a model are similar to those fictional sentences that are true in a story, such as, “Robinson Crusoe lived 28 years in a desert island”, which is a different kind of truth from a factual truth like “Daniel Defoe lived in London”. There have been two main different lines of thought concerning how fictional models represent their targets. The first one is the indirect account of representation developed by Frigg and Nguyen (2016, 2017, 2020). In their view, models are imaginary entities that represent real-world targets in virtue of having a relationship of similarity or resemblance with them. However, the representation is mediated by

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what they call model systems, which are generated by model descriptions (together with some accepted background knowledge). Those imaginary objects represent a given target if they satisfy some specific conditions: they must denote the target, exemplify certain properties of the model, provide a key to interpret them, and impute those interpreted properties of the model system to the target (that is why they call it the DEKI account of representation). The second conception is the direct account of representation put forward by Toon (2012) and Levy (2012, 2015). According to this view, there is no mediating entity between imaginary descriptions and realworld targets, and, in a certain sense, there are no models as mediators. Models are then characterized as imaginative descriptions of phenomena that prescribe us how to imagine that the real systems are or behave. As a consequence, models represent directly their targets. Both accounts of scientific representation are still under construction and for that reason, they have not been elaborated in all their details. Many difficulties remain unsolved and this book is intended as a modest contribution to the debate on the fictional views of models, which is a family of different positions, rather than a unified theory of models.36 The fiction view of models is often understood as an account of the ontology of models exclusively. The general answer is that models are imaginary entities but this, in turn, poses the question of how should we conceive of that kind of entities. This answer may suggest that models are mental entities that exist only in the mind of individual agents. Although we can agree on the fact that models, as well as other scientific constructs, are the products of human imagination, it does not follow from this that they are purely imaginary objects, much the less that they are mental entities. On the other hand, it is not easy to explain how imaginary entities could be compared to real-world phenomena in order to determine whether they are similar or not to them. Whatever the answer to the ontological problem, it seems that a fictional conception of models does not imply a definite theory of representation. Frigg and Nguyen’s DEKI account, for instance, is in principle independent of the fiction view of models and compatible with other non-fictional conceptions. If so, the fiction view of models can be supplemented by different theories of representation. Regarding the question of the pragmatics of models, the fiction view is completely silent because it does not aim at providing an account of how models are used in scientific practices. And surely, the positive analogies between scientific models and literary works of fiction do not have very much heuristic value to explore the many functions of scientific models. Possibly, the negative analogies between artistic and scientific representations are more significant than the positive ones. It seems quite reasonable to assume that literary fictions and scientific models are used for very different purposes. For example, we do not employ the fictional characters or events of literary narrations to make calculations or predictions concerning the structure or the behavior of physical phenomena. From a functional point of view, one that is not interested in what scientific models are but rather in what we do with them, 36 This brief section does not make justice to the complexities of Walton’s make-believe theory or the direct and indirect accounts of scientific representation. For a detailed discussion and defense of the fiction view of models see Frigg and Nguyen’s contribution to this book.

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the analogy between artistic and scientific representations may not be particularly enlightening. In any event, the fiction view of models could be supplemented with an empirical study of the pragmatics of models. In principle, it may be compatible with different accounts of scientific practices. However, from a descriptive point of view, we can observe that most scientists are highly reluctant to qualify their models as works of fiction; they always prefer to describe them as idealizations that approximate the phenomena to a certain degree.

1.10 Models as Epistemic Artifacts The standard conception of knowledge as involving some sort of representation of the real-world phenomena was questioned by different philosophical traditions, such as pragmatism. According to pragmatist philosophers, science does not aim at representing the world; rather it provides useful instruments to solve the problems of our existence, the problems we have to face in our everyday experience to survive and improve our life. Some variety of instrumentalism concerning scientific theories has been the standard epistemology of pragmatist philosophers. In the domain of the philosophy of science, Hacking (1983) was influential in disregarding representationalism as the proper conceptual framework within which epistemological problems (such as the one of the approximate truth of theories) should be posed and solved. As is known, he emphasized the dimension of the intervention as a fruitful alternative. Instead of asking how models, as well as other scientific products, represent the phenomena, Hacking claims, we should better ask how we use models to interact with the world. The artifactual account of models, which was put forward by Knuuttila (2005, 2011, 2017), can be understood as an attempt at a non-representational conception of scientific models. It has a pragmatist spirit and, consequently, an instrumentalist slant. It does not aim at finding a substantive theory of scientific representation, that is, a general account that could explain how we obtain knowledge from models. Instead, artifactualism proposes to shift the focus of the philosophical analysis towards the study of how models are constructed and employed by their different users. That is what Knuuttila (2011: 263) calls “the productive point of view”, an approach that is in line with what we have called the pragmatic approach to the study of scientific models. Because models are artifacts, they should be conceived of as “human-made objects intentionally produced for some purposes within the sphere of particular human activities.” (Knuuttila 2017: 11) There cannot be any doubt that scientific models are intentional products designed by scientists with specific purposes in mind. What distinguishes scientific models from other types of artifacts, say, airplanes, is that they are epistemic tools constructed to obtain knowledge concerning some definite domain of phenomena. From this point of view, we can gain epistemic access to a given domain of phenomena when we interact with the models of those phenomena. We obtain knowledge, literally, by constructing and manipulating models. In principle,

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the artifactualist does not deny that models can provide successful representations of real-world phenomena. However, such representational relations are considered as accomplishments of scientific work typically involving different epistemic means. Moreover, artifactualism stresses the fact that models are employed in very different scientific practices in which the main purpose of the users may not be to represent a definite target. Models can be used, for instance, to explore new domains of phenomena, to design experiments or computer simulations, or to build other models or theories. The artifactual conception of models has a straightforward answer to the ontological question: models are artifacts of the same kind as other human products, such as microscopes, cell phones, or literary fictions. In principle at least, modeling, as an intersubjective human activity, must have a material dimension. We cannot dwell here on the question of how to distinguish natural from artificial objects or on other specific questions concerning the ontological status of artifacts. However, it follows from this point of view that all models possess some sort of materiality, that is, either they are physical objects or they have some physical support. This is true even for models that are usually called abstract entities. Mathematical and theoretical models generally cannot be used without being presented in some physical format; after all, every software needs some hardware if it has to be applied to solve a definite problem. As purely abstract entities, models cannot be employed to solve any specific problem for the simple reason that we cannot interact with them. Even mathematical equations require some material support if their solution is to be made public. The artifactual conception of models is still a program under construction and for that reason, it leaves us, for the moment, with many unanswered questions. They have to do both with the ontology of artifacts and with how epistemic tools are used to represent the phenomena or, more generally, to obtain reliable knowledge about the real world. The last one of the aforementioned questions seems most urgent for the artifactual approach, as it leans toward the pragmatist approach to scientific representation, and does not primarily focus on ontological questions.

1.11 The Place of Models in Science Most philosophers of science of the twentieth century assumed that scientific theories were the main vehicles of scientific knowledge and for that reason, that they had to be regarded as the main subject of study for the philosophy of science. By the beginning of the twenty-first century, the turn of the tide was reversed in favor of models. For instance, Suppe (2000: 109) wrote that “today, much of science is atheoretical” and that “today, models are the main vehicles of scientific knowledge”. This was certainly an overstatement. Twenty years later, when the distinction between models and theories has become problematic, we should acknowledge that both, models and theories, are the vehicles of scientific knowledge. In the first place, theories are taught as a part of the basic knowledge of many disciplines in most science courses. In the second place, science students learn to build models, primarily, as models

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of some established theories. Some domains of science are indeed atheoretical but this cannot be generalized to all scientific disciplines. Theories are still the basis of scientific education and science students learn to apply theories to solve well-defined standard problems. In some cases, but not always, solving those problems requires the construction of a model that makes possible the application of a general theory to a concrete particular situation. Certainly, the practice of normal science does not consist in inventing wide-ranging theories; the invention of such theories, say, Einstein’s special relativity is an exceptional episode in the history of science. Nonetheless, it is not true that the whole practice of normal science consists in building models, at least in the restricted sense of theoretical models of some generality. A good deal of practice in the field of experimental sciences involves constructing and calibrating instruments and employing them to collect data. Those data are later submitted to statistical analysis, which usually ends with the construction of a model of such data. In turn, theoretical models are built, as we have remarked, on every occasion in which a definite problem is not tractable with the theoretical resources available at that moment. This is certainly the prevalent situation in many sciences but it is not characteristic of all scientific practices at any time. The pervasiveness of idealized models has been acknowledged by most presentday philosophers of science and, because of the extensive use of computer simulations in all sciences, we can expect that the practice of modeling will gain even more significance in the future. The consequences of idealizations, which cannot be completely removed from any model, are also well understood: all scientific models are at best partial and inaccurate representations of some domain of complex phenomena. They can yield just approximate predictions or explanations of those phenomena that are regarded as adequate regarding some definite purposes. In principle, there are many different ways of modeling the same phenomena. Which model has to be regarded as the best representation of the phenomena (if it aimed at representing them) is something that cannot be answered in the abstraction of the problem that originated the model, the theoretical context in which it was built, and the purposes for which it was built. As Giere (2001: 1060) has remarked, “there is no best scientific model of anything; there are only models more or less good for different purposes”. This means that we may have different, and even mutually incompatible, models of the same phenomena, which are all equally good for different purposes, above all, because they aim at solving different problems. The adequacy for a given purpose has been recently acknowledged as the main criterion for evaluating models. Writing about climate models, Winsberg (2018: 33) has claimed that to be a good model is not only purpose relative but also relative to a standard of accuracy and a given domain of application. This pragmatic approach tends to deflate the significance of the problem of how models represent the phenomena. Whatever the answer one may offer to the question of what is an adequate representation, everybody should admit that in practice a model works as a solution to a definite problem when it is adequate for the purposes of the model builders or users. At the end of the day, the problem of scientific representation, the specific question concerning how models represent the phenomena, is just one of the many questions that can be posed concerning the nature and function of models in science. Even

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representationist philosophers often acknowledge that not every scientific model aims at representing the phenomena and that representation is not “the only (or even primary) function of models” (Frigg and Nguyen 2017: 50). There is, then, a broad room for the study of the ontology and the uses of scientific models independently of the issue of representation. In turn, because we do not have an entirely satisfactory theory of scientific representation yet, there is much more work to do in that field of research. Predicting the future development of our knowledge is always an impossible venture, however, given the pervasive use and the growing importance of computer simulations in the construction and application of models, we can guess with reasonable certainty that the study of simulations will become more and more relevant to the philosophy of scientific modeling. Acknowledgments We are grateful to Otávio Bueno and Tarja Knuuttila for reading and commenting the sections on structural representation and artifactualism, respectively.

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Duhem, P. (1906). La théorie physique. Son objet et sa structure. Paris: Chevalier et Rivière. [Second edition: 1914]. [English translation from the 1914 second French edition: The aim and structure of physical theory. Princeton: Princeton University Press, 1954]. Einstein, A. (1918). Prinzipielles zur allgemeinen Relativitätstheorie. Annalen der Physik, 55(4), 241–244. Einstein, A. (1919). Induktion und Deduktion in der Physik. In M. Janssen et al. (Eds.), The collected papers of Albert Einstein, Vol. 7: The Berlin years. Writings (pp. 218–220). Princeton: Princeton University Press, 2002. Field, H. (2016). Science without numbers: A defence of nominalism. Second edition. Oxford: Oxford University Press [First edition: Princeton: Princeton University Press, 1980]. Fine, A. (1993). Fictionalism. Midwest Studies in Philosophy, 18(1), 1–18 [Reprinted in M. Suárez, (Ed.), Fictions in science: Philosophical essays on modeling and idealization (pp. 19–36). New York: Routledge, 2009]. Fine, A. (1998). Fictionalism. In E. Craig (Ed.), The Routledge encyclopedia of philosophy (Vol. 3, pp. 667–668). London: Routledge. Friend, S. (2007). Fictional characters. Philosophy Compass, 2(2), 141–156. Frigg, R. (2010a). Models and fiction. Synthese, 172(2), 251–268. Frigg, R. (2010b). Fiction and scientific representation. In R. Frigg & M. Hunter, (Eds.), Beyond mimesis and convention: Representation in art and science (pp. 97–138). Dordrecht: Springer. Frigg, R. (2010c). Fiction in science. In J. Woods (Ed.), Fiction and models: New essays (pp. 247– 287). Munich: Philosophia Verlag. Frigg, R., & Nguyen, J. (2016). The fiction view of models reloaded. The Monist, 99(3), 225–242. Frigg, R., & Nguyen, J. (2017). Models and representation. In L. Magnani & T. Bertolotti (Eds.), Springer handbook of model-based science (pp. 49–102). Cham: Springer. Frigg, R., & Nguyen, J. (2020). Modelling nature: An opinionated introduction to scientific representation. Cham: Springer. Gerlee, P., & Lundh, T. (2016). Scientific models: Red atoms, white lies, and black boxes in a yellow book. Cham: Springer. Giere, R. (1988). Explaining science: A cognitive approach. Chicago and London: The University of Chicago Press. Giere, R. (1997). Understanding scientific reasoning. Fourth edition. Sea Harbour Drive, FL: Harcourt Brace College Publishers. [First edition: 1979]. Giere, R. (1999). Science without laws. Chicago and London: The University of Chicago Press. Giere, R. (2001). The nature and function of models. Behavioral and Brain Sciences, 24(6), 1060. Giere, R. (2004). How models are used to represent reality. Philosophy of Science, 71(5), 742–752. Giere, R. (2006). Scientific perspectivism. Chicago and London: The University of Chicago Press. Giere, R. (2009) Why scientific models should not be regarded as works of fiction. In M. Suárez (Ed.), Fictions in science: Philosophical essays on modeling and idealization (pp. 248–258). London: Routledge. Giere, R. (2010). An agent-based conception of models and scientific representation. Synthese, 172(2), 269–281. Godfrey-Smith, P. (2006). The strategy of model-based science. Biology and Philosophy, 21(5), 725–740. Godfrey-Smith, P. (2009). Models and fictions in science. Philosophical Studies, 143(1), 101–116. Hacking, I. (1983). Representing and intervening: Introductory topics in the philosophy of natural science. Cambridge: Cambridge University Press. Heidegger, M. (1954). Was heisst Denken? Tübingen: Max Niemeyer Verlag. Heidegger, M. (1961). Die Frage nach dem Ding: Zu Kants Lehre von den transzendentalen Grundsätzen. Tübingen: Max Niemeyer Verlag. Hesse, M. (1963). Models and analogies in science. London: Sheed and Ward. [Revised edition: Notre Dame: Notre Dame University Press, 1966].

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Hilbert, D. (1899). Grundlagen der Geometrie. In Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen (pp. 9–32). Leipzig: Teubner. [Seventh corrected edition: Grundlagen der Geometrie. Leipzig: Teubner, 1930]. Hughes, R. I. G. (1997). Models and representation. Philosophy of Science, 64(4) (Supplement): S325–S336. [Reprinted in R. I. G. Hughes, The theoretical practices of physics: Philosophical essays (pp. 150–163). New York: Oxford University Press, 2010]. Knuuttila, T. (2005). Models, representation, and mediation. Philosophy of Science, 72(5), 1260– 1271. Knuuttila, T. (2011). Modelling and representing: An artefactual approach to model-based representation. Studies in History and Philosophy of Science Part A, 42(2), 262–271. Knuuttila, T. (2017). Imagination extended and embedded: Artifactual versus fictional accounts of models. Synthese. https://doi.org/10.1007/s11229-017-1545-2. Laymon, R. (1985). Idealizations and the testing of theories by experimentation. In P. Achinstein & O. Hannaway (Eds.), Observation experiment and hypothesis in modern physical science (pp. 147–173). Cambridge, MA: The MIT Press. Levy, A. (2012). Models, fictions, and realism: Two packages. Philosophy of Science, 79(5), 738– 748. Levy, A. (2015). Modelling without models. Philosophical Studies, 172(3), 781–798. Levy, A., & Godfrey-Smith, P. (Eds.). (2020). The scientific imagination: Philosophical and psychological perspectives. New York: Oxford University Press. McMullin, E. (1985). Galilean idealization. Studies in History and Philosophy of Science, 16(3), 247–273. Morgan, M. (2012). The world in the model: How economists work and think. New York: Cambridge University Press. Morgan, M., & Morrison, M. (Eds.). (1999). Models as mediators: Perspectives on natural and social science. New York: Cambridge University Press. Morrison, M. (2015). Reconstructing reality: Models, mathematics, and simulations. New York: Oxford University Press. Nagel, E. (1961). The structure of science: Problems in the logic of scientific explanation. New York: Harcourt, Brace & World. [Second edition: Indianapolis: Hackett, 1979]. Reichenbach, H. (1924). Axiomatik der relativischen Raum-Zeit-Lehre. Braunschweig: Vieweg & Sohn. Shapiro, S. (1991). Foundations without foundationalism: A case for second-order logic. New York: Oxford University Press. Sneed, J. (1971). The logical structure of mathematical physics. Dordrecht: Reidel. Stegmüller, W. (1973). Theorienstrukturen und Theoriendynamik. Berlin: Springer. [English translation: The Structure and Dynamics of Theories. Berlin: Springer, 1976]. Suárez, M. (2003). Scientific representation: Against similarity and isomorphism. International Studies in the Philosophy of Science, 17(3), 225–244. Suárez, M. (2004). An inferential conception of scientific representation. Philosophy of Science, 71(5), 767-769. Suárez, M. (2010). Scientific representation. Philosophy Compass, 5(1), 91–101. Suárez, M. (2015). Deflationary representation, inference, and practice. Studies in History and Philosophy of Science, 49(1), 36–47. Suárez, M. (2016). Representation in science. In P. Humphreys (Ed.), The Oxford handbook of philosophy of science (pp. 440–459). New York: Oxford University Press. Suárez, M., & Solé, A. (2006). On the analogy between cognitive representation and truth. Theoria. Revista de Teoría, Historia y Fundamentos de la Ciencia, 21(1), 39–48. Suppe, F. (Ed.). (1977). The structure of scientific theories. Second edition. Urbana and Chicago: University of Illinois Press. (First edition: 1974). Suppe, F. (1989). The semantic conception of theories and scientific realism. Urbana and Chicago: University of Illinois Press.

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Suppe, F. (2000). Understanding scientific theories: An assessment of developments, 1969–1998. Philosophy of Science, 67(3) (Proceedings), S102–S115. Suppes, P. (1957). Introduction to logic. New York: Van Nostrand. Suppes, P. (1960). A comparison of the meaning and uses of models in mathematics and the empirical sciences. Synthese, 12(2/3), 287–301. Suppes, P. (1969). Studies in the methodology and foundations of science: Selected papers from 1951 to 1969. Dordrecht: Reidel. Suppes, P. (1993). Models and methods in the philosophy of science: Selected essays. Dordrecht: Reidel. Suppes, P. (2002). Representation and invariance of scientific structures. Stanford: CSLI Publications. Swoyer, C. (1991). Structural representation and surrogative reasoning. Synthese, 87(3), 449–508. Teller, P. (2009). Fictions, fictionalization, and truth in science. In M. Suárez (Ed.), Fictions in science: Philosophical essays on modeling and idealization (pp. 235–247). New York: Routledge. Teller, P. (2012). Modeling, truth, and philosophy. Metaphilosophy, 43(3), 257–274. Teller, P. (2014). Representation in science. In M. Curd & S. Psillos, (Eds.), The Routledge companion to philosophy of science. Second edition (pp. 490-496). London and New York: Routledge. [First edition: 2008]. Tent, K., & Ziegler, M. (2012). A course in model theory. New York: Cambridge University Press. Thomasson, A. (1999). Fiction and methaphysics. New York: Cambridge University Press. Thomasson, A. (2020). If models were fictions, then what would they be?. In A. Levy & P. GodfreySmith (Eds.), The scientific imagination: Philosophical and psychological perspectives (pp. 51– 74). New York: Oxford University Press. Toon, A. (2012). Models as make-believe: Imagination, fiction, and scientific representation. New York: Palgrave Macmillan. Vaihinger, H. (1927). Die Philosophie des Als Ob. Tenth edition. Leipzig: Felix Meiner [First edition: 1911]. [English translation from the 1920 sixth German edition: The philosophy of ‘as if’. New York: Harcourt, Brace, and Company, 1925]. Van Fraassen, B. (1980). The scientific image. Oxford: Clarendon Press. Van Fraassen, B. (1989). Laws and symmetry. Oxford: Clarendon Press. Von Neumann, J. (1932). Mathematische Grundlagen der Quantenmechanik. Berlin: Springer. Walton, K. (1990). Mimesis as make-believe: On the foundations of the representational arts. Cambridge, MA: Harvard University Press. Weisberg, M. (2013). Simulation and similarity: Using models to understand the world. Oxford and New York: Oxford University Press. Winsberg, E. (2018). Philosophy and climate science. Cambridge: Cambridge University Press. Woods, J. (2007). Fictions and their logic. In D. Jacquette (Ed.), Philosophy of logic (pp. 1063–1126). Amsterdam: Elsevier. Woods, J. (2018). Truth in fiction: Rethinking its logic. Cham: Springer.

Alejandro Cassini is Professor of Philosophy and History of Science at the University of Buenos Aires and Senior Researcher at the Consejo Nacional de Investigaciones Científicas y Técnicas of Argentina. He served for twenty years as editor of the Revista Latinoamericana de Filosofía. He is the author of El juego de los principios: Una introducción al método axiomático (Buenos Aires: A-Z Editora, Second edition, 2013. First edition: 2007). He is the editor (with Laura Skerk) of Presente y futuro de la filosofía (Buenos Aires: Ediciones de la Facultad de Filosofía y Letras de la Universidad de Buenos Aires, 2010). Juan Redmond is Professor of Philosophy at the Universidad de Valparaíso (Chile). He works on different topics of logic, philosophy of logic, and epistemology. He is Editor-in-Chief (with Shahid Rahman) of the Springer collection Logic, argumentation, and reasoning: Perspectives from the social sciences and the humanities. He serves as editor of the Humanities Journal of Valparaíso.

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He is the author (with Matthieu Fontaine) of How to play dialogues: An introduction to dialogical logic (London: College Publications, 2011). He is the editor (with Olga Pombo Martins and Ángel Nepomuceno Fernández) of Epistemology, knowledge and the impact of interaction (Cham: Springer, 2016).

Chapter 2

An Artifactual Perspective on Idealization: Constant Capacitance and the Hodgkin and Huxley Model Natalia Carrillo and Tarja Knuuttila

Abstract Natalia Carrillo and Tarja Knuuttila claim that there are two traditions of thinking about idealization offering almost opposite views on their functioning and epistemic status. While one tradition views idealizations as epistemic deficiencies, the other one highlights the epistemic benefits of idealization. Both of them treat idealizations as deliberate misrepresentations, however. They then argue for an artifactual account of idealization, comparing it to the traditional accounts of idealization, and exemplifying it through the Hodgkin and Huxley model of the nerve impulse. From the artifactual perspective, the epistemic benefits and deficiencies introduced by idealization frequently come in a package due to the way idealization draws together different resources in model construction. Accordingly, idealization tends to be holistic in that it is not often easily attributable to some specific parts of the model. They conclude that the artifactual approach offers a unifying view into idealization in that it is able to recover several basic philosophical insights motivating both the deficiency and epistemic benefit accounts, while being simultaneously detached from the idea of distortion by misrepresentation. Keywords Scientific modeling · Idealization · Artifactual account · Hodgkin and Huxley model

This project received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 818772) and from the UNAM PAPIIT project “Cognici´on, artefactualidad y representaci´on en la ciencia.” IN402018. Natalia Carrillo and Tarja Knuuttila contributed equally to this work. N. Carrillo (B) · T. Knuuttila University of Vienna, Vienna, Austria e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Cassini and J. Redmond (eds.), Models and Idealizations in Science, Logic, Epistemology, and the Unity of Science 50, https://doi.org/10.1007/978-3-030-65802-1_2

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2.1 Introduction There are two traditions of thinking about idealization offering almost opposite views on their functioning and epistemic status. While one tradition views idealizations as epistemic deficiencies, the other one highlights the epistemic benefits of idealization. The deficiency account of idealization focuses on how modelers idealize for the purpose of tackling complicated real-life phenomena and achieving tractable representations. The hope is that the advancement of science and the availability of better modeling methods could eventually deliver more accurate representations of worldly target systems. In contrast, the epistemic benefit account of idealization emphasizes the fact that in scientific modeling a detailed depiction is not often sought for. Instead, idealization facilitates more efficient explanations, and better understanding of phenomena that would not be possible without it, and so the justification of idealization does not lie in its future eliminability (see Batterman 2009, 16). Indeed, the crucial difference between the two accounts boils down to whether de-idealization is desirable or not (irrespective of whether it would be possible) (see Knuuttila and Morgan, 2019). While the deficiency accounts aim for de-idealization, the epistemic benefit accounts offer reasons for why scientists might be justified in not de-idealizing their models. Although the deficiency and epistemic benefit accounts appear thus diametrically opposed with regard to the status of idealization, there is still something that many, if not most of these accounts agree upon: idealizations introduce distortion into models with respect to our knowledge of worldly target systems. In other words, idealizations deliberately misrepresent. Distortions of these kinds are not difficult to find: the classic examples concern limiting concepts, e.g. when assuming that a thermodynamic system has an infinite number of particles, or treating populations of discrete individuals as continuous. In these kinds of cases, the model world undoubtedly involves features that are known not to hold in worldly target systems. But in many other cases the model is such an elaborate construct that it is difficult to tell how exactly it is supposed to differ from the worldly systems of interest—even in cases in which we had a lot of knowledge of them already. Consider, for example, assuming in an economic model that people form their beliefs of a value by drawing a value from a probability distribution (Alexandrova 2006), or modeling biochemical networks in an analogy to electric circuits (Knuuttila and Loettgers 2014). We suggest that in these and many other cases, idealization is better understood from an artifactual perspective that does not take the representational model-world relationship as a point of departure, presupposing the possibility of some straightforward comparisons between models and their supposed target systems. From the artifactual perspective, idealization can be treated holistically, as a set of interrelated assumptions emerging in, and entailed by, the model-building process. In focusing on model construction, the artifactual approach pays attention to the characteristics of actual tools of representation, and how they shape the target system. In this article, we approach idealization from the artifactual perspective (Knuuttila 2005, 2011, 2017), comparing it to the distortion-to-reality accounts of idealization,

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and exemplifying it through the case of the Hodgkin and Huxley model of nerve impulse. This early modeling achievement within neurophysiology has engendered a lively discussion within mechanistic philosophy of science. In this discussion, Craver (2006, 2007) and Levy (2013) have offered opposite interpretations of the epistemic value of the Hodgkin and Huxley model. While they do not explicitly discuss the model in terms of the notion of idealization, they nevertheless address its schematic and simplified nature. In evaluating the epistemic character of the Hodgkin and Huxley model, Craver gives it a deficiency reading, while Levy highlights the epistemic benefits its “aggregative” abstractions offer. From the artifactual perspective, the epistemic benefits and deficiencies introduced by idealization frequently come in a package due to the way idealization draws together different resources in model construction. Accordingly, idealization tends to be holistic in that it is not often easily attributable to just some specific parts of the model (even though it might seem so at first glance). Instead, the idealizing process tightly embeds theoretical concepts and formal tools into the construction of a model. Frequently, analogies are employed to recruit epistemic resources from other areas of research. In this process, idealization enables coordination between theoretical concepts, formal tools, measuring apparatus, diagrams, and experimental preparations (Carrillo 2018, 2019). The Hodgkin and Huxley model provides a good example of such intersection of analogical reasoning and idealization.

2.2 Galilean and Minimalist Idealization The contrast between the deficiency and epistemic benefit accounts of idealization occupies a center stage in Weisberg’s seminal discussion of Galilean and minimalist idealization.1 While Galilean idealizations make representations deficient, minimalist idealization brings epistemic benefits. In Galilean idealization, according to Weisberg, “[o]ne starts with some idea of what a nonidealized theory would look like. Then one mentally and mathematically creates a simplified model of the target.” (Weisberg 2007, 640). In other words, through mental and mathematical effort scientists seek to translate the existing knowledge into a computationally tractable and in principle corrigible simplified model. The notion of Galilean idealization, in Weisberg’s construal, draws attention to both the complexities of real-world situations, and the difficulties of representation that require simplification—those simplifications being subsequently alleviated by the advancement in mathematical techniques and computational power. Minimalist idealization, in contrast, focuses on causal factors instead of the challenges of representation. It seeks to single out “only the core causal factors” that “make a difference to the occurrence and essential character of the phenomenon in

1 Frigg

and Hartmann (2012) have introduced a somewhat similar distinction between Aristotelian and Galilean idealization.

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question” (Weisberg 2007, 642). As the quote shows, Weisberg characterizes minimalist idealization in line with the difference-making account of explanation put forth by Strevens (2008), although his construal of minimalist idealization covers also Cartwright’s, Mäki’s and Batterman’s accounts. But even though the aforementioned authors have argued for minimalist idealization, their accounts differ substantially. What Weisberg glosses over is that the accounts put forth by Cartwright, Mäki and Batterman cannot easily, if at all, be characterized as variants of the difference-making account.2 Mäki (1992) and Cartwright (1999) rely on the notion of isolation: in idealizing, scientists isolate some causal factors in an analogy to an experimental setup. Mäki builds his account on the idea of how various unrealistic model assumptions are used to theoretically “seal off” a set of relations from the influence of others. Cartwright (1999) invokes what she calls a “Galilean experiment” that studies the effect of one cause operating on its own by eliminating all other possible causes. Mäki’s and Cartwright’s accounts seem more modest in addressing the contributions of separable causal factors instead of aiming to pick out the causal difference-makers. In other words, idealization may only aim at studying causal capacities of a system. Given that Cartwright refers to “Galilean experiments” in delineating her isolationist account, and McMullin’s classic essay on “Galilean idealization” (1985) gives a much broader account of Galilean idealization than what Weisberg does, the term Galilean idealization seems to be a partially ambiguous label for deficiency accounts of idealization. Furthermore, the accounts of Strevens, Mäki and Cartwright differ substantially from that of Batterman, although all four of them have addressed the benefits of minimal modeling that make de-idealization undesirable. What the three former accounts have in common is that they cast idealization in terms of singling out a few causally effective factors: idealization makes a positive epistemic contribution in identifying the contributions of these causal factors/difference-makers. Batterman, in turn, has explicitly argued against isolationist and other representational accounts that rely on “veridical representation of difference-making features within the model”, or more generally “common features” between the model and a real-world system (e.g. Batterman and Rice 2014, 355). Batterman and Rice argue that idealization can make salient how diverse real-world systems, despite the differences in their micro-causal make-up, can “exhibit the same patterns of behavior at much higher scale” (Batterman and Rice 2014, 350). Idealization thus functions as a device for coarse graining, enabling the recognition of multiply realizable patterns across various phenomena. Such idealizations may show which features are irrelevant, yet Batterman’s account does not boil down to a difference-making account. The point is that idealizations themselves perform positive explanatory work, instead of assigning the explanatory task only to causal difference makers (through separating their contributions from 2 Potochnik

(2017) considers idealization rampant and unchecked in science. In her account idealizations help limited cognitive agents to set aside complicating factors to identify causal patterns. Consequently, Potochnik’s notion of idealization contains features of both deficiency and epistemic benefit accounts.

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those of non-difference makers, see Rice 2018). Although Batterman thus emphasizes the epistemic benefits of idealization, his version of minimalist idealization does not share the representationalist commitments of the other minimalist accounts. Indeed, despite their opposite approaches to the epistemic status of idealization, the traditional deficiency and epistemic benefit accounts of idealization implicitly apply the criterion of representational accuracy in their analysis of idealization. At the bottom there is the idea of idealization as a distortion that already suggests a model-world comparison, i.e. representing the worldly systems differently from how they actually are, and ascribing to them properties they do not have (see GodfreySmith 2009, 47). The criterion of representational accuracy leads in the case of deficiency accounts to the quest for de-idealization. The benefit accounts, in turn, both in their difference-making and isolationist guises, suppose that some parts of the model accurately describe some causal factors and their functioning. Yet, from the perspective of scientific practice, the criterion of representational accuracy does not seem to adequately capture the way scientists employ idealization in their modeling practices. The deficiency account presumes that successive rounds of modeling could bring the model more accurate and realistic. Yet this presumption may be unwarranted, already because of the way the model was constructed. Such gradual de-idealization is often not achievable, or feasible, firstly, due to the affordances and limitations of the representational tools employed. The specific mathematical and computational techniques chosen, for example, shape the modeled phenomenon in particular ways (just think of how heterogeneous phenomena are rendered similar through the application of network methods). Secondly, and relatedly, in scientific practice mathematical and computational techniques are often coupled with particular theoretical concepts and perspectives and so intersected in the actual construction of a model. Consequently, any representation-independent correspondence to a target system seems an unattainable goal at the outset, and not just for practical reasons (see Knuuttila and Morgan 2019). Some discussions of idealization affirm this difficulty of de-idealization in a roundabout way. For instance, Sklar distinguishes between controllable and uncontrollable idealizations on the basis of how tractable they are and whether scientists know how to compensate for them (e.g. Sklar 2000). In other words, in the case of uncontrollable idealizations it seems misguided to talk about distortion of reality when scientists do not even know how they might be corrected. Minimalist idealization, in its difference-making, and isolationist variants appear to make less taxing demands than the deficiency view, but a closer examination shows that this is not the case. The underlying commitments of these accounts concerning both the decomposability of models and the causal structure of the world make them more heavily dependent than the deficiency account on representational and conceptual transparency. Though the deficiency account presumes that models could be de-idealized such that they would better approximate the real situations, they at least acknowledge the challenges of representation. The way to move forward, we suggest, is to detach the discussion of idealization from the ideas of distortion and misrepresentation. Idealization may distort but it also does something else: it keeps the model together. We will claim that two dimensions

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of the artifactual account, namely, its focus on the actual representational tools on one hand, and on the constrained construction of the model on the other hand, can recover many basic insights of the deficiency and epistemic benefit accounts. However, the artifactual account does not primarily offer a new account of idealization, but rather an alternative metalevel perspective to modeling. There is no need to throw many of the insights of the traditional discussions away—once they are freed from their traditional representational and realist assumptions.

2.3 The Artifactual Perspective on Idealization The artifactual approach sees models for what they are; human made, altered or engendered objects3 , whose affordances are being utilized for epistemic purposes, in the context of specific scientific practices. Although most models can be considered as human made objects, there are other kinds of entities, such as model organisms and laboratory populations that might better be characterized as human altered and/or human engendered. Knuuttila (2005, 2011, 2017) develops the artifactual account of models as an alternative to the representational approach, which does not duly recognize the epistemic value of modeling: The characteristic unit of analysis of the representational approach, the relationship of a single model and its supposed real target system, is too limiting in that it pays no attention to the models themselves as unfolding, constructed entities, or to the model-based theoretical practice that typically proceeds on the basis of many related, and also complementary, models (Knuuttila 2011, 263).

Knuuttila suggests that models, understood as particular kinds of epistemic artifacts, can be approached from two intertwined perspectives. First, scientific models are objects, whose construction is constrained in view of some epistemic goals.4 The traditional answer to the question of how models are able to give us knowledge appeals to the notion of representation. In contrast, perceiving a model as an artifact pays heed to its constrained arrangement that renders a certain scientific problem more accessible and manageable, helping scientists to address it in a systematic manner. That idealization enables an epistemic access to complicated phenomena is recognized in the literature (though with representationalist overtones). But idealization is also inherently related to the second dimension of the artifactual approach: the focus on the actual representational tools made use of in modeling. The constrained constitution of a model is not only due to its purposeful construction, but also to the representational tools employed in the theoretical, mathematical 3 Artifacts do

not have to be human made objects. A rock used to open a clam may be thought of as an artifact even though it is not constructed by humans for that purpose. Artifactuality in general can rather be defined through the roles an object plays in some human (or animal) activity. 4 Models are typically multipurpose tools: the aims of modeling may and do change, and multiple aims may co-exist as models are being reused, reconstructed and repurposed by different groups and stakeholders.

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and computational framing of the research question. These tools enter their own specific constraints into models. It is important to note that such constraints are both enabling and limiting. For example, mathematical and computational methods allow particular inferences and solutions, but not others. Different representational tools are suitable for expressing different things. From the artifactual perspective idealization emerges as an ineliminable part of model-building, sitting at the intersection of various constraints—theoretical, conceptual and representational—both affording and narrowing down. The artifactual approach lends a unifying view into idealization in that it is able to recover several basic philosophical insights motivating both the deficiency and epistemic benefit accounts, being simultaneously detached from the idea of distortion by misrepresentation. It can accommodate both the importance of tractability stressed by the deficiency account, and the epistemic benefits delivered by minimalist idealization. These insights can be elicited from the twofold character of models as unfolding objects constructed by employing already established representational tools in view of some epistemic aims. Although the focus of the artifactual account on representational tools highlights the importance of tractability, it also proves richer in this regard. As the artifactual account is not based on misrepresentation but addresses model construction instead, idealization can be seen to fulfill many other tasks than that of rendering the model tractable. In fact, idealization makes the model possible in the first place. It enables modelers to apply different mathematical and computational techniques as well as measuring and intervening apparatus to investigate the scientific question at hand. While idealization has traditionally been understood as deliberate misrepresentation of a feature of the target system, the artifactual approach is not hung up on the accuracy/distortion of a model or its parts. In contrast, the artifactual approach views idealization as a set of assumptions that aligns different representational tools in the pursuit of constructing a model capable of answering pertinent research questions. Thus, idealization enables the application of mathematical and computational tools such that the model holds together, allows manipulations and generates useful results. Whereas the focus on the use of representational tools addresses the tractability concerns of the deficiency account, the focus on the constrained construction of a model highlights the benefits of idealization. And it does so without the (too) heavy realist and representationalist commitments of the traditional benefit accounts. The artifactual account perceives theoretical models as highly constrained objects that are frequently built to study some pending theoretical or empirical questions. Modelers study and manipulate models in order to better understand some interdependencies lying behind actual and possible phenomena. Although such understanding-bearing models typically are minimal, they also are often too artificial, preliminary and hypothetical to really pick any difference-making factors. And if they seem to do so, it seems rather a result of a successful modeling endeavor than a feature of the modeling heuristic itself. More can be said in favor of the isolationist accounts that are causally more cautious in not expecting the isolated causal factors to be actual difference makers. But, as already mentioned, such isolation accounts make strong

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(de)decomposability assumptions concerning both the modularity of worldly target systems and models themselves.5 While thus the artifactual approach to idealization accommodates many important insights of deficiency and epistemic benefit accounts, it does not presume that idealization boils down to deliberately misrepresenting “what the world is actually like” by introducing distortion to scientific representations through “known falsehoods” (Levy 2018, 8). One central problem of the idea of distortion is that it supposes too much to be known. It is as if in modeling scientists were representing systems of which they already had well-articulated and certified knowledge. Second, viewing idealization as distortion disregards the modal dimension of science. Namely, scientific modeling is an explorative endeavor that aims to probe how certain phenomena could be produced, instead of just studying actual phenomena (Gelfert 2016).6 The focus is on genuine possibilities and not just some counterfactual scenarios within the range of known behavior of some actual systems. Minimal modeling, from the artifactual perspective, frequently involves narrowing down to the minimal elements and interactions that could be sufficient to produce a behavior or a pattern of interest. Yet, the possibilities that can be explored are constrained by the tools available (i.e. what mathematical methods can be used, what analogies are available, what can be measured). These constraints point towards the third important problem of distortion accounts: their neglect of the challenges of representation. The distortion view implicitly assumes that the representational tools used in science would furnish scientists malleable and transparent enough means for choosing how to misrepresent the already known world. This last criticism concerns especially the minimalist variant of the distortion accounts of idealization. Finally, and perhaps paradoxically, attending to the actual model construction highlights the conceptual dimension of modeling. The strange consequence of the notion of distortion by misrepresentation is the virtual disappearance of theoretical and conceptual activity from our notion of modeling. It becomes a matter of at least partial accurate representation of the world (as it is actually like), instead of providing theoretical perspectives for understanding of actual or possible phenomena. The artifactual perspective witnesses how theoretical, conceptual, interventional and representational resources become intertwined in the modeling process. In the following sections we examine the Hodgkin and Huxley model from the artifactual vantage point, highlighting the role various kinds of idealizing assumptions have played in its construction. We pay particular attention to the assumption of constant membrane capacity, and how it enabled the analogical transfer of the toolbox of electrical engineering to nerve signal research.

5 The epistemic benefit account by Batterman and Rice (2014) seems to be in line with the artifactual

account, since they focus on the epistemic work done by limiting operations. has recently argued that applying mathematical methods involves holistic distortion that enables researchers to “to extract various kinds of modal information” (2018, 2802).

6 Rice

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2.4 The Hodgkin and Huxley Model The Hodgkin and Huxley model consists of a set of differential equations reproducing the behavior of nerve cells as registered in experiments on squid giant axons. These experiments were performed with voltage-clamp techniques invented in the 1940s after the discovery that squids have a giant axon that is suitable for nerve signal experimental research. The Hodgkin and Huxley model (HH model) is based on an analogy between an electrical circuit and the nervous membrane, equating ionic currents across the membrane with electrons flowing across resistances in an electric circuit. Drawing an analogy between the nerve cell and electrical circuits, Hodgkin and Huxley were able to derive equations that establish relations between current, voltage and ionic permeability at one point in the membrane. Before the consideration of the nerve cell membrane as an electrical circuit, the electrophysiological program in which Hodgkin and Huxley were trained had already benefitted from other analogies. The analogy between galvanic cells and nerve cells, in particular, played a vital role. Galvanic cells were used in physical chemistry at the end of the 19th century to study the relationship between chemical and electrical gradients in ionic solutions. They consist of two compartments separated by a semipermeable membrane (see Fig. 2.1). Semi-permeable membranes allow some species of ions to cross but not others. The experiments in these devices involve dissolving different amounts of salt in each compartment and measuring differences in electrical potential between the compartments with a voltmeter. The ions in the salt diffuse, and the ion species that the membrane is permeable to, can move across to the other compartment. If more salt is placed in one compartment, then there will be a diffusion gradient from one compartment to the other. Since ions are electrically charged, movement of ions from one compartment to the other breaks the electrical balance, generating an electrical potential amongst the two compartments. Eventually, the concentration of ions of the species that can travel across the semipermeable membrane is stabilized. This is called electrochemical equilibrium.

Fig. 2.1 Diagram of a galvanic cell Source Natalia Carrillo

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As this physicochemical balance of forces came to be understood better, the idea that a similar mechanism could lie behind nervous transmission was proposed (Ostwald 1890). This approach defined the research agenda for many physiologists in the next decades. An important resource for this research was the Nernst equation. In his famous work “The electromotive action of ions” (1889), Nernst developed a mathematical expression for the electromotive force of ions in a galvanic cell in terms of their concentrations and voltage across membranes. With this equation it is possible to calculate the electric potential difference between two solutions of a uni-univalent electrolyte7 at different concentrations separated by a semi-permeable membrane. Based on Nernst’s work, the excitatory process of nerves began to be associated to the “nerve membrane” on physicochemical grounds. A particularly influential development of this sort was Bernstein’s membrane theory (1902). Bernstein had detected an “action current” in nerves, associated to muscle contraction. To explain the rise and fall of the current, Bernstein assumed that a difference in concentration exists between the inside and outside of the membrane and suggested that the observed current was due to a “collapse” of the membrane. Thus, in the excited state the cell becomes permeable to all ions such that the ions are momentarily free to cross according to the diffusion and electrical gradients. That was the main theory at the time when Hodgkin and Huxley started their research. Giant axons of squid had been discovered in 1936 by J. Z. Young, and were immediately recruited by neurophysiologists for their wonderfully advantageous proportions (they can be up to 1 mm in diameter). This experimental material was susceptible to measurements that were impossible in other tissues with the technology of the time. The prospect of intervening electrically in the giant axon of squid motivated researchers to investigate the electrical features of the nerve cell membrane in order to understand the interactions between the electrical devices and this newly discovered material. This allowed the development of apparatus such as the voltage clamp that injected and recorded electrical currents in the nerve cell.8 With this equipment, new empirical discoveries were made. After performing experiments on squid giant axons, Hodgkin and Huxley realized that Bernstein’s account was not entirely correct, since the membrane did not “collapse” but briefly became first permeable to potassium and later permeable to sodium (Hodgkin and Huxley 1952a, Huxley 1999). As these measuring artifacts were developed and the nerve cell was rendered in electrical terms, various theoretical and representational tools traditionally associated to electrical engineering and electromagnetism became available for theorizing about nerve impulse generation and transmission. Electric circuit diagrams were used to depict the dynamics of charge distribution around and across the membrane. Cole and Baker were the first to model the membrane in terms of what they described as 7A

solution in which each ion has a valence of 1, and produces two ions when dissociated. voltage clamp fixes the transmembrane voltage of the membrane to a value set by the experimenter, and records the current that had to be injected in order to maintain voltage steady at that value. That recording is the inverse of the actions that took place in the membrane, and was interpreted by Hodgkin and Huxley and others as the inverse of the transmembrane currents.

8 The

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an “approximate equivalent” electrical circuit (Cole and Baker 1941). Hodgkin and Huxley later on exploited this idea successfully, obtaining a system of equations that can simulate the electrical recordings in giant axons. Hodgkin and Huxley contributed to the aforementioned line of research that was trying to bring together the previous physicochemical model with resources from the field of electrical engineering. Whereas Cole and Baker focused on a circuit with a resistor and an inductor, Hodgkin and Huxley analyzed the dynamics of ionic currents around the membrane in terms of currents in a resistor-capacitor circuit with a constant capacitance and ohmic variable resistances. The HH equivalent circuit interprets the insulating features of the membrane as the behavior of a capacitor,9 the difference in ionic concentration across the membrane as an electric potential source,10 and the mechanism of permeability as variable resistances. In the equivalent circuit the voltages of the different batteries were set to the value of the electrochemical equilibrium of the corresponding ionic species (sodium and potassium), calculated with Nernst’s equation. The resistances11 were the most difficult to characterize, and were thought of as variable resistances with first order dynamics that were fine-tuned to empirical data. Hodgkin and Huxley consider the capacitor of their circuit as a one with constant capacity. A capacitor would change its capacity to store charge if the distance between its plates changes, or if the area of the plates changes. In the devices used in electrical circuits this is seldom the case, that is, the capacitance of capacitors is usually fixed. Although the membrane is materially quite different from a capacitor like the one used in circuits, Hodgkin and Huxley assumed that the capacitance of the capacitor in the “equivalent” circuit is constant (Hodgkin and Huxley 1952b, 505; Hodgkin et al. 1952, 426). This was an important assumption for Hodgkin and Huxley for a number of interrelated reasons that include but also go beyond the formulation of the mathematical model. By conceiving the nerve cell as an electric circuit, it was possible to obtain a mathematical expression for the ionic currents across the membrane by applying Ohm’s Law and Kirchhoff’s laws to describe the movement of electric currents in the equivalent circuit. In view of the discussion of idealization, it is important to note that Hodgkin and Huxley’s mathematical model retains the already imported set of theoretical and representational resources from physical chemistry, through setting the electromotive force (i.e. the voltage of the batteries in the circuit) to the value given by the Nernst’s equation. Thus, the analogy to an equivalent circuit is not independent 9A

capacitor is usually constructed of two plates of conducting material separated at a distance small enough so that charges on one side will feel electric repulsion or attraction from charges on the other plate. If connected to a battery, one plate becomes negatively charged with respect to the other. 10 This is the equivalent of a battery. When a cable connects one pole of the battery to the other, the charges tend to move from the pole with excess negative charge to the one with less negative charge. 11 Hodgkin and Huxley used the term “conductance,” which is the multiplicative inverse of the resistance, but for simplicity we describe them as resistances since both terms retain the same underlying concept.

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of the analogy to galvanic cells, they rather become integrated in the Hodgkin and Huxley model. In other words, the recruitment of epistemic resources from electrical engineering to nerve signal research was related to the previous idealized physicochemical rendering of the workings of the nerve cell as a rigid barrier with changes in permeability (Carrillo 2018). One might argue that the assumption of constant capacitance is an idealization in the sense of distortion because there are no perfect capacitors in nature. Or one could wonder whether it amounts to omission instead, since some minor variations in the membrane’s width are discarded when conceiving the membrane’s capacity as constant. But the artifactual perspective underlines how the idealization of constant capacitance allows for coordination of an electrical interpretation of the nerve cell with the previous physicochemical elements that were already playing a theoretical role. This role is neither entirely detrimental nor entirely beneficial, highlighting that such idealization is less understandable through model-world comparison than through its integrative role in drawing together particular representational tools and their already established uses. Let us now comment on the role that this idealized approach to the nerve signal played in the experimental part of Hodgkin and Huxley’s research. One important empirical result Hodgkin and Huxley obtained concerned the separate contributions of sodium and potassium to the overall current (Hodgkin and Huxley 1952a, Fig. 5). To perform this experiment, they changed the composition of the extracellular solution such that sodium ions would be in electrochemical equilibrium. This amounts to the experimental equivalent of taking away the sodium battery from the equivalent circuit. They then stimulated the axon, generating a signal that would, in their interpretation, only reflect potassium currents, since sodium ions were not subject to any electromotive force. The interpretation of this experiment as the isolation of potassium current relies crucially on the assumption that once the capacitor is charged, the currents in the membrane are all transmembrane currents. This reading of the experiment by Hodgkin and Huxley makes important assumptions regarding the nature of the membrane. From the viewpoint of the equivalent circuit, the idea is that through the process of nervous transmission the capacitor does not change shape or otherwise change its capacity to store charge. If the membrane would change shape, then the neighboring charges would be displaced, generating capacitive currents that could contribute to the recordings. This would imply that the recorded currents could be due to potassium ions crossing the membrane or because of the displacement of charges neighboring the membrane. Assuming that there are no variations in the membrane’s capacitance, Hodgkin and Huxley were able to conclude that the current they registered was due to potassium ions crossing the membrane. They then subtracted this curve to the total current in normal conditions, and obtained the sodium current.12 For this reason, the assumption of constant capacitance plays a role in the interpretation of experimental results. 12 Hodgkin and Huxley also considered the contribution of a “leak” current, which accounted for minor errors in prediction and measurement. We are not discussing this for simplicity since it does not contribute substantially to the excitable behavior of the axon.

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The assumption of constant capacitance was also mathematically convenient. In order to derive the equations, it is necessary to describe the currents in each of the electronic devices of the equivalent circuit (Kirchoff’s Law). The equation that describes the charge in the capacitor is Q = CV, where C is capacitance, V is voltage and Q is charge. In order to get the current in the capacitor (I c) one derives the equation, obtaining Ic =

d dC dQ dV = (C V ) = C +V dt dt dt dt

disappears (since the derivative If the capacity is a constant function the term V dC dt of a constant function is zero). If capacitance is not constant, this term would of course remain, and the equations describing the system would be different than those obtained by Hodgkin and Huxley (1952b). In this manner, the assumption of constant capacitance is not only present in the interpretation of the experiments but also in the derivation of the equations. Ultimately, the idea that there are no significant contributions from capacitive currents to the dynamics of transmembrane voltage during nervous transmission became deeply ingrained into the electrophysiological tradition (Takashima 1979, 133). But some researchers have stressed that this assumption does not have the empirical support it would require to be considered as unproblematic (Iwasa and Tasaki 1980, Heimburg and Jackson 2005). As we will see in more detail in the following sections, the issue goes beyond the question of whether constant capacitance is an eventually de-idealizable falsity or a beneficial lie, since it is not clear what it would mean for the membrane to truly be a constant capacitor or not to be one. The point is that the assumption of constant membrane capacity is foundational for the research program in question in a way that eludes any idealization-free method of finding out whether the membrane is a constant capacitor. The evaluation of the epistemic value or status of constant capacitance cannot rely on model-world comparisons in any straightforward way. We expect this finding to apply to many other idealizations as well, at least on closer inspection, and taking into account the role they play in model construction.

2.5 Discussion of Idealizations in the Hodgkin and Huxley Model The lively philosophical discussion of the Hodgkin and Huxley model furnishes a fruitful vantage point for delineating the differences between the artifactual and idealization-as-distortion accounts of idealization. Philosophers participating in this discussion have presented widely different interpretations of the epistemic contributions of the Hodgkin and Huxley model to neuroscience. The first to address the Hodgkin and Huxley model from a philosophical perspective was Weber, who

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claimed that the HH model explains in the same way as many physical explanations do: it entwines the experimental regularities and general physical laws (that are invariant under some interventions) (Weber 2005, 2008). In contrast to Weber, Craver (2006, 2007) argued that the HH model should not be understood as being derived from the laws of physics, but rather as a how-possibly sketch of a mechanism that sustains nerve impulses.13 For Craver, the HH model is only of a how-possibly character, because it does not give an account of the “nuts and bolts” of the mechanism by which ions cross the nervous membrane. Based on this fact—recognized by Hodgkin and Huxley themselves—Craver claimed that the explanation of the nerve impulse was not truly given until the proteins that form ionic channels across the membrane were discovered, thereby completing the explanatory sketch.14 According to Craver, it was only at this stage that a complete mechanistic explanation of the nerve impulse was delivered. Ultimately, Craver claims that in order for mechanistic models to explain, they would need to “account for all aspects of the phenomenon by describing how the component entities and activities are organized such that the phenomenon occurs” (Craver 2006, 374). Levy (2013) picked up this requirement of completeness arguing, contra Craver, that the explanatory achievement of the HH model is in fact due to its abstract character. Because the HH model abstracts from the individual movement of ions, it is able to more generally account for the ionic currents—without having to open the “black box” of the mechanism of ion transport. For Levy, the contribution of the model is due to its characterization of regularities at an aggregative level: “the discrete-gating picture relates whole-cell behavior to events at a lower level via aggregation: the system’s total behavior is the sum of the behaviors of its parts.” (Levy 2013, 15). He goes on to explain that such “aggregative abstraction” could be “truer to the mechanistic ideal, because it explains the relationship between lower-level mechanisms and higher-level ones” (20).15 Although this philosophical discussion of the epistemic contribution of the Hodgkin and Huxley model is not framed in terms of idealization, the way it focuses on the simplified/schematic nature of the HH model certainly allows for such an interpretation. Indeed, the contrast between Levy and Craver revolves around the 13 Craver’s criticism of Weber is a part of his more comprehensive mechanistic account of explanation, whose main target of criticism is the covering law account of explanation. 14 Discrete ion fluxes were detected by Neher and Sackmann in the 1970s, supporting the idea of a passive mechanism of ionic transport. Later, in the late 1990s, evidence of the existence of the potassium ion channel was obtained with x-ray crystallography. These results were considered as sound evidence for the hypothesis that it is voltage-sensitive protein ion channels that change the permeability of the membrane during a nerve impulse. 15 We consider aggregative abstraction an idealization, since we take that omissions and distortions cannot sharply be distinguished, in contrast to Levy (2018), who holds that we ought—and can—differentiate between them. Another justification for considering aggregative abstraction as an idealization is due to the distinction between Galilean and minimalist idealization. If minimalist idealization is understood as removing non-difference-making factors (or their contributions), it would amount to the conventional abstraction-by-omission account. So, by the standards of already established discussion of idealization, it is legitimate to regard Levy’s “aggregative abstraction” as idealization.

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question of whether the HH model should be de-idealized in order to be explanatory, or if the model was explanatory because it ignored such detail. Consequently, Craver gives a deficiency reading of the HH model, as it does not make explicit which are the nuts and bolts of the mechanism of ionic transport. Levy, on the contrary, views the simplified nature of the HH model as an epistemic benefit, since in abstracting from the mechanism of ion transport the HH model is able to account for the ionic currents. Note how both authors, despite their differences, nevertheless ascribe to the representational approach in line with what we claimed above concerning the common supposition shared by both deficiency and epistemic benefit accounts of idealization. Both authors evaluate idealizations in terms of how they are able to pick out what is relevant for the explanation of the target phenomenon (and leave out what is not), and how such choices could be detrimental should they ignore relevant parts (or levels of abstraction). In other words, both authors implicitly agree that the assessment of idealizations should be done on the basis of model-world comparisons. The analogical bedrock of the HH model casts doubt on the shared reliance of both Levy and Craver on the ability of scientists to hand-pick relevant factors or levels, or omit them in some representation-neutral manner. Interestingly, other philosophical discussants have paid attention to the role different formal and theoretical tools play in the derivation of the HH equations. Bogen (2008) suggested that the laws, such as Kirchhoff’s Laws and Ohm’s Law, are used in the HH models as calculation tools. That is, they function as formal relations used to derive the equations, and to hypothesize about electrical quantities of interest (Bogen 2008, see also Schaffner 2008). We take Bogen to be pointing at the artifactual role of the equivalent circuit and the laws used to describe the currents in it. Under this interpretation, rather than considering the electric circuit as a (mis)representation of the membrane, it is better viewed as an analogy that is associated with representational tools allowing the study of the nervous membrane’s excitable behavior. Indeed, Hodgkin and Huxley themselves underlined the fact that the equations do not point towards specific mechanisms but provide a way of mathematically describing the overall dynamics (1952b, p. 541, also discussed in Craver 2008, 1023; and in Levy 2013, 8). Regarding the constant capacitance idealization, it is interesting to note that the previous philosophical discussion of the HH model has not addressed it at all. This may be due to the fact that this particular idealization is not easily rendered representationally. If considered in terms of the deficiency account, one would expect that further improvements of the HH model might be able to correct this idealization. But correcting this idealization would mean changing the interpretation of the empirical results, and ultimately debunking the whole research program (something that is attempted by some neuroscientists, see Tasaki 1982, Lowenhaupt 1996, Heimburg and Jackson 2005, El Hady et al. 2015). On the other hand, from the epistemic benefit perspective, this idealization could be viewed as a distortion that ignores details that are not difference makers. This interpretation would be more suitable than the deficiency view with regard to the assumption of constant capacitance. It exhibits the fact that, to the best of their knowledge, Hodgkin and Huxley thought that capacitance was constant, and therefore did not consider it as a difference maker. However, this is not the whole story of why

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they made this assumption. The evidence for constant capacitance at the time was insufficient for it to be considered along the lines of the difference-making account, i.e. there was no conclusive evidence that capacitive currents would not contribute to the overall measured current (see Takashima 1979). So why did Hodgkin and Huxley make this assumption? The artifactual account, we claim, can give an answer. The assumption of constant capacitance did not emerge just for tractability reasons, and neither can it be cast solely in terms of difference-making. The isolation of a causal factor story by Cartwright and Mäki might fare better, yet it relies too much on decomposability. The point is that the previous renderings of the nerve signal were operating on assumptions that, when forced to be thought of in electrical terms, would translate into constant capacitance. Consequently, the assumption was already shaped by previous modeling attempts and conceptualizations of the nerve signal that both enabled and bounded the way the HH model was achieved. Last but not least, the research program developed by Hodgkin and Huxley (and many others) would not have made sense if the capacitance were allowed to vary. Such acknowledgement would have obliged scientists to consider that the currents could be due to either capacitance changes or permeability changes (in regard to both the experiments and the equations!). As a result, the two accomplishments of Hodgkin and Huxley, the experiments and the model, would have been nullified. And it is highly likely that the whole research program would have been led into an entirely different direction. The artifactual perspective on Hodgkin and Huxley’s achievement attests, then, to the role of idealizing assumptions in the intertwinement of different theoretical, mathematical and empirical considerations that cannot be related to misrepresentation alone. From the artifactual viewpoint, the idealization of constant capacitance (or of the membrane as a rigid semipermeable membrane) emerged from the effort of aligning and integrating diverse epistemic resources that had previously been exploited. These resources include the galvanic cells, Nernst’s equation, electric circuit diagrams transferred from electrodynamics (and the laws applied to them), and the actual electrical devices implemented in the recording and intervening apparatus. If capacitive currents had not been assumed to disappear after the initial rearrangement of charges, the scientific problem itself would have changed dramatically, both empirically and also from the perspective of the construction of the target to be explained. For Hodgkin and Huxley this “simplification” (Hodgkin and Huxley 1952b, 505) seemed harmless, and without it the research program would probably have been paralyzed. It seems, then, that Hodgkin and Huxley, as well as many electrophysiologists before and after them, were exploring the explanatory potential of a series of assumptions that could not be de-idealized without corrupting the research program as a whole. The study of the nerve signal as a phenomenon in which there are no capacitive currents contributing to the global measured current in voltage clamp16 delivered important explanatory benefits.

16 Except for the initial charging of the capacitor at the instant when the voltage is fixed to a particular

value with the voltage-clamp.

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From the artifactual perspective, idealization-as-distortion view results in a naïve account of model-building, if only because the contact with the world is through various kinds of epistemic artifacts that are recruited and integrated in model construction. The upshot is that there are holistic idealizations for the evaluation of which there are no theory or representation free model-world comparisons (cf. Teller 2008). This is clearly the case with the HH model, where the model simulates experimental results that are interpreted under the assumption of constant capacitance—committing to the same idealization as required for the derivation of the equations. This means that the experimental results cannot arbitrate between the model and the world regarding that idealization.

2.6 Conclusions In this paper we have argued for an artifactual approach to idealization by showing how it makes salient some important features of modeling that we analyzed through the case of nerve signal modeling. Most philosophical accounts approach idealization as distortion and consequently presume, either explicitly or implicitly, the possibility of establishing determinable representation free model-world comparisons for evaluating idealizations and their epistemic roles. In contrast, we have focused on how models are achieved by using actual representational tools and other epistemic resources. The artifactual perspective emphasizes that even those idealizing assumptions that would traditionally be rendered as misrepresentations may have intricate relationships to various epistemic resources exploited in model construction. Accordingly, we have examined idealization by focusing on the different renderings of the nerve signal in the unfolding modeling process. In this process, modeling of the nerve cell with representational tools from physical chemistry led to the idealization of the nerve cell membrane as a semipermeable membrane like those in galvanic cells. Then we examined how this assumption was further developed into the assumption of constant capacitance in the equivalent circuit that Hodgkin and Huxley used to model the nerve cell membrane. The resulting set of assumptions is an example of holistic idealization that aligns and integrates different empirical, theoretical and representational resources. It is our claim that the relationship between these assumptions, and their role in coordinating diverse representational tools, only becomes salient when we adopt an artifactual approach instead of viewing idealizations as distorting misrepresentations. As a result, the artifactual account recognizes the often holistic nature of idealization: idealizing assumptions hold the model together in configuring different epistemic artifacts. Such holistic nature of idealizing assumptions means that they are both enabling and limiting in a manner that is not dissectible into either one of them, as the deficiency and epistemic benefit accounts imply. The artifactual account both occupies a middle ground between the traditional deficiency and epistemic benefit accounts, and goes beyond them in attending to features of idealization that have

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largely been passed by in the contemporary discussion. In particular, the engagement of the artifactual approach with actual representational tools highlights the role of idealization in aligning and integrating mathematical, statistical, or computational methods with theoretical notions, concepts, and results from recording and intervening apparatus. Another important dimension of the artifactual account, the focus on the constrained construction of a model, is able to account for the epistemic benefits of idealization without too heavy realist baggage or too demanding decompositionality requirements that riddle the difference-making and isolation accounts. In regard to the decomposability issue, the artifactual approach is in agreement with Rice (2019) in that scientific models are not modular arrays, whose parts could, in some straightforward way, be compared to matters of fact about a target system. Scientific models are commonly de-idealized, of course, but the challenges of such processes are many, and partly overwhelming, as would only be expected should one take notice of the artifactual dimension of modeling (see Knuuttila and Morgan 2019). Finally, our analysis of the Hodgkin and Huxley model reveals an intimate link between analogical reasoning and idealization that has been overlooked in the present discussion of idealization. Once the nerve impulse was modeled in terms of an electric circuit, it became possible to establish relations between formal laws and theoretical concepts, calculation methods and measuring techniques. The nerve signal research shows how central such an idealization as constant capacitance can be in drawing together different resources, and in establishing links between different research fields. In view of the distortion-to-reality accounts of idealization it is important to notice that if a membrane is cast as a capacitor, what is being assumed is that the whole of the circuit and the membrane behave in the same way. Since the artifactual account does away with the representationalist commitments that require us to decide whether models or their parts accurately describe their targets or not, it enables us to better understand how analogical reasoning routinely exploits idealization.

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Cartwright, N. (1999). The vanity of rigour in economics: Theoretical models and Galilean experiments. Centre for Philosophy of Natural and Social Science. Discussion paper series 43/99. Cole, K., & Baker, R. (1941). Longitudinal impedance of the squid giant axon. The Journal of General Physiology, 24(6), 771–788. Craver, C. (2006). When mechanistic models explain. Synthese, 153(3), 355–376. Craver, C. (2007). Explaining the brain: Mechanisms and the mosaic unity of science. New York: Oxford University Press. Craver, C. (2008). Physical law and mechanistic explanation in the Hodgkin and Huxley model of the action potential. Philosophy of Science, 75(5), 1022–1033. El Hady, A., & Machta, B. (2015). Mechanical surface waves accompany action potential propagation. Nature Communications, 6(6697), 1–7. Frigg, R., & Hartmann, S. (2012). Models in science. Stanford Encyclopedia of Philosophy. Retrieved January 26, 2020 from https://plato.stanford.edu/entries/models-science/. Gelfert, A. (2016). How to do science with models. Cham: Springer. Godfrey-Smith, P. (2009). Abstractions, idealizations, and evolutionary biology. In A. Barberousse, M. Morange, & T. Pradeu (Eds.), Mapping the future of biology: Evolving concepts and theories (pp. 47–56). Dordrecht: Springer. Heimburg, T., & Jackson, A. (2005). On soliton propagation in biomembranes and nerves. Proceedings of the National Academy of Sciences, 102(28), 9790–9795. Hodgkin, A. L., Huxley, A., & Katz, B. (1952). Measurement of current-voltage relations in the membrane of the giant axon of Loligo. The Journal of Physiology, 116, 424–448. Hodgkin, A. L., & Huxley, A. (1952a). Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo. The Journal of Physiology, 116, 449–472. Hodgkin, A., & Huxley, A. L. (1952b). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology, 117, 500–544. Huxley, A. (1999). Overton on the indispensability of sodium ions. Brain Research Bulletin, 50(5–6), 307–308. Iwasa, K., & Tasaki, I. (1980). Mechanical changes in squid giant axons associated with production of action potentials. Topics in Catalysis, 95(3), 1328–1331. Knuuttila, T. (2005). Models, representation, and mediation. Philosophy of Science, 72(5), 1260– 1271. Knuuttila, T. (2011). Modelling and representing: An artefactual approach to model-based representation. Studies in History and Philosophy of Science Part A, 42(2), 262–271. Knuuttila, T. (2017). Imagination extended and embedded: Artifactual versus fictional accounts of models. Synthese. https://doi.org/10.1007/s11229-017-1545-2. Knuuttila, T., and Morgan, M. (2019). Deidealization: No easy reversals. Philosophy of Science, 86(4), 641–661. Knuuttila, T., and Loettgers, A. (2014). Varieties of noise: Analogical reasoning in synthetic biology. Studies in History and Philosophy of Science, 48, 76–88. Levy, A. (2013). What was Hodgkin and Huxley’s achievement? The British Journal for the Philosophy of Science, 65(3), 469–492. Levy, A. (2018). Idealization and abstraction: Refining the distinction. Synthese. https://doi.org/10. 1007/s11229-018-1721-z Lowenhaupt, B. (1996). Gated, ion-selective sodium and potassium channels of the giant axon: Do they have a role in bioelectric excitation? Electro-and Magnetobiology, 15(2), 151–157. McMullin, E. (1985). Galilean idealization. Studies in History and Philosophy of Science Part A, 16(3), 247–273. Mäki, U. (1992). On the method of isolation in economics. Poznan Studies in the Philosophy of the Sciences and the Humanities (Special Issue Idealization IV: Intelligibility in Science, edited by Craig Dilworth). 26, 319–354. Nernst, W. (1889). Die elektromotorische Wirksamkeit der Jonen. Zeitschrift für Physikalische Chemie, 4(1), 129–181.

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Ostwald, W. (1890). Elektrische Eigenschaften halbdurchlässiger Scheidewände. Zeitschrift für Physikalische Chemie, 6(1), 71–82. Potochnik, A. (2017). Idealization and the aims of science. Chicago: The University of Chicago Press. Rice, C. (2018). Idealized models, holistic distortions, and universality. Synthese, 195(6), 2795– 2819. Rice, C. (2019). Models don’t decompose that way: A holistic view of idealized models. The British Journal for the Philosophy of Science, 70(1), 179–208. Schaffner, K. (2008). Theories, models, and equations in biology: The heuristic search for emergent simplifications in neurobiology. Philosophy of Science, 75(5), 1008–1021. Sklar, L. (2000). Theory and truth: Philosophical critique within foundational science. Oxford: Oxford University Press. Strevens, M. (2008). Depth: An account of scientific explanation. Cambridge, MA: Harvard University Press. Takashima, S. (1979). Admittance change of squid axon during action potentials: Change in capacitive component due to sodium currents. Biophysical Journal, 26(1), 133–142. Tasaki, I. (1982). Physiology and electrochemistry of nerve fibers. New York: Academic Press. Teller, P. (2008). Of course idealizations are incommensurable! In L. Soler, H. Sankey & P. Hoyningen-Huene (Eds.), Rethinking scientific change and theory comparison (pp. 247–264). Dordrecht: Springer Netherlands. Weber, M. (2005). Philosophy of experimental biology. New York: Cambridge University Press. Weber, M. (2008). Causes without mechanisms: Experimental regularities, physical laws, and neuroscientific explanation. Philosophy of Science, 75(5), 995–1007. Weisberg, M. (2007). Three kinds of idealization. Journal of Philosophy, 104(12), 639–659.

Natalia Carrillo is Postdoctoral Fellow at the Department of Philosophy, University of Vienna. She works in the general philosophy of science, with a particular interest in epistemological questions regarding scientific modeling, abstraction and idealization in science, and the relationship between history of science and philosophy of science. Tarja Knuuttila is Professor of Philosophy of Science at the Department of Philosophy, University of Vienna. Earlier she was appointed as an Associate Professor at the University of South Carolina. She served 2007-2010 as the Editor-in-Chief of Science & Technology Studies. She has specialized in the study of scientific representation and modeling. Her approach is comparative and interdisciplinary; she has studied modeling in economics, ecology, systems and synthetic biology, computational linguistics and neuroscience. She has pioneered the artifactualist account of models in several of articles published since 2003 in leading philosophy of science journals.

Chapter 3

Informative Models: Idealization and Abstraction Mauricio Suárez and Agnes Bolinska

Abstract Mauricio Suárez and Agnes Bolinska apply the tools of communication theory to scientific modeling in order to characterize the informational content of a scientific model. They argue that when represented as a communication channel, a model source conveys information about its target, and that such representations are therefore appropriate whenever modeling is employed for informational gain. They then extract two consequences. First, the introduction of idealizations is akin in informational terms to the introduction of noise in a signal; for in an idealization we introduce ‘extraneous’ elements into the model that have no correlate in the target. Second, abstraction in a model is informationally equivalent to equivocation in the signal; for in an abstraction we “neglect” in the model certain features that obtain in the target. They conclude that it becomes possible in principle to quantify idealization and abstraction in informative models, although precise absolute quantification will be difficult to achieve in practice. Keywords Information · Idealization · Abstraction · Content

3.1 Introduction Scientific models are often employed to gain information regarding their targets. The building of a model is guided by preliminary knowledge of some phenomenon, and the model aims to provide further information regarding unknown aspects of the phenomenon, or its underlying causes. This is evident in most models with predictive power that we know, but it applies to most other models whenever some cognitive gain in understanding is sought (for compilations of an array of case studies in physics, M. Suárez (B) Faculty of Philosophy, Department of Logic and Theoretical Philosophy, Complutense University of Madrid, 28040 Madrid, Spain e-mail: [email protected] A. Bolinska Department of Philosophy, University of South Carolina, Columbia, SC 29208, USA e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Cassini and J. Redmond (eds.), Models and Idealizations in Science, Logic, Epistemology, and the Unity of Science 50, https://doi.org/10.1007/978-3-030-65802-1_3

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economics, and other sciences, see e.g. Morgan and Morrison 1999; Thomson-Jones and Cartwright 2005; Suárez 2009). A key feature of scientific representations is their capacity to enable users to draw informative inferences from what we may call representational vehicles, or sources, to their target systems. This has been studied in depth and is by now widely accepted (Boesch 2017; Bolinska 2013; Contessa 2007; Suárez 2004).1 What is not so established is the nature of the information that is conveyed by models. In this paper we focus on the nature of the information provided, and apply the tools of communication theory in order to draw a few interesting lessons regarding some approximation techniques typical in scientific modeling. When a model is represented as analogous to a communication channel, the sources of informational noise and equivocation have correlates in different forms of idealization and abstraction in modeling practice. This sheds some light on some common methods for minimizing idealization and abstraction, as well as their rationale. The relevant information theoretical concepts are introduced in Sect. 2: They originate in Shannon’s classic (1948) and have been discussed in an epistemological context by Dretske (1981).2 The central analogy between communication channels and scientific models is laid out in Sect. 3. The following Sect. 4 expounds on a case study within the kinetic theory of gases, and it argues for a role for informational noise and equivocation in standard understandings of idealization and abstraction. The concluding Sect. 5 wraps up the main claim, and raises some questions prompted by the analogy that deserve further exploration.

3.2 The Mathematical Theory of Communication (MTC) In a communication system, a message travels from an emitting source to a receiver through some medium, such as a radio signal. Communication channels are noisy: part of the message can be obscured, for instance, through crosstalk in the radio signal, interference, or a random noise generator mixing into the signal. On the other hand, there is always less than perfect quality in the transmitted message, i.e. every signal suffers from loss, also known as equivocation, due to impurities in the transmission channel, or in the coding and decoding of the message. In other words, no communication channel is ever 100% efficient. The goal of effective communication is thus rarely, if ever, to completely eliminate or eradicate the inefficiencies in the form or either noise or equivocation—since to bring those inefficiencies to 0% in practice is an impossible task. Rather the goal of effective communication is to maximize the efficiency of the signal within the bounds of what is in fact possible for any given channel. For a complete array of possible signals, every channel will have some limit to what is capable of transmitting from emission to reception. The limit 1 So

is the concomitant terminology of ‘sources’ and ‘targets’ (Suárez 2004); yet, for reasons that will become apparent, it is best for the purposes of this article to refer to representational vehicles and targets, to aptly distinguish them from the terms employed in information theory. 2 See Bolinska (2015) for an application of these notions in the philosophy of science.

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is the average efficiency of the channel. In other words, the informational efficiency of any communication channel is an average property of the channel relative to all possible transmissions through that channel. The goal of communication efficiency is to maximize this quantity—and it entails choosing a channel with as great a ratio of information to noise, or equivocation, as is possible for transmissions of information from a given source to a receiver. A communication system comprises minimally five separate parts: an emitting source; an encoder or transmitter; a signal; a decoder or receptor; and a receiver. (See Fig. 3.1: Shannon’s information theory). The source possesses or generates certain quantitative properties, which the encoder or transmitter (some sort of machine or mechanism) codifies in a signal. The signal carries the information over to a decoder that extracts the relevant information regarding the source in a form that is adequate for the purposes of a receiver. In practice there is a degree of information loss at every stage (e.g. Pierce 1961): the source properties may not all get adequately codified, or not codified at all in the signal. The signal may lose some of its properties or resolve. The decoding may be deficient and fail to extract all the information in the source. The receiver may in principle be inefficiently geared towards the information received. In other words, there are many sources of what we call equivocation: relevant information regarding the source that is not transmitted over to the receiver. Dretske (1981, pp. 16ff.) represents equivocation formally as follows. Let us refer to the source as s and the receiver as r. And let us refer to the total amount of information contained in s as I(s), and to the total amount of information received at r as I(r). Then we can denote the total amount of information about s that is received at r as Is (r). A straightforward measure of equivocation is then given as: E(r ) = I(s) − Is (r )

(Equivocation)

That is, the equivocation of a communication system is the amount of information that gets lost in the transmission from the source to the receiver, i.e. the amount of

Fig. 3.1 Shannon’s mathematical theory of communication (Shannon 1948)

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information generated at the source that fails to be transmitted. Dretske’s formula applies to average equivocation in a particular communication channel between a given source s and a given receiver r. It is clear that in order to compute it quantitatively we need to possess measures of the information contained in the source I (s), and of the part of this information that is in effect transmitted to the receiver, I s (r). Dretske points out that in communication theory we can quantify the information contained in any system by calculating the reduction in the number of possible states of the system, as: I (s) = log n, where n is the number of possibilities that get reduced to 1. (This is sometimes known as the information entropy of the source, and assumes that the source is some kind of process or phenomenon endowed with some dynamics that reduce a large space of possibilities into one—an assumption that I shall return to later on in the discussion of the case study). Reducing the equivocation in a channel increases completeness—a channel that equivocates a lot provides an impoverished rendition at reception of the qualities of the source. Dretske suggests (1981, p. 25) that we can calculate the equivocation in a communication system as follows. Suppose that there are only eight possible events at the source {s1 , s2 , …, s8 }, and correspondingly eight possible events at the receiver {r 1 , r 2 , …, r 8 }. Now, suppose that the signal is such as to generate each event at the receiver with a given probability given each event at the source. That is, there are well defined values of P (r i /sj ) for every couple {r i , sj }. We may focus on a particular event, say r 7 , and work out the equivocation for that event as follows: Es (r 7 ) = – P (si /r 7 ) log P (si /r 7 ). This is just the probabilistically weighted average of the equivocation for each of {si } with respect to r 7 . To compute the average equivocation for the channel we need only sum up the contributions made to it by each of the events, weighted by their probabilities:     E(r ) = S j P r j E r j

(Average Equivocation)

In other words, the equivocation of a communication channel is an average quantity computed over each of the possible values of the properties of the source, weighted by the conditional probability that each of those values generates a particular value of some property in the receiver. We calculate the equivocation for each value of the receiver property by estimating the probability that information loss may occur for this value of the receiver property. And then we sum over all the values weighted by their corresponding probability. Whenever applicable, the procedure yields quantitative values for the equivocation and this constitutes a measure of the channel’s efficiency. The larger the equivocation, the larger share of information is lost in the transmission from source to receiver, since: I(r) = I(s) – E(r). The smaller the equivocation, by contrast, the larger proportion of the information contained in the source is transmitted to the receiver. At the limit, when the equivocation is zero, all the information contained at the source is completely transmitted: I(s) = Is (r). The other important source of inefficiency in a communication channel is noise. Roughly, the noise of a communication channel is whatever extraneous information is picked up, and thus added to the signal as it travels from source to receiver. It may be added at the stage of coding, e.g. at the source, or it may get added later on during

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the transmission or at the stage of decoding at the receiver’s end. At any rate, the noise in a communication channel is defined as whatever information is transmitted to the receiver which does not originate in the source: N(s) = I(r ) − Is (r )

(Noise)

Hence the greater the noise the larger share of the information received was not actually generated at the source; the smaller the noise the greater proportion of the information did actually originate at the source. At the limit where noise is null, all the information received originates at the source: I(r) = Is (r). Reducing the noise over a communication channel increases faithfulness, or reliability. We can calculate the average noise over a communication channel in converse fashion to equivocation as follows. First calculate noise for every possible event in the source. Thus, for instance, for event s7 at the source, its contribution to the average or overall noise is given as: N (s7 ) = – i P (r i /s7 ) log P (r i /s7 ). Then the noise of the channel is simply the statistical average of each of these contributions, i.e. the contribution of each weighted according to its probability:     N(s) = S j P s j N s j

(Average Noise)

In many cases, noise detracts from signal transmission, preventing some of the information from the source to be transmitted to the receiver, i.e. increasing equivocation. But this need not be the case: it is possible for the receiver to contain additional information, information that didn’t originate in the source, which doesn’t detract from signal transmission. Thus, an increase in noise need not logically or conceptually imply a corresponding increase in equivocation (Dretske 1981, pp. 20–21). Most communication channels operate on signals of much greater complexity, with a much larger and more complex space of possibilities at the source than here described. But the basic notions stand, and they will suffice for the purposes of this paper. It is particularly relevant that informational quantities like equivocation and noise are averages and therefore properties of the communication channel, not of any particular signal. While this averaging feature of informational notions is not particularly suited to Dretske’s purposes (he chooses to concentrate instead on concrete signalling actions), it is well suited for the purposes of the analogy with scientific modeling that shall be explored here. A complex communication channel comprises five elements, and its most relevant informational quantities must be computed as averages pertaining to the channel as a whole.

3.3 An Analogy: Models as Communication Channels The practice of model building displays considerable sensitivity to some notion of fit between a representational vehicle and its target. The fit, however, very rarely involves matching pairwise structural properties; more often than not what is at stake is how

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relevantly informative the source is as an inferential whole about certain aspects of the target.3 However, the notion of information at work here is suitably thin, for any kind of model. A richly informative model describes its target in great detail; yet not all the detail need be a guide to what the target really is like—sometimes the detail serves pragmatic purposes in understanding or predicting the ensuing phenomena. Many models idealize their targets a great deal, presupposing point mass particles, frictionless planes, and the like. These are details and are in some sense informative, but for all we know they are not faithful to their targets. By contrast, a powerful or deep model may not be rich in detail but is nonetheless informative in some sense about the main or critical characteristics, those that are more central to the production of the phenomena. What is the sense of informativeness that is involved in these models? From an inferential point of view, the richer model is most informative in the sense that it licences a large number of inferences to many different aspects of the phenomenon. A deep model may licence fewer inferences, and to fewer aspects of the phenomenon, but they are inferences to aspects that we have reason to suppose are more central, including some of its putative causes. In other words, models encode information. The kind of information will differ in different types of model, and the nature of the information involved will also differ depending on the account of representation endorsed. But in any case, information transmission is often a main aim of modeling.4 This observation suggests an analogy: the target of the model acts often as an information source for the representational vehicle to transmit to the user of the model. In other words, modeling often is in some sense a communication practice, and a model may be thought of, at some level of abstraction, as a communication channel. It could help understand the practice of modeling to make it more precise what this sense is. Nevertheless, a cogent application of the analogy meets some challenges. In a communication channel as described in Shannon’s theory (Fig. 3.1), the source’s information is first encoded into a transmitter, which emits a signal. The signal is carried by some means to a receiver, which decodes the information and provides it over to its destination. There are thus five objects laid out in a communication system, and there is need to fix on the respective analogues in the case of modeling if Shannon’s theory is to apply. Now, we are not suggesting that models are just communication channels, but only that there are helpful analogies that may be exploited to better understand modeling practice. There may well be different ways of helpfully laying out the analogy, depending on both types of models, and the underlying account of representation. In other words, we make no claim that the proposal advanced here will always be applicable or helpful, in every instance of modeling, regardless of how the case is understood or interpreted. And we are certainly not claiming that the analogy presented here exhausts everything that may be claimed 3 In

fact, a 1–1 copy of the target—as in Borges’ (1954) beautiful parable—would be a 100% informative ‘channel’ and, yet, a perfectly useless model. 4 Indeed one of us (Bolinska 2016) has explicitly understood models as tools for conveying information, identifying the features responsible for their informativeness. See also the discussion of ‘objectivity’ in Suárez (2004, forthcoming).

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about modeling practice, even from an informational point of view. Our more modest claim is that the analogy provides some tools to better understand the kind and nature of the information transmission that takes place in some instances of modeling, as pertains their inferential function in particular. The proposal is thus to treat the system, process, or phenomenon of interest (the representational ’target’) as the informational source in a communication channel. The system or phenomenon of interest is typically a dynamical process, or it involves one, and it is often represented to us already in some preliminary or antecedent description. (The apparent circularity is a well-known issue in the representation literature, which there is no space to broach here—see Van Fraassen 2008, Ch. 11 and part IV, for further discussion). The information about this dynamical process is then encoded in at least either of two ways. It can be studied empirically, on the basis of the static data that it elicits in some experimental trial, and the resulting information can be built into what is known as a data-model (in the sense of e.g. Suppes 1962). Or it can be modeled dynamically, in terms of a given parameter set, in what is a phenomenological model. Different ways of encoding the same information may be more or less appropriate for the purposes at hand. In either case, the model then acts as a courier of information, a tool to compactly convey codified information regarding the system of interest. At the other end of the process, the model needs to be interpreted in ways that make the information salient for the purposes of prediction, understanding, explanation, or generalization. This will typically require a scientist to employ a theory or sets of theories (sometimes high-level theories, such as, in physics, the kinetic theory of gasses, the theory of general relativity, or quantum mechanics; other times mid-level theory, such as diffraction optics, radiation theory, or the molecular hypothesis for gases) to interpret the phenomenological or data model. The information at the source is thus finally transmitted over to its final destination, the model user. It is important to note the multiple steps involved in this communicative act, and the concomitant judgements along the way: The source has to be competently and aptly described, the information it contains must then be codified /transposed into an appropriate model that will act as an information-carrier, and this model must be correctly interpreted in the light of some theoretical knowledge for the information to be relevant, comprehensive and/or apt for the purposes of the model user. The overall communication channel may be schematically described as follows: Phenomenon (source) → Phenomenological/data model (coding) → → [Transmission] → →Theoretical model (decoding) → User (receiver)

In this picture the communication system also has five steps, the middle transmission step being the immersion of the phenomenological description or model into the theoretical explanation or interpretation of the phenomenon. Information loss, in the shape of either noise or equivocation, is possible in every step in this chain. There is bound to be information loss in the choice of data or phenomenological model (here understood in informational terms as the selection of the coding system); in the actual embedding of the data or phenomenon into the theoretical description (understood

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informationally as the transmission of a signal); and in the choice of the theoretical model that interprets the phenomenon (the choice of the information decoding system). There are no doubt judgements in all three cases as to what is most likely to preserve the largest amount of information or the most critical kind of information. But such judgements are also involved in any communication channel (for some insight into the kinds of practical choices those working in information theory have to routinely make, see MacKay 2003). All these choices must be fit for purpose, so a lot will depend on the actual goals pursued by the user of the model—and these may vary greatly depending on the context of use. The kind of contextuality of use involved in modeling practice is by now widely accepted (cf. Bailer-Jones 2003; Giere 2004, 2009; Mäki 2009; Teller 2001), and it cannot be easily algorithmically or automatically done away with, if at all. This severely constrains the analogy in at least two different ways—which explain why modeling practice cannot be simply reduced to the building of effective communication channels. In a communication system, the goal is for the message at the receiver end to identically reproduce the message at the source, or at least to do so with minimal information loss. Yet, as noted, a scientific model rarely aims to reproduce the target system in its entirety exactly. More often than not a scientific model aims to capture certain consequences of central features of the target system—those consequences that are of importance for the purposes of prediction, explanation, generalization, etc. There are at least two forms of helpful ‘distortion’ in modeling practice that need to be explained from an informational point of view. They correspond roughly to abstraction, and idealization.5 Consider, for instance, the oft-discussed example of the simple harmonic oscillator as a model of a pendulum subject to no friction (Giere 1988). There are two ways to understand the absence of friction in these models. The first one assumes the model is an abstract rendition of the phenomenon that ignores some of the complexity: the real target phenomenon possesses friction, but the model does not. In informational terms, this means that some of the information available at the source is not represented in the model of the phenomenon in any way that can be interpreted theoretically as friction. As a result, the information does not get transmitted over to the receiver. Thus, in informational terms, there is ‘equivocation’ involved in the simple harmonic oscillator model of a real pendulum. Now consider the alternative reading of ‘frictionless’ according to which the simple harmonic oscillator model is an idealization. On this reading the model includes a property (‘lack of friction’, or ‘frictionlessness’) that the phenomenon that it models does not in fact possess. The model idealizes the phenomenon by introducing properties that are not there in the source phenomenon. This understanding is of interest too, and rather typical in modeling: it entails introducing properties in the model description for ease of calculation, manipulation, prediction, etc. The ‘frictionlessness’ of the ideal pendulum is informationally akin to ‘noise’—it is present in the model yet does not originate in the source but is ‘extraneous’. One can thus see, 5 Our use of the terms is inspired by the definitions in Thomson-Jones (2005) and Weisberg (2007);

and is reflected most precisely in those used in Pero and Suárez (2016).

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following these two interpretations of ‘frictionless’, that abstraction will typically be analogous in informational terms to ‘equivocation’, while idealization is analogous to informational ‘noise’. It is clear that a lot in the building and applying of models depends on controlling and monitoring both ’equivocation’ and ’noise’. As regards equivocation, reducing its presence in a model is essential to the aim of comprehensiveness—a model with 100% equivocation is a model where none of the information originating in the source is transmitted (Is (r) = 0) and is hence perfectly useless. As regards noise, a model is helpful only in as much it transmits faithfully the features of the source. Otherwise we run the risk of incorrectly ascribing to the phenomenon properties that it does not possess—a model where 100%, of the information received is ‘extraneous’ is again useless, since Is (r) = 0, i.e. none of the information received originates in the source.

3.4 Modeling and the Transmission of Information: Some Examples One of the paradigmatic case studies in the contemporary literature is the billiard ball model of gases. It is first treated as part of an extensive discussion of the kinetic theory of gases in Campbell (1920) and thereafter in great depth in Mary Hesse’s masterly (1966).6 In the kinetic theory of gases, the dynamics of molecules in a gas is modeled as if it were a system of perfectly elastic microscopic balls in collision—a set of miniature ‘billiard balls’ in constant motion. Let us refer to the real properties of actual gas molecules as {G1 , …, Gn } and those of billiard balls as {B1 , …, Bm }. Then the model sets up a correspondence between a subset ({G1 , …, Gi }, with i < n) of the properties of gas molecules and a subset ({B1 , …, Bi }, with i < m) of the properties of billiard balls. We say that gas molecules are billiard balls as regards their collision dynamics, but this does not mean that they share all the properties of billiard balls. Billiard balls are coloured and shiny and reflect light, but gas molecules possess none of these properties. Even as regards their dynamical properties there are significant differences. Billiard ball motion is subject to limited friction against the surface on which they move, while gas molecules presumably interact freely.7 And conversely, a system of gas molecules exhibits macro-properties that no system of billiard balls can ever display, such as viscosity or free expansion. In other words, the model omits certain properties of gas molecules, while including others that molecules don’t in fact have. In accordance with the analogy laid out in this paper, the model may for the purposes of information be taken to be a 6 More recently, Pero and Suárez (2016), Suárez and Pero (2019) and Suárez (forthcoming) contain

an extended historical discussion of this example that corrects some philosophical misconceptions. 7 It is actually worse than that: contrary to assumptions billiard balls are not perfectly elastic, but of

course experience minor energy loss in the form of heat in collisions; but then again neither are gas molecules perfectly elastic, since they also experience loss of (kinetic) energy in collision. See the discussion in Pero and Suárez (2016, pp. 75–76).

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communication channel transmitting information from the source—a gas—encoded and transmitted in accordance with some phenomenological model (in the form of the billiard ball model), interpreted in the terms of kinetic theory of gases, for the sake of the receiver’s information (the physicist able to interpret or decode the information in the signal). Properties of the gas that are ignored in the model will contribute equivocation in the signal, impoverishing the communication, while features in the model that do not correspond to properties of the gas will contribute noise. In line with the discussion in the previous section, while we would ideally reduce both noise and equivocation to zero, if we could, their introduction turns out to be unavoidable for any practicable model at all. The main difficulty in applying the informational framework to the billiard ball model is in laying down the appropriate probability distributions. We shall have to make a number of assumptions at this point—including some prior distributions for the different combinations of values. Fortunately, however, the assumptions are warranted by the physical model itself as well as the philosophical discussions regarding the billiard ball model.8 We shall therefore assume a normal distribution over the values of the properties of the source system (the gas molecules). This is warranted by the physics, since the standard assumption regarding the velocity of free molecules in a gas is that they are distributed in accordance to Maxwell-Boltzmann statistics. To be more precise, this says that, in a vessel containing m particles in equilibrium, the proportion of particles with a particular velocity v is n (where n < m) and is given by the Maxwell-Boltzmann distribution as:  f (v) =

   m 3 −mv 2 4π v 2 ex p 2π kT 2kT

(Maxwell-Boltzmann distribution)

This probability distribution function depends only upon the initial velocities of the gas molecules, since the so-called Boltzmann constant k and the thermodynamic temperature of the gas T are both constants of motion. We shall assume that all these velocities have correlative properties in the velocities of the billiard balls in a system of billiard balls. However, there is no reason in principle why the velocities of an equally large group of billiard balls should also obey a Maxwell-Boltzmann distribution. In fact, the notion of equilibrium itself makes no sense for billiard balls. We may assume by fiat that the set of billiard balls in our model obeys the MaxwellBoltzmann distribution, and this is indeed commonly done. What this means, from the point of view of information theory, is that we assume that there is no information loss in the description of the molecules’ velocities in the billiard ball model.

8 Although, certainly, these assumptions can be contested. But nothing much hinges on the particular

values. The only claim needed for the present proposal to go through is that there are some values for these probability distributions—whether they are within our reach to know is not essential. As Dretske points out (1981, p. 55) the probability distributions that go into communication theory, and in particular the conditional probabilities in the definition of equivocation, are objective, and may be very hard to get to know.

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But, of course, these are not the only properties. Let us begin with all the noise introduced into the system: the properties of billiard balls that are irrelevant to the kinetic theory of gas, such as their colour, their shine, and of course their rigid solid structure. Take colour, and assume for the sake of argument that billiard balls can be any of seven colours from the deep red to the violet end of the spectrum, and that each colour is as likely as any other (this is just an assumption about the entities in the model, for which it is not relevant whether or not it contradicts standards in e.g., ordinary sets of billiard balls in the game of pool!). This already entails that, in the equation for the average noise in the channel, the value of the probabilistic distribution is uniform over all the colours:       N(s) = S j P s j N s j , is such that P (si ) = P s j = a for all i, j. But now, a further substantial assumption must be made, namely that the colour of the balls is not correlated with any of the properties of the gas molecules that they represent. The assumption seems intuitive, and it would be a strange model that correlated the properties across in this way, but it is not an in principle impossible model.9 So we shall just have to assume that the model does not work that way and that the colour of the balls is completely uninformative with respect to any of the physical properties of the gas molecules. There is no correlation. This entails that the cross or conditional probabilities in the expression for the noise are equal, and the conditional probability distribution is flat: P (r i /sk ) = P (r j /sk ) = b, for any i, j, and any property sk of the source. Now, once we have established that the probabilities are constant numbers across the average, we can easily see that the contribution to the noise for each event at the source is: N (sk ) = – i P (r i /sk ) log P(r i /sk ) = –b  i log P(r i /sk ). But as we had already found the noise to be a constant of the average of the noise contribution from each value, we obtain: N (s) = a  j N(sj ) = a. b  ik – log P(r i /sk ). In other words, the noise is just a constant function of each of the conditional events. If those are zero, then the noise goes down to zero and the signal achieves maximal efficiency. What does this mean for our billiard ball model? It means that the information transmitted by the model about the system of gas molecules is only as efficient as the ‘noise’ by spurious variables in the models is low—and this depends only on how much every one of the possible values of any of the spurious variables is correlated with the relevant physical variables in the source. Bear in mind that the correlation is objective, so even if we lack any knowledge—even if we assume there to be no correlation—the actual noise in the signal depends on the existence of the correlations, independently of our knowledge. So, we are never in a position to rule out informational noise (because we can never completely de-idealize the model). 9 For

instance, one could imagine a model where the colour of the elastic ‘billiard’ balls is taken to represent the initial velocity of each corresponding molecule in the gas, with purple representing the larger speeds and red representing the smaller speeds and all the other colours representing intermediate speed ranges in the prescribed order in the electromagnetic visible spectrum. Such model would not be very useful, but it is perfectly possible, for any given gas.

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Let us now consider equivocation. Very similar considerations will apply, even though we are now interested in the properties in the source that are ignored or abstracted away in the model. As mentioned, all the macroscopic properties of the gas (free expansion, viscosity) are derived from theory on the basis of the model, but do not appear in the analogy itself: systems of billiard balls exhibit neither viscosity nor free expansion. However, the same reasoning we applied to noise will also apply to the equivocation function: E(r) =  j P (r j ) E(r j ), even if equivocation depends upon the probabilities of the receiver, not the source. If the model is deterministic, we can assume all the probabilities to be zero or one for all the values of all the dynamical variables of interest.10 Then the equivocation depends only on the contribution made by the value that actually obtains, suppose r7 . This in turn is given by E(r 7 ) = – P (si /r 7 ) log P (si /r 7 ). Yet, if the value at reception is uniquely picked out, it should then not depend statistically on the values of any other variables at the source (the position of a billiard ball in the model ought to depend only on the position of the corresponding gas molecule in the gas). So once again the conditional probability function is one or zero (P (si /r 7 ) = 1 if i = 7, and = 0 otherwise). So, we obtain the result that the equivocation can only go to zero if every variable in the source has a correlated variable in the model-signal. Patently this is not the case for the macroscopic variables, in which case E(r) = 0 and the model displays a degree of inefficiency. While it is not possible to quantify the inefficiency in detail, it should be clear that the way to reduce equivocation is to tightly correlate every dynamical variable in the source to a variable in the model. This ensures completeness in the description the model provides of the phenomena (e.g. by building additional properties into billiard ball systems that account for the macroscopic properties of the gas), so not surprisingly equivocation inefficiency goes down. In other words, from an informational point of view, idealization introduces noise into the signal transmission provided by the model; while abstraction introduces equivocation; and the way to reduce the inefficiency generated by both is to tightly match the properties in the model to those in the source, and vice versa. No model is ever perfect, in the sense of ever achieving this kind of one-to-one matching. On the contrary every model contains some degree of each kind of inefficiency. The art of modeling, in informational terms, involves a trade-off between noise and equivocation.

3.5 Conclusion Many scientific models aim at conveying information regarding their targets. When a model does so—and in so far as it does so—a model functions as a communication 10 Determinism

is also arguably an assumption for systems of billiard balls, if we assume an initial probability distribution over the dynamical variables of interest (as is done, e.g. in the tradition of the method of arbitrary functions). We shall ignore this complication and assume deterministic Newtonian dynamics.

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channel. We have in this paper endeavoured to take this insight seriously, and to apply the rudiments of communication systems theory to scientific modeling. The result is no doubt just an analogy—since models are per se not built as communication channels, nor may they be entirely treated as if they were. But it is an instructive analogy, which already at a preliminary stage sheds considerable light on some aspects of modeling practice. In particular, we have shown that some typical modeling techniques have correlates in informational terms. The analogy shows that often tractability is gained at the expense of informational faithfulness or completeness. There is therefore a certain trade off taking place in modeling which is best explained in terms of informational cost. Philosophers interested in modeling have not so far appreciated this informational cost, nor have they considered its diverse forms. We try to provide a first approximation in terms of informational equivocation and noise. These are technical terms—and we have employed the definitions in Shannon’s mathematical theory of communication. On this informational analogy, roughly, what is known as idealization can be understood as introducing noise; while abstraction introduces a form of informational equivocation. Shannon’s theory moreover provides precise ways to quantify over informational loss, by measuring the informational content in the source, and detracting noise and equivocation in the signal. It then becomes possible to derive a quantitative measure of information effectively transmitted. The application of such severe quantitative methods to scientific modeling is limited, and this shows some of the limitations of the analogy. Measuring the informational content of a dynamical system or phenomenon is far from trivial, as it depends on the description of the parts and their interrelation. In other words, the informational content of a system or phenomenon often sensitively depends upon what we call the phenomenological model. Yet, once this model is in place, it becomes possible to establish relations between its parts (which are genuinely probabilistic correlations in the case of statistical physics models) and compare them to those in a higher-level theoretical description. We have illustrated how this would work in the case of the billiard ball model of gases—a phenomenological description of a gas within Maxwell’s kinetic theory. The result is a rendition in informational terms of the idealizations and abstractions that operate in the model. This analogy is sufficiently robust to allow us to draw some conclusions regarding what a more realistic (either less idealized, or more concrete) description would involve; it also provides a better understanding of the trade-offs involved between tractability and information efficiency. Furthermore, the analogy between scientific models and communication channels is suggestive of a number of further methodological and epistemological issues that deserve to be explored—although there is no space in a single article to address them in any detail. The more obvious methodological questions concern the goals that trump informational efficiency—and, in particular, whether they carry an expectation of greater informative efficiency down the line. If so even the divergences from the goal of informational efficiency described here—in terms of idealization and abstraction in scientific models—would ultimately be accountable by recourse to presumed informational gains further on. The analogy would become more than just suggestive: It would provide the rudiments of an account of modeling as a branch of

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information theory. From an epistemological point of view a question that deserves to be studied is the status of the so-called veridicality thesis, which assumes all information to be true. The thesis has as many defenders as detractors (Floridi 2007; Scarantino and Piccinini 2010), and I have attempted to keep all our claims neutral in this paper. (In other words, the hope is that every claim in this paper is acceptable to epistemic realists and antirealists alike). Yet, it is reasonable to suppose that the veridicality thesis will apply in some cases of informative modeling, but not in all cases. If so, it would be worth figuring out a principled way to draw the relevant distinctions within modeling practice itself—thus rendering the realism-antirealism debate about scientific models an internal issue in science itself. Acknowledgements Mauricio Suárez would like to thank Tarja Knuuttila and the audience at the 2016 Valparaiso workshop on Models and Idealizations in Science for their comments and reactions, as well as financial support from the Spanish Ministry of Science and Innovation project PGC2018 099423. Agnes Bolinska would like to thank Anna Alexandrova and Joseph D. Martin for helpful feeback.

References Bailer-Jones, D. (2003). When scientific models represent. International Studies in the Philosophy of Science, 17(1), 59–74. Boesch, B. (2017). There is a special problem of scientific representation. Philosophy of Science, 84(5), 970–981. Bolinska, A. (2013). Epistemic representation, informativeness and the aim of faithful representation. Synthese, 190(2), 219–234. Bolinska, A. (2015). Epistemic representation in science and beyond. PhD Dissertation. Toronto: University of Toronto. Bolinska, A. (2016). Successful visual epistemic representation. Studies in History and Philosophy of Science Part A, 56, 153–160. Borges, J. L. (1954). Del rigor en la ciencia. In Historia universal de la infamia: Obras completas 3, Buenos Aires: Emecé Editores, pp. 131–132. Translated as On Exactitude in Science. In Jorge Luis Borges: Collected Fictions, London: Penguin, 1998, p. 325. Campbell, N. (1920). Physics: The elements. Cambridge: Cambridge University Press. Contessa, G. (2007). Scientific representation, interpretation and surrogative reasoning. Philosophy of Science, 74(1), 48–68. Dretske, F. (1981). Knowledge and the flow of information. Cambridge: Cambridge University Press. Giere, R. (1988). Explaining science: A cognitive approach. Chicago: The University of Chicago Press. Giere, R. (2004). How models are used to represent reality. Philosophy of Science, 71(5), 742–752. Giere, R. (2009). An agent-based conception of models and scientific representation. Synthese, 172(2), 269–281. Hesse, M. (1966). Models and analogies in science. Notre Dame: University of Notre Dame Press. Luciano, F. (2007). In defence of the veridical nature of semantic information. The European Journal of Analytic Philosophy, 3(1), 1–18. MacKay, D. (2003). Information theory, inference, and learning algorithms. Cambridge: Cambridge University Press.

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Mäki, U. (2009). Missing the world. Models as isolations and credible surrogate systems. Erkenntnis, 70(1), 29–43. Morgan, M., & Morrison, M. (Eds.). (1999). Models as mediators: Perspectives on natural and social science. Cambridge: Cambridge University Press. Pero, F., & Suárez, M. (2016). Varieties of misrepresentation and isomorphism. European Journal for the Philosophy of Science. 6(1), 71–90. Pierce, J. (1961). An introduction to information theory: Symbols, signals and noise. New York: Dover. Scarantino, A., & Piccinini, G. (2010). Information without truth. Metaphilosophy, 41(3), 313–330. Shannon, C. (1948). The mathematical theory of communication. The Bell System Technical Journal, 27(3), 379–423; 27(4), 623–656. Suárez, M. (forthcoming). Inference and scientific representation. Chicago: University of Chicago Press. Suárez, M. (2004). An inferential conception of scientific representation. Philosophy of Science, 71(5), 767–779. Suárez, M (Ed.). (2009). Fictions in science: Philosophical essays on modeling and idealization. London: Routledge. Suárez, M., & Pero, F. (2019). The representational semantic conception. Philosophy of Science, 86(2), 344–365. Suppes, P. (1962). Models of data. In E. Nagel, P. Suppes, & A. Tarski (Eds.), Logic, methodology and philosophy of science: Proceedings of the 1960 international congress (pp. 252–261). Stanford: Stanford University Press. Teller, P. (2001). Twilight of the perfect model model. Erkenntnis, 55(3), 393–415. Thomson-Jones, M. (2005). Idealization and abstraction: A framework. In M. Thomson-Jones & N. Cartwright (Eds.), Idealization XII: Correcting the model-idealization and abstraction in the sciences (pp. 173–218). Amsterdam: Rodopi. Van Fraassen, B. (2008). Scientific representation: Paradoxes of perspective. Oxford: Oxford University Press. Weisberg, M. (2007). Three kinds of idealization. Journal of Philosophy, 104(12), 639–659.

Mauricio Suárez is Professor of Logic and Philosophy of Science at Complutense University of Madrid and a Research Associate at the London School of Economics. He has put forward and developed the inferential conception of scientific representation. He is the author of Filosofía de la ciencia. Historia y práctica (Madrid: Tecnos, 2019), and Philosophy of probability and statistical modelling (Cambridge: Cambridge University Press, 2020). He is the editor of Fictions in science: Philosophical essays on modeling and idealization (New York: Routledge, 2009) and Probabilities, causes, and propensities in physics (Dordrecht: Springer, 2011). Agnes Bolinska Assistant Professor of Philosophy at the University of South Carolina. Her research interests include the nature of scientific representation, the methodology of macromolecular structure determination, and the relationship between history of science and philosophy of science.

Chapter 4

Deidealized Models Alejandro Cassini

Abstract Alejandro Cassini analyzes how highly idealized theoretical models can be deidealized. He argues that idealized models are built with a definite purpose and for that reason, the advantages and disadvantages of idealizing depend essentially on the specific purpose for which a given model is designed. As a consequence, even when deidealization may be feasible, a cost–benefit analysis may suggest avoiding it. He exemplifies those circumstances with a study of deidealized models of the Solar System and physical pendula. He concludes that deidealization has not to be conceived of as an end in itself, or as aiming at a veridical representation of the phenomena, but rather as a means to other ends, such as obtaining better explanations or predictions, or more generally, improving the expediency of our models to solve the problems that originated their construction. Keywords Theoretical models · Deidealization · Representation · Prediction

4.1 Introduction Scientists and philosophers of science call models an astonishing variety of different human constructs that are employed in the different sciences to obtain epistemic access to real-world phenomena. Even a casual look at the scientific practices in any science shows that there are many kinds of models, from physical objects such as scale models to set-theoretical structures and systems of mathematical equations. On the other hand, models are used for many different purposes, among others, to explore, describe, explain, and predict different types of phenomena. Due to this diversity, it is unlikely that a philosopher of science will be in a position to obtain wide-ranging -but non-trivial- generalizations concerning the nature and the function of scientific models. Something similar can be said of the concepts of idealization and deidealization, which I will address here. Although they have not been extensively A. Cassini (B) University of Buenos Aires, Department of Philosophy, Puán 480, 1406, Buenos Aires, Argentina e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Cassini and J. Redmond (eds.), Models and Idealizations in Science, Logic, Epistemology, and the Unity of Science 50, https://doi.org/10.1007/978-3-030-65802-1_4

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elucidated yet, certainly there are several kinds of idealizations and, as a consequence, different ways of deidealizing a model. I then want to limit my account of deidealization to a specific variety of idealized models. To begin with, I will consider models in the field of classical physics exclusively. I will not intend to generalize or to extend my conclusions to other sciences, although I do not discard the possibility that some of them will also hold in different natural sciences. In the second place, I will consider mainly theoretical models in physics and I will leave aside other kinds of models, such as data models, models of experiments, or computer simulation models. In the third place, among theoretical models, I will deal with those models that embed the so-called Galilean idealizations. Finally, I will take into account mainly the predictive use of models. In particular, I will not consider minimal models, that is, those models designed to provide causal explanations of physical events and processes. More generally, I will leave aside all explanatory models even though the same model may be used to perform predictive and explanatory tasks alike. I will argue that deidealizing models is sometimes possible, at least in principle, but it is not always a useful or convenient strategy. Idealized models are built with a definite purpose and, as a consequence, the advantages and disadvantages of idealizing depend essentially on the specific purpose for which a given model is designed. As I will show, deidealizing a model often have costs and benefits and, even when deidealization may be feasible, a cost–benefit analysis may suggest avoiding it. I will conclude that deidealization has not to be conceived of as an end in itself but rather as a means to other ends, such as obtaining better explanations or predictions, or more generally, improving the expediency of our models to solve the problems that originated their construction.1

4.2 What Should We Understand By Idealization? There is a widely extended agreement among philosophers of science on the fact that all scientific models are idealized to some degree. When models are conceived of as representations of some phenomena or domain of phenomena, the concept of idealization can enter into the very definition of model: in a broad sense, models are idealized representations of the phenomena. Needless to say, this general characterization presupposes a representationalist conception of models. I will not intend to elucidate the notion of representation here, which I will take as a primitive notion embedded into the representationalist conception of models. For the sake of the argument, I will start by assuming that at least many scientific models do represent the phenomena. Representationalism has been the mainstream account of scientific models in the last 1 The

idea of deidealizing a model was introduced into the philosophical literature by McMullin (1985: 259, 261). Laymon (1985: 159) employed the notion of “better approximations”, characterized as “approximately true and more realistic” idealizations, to express a similar notion, although not applied to models but to theories.

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decades, although several different accounts of scientific representation have been put forward by philosophers of science.2 Despite being accepted as one of the essential features of scientific models, the concept of idealization has received less attention by the philosophers of science than that of representation. By and large, the discussions on idealization and representation have proceeded apart from each other. As a consequence, we still lack a standard or mainstream account of idealization and to some extent a well-defined vocabulary to refer to this feature of representational models. Scientists, in turn, make frequent use of the term but mostly in a broad and rather loose sense. Given the present situation of the philosophical studies on idealization, we cannot expect very much terminological precision or agreement in distinguishing different meanings of the term. This unavoidably leads to the necessity of stipulating a more or less welldefined meaning for some concepts. The predicament in which each philosopher coins his or her own terminology or stipulates new meanings for the old terms is not to be recommended as a matter of principle. However, it is not possible to avoid it entirely given the different uses of the concepts involved in the elucidation of the concepts of idealization and deidealization. The concept of idealization has been related in the philosophical literature with those of abstraction, approximation, distortion, incompleteness, and simplification, among others. In turn, those concepts are not well-defined either and have been characterized in different ways. Scientists who build or use models often agree in describing them by using those terms but they generally do not provide a precise elucidation of them. A model is sometimes described as a simplified representation or account of a complex phenomenon. It is said that a simplified model is obtained by abstracting some properties of the modeled phenomena, that is, by selecting some reduced set of properties of the target regarded as relevant or significant for some purposes and neglecting all its remaining properties. A model is then simplified because it is built by using a few properties that belong to the target. For the same reason, it is often said that models are abstract representations of the phenomena. On many occasions, scientists attribute to models certain properties they believe the target does not possess, or at least that it does not possess to some degree. For that reason, it is said that the model is (to some degree) a distorted representation of the target. As a consequence of being abstract and simplified, models are said to be incomplete. Because they are distorted, models often provide only descriptions of the target’s structure or predictions of its behavior which are not entirely exact or precise and, for that reason, they are said to be approximations to the phenomena. I will not intend to describe here the different accounts of idealization available in the philosophical literature. The topic has been addressed from different points of view because, besides modeling, the concept of idealization is relevant to different issues in the philosophy of science, such as realism, explanation, laws of

2 See Chap. 1 of this book for a more detailed account of the different theories of representation. See also Frigg and Nguyen (2017, 2020) for an extended discussion with many references to the literature.

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nature, and the aims of scientific inquiry.3 On the other hand, those philosophers of science who have addressed the specific issue of idealized models have emphasized different components of idealization and have assigned partially different meanings to them. The concept of abstraction is the paradigmatic example of this diversity. There have been several accounts of abstraction, which sometimes has been considered as something different from idealization or, alternatively, as a component of every idealization.4 For those reasons, at least a preliminary characterization of idealization is required to discuss deidealized models. In my view, idealized models are built through of double process of abstraction and distortion. There is no agreement among philosophers of science on how these two concepts relate to the concept of idealization. Sometimes, abstraction is seen as something different from idealization and sometimes as a kind of idealization. Something analogous can be said about distortion. I will regard idealization as involving both abstractions and distortions. I will understand by abstracting the process that consists in ignoring or neglecting some features or properties that the modeled phenomena do possess, or better, that we believe they possess. The reason for neglecting those properties is that we, the model builders, believe that they are not relevant to the purposes for which a given model is built. Roughly speaking, scientific models are built to solve a definite problem, even when their purpose is purely exploratory of a new domain of phenomena. Consequently, in building a model certain properties of the modeled phenomena are abstracted because the model builders think that they are not relevant to solve the problem in question. This does not necessarily mean that those properties have no relevance at all for the solution; rather, it means that the model builders regard them as negligible because their influence is sufficiently small as to be neglected for all practical purposes. This is, of course, a matter of degree. For instance, in calculating the trajectory of a short-range and very fast missile, the direction of a slow wind current can be neglected because it cannot disturb very much its trajectory. By contrast, if we were to calculate the trajectory of a long-range and low-speed projectile, the direction of the wind current becomes a very significant causal factor that could not be neglected, if we have the purpose of reaching a target with a certain precision. The properties of the modeled phenomena are generally represented in theoretical models by some variables or parameters. Abstracting then amounts to not including in the model any variable or parameter that could represent some properties of the modeled phenomena. 3 For

instance, two recent books devoted specifically to scientific idealization, those by Potochnik (2017) and Wheeler (2018) deal mainly with the aims of science and the laws of nature, respectively. Older books, such as Niiniluoto (1999) and Sklar (2000) deal with idealization within the context of a realist construal of theories. Some general books on scientific models, such as Toon (2012) or Gelfert (2016), do not discuss the concept of idealization. By contrast, Weisberg (2013) and Morrison (2015) deal extensively with it. 4 In this way, Jebeile (2017: 216) writes that “abstractions differ from idealizations in that they are omissions of some aspects in the target system which are not relevant for the problem being studied […], whereas idealizations are distortions”. For different accounts of the concept of abstraction and its relations to that of idealization see Cartwright (1989, 1999), Thomson-Jones (2005), Nola (2005), Godfrey-Smith (2009), Morrison (2015), and Levy (2019).

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I will understand by distorting the process by which the model builders introduce into a model some variables or parameters they believe do not represent a property that the modeled phenomena possess, or at least they do not possess to some degree. The first case amounts to introduce into the description of the model some terms we regard as fictional or, at least, as non-referring terms. The latter case amounts to fix the values of some parameters to certain values we believe are not the correct values regarding the phenomena. For instance, it is a common practice to set the values of some parameters to round numbers when we believe they are not, or even to set them to specific values (such as 0, 1, or infinite, often as limiting values) which permits simplifying the calculations. Again, this is a matter of degree and depends on the purposes for which the model is built. For instance, if the purpose is to predict the future value of a variable with a certain precision, numbers can be rounded to a certain degree, say, the third decimal place; for a different purpose, if higher precision is required, this distortion would be unacceptable. This is a well-known situation in mathematics, where exact calculations or exact solutions to the equations are often impossible (just think about using the numbers π or ε in calculations). As a consequence, all mathematical models, if they are to be tractable, have to introduce some distortions of this kind. Computer simulation models are a good example: discretizing continuous equations or introducing spatiotemporal grids are necessary to make them computationally tractable. We use those distorted models (in a spite of believing that spacetime is continuous) because they are expedient in solving problems, although we know that the solutions they can provide are only approximations and sometimes just coarse-grained approximations. Those idealizations that proceed mainly by abstraction were called Aristotelian idealizations, whereas those that proceed mainly by introducing distortions were called Galilean idealizations (perhaps without very much historical accuracy).5 I will not follow this terminology here. I will consider that most scientific models are both abstract and distorted to some degree although in some specific cases it could be not easy to distinguish each kind of idealization or to separate one from the other or even to count the exact number of idealizations embedded into a given model.

4.3 How Idealized Models Are Built There are certainly very different kinds of models in science but, for the sake of the argument, I will distinguish just two kinds: physical models and theoretical models. By physical models I mean single physical objects or aggregates of objects that are used to model some phenomena, such as scale models, icons, drawings, or maps. By theoretical models I mean above all mathematical or computational models. Sometimes, physical models are called concrete and theoretical models are called abstract but I do not want to commit myself to that terminology. This is not intended as a classification of scientific models or as an exhaustive account of the different 5 McMullin

(1985) is the classic paper that originated this terminology.

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kinds of models. It is just an expedient distinction I regard as convenient to analyze the issue of deidealized models. On the other hand, I do not intend to address here the question of the ontology of models. Let us start with some simple physical models. Engineers and architects employ all kinds of scale models in his or her projects of buildings, which are generally not built yet. The scale models are made of such materials as cardboard, plaster, balsa wood, and the like. Sometimes the scale is so small that not very much detail can be represented in the model. Those scale models are obviously abstract because they do not represent many features of future buildings. For instance, each building will contain perhaps thousands of bolts and nuts, which are not represented in the scale models. It would be possible in practice to include in the scale model the location of every brick of every wall of a house but that would be regarded as irrelevant to the purpose of designing that house or to the purpose of showing to the buyers how their living room will look like. On the other hand, those scale models are distorted because, for instance, they use materials, such as cardboard, which will be not part of the real buildings. Models that intend to represent the structure of molecules, usually made of plastic balls and rods, are good examples of physical models that are not made at a scale. The sizes of the balls and the rods do not represent the sizes of the atoms or the distances between them when they form molecules. These idealized models are essentially distorted. Each plastic ball represents an atom, to which we do not assign a definite form or color. We do not believe anymore that atoms are tiny spinning spheres. Each rod represents a chemical bond between the different atoms of the same molecule but we do not believe that the chemical bond is a material thing; rather, we conceive of it as a force, such as the electrostatic attractive force between charged particles. In spite of all the distortions they involve, physical models of the molecules are useful to understand the three-dimensional structure of complex molecules.6 Theoretical models are generally less simple and intuitive than physical models. There are many different reasons for building theoretical models, even when we have at our disposal principled and wide-ranging theories. One of the most pervasive reasons is the mathematical or computational intractability of the equations of a given theory. Many abstractions and distortions are introduced into a model just to make it tractable. Let us take as an example the classical, Newtonian, theory of gravitation and suppose that what we want to do is to apply it to the Solar System to calculate the trajectories of the different planets. We know that the gravitational interaction between three or more bodies is mathematically intractable (except for some higly symmetrical configurations) and, as a consequence, that Newton’s law of gravitation cannot be applied to a many-body system such as our Solar System. That is the reason why, despite having an exact gravitational law, physicists are forced to build simplified models in order to apply that law to complex physical systems. The simplest gravitational model is a two-body system composed of two massive bodies separated by a large distance (relatively large when compared to the dimensions of the bodies). This model is built by abstracting and distorting many properties 6 Molecular

models and their different uses are discussed in detail by Toon (2012).

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we believe planets, or massive bodies generally, do possess. Let us consider the distortions they introduce. First, the bodies are regarded as perfect homogeneous spheres, what we know planets are not. More generally, if two irregular and inhomogeneous bodies, such as asteroids, are separated by a very large distance-as compared to their sizes- their dimensions can be neglected and they can be treated as point-like massive particles. Second, those spheres are treated as geometrical points, ignoring their different dimensions, on the basis of a theorem of Newtonian mechanics, which proves that perfect homogeneous spheres gravitate as if all their masses were concentrated in their centers of gravity. Let us see now the abstractions. First, all other bodies are neglected, so that we assume that the two-body system is located in an empty space. Of course, we know that every planet in the Solar System exerts a gravitational force on any other but we consciously choose to ignore the many-forces system and to reduce them to only one gravitational force. Second, we assume that there are no other types of forces beyond gravitation acting on the two-body system. We choose to neglect, for instance, the electrostatic force between the planets because we believe it is too small to be of any significance. We can do that because we believe that planets are almost electrically neutral bodies, which are made of many particles whose negative and positive electric charges compensate each other. Finally, we neglect the gravitational influence of the stars because we know they are far apart from the planets to exert a significant force on them (given that the gravitational force decreases with the square of the distance). The two-body gravitational model is highly idealized but it has a limited domain of applicability. It cannot be applied to every system of two masses. For instance, if we have two highly irregular and inhomogeneous bodies separated by a relatively short distance, the model will deliver wrong predictions. Likewise, if we have two perfect homogeneous spheres endowed with a small mass but electrically charged, the electrostatic force could not be neglected. The two-body gravitational model can be applied to different two-planet physical systems, say, Sun-Earth or EarthMoon, because those systems approximate the conditions embedded into the model. The planets are very massive bodies, almost spherical and almost homogeneous (at least, at a large scale), electrically neutral, and separated between them by very large distances compared to their sizes. In any event, a two-body gravitational model has limitations, even when applied to the Solar System. If we were interested in calculating the orbit of Mars to a high degree of accuracy, we could not use a Sun-Mars two-body model; the gravitational influence of Jupiter, which has a large amount of mass and it is relatively near to Mars, cannot be neglected. Jupiter, whose distance to Mars varies very much, exerts a strong gravitational perturbation on Mars’ orbit, whose irregular trajectory was known from ancient times. The less idealized Sun-Mars-Jupiter three-body model is mathematically intractable but an approximate and quite complicated calculation can be performed by introducing further idealizations into the model. One of the most impressive achievements of the science of celestial mechanics in the nineteenth century was to develop the required mathematical techniques to solve approximately the equations of many-body systems. As the experience shows, the three-body model permits to approximate the observed trajectory of Mars very much better than the

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two-body model. Nonetheless, both models deliver mere approximations. An exact calculation of the trajectory of a given planet, one that would take into account the gravitational influence of all the bodies that compose the Solar System (not to mention the stars), is far beyond our possibilities. Notice that the two-body gravitational model is a theoretical construction that can be applied to many different physical systems: a system composed of two planets, or of a planet and a star, or of a planet and a satellite, or of two stars, and others. This means that the model has many possible domains of application. Whether a physical system that has been selected as the target of the theoretical model has the conditions required by the model is something that we cannot know a priori; it is something that has to be determined by appealing to large portions of the available empirical knowledge we have of each physical system or physical phenomena. We know by experience that the two-body gravitational model works for the Sun-Earth system but does not work very well for the Mars-Jupiter system. In principle, the domain of application of a given model is always open and subject to constant changes to the extent that our knowledge of the physical phenomena evolves. Notice also that a theoretical model, being abstract, does not represent a particular object or physical system. Theoretical models are universal in the usual sense that they can be applied to many different particulars. A scale model of an actual bridge represents a particular entity; by contrast, a mathematical model of the structure of a suspension bridge does not represent a particular bridge but rather the structural properties common to all bridges of that kind.

4.4 Costs and Benefits of Idealization Idealizations have produced mixed feelings among philosophers of science. On the one hand, idealized models seem to be misrepresentations of the real-world phenomena; on the other hand, they bring us many epistemic benefits and allow us to obtain genuine knowledge of those phenomena. It is generally accepted that scientific models cannot be built without idealizations and, consequently, that idealizing is an essential procedure in model-construction. Likewise, it seems manifest that idealization is a matter of degree in the sense that we can build many different models of the same phenomena (say, a clock pendulum or a gas inside a bottle), some of them more or less idealized than others. And it also seems clear that idealizations are useful for many purposes, such as simplifying complex phenomena or making mathematically tractable the equations of a model. Beyond these facts, there is scarce consensus among scientists or philosophers of science on how to assess the costs and benefits of idealizing. One of the reasons for this predicament is the fact that the assessment of those costs and benefits depends essentially on several broad philosophical assumptions concerning, for instance, the aims of science or the very definition of knowledge. The main justification for building highly idealized models consists in appealing to pragmatic reasons. In the broad sense of this word, those reasons go from the

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pedagogical advantages of using a visual representation of unobservable entities to the requirement of mathematical and computational tractability of the equations of theoretical models. Many of the so-called epistemic virtues of scientific theories can be applied to idealized models, among others, simplicity, generality, and explanatory and predictive power. Those epistemic virtues are generally regarded as non-factual and as pragmatic criteria for theory choice. They can also be viewed as desiderata, or “representational ideals”, as Weisberg (2013: 105) has called them, which guide the construction of scientific models, in particular, the selection of the kinds of idealizations that a given model should embed. There is a broad consensus on the fact that epistemic virtues are not truth-conducive, that is, that a very simple or general theory is not more verisimilar or truthlike than a less simple or a less general rival theory. Something similar can be said of representational models: a simple (explanatory or predictive) model is not necessarily a better representation of some phenomena than a more complicated model. There is also a broad consensus on the fact that epistemic virtues often point in different directions and cannot be maximized at the same time. For that reason, some compromises and tradeoffs between different virtues are unavoidable. This is also the case with models. The simplest model sometimes is not the best at predicting accurately the values of a variable in which we are interested. To obtain more accurate predictions, we often need to build a more complicated model. The standard representationalist conception of models has often stressed the deficiencies of highly idealized models. Such models are regarded as just an unavoidable means of accessing complex phenomena, given our epistemic limitations and our incomplete knowledge of the world. According to this view, models are incomplete and distorted precisely because they are idealized. As a consequence, they can deliver, in the best cases, partial and inaccurate representations of the phenomena. Strictly speaking, every model misrepresents the phenomena in at least some respects, although some models can be said to be better representations than others. In principle, richer and complex models, those that include a lesser number of idealizations, provide better representations of the phenomena than simple and highly idealized models. From these premises, it follows a positive assessment of deidealization: deidealized models are always epistemically superior to highly idealized models because they necessarily provide less distorted representations of the phenomena. This kind of “deficiency” conception of idealizations is rooted, one way or another, in some variety of epistemological realism. The realist stance is reflected in the many expressions applied to models in which it is assumed that the fundamental aim of science is to provide a “truthful” or “realistic” representation of the real world. It is not the case that every representationalist has to be an epistemological realist but most of them, sometimes implicitly, are committed to realist assumptions.7 In most of its formulations, this conception assumes -usually in an implicit way- that models aim at providing us with approximately true descriptions of the real-world phenomena. 7 Laymon (1985, 1995), McMullin (1985), Niiniluoto (1999), Sklar (2000), Giere (2006, 2009), and

Teller (2001, 2008, 2009, 2012) are overtly realist concerning the question of how idealizations relate to the real world. In turn, Thomson-Jones (2005), Wimsatt (2007), Godfrey-Smith (2009), Morrison (2015), and Strevens (2008, 2016) have dealt with idealizations in terms that show some commitments to realist assumptions on ontological and epistemological matters.

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Although models are not truth-bearers (at least in a fundamental or primary sense) idealized models and idealizations alike are often called “false” representations. It is said, for instance, that “idealized models can represent despite their false assumptions” (Potochnik 2017: 50). Models then represent the real world as it is but only in an approximate, inaccurate, imprecise, incomplete, and distorted way. As Teller (2012: 258) summarized, “models succeed in representing things as they are always to limited degrees of accuracy”. Idealizations are problematic for the realist tradition because they appear to involve deliberate simplification and distortion of the phenomena or, more generally, as Godfrey-Smith (2009: 49) has put it, because “we saw that idealization involves a departure from reality”. Idealizations are then regarded as “deliberate falsifications” and sometimes as involving “a dramatic distortion of reality” (Strevens 2016: 37). Similar expressions can be easily found in the literature on modeling and idealization. A partially different view of models, such as those endorsed by Strevens (2008, 2016) and Potochnik (2017), has stressed the benefits of idealization. Leaving apart the idea of distortion, it has pointed out the many virtues of highly idealized models. Models provide explanations or predictions that would have been not possible without the idealizations they embed. Precisely because theoretical models are abstract, they can be applied to many different physical systems or phenomena. They are extremely versatile constructs that permit us to explore, describe, and explain a wide variety of different domains of phenomena. Idealized models permit us to make tractable what is intrinsically intractable for mathematical or computational reasons. They give us epistemic access to complex phenomena by simplifying the representations we construct of them. From this point of view, a highly idealized and simple model may be more flexible and versatile than a less idealized but complex model. If all this is true, what is the point of deidealizing models?

4.5 Deidealizing Models Deidealization may be a deceptively simple process if we focus exclusively on scale models and other physical models of particular objects. Adding more details to the models, such as colors, textures, and other properties seems enough to obtain a more realistic representation of the targets, for instance, a house or a bridge. Nonetheless, these kinds of deidealized models are not always adequate for the purposes for which the model was built in the first place. If we want to sell a weekend house, a more realistic scale model, one that includes, for instance, the actual colors of the future house, may be very convenient to persuade a potential buyer. But if we want to study the structural features of a large building, adding more details to a scale model may be counterproductive because it may obscure the structural features we are interested in. Maps are even better examples of the practical disadvantages of deidealization. If all we need is a map that permits us to find the way out from a forest, a very detailed map, one that represents every minor accident of the territory, could be much less useful for that purpose than a simple map that only represents the major ways that traverse

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the forest. So, even when deidealization is feasible, it is not always convenient from a practical point of view. How much idealization should we embed in a given model is something that depends essentially upon the purposes for which the model is to be employed. Notice that the possibilities of deidealizing a physical model are limited by the representational means selected in building that model. For instance, a scale model of an aircraft set to a 1:100 scale allows for more deidealizations than a model of the same object set to a 1:1000 scale. Something similar can be said of maps. A world map of the size of an office paper sheet does not permit to add very much detail to the representation of the territory. Deidealization is more difficult, and not always feasible, in the case of theoretical models. Let us take the standard example of the models of the pendulum.8 The model of the ideal or simple pendulum is, as its name indicates, highly idealized. Among the distortions it contains, it assumes that (i) the support does not move, (ii) the cord is massless, inextensible, and always remain taut, (iii) the bob is a point mass, (iv) the motion of the pendulum occurs in two dimensions, and (v) the gravitational field is uniform. Among the abstractions, it assumes that (i) the pendulum moves in a vacuum so that there are no effects of air friction, and, as a consequence, the motion does not lose any energy, (ii) there are no changes in the temperature or pressure that could affect the pendulum, (iii) the position of the pendulum is not altered when the bob is in motion. The model delivers approximate predictions for the period T of a real pendulum for  small oscillations of the bob. The fundamental equation of the pendulum T = 2π gl (where l is the longitude of the cord and g is the acceleration of gravity) can be applied when the angle of oscillation θ is not larger than 0.1 radians or 6°. This formula is in itself the outcome of an approximation when the angle θ is 2 equated to sinθ. The exact formula for the simple pendulum is ml δδt θ2 = −mg sin θ . Given that for small angles θ ∼ = sinθ (when the angles are measured in radians), if we take θ = sinθ, we obtain the fundamental simpler equation.9 The assumption θ = sinθ is then another idealization of the model introduced to simplify the calculations. The compound pendulum model is less idealized than the simple pendulum model because it takes into account the mass of the pendulum, its moment of inertia, and the distance  from the pivot point to the center of mass. The corresponding equation T = 2π mgI R (where I is the momentum of inertia of the pendulum, m is the total mass of the pendulum and R is the distance from the pivot point to the center of mass of the pendulum) can be regarded as a deidealization of the simple pendulum model (actually, the equation of the simple pendulum can be deduced from the equation of the compound pendulum as a limiting case).

8 The

pendulum model was regarded as a paradigm for the deidealization process by Morrison (1999: 48–53, 2015: 230–233). The same example is often employed in the debates concerning realism and approximation to truth, for instance, in Rowbottom (2019: 8–14). 9 For instance, if θ = 1° (= 0.17453 radians), sin θ = 0.17452 radians; if θ = 6° (= 0.10472 radians), sin θ = 0.10452 radians. All numbers are approximate.

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This model can be used for different purposes, which go from building precision clocks to measuring the acceleration of gravity. Depending on our purposes, the compound pendulum model may be more or less adequate or utterly inadequate. For instance, the model is not useful for constructing precision pocket watches. Christian Huygens, the discoverer of the pendulum clock, noticed that his clocks ran slower in summer due to the expansion of the rod produced by the higher temperatures. Temperature-compensated pendula, such as the mercury pendulum, were invented during the eighteenth century to improve the precision of the clocks in use. By the end of that century, the famous clockmaker Abraham-Louis Breguet discovered that a pendulum watch ticks at a different rate when it is located in a horizontal or vertical position. To overcome the differential effects of gravity he invented the complex tourbillon pendulum -still in use in luxury mechanical watches- whose precision could not be surpassed until the invention of the quartz mechanism.10 The compound pendulum model is still highly idealized and cannot deliver very good approximations to the behavior of real pendula. A real pendulum can be approximated by the compound pendulum model when it is artificially isolated from the environment, for instance, when it is put inside an empty glass box and located on a steady platform that is free of vibrations, when the temperature and the pressure are held constant, and so on. If we want to get a better approximation to the behavior of a real pendulum in non-controlled conditions, we need to build a less idealized model. There are many possible deidealizations of the compound pendulum model. They amount to introduce many parameters into the model, called generically corrections by physicists. These include (i) finite amplitude corrections for different angles of oscillation; (ii) mass distribution corrections, where the finite mass of the bob and the cord are taken into account; (iii) air effects corrections; and (iv) elasticity corrections, in which the stretching of the chord and the motion of the support are taken into account. Each of those kinds of corrections, in turn, consists of several different factors. I will briefly comment on the air corrections, which are the most complicated and mathematically sophisticated.11 If we want to account for the effects of the air on the motion of the pendulum, we need to take into account four different corrective factors. First, we have the buoyancy of the bob, produced by the so-called Archimedes’ force, which reduces its apparent weight by the weight of the displaced air. Second, there are the damping effects produced by the resistance of the air on the wire and the bob of the pendulum; these effects decrease the amplitude and increase the period of the oscillations. Third, we have the added mass effects, which are produced by the changes in the motion of the air that surrounds the bob; as a consequence, the kinetic energy of the pendulum is partly that of the air and the effective mass of the system is larger than that of the bob. Fourth, there are several theoretical damping constants, which take account of other sources of damping, such as, for instance, the friction at the point of support. All those corrections are relatively small and mathematically complicated, but they 10 Baker

and Blackburn (2005), Chap. 10, contains a fascinating history of the pendulum clock. a detailed mathematical treatment of this topic see Nelson and Olsson (1986) and Baker and Blackburn (2005), Chap. 3.

11 For

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are indispensable if we want to build a high-precision clock or to perform an accurate measurement of the local acceleration of gravity.12 The addition of just one new parameter to the theoretical model of a pendulum, say, air buoyancy, is by itself a deidealization of that model. We can regard some sequences of deidealized models of the pendulum as better approximations to the behavior of a real physical pendulum oscillating in non-controlled conditions. Models, of course, cannot be compared to the things in themselves, but rather to the data we have collected about physical phenomena. In this way, we can compare the predictions derived from a given model of the pendulum concerning its period to our measurements of the periods of real pendula. We can then determine that the predictions of less idealized models fit better, within the margins of the measurement error, with the available data. The match between any theoretical model and the data will be always approximate, and this for two different reasons: in the first place, because we cannot have error-free measurements of continuous quantities, such as length, mass, or period; in the second place, because we cannot derive exact predictions from the different deidealized theoretical models of the pendulum. The latter fact is of particular importance for the assessment of idealizations. From a purely mathematical point of view, the deidealization process often amounts to introducing equations without exact analytical solutions and, as a consequence, to the unavoidability of employing different approximation techniques. This fact is another constraint to the possibilities of deidealization for mathematical models and theoretical models in general. The question of whether a deidealized model that is obtained by correcting a previously available model should be regarded as a different model is a matter of convention. Usually, scientists think that a deidealized model must be endowed with its own identity and, consequently, it has to be treated as a new model and not just as a version of the old model. The simple pendulum and the compound pendulum are thus two different models, and any deidealization of the compound pendulum model is another model of the pendulum. If we were to compare those different models concerning their degree of idealization, we would quickly find problems of incommensurability. There cannot be any doubt that a model that includes all the corrections of another model and adds other corrections is less idealized than the former. For instance, the compound pendulum model with finite amplitude corrections is more idealized than the compound pendulum model with finite amplitude and mass distribution corrections, and so on. However, when two models embed different kinds of idealizations, there is no general criterion to determine which one is more or less idealized than the other. Let us consider the case of three deidealized models of the compound pendulum with finite amplitude corrections but with different kinds of air corrections: buoyancy effects, damping effects, and added mass effects. There is no way of ordering them concerning their degree of idealization. Sometimes it will 12 The

effects of buoyancy and added mass are significant for long times. As Nelson and Olsson (1986: 117) point out: “If the pendulum were a clock, it would lose 8.6 seconds in one day on account of added mass (and another 7.3 seconds due to buoyancy) compared to a similar pendulum swinging in a vacuum”.

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be possible to establish a partial order between different deidealized models but only in some special cases, this order will be linear.13 Deidealization has often been a desideratum for the realist view of models. A sequence of models, in which each one is obtained by deidealizing its immediate predecessor is regarded by realist philosophers as if it were a progressive sequence of increasing truthlike models, that is, as a progress towards a more complete and realistic representation of the modeled phenomena. Morrison (1999: 51), for instance, refers to the deidealization of the physical pendulum models as an exemplar case of model building in which “we start with a background theory [Newtonian mechanics] from which we can derive an idealized model that can be corrected to provide an increasingly realistic representation (model) of a concrete physical phenomenon or system” (my emphasis). Nonetheless, there is a general agreement on the fact that all effective models are doomed to be forever incomplete and, for that reason, we cannot hope for a perfect model, that is, one that provides a complete, and possibly definitive, representation of the real-world phenomena.14 We cannot expect to obtain the truth, the whole truth, and nothing but the truth from our models. Nonetheless, according to the realist construal, we can do expect to obtain from them progressive approximations to a true description of the world. In this way, deidealized models, as the pendulum case shows, provide better approximations to the truth, at the evident cost of losing simplicity and scope. Realists take an apparent paradoxical stand to idealizations. They regard them as useful but, in principle, dispensable. The advancement of science, in their view, must bring with it the elimination of most idealizations or, at least, that is the ideal to which science must approach. Vaihinger took a similar attitude towards all the fictions employed in science. He thought that full fictions were self-inconsistent, what it is not what we think of the idealizations embedded in a model. What he called semi-fictions, which are empirically false assumptions introduced in science with the explicit awareness of their falsity, are now for the most part called idealizations. According to Vaihinger (1927: 98), all scientific fictions are provisional devices that will “disappear in the course of history”. However, the use of fictions is justified to the extent that they are expedient devices, useful means to some definite ends. Expediency is the only justification for using them. Idealized models have to be justified the same way. A highly idealized and simplified model is not built in the first place in order to be deidealized in the future. It is built to solve a definite problem in the present. The costs and benefits of deidealization have to be assessed on this pragmatic basis. For that reason, it cannot be claimed that all the idealizations embedded in a given model are provisional. Perhaps, some are there to stay. The main reason to use highly idealized models is that they work, that is, they are useful means to provide a solution to a given problem, a solution that is adequate to the purposes of the model builders (for instance, to a defined standard of accuracy). More precisely, models are indispensable tools to solve problems, which implies that the 13 This

was noticed in passing by Laymon (1995: 372, footnote 5). (2001) has stressed this central tenet, which is common to all realist construals of models, such as Giere´s (2006) perspectival realism.

14 Teller

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required solution cannot be obtained by other means, in particular, by applying theories directly to specific problems, as the example of Newtonian gravitational theory shows. When principled theories are not available for some domain of phenomena (say, in nuclear physics), the necessity of building models is even more evident.

4.6 Cost and Benefits of Deidealization From the standpoint of a representationalist conception of models, idealizations are intentional misrepresentations. They misrepresent the phenomena because they distort them. By contrast, unintentional misrepresentations are simply errors, which, when detected, should be corrected. Deidealization does not consist in correcting errors but in removing at least some idealizations from a given model. On every account of modeling in science, idealizations are not entirely removable from models, that is, we cannot conceive of a model without any idealization, because that would not be a model anymore. The best we can hope for deidealized models is to suppress at least some idealizations by adding more details to a given model and enriching it, for instance, introducing new parameters and variables into its equations. Which is then the aim of deidealizing a model? In the first place, we should discard every notion of completeness as the aim of deidealization. Models are often said to be incomplete, but with respect to what are they incomplete? It seems obvious that a complete representation of the realworld phenomena, in which every property is represented accurately, is by principle impossible. We cannot even know how this complete representation would look like and, a fortiori, we are unable to determine whether a given model approximates or not to that representation. Completeness, in my view, cannot be endorsed as a Kantian regulative ideal useful to guide the theoretical research, as Weisberg (2013, p. 106) claims. From a practical point of view, a purported complete representation of the phenomena may be as useless as a one-to-one scale map of a territory. Models can be regarded as relatively incomplete representations concerning our available knowledge of the modeled phenomena. But given that this knowledge is constantly changing, the incompleteness of a model has to be assessed in each particular context. In light of these reasons, it should be convenient to abandon the talk about incomplete models and also about complete representations, even in the case of comparisons between different models. Instead, we can talk of richer and more complex models, by which we should understand models that provide more detailed representations of the phenomena. Deidealized models can then be said to provide richer representations in this sense. Why these kinds of representations should be desirable? Deidealized models sometimes have theoretical and practical advantages. As the physical pendulum example shows, a richer model can greatly improve its predictive capacities. A reasonably accurate pendulum clock cannot be constructed without a complex deidealized model of its pendulum. No mathematical model of a pendulum would be able to predict the exact real value of the period of a real physical pendulum.

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The real values of continuous physical quantities, as is acknowledged, are unknowable. Nonetheless, a less idealized model can give us better predictions, both more precise and accurate, that fit better with the outcomes of our measurements. If we want to use a pendulum model to deduce a precise value for the local acceleration of gravity g, the simple pendulum model or the physical pendulum model turn out to be inadequate for that purpose. To measure the parameter g with an accuracy of one part in 104 , a highly complex deidealized model is needed. We have to sacrifice the simplicity of the physical pendulum model in order to reach that predictive power. This is a good example of the tradeoffs between different desiderata: a model of a physical pendulum, being at the same time very simple and highly predictive, is simply not possible. The disadvantages of deidealized models of the pendulum are manifest from the pedagogical point of view. No basic course in classical mechanics would make any use of them because they are too complicated to learn the fundamental physics of the pendulum. The equation of the compound pendulum, employed by most textbooks, suffices for that purpose. This is generally true of all deidealized physical models. They can be addressed once we have learned to use simpler and more idealized models. Deidealized models are designed for a diversity of specific purposes, not just to get a richer representation of the phenomena. A deidealized model of the pendulum that allowed us to calculate T or g with an accuracy of, say, one part in 1030 would be practically useless because our measurements of length and duration are not able to reach such a degree of accuracy. In the far future, such a model, if we were able to build it, perhaps might find some domain of application. The main disadvantage of deidealized models lies in their intrinsic mathematical intractability. If we introduce several idealizations in a model of some phenomena just to make its mathematical equations tractable, there is no point in deidealizing the model if the result is intractable. Something similar occurs with computational tractability. Computer simulations appeal to many idealizations with the explicit purpose of gaining tractability. If a deidealized model is not computationally tractable, it is of no use to be implemented as a computer simulation. When deidealized models are feasible, that is, when they are still mathematically or computationally tractable, different practical reasons determine their utility and possible use. Here, again, scientists appeal to different tradeoffs between various desiderata and factual constraints. For instance, if a deidealized complex model is excessively expensive in terms of the time of its computation and it does allow just a small increase in predictive accuracy, a cost–benefit analysis may drive us to discard it if we have at our disposal other inexpensive and reasonably predictive models. In the last resort, the justification of every deidealization is rooted in pragmatic reasons. A model has to be deidealized when it is not anymore useful to solve a definite problem, for instance, when it is unable to deliver the predictions we are interested in with the required degree of precision or accuracy. Corrective deidealizations are applied to the extent that they contribute to achieving a better solution to that problem. The pendulum example is again relevant to show this. If we want to calculate the local acceleration of gravity with an accuracy of four figures, some deidealizations are useful means to do it and others are not. In principle, we could wish to calculate

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the effect of the collision of a single molecule of air, or the effect of any small number of air molecules, with the pendulum’s bob. Of course, we do not know how to do that, although we do believe that this extremely small effect exists (and must have a precise value, from the point of view of God’s eye). We simply admit that this calculation is irrelevant to the purpose of calculating g up to measurable values. Likewise, we could wish to calculate the gravitational action of all the minor bodies in the Solar System on the orbit of Mars or to build a general relativistic model of the celestial mechanics for the Sun-Mars-Jupiter system. We do not even intend to do those things because we regard them as excessively complicated and expensive, if not entirely irrelevant, to solve the problems we have posed concerning celestial mechanics.15

4.7 Can All Models Be Deidealized? Models cannot be conceived of without idealizations so that wherever scientists build and use models they must appeal to idealizations. The question, then, is not whether we can dispose of idealizations in doing science, but rather whether it is always possible to replace highly idealized models with less idealized models, or more specifically, whether it is always possible to deidealize a given model. This question must be sharply distinguished from other related questions. One of them is whether we can always know how to deidealize a model. The other is whether it is useful or convenient to deidealize a given model. The last one is whether deidealization must be conceived of as one of the aims of science. I will briefly address these questions one by one in the remainder of this section. Before doing it, some general remarks are in order now. First, although in practice these three questions are closely related, from a logical point of view they are mutually independent, so that a positive or negative answer to one of them does not imply any definite answer to the others. Second, a negative answer to the three questions does not imply that models cannot be deidealized in principle. And third, we have to be cautious in generalizing on scientific models. Given the diversity of scientific models, it is more likely to draw conclusions about a restricted class of models belonging to the same discipline. As I said, I will restrict my analysis to predictive models in classical physics. The physical pendulum example shows beyond any doubt that at least some models can be, and have been, deidealized. Models in celestial mechanics point to the same conclusion. Of course, nothing follows in general from two examples, but they prove at least that deidealization is possible, that in some cases we do know how to perform it, and that in some circumstances and for certain definite purposes deidealizing is a useful and convenient task. 15 This

does not mean, of course, that a relativistic three-body model lacks any interest in itself nor that such a model may not be useful to solve some definite problem in celestial mechanics. In principle, we cannot assure that a model, no matter how complicated could be, will not find a domain of application.

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From the fact that deidealization is possible, it does not follow that we always know how to do it; much the less, that building deidealized models is an easy task. The celestial mechanics’ example shows that in some cases deidealizing a model may demand very hard labor. A three-body gravitational model required the development of complex mathematical techniques, which only allowed for approximate solutions to the equations. In principle, it is possible to deidealize the three-body model and to extend it to an n-body model, but we simply do not know how to do it. The nbody problem, even for small values of n, is mathematically intractable with our approximation techniques. A general relativistic n-body model is even much more difficult, although we believe that a model of this kind would be the more accurate one for the Solar System dynamics. Nonetheless, this does not mean that the n-body problem will remain unsolved forever. Perhaps in the far future, we will be able to develop stronger mathematical techniques that permit us calculating the gravitational effects of all the planets on Mars orbit. Beyond any doubt, many scientific models are far more complex than the pendulum models. It seems obvious that in most cases deidealization is not easy at all and that for a broad variety of idealized models we do not have the required knowledge to deidealize them. However, nothing follows from this state of ignorance about the impossibility of deidealizing models generally. By itself, the fact that we do not know how to deidealize a given model does not prove that the model in question cannot be deidealized. Some recent works have claimed that some idealizations are irreversible and cannot be removed from certain types of models. An example of ineliminable idealizations is that of the so-called minimal models (Batterman 2002, 2009) or minimal idealizations (Weisberg 2007, 2013). These kinds of models aim at finding explanations of the occurrence of some physical regularity by isolating the dominant causal factors that are responsible for the observed regular behavior. The process of idealization consists here in abstracting all the inessential details, that is, all the details that might change without affecting the regular repeatability of the phenomena we are interested in explaining and understanding. As Batterman (2009) has argued, those idealizations are ineliminable because they play an essential role in the proper explanation of the phenomena of interest. If we were to deidealize a minimal model by adding more details to it, for instance, new parameters, we would lose the explanatory power of the model. In those cases, deidealization is not only counterproductive but it can even be self-defeating. As a general rule, more idealized minimal models better explain the phenomena than less idealized models. We do not get a deeper understanding of a natural regularity by adding more details to a minimal model. On the contrary, those details may obscure or screen off the dominant causal factors that produce that regularity. In Batterman’s words: “adding more details counts as explanatory noise -noise that often obscures or completely hides the features of interest” (Batterman 2010: 17).16 16 The questions concerning how minimal models explain and what kind of understanding they produce have been addressed, with some significant differences, by Strevens (2008, 2016) and Batterman and Rice (2014); their answers are not relevant to our purposes here.

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A stronger argument against the possibility of deidealization was put forward by Batterman and Rice (2014)and elaborated by Rice (2018, 2019). According to Batterman and Rice, models represent their targets in a rather holistic way and not through separable components. That is why idealized models have to be conceived of as holistic distortions of the phenomena, in which the idealized components cannot be isolated from the non-idealized components. As a consequence, a model cannot be gradually deidealized by removing its idealizations one by one. Models represent holistically their targets and often distort both relevant and irrelevant causal factors alike. More generally, idealizations are introduced globally into a model in order to allow for the application of mathematical modeling techniques that would be otherwise inapplicable. Such global idealizations then cannot be removed without impairing the explanatory power of the model, or even without destroying the model as a whole. For instance, when fluid mechanics models a current of water as a continuum, this idealization cannot be removed from the model. We believe that water and other fluids are discrete and that any volume of a fluid is composed of a finite number of molecules. Nonetheless, the model that represents water as a continuum cannot be deidealized to get a model in which water is represented as an aggregate of billions of molecules separated by an empty space. If we were to do that, we would lose the explanations that fluid mechanics provides because the mathematical apparatus of standard fluid mechanics could not be applied to a discrete model of a current of water. That is the reason why at least some idealizations are ineliminable and have to be conceived of as permanent or essential features of a given model. Another very much-discussed example is that of the thermodynamic limit, which is employed in models of statistical mechanics that explain, among other things, phase transitions in physical systems.17 Those idealized models assume that the number of particles of any physical system approaches infinity. More precisely, the thermodynamic limit consists in assuming that N → ∞, V → ∞, and N/V remains constant and approaches a finite quantity (where N is the number of particles of a physical system and V is its volume). We believe that this is not true of any physical system, however, the idealization is essential to explain all phase transitions because, according to statistical physics, those transitions cannot occur in a system composed of a finite number of particles. As Styer (2004: 27) puts it: “phase transitions do not exist at all in finite systems! They appear only in the thermodynamic limit”. This idealization cannot be conceived of as an approximation grounded on the fact that the number of particles in any physical system is very large (of the order of 1023 or higher) because no finite number can approximate to infinite. The thermodynamic limit is an idealization that cannot be removed from models of statistical physics. There is no gradual transition from an infinite number of particles to a finite number, and, for any finite number of particles, phase transitions are not possible.

17 Philosophical

discussions of the thermodynamic limit can be found in Sklar (1993, 2000) and Batterman (2002a, b, 2010). Styer (2004) is a concise mathematical analysis that shows the many uses of this limit in statistical physics, beyond explaining phase transitions.

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A related argument against the possibility of deidealization appeals to the epistemic opacity of highly complex models. The typical example is provided by global climate models. Winsberg (2010, 2018) has claimed that those models have an architecture that does not permit to decompose them in separately manageable pieces. They are holistic in a strong sense of the term. Climate models are built from many different modules and submodules and involve as much as a hundred parameter options. The interaction between the different modules is itself very complex and the process of coupling some submodules, which include their own parametrizations, is often a very difficult problem. According to Winsberg (2018: 142), this complex architecture, which he calls “fuzzy modularity”, has the consequence that “the overall dynamics of one global climate model is the complex result of the interaction of the modules -not the interaction of the results of the modules”. Moreover, climate models are the outcome of a piecemeal construction that usually requires many years of patient assemblage and mutual adjustment of their component modules. As a consequence, no single scientist or engineer could be in a position to know the construction process of the model or the minute details of its architecture. Those complex models are “analytically impenetrable”, as Winsberg (2010: 105) has called them. It is practically impossible to track the sources of successes and failures of this kind of models up to single separable modules or submodules. This results in a sort of epistemic inscrutability of all complex models of global climate. If those models are epistemically opaque and not decomposable, we cannot expect to know precisely which are the idealizations embedded into them. A fortiori, we could not hope to deidealize such models by removing their idealizations one by one. In a similar vein, Knuuttila and Morgan (2019)have argued that the idealizations embedded in several economic models cannot be reversed because they cannot be separated from each other. In many cases, economic models are not decomposable into independent parts, which could eventually be controlled, edited, and corrected. These authors then claim that “it may be not possible to add back in certain causal factors without consequences, as these factors are related to others the scientist also wants to keep in the model” (Knuuttila and Morgan 2019: 658). As a consequence, it may not be possible to deidealize a definite assumption without collapsing the functionality of the model. Models cannot be deidealized step by step because they were not constructed this way, rather all the idealizations were jointly embedded when the model was built.18 All the above arguments point in the same direction. They stress the fact that many scientific models represent holistically their targets and, consequently, are functional as inseparable wholes. In my view, this does not prove that deidealization is impossible in principle. After all, if models are non-decomposable wholes, a highly idealized model can be entirely replaced by a less idealized model of the same phenomena. What these arguments do show is that some models cannot be deidealized step by step, as some philosophers of a realist slant have assumed. Unlike the physical pendulum models, sometimes it would not be possible to build a sequence of

18 I

acknowledge Tarja Knuuttila for pressing this point in a personal communication.

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gradually less and less idealized models. Some models are then subject to an all-ornothing choice: either we use them as unities that represent holistically the intended phenomena or we have to replace them with entirely different models. While we use them for the purposes for which they were built, none of their idealizations can be corrected or removed.19 Angela Potochnik (2017), in turn, has claimed that idealizations are not only “rampant” in science, but also “unchecked”. By this last term, she understands that “there is little focus on eliminating idealizations or even on controlling their influence” (Potochnik 2017: 58). This is not generally true. On many occasions, scientists are very well aware of the idealizations they introduce into a model, of the reasons why they are necessary, of the consequences that follow from each specific idealization, and of the possibilities of removing or correcting them. Besides, on some occasions, as the example of the models of celestial mechanics shows, they are also aware of the theoretical and practical difficulties of deidealizing a given model, and of the costs and benefits that a particular deidealization might imply. Idealized models are always built with a definite purpose in mind -to solve a well-posed problem- and are deidealized for the same reason. More precisely, they are deidealized when the available models are not anymore adequate to solve a new problem. The compound pendulum model has been deidealized either to measure the local acceleration of gravity with a high level of accuracy or to construct high precision clocks. The model was deidealized to a degree that was regarded as sufficient for the purposes to which it has to be used, say, a measurement of g up to for figures of accuracy. The compound pendulum model was simply not adequate for that purpose. In turn, the deidealized model might be utterly inadequate for a different purpose, even for the measurement of g with a higher degree of accuracy. Deidealizing a model may be an interesting theoretical exercise in mathematics or physics, but by itself is not an enterprise designed to solve a problem that could originate a new model. Models are generally deidealized when a richer model is regarded as necessary to solve a well-defined problem. In any event, all deidealized models are still idealized models. Models are not deidealized with the aim of obtaining a complete or perfect representation of the physical phenomena. The degree of deidealization that is regarded as satisfactory depends on the context in which the model is to be used and, above all, on the purposes of the modelers. It is then possible that deidealized models were not adequate for the modelers and, as a consequence, that deidealization may be regarded as counterproductive in a given context. All this depends on the problem in question and the desired solution. Accuracy and precision, for instance, are not always desirable. Every physicist or engineer knows that building a relativistic model of the collision of two cars to learn how to improve the safety of the interior seats is hardly convenient. As a general rule, an excess of theoretical accuracy is counterproductive in practical contexts. Even when a high level of accuracy is desired, it could be sacrificed in order 19 Of course, models can be applied to different domains of phenomena and used for different purposes than those intended by the original modelers. This is an interesting issue I cannot address here.

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to gain other virtues, such as simplicity or expediency. In conclusion, a deidealized model is not always useful and deidealization by itself is not always desirable. There is no agreement among scientists or philosophers on the ultimate aims of science. For that reason, it is difficult to determine how the construction of idealized models and their possible deidealization may contribute to reach those aims. It is generally taken for granted that prediction is one of the main aims of physics and that models that allow for more accurate and precise predictions are prima facie better than other less predictive models. However, for most philosophers of science prediction is not the only aim of science; explanation and understanding are widely regarded as genuine aims too. When several different aims are taken into account, conflicts are unavoidable and no general criteria of choice are available. Should we prefer a visual model of a physical phenomenon over a mathematical model? Should we prefer a more explanatory or a more predictive model of the same phenomena? This is a well-known predicament concerning the epistemic virtues of theories and it is not essentially different regarding models. There are no general and contextfree rules for making decisions about the different possible tradeoffs in building a model. Deidealization is often one of the elements at stake in a tradeoff but it can be discarded or partially sacrificed as a result of a cost-benefit analysis. All realist philosophers would agree on the fact that truth is the ultimate aim of science and that our theories and models are capable of approximating the truth to some degree. Although the concept of truthlikeness has been refractory to a precise elucidation, there is an appealing intuitive sense in which we can say that deidealized models provide better approximations to a true description of the realworld phenomena than highly idealized models. The physical pendulum is again a good example: we surely acknowledge that, if we assume that the model intends to describe the behavior of real pendula in non-controlled environments, deidealized models are better approximations to it than the ideal pendulum or the compound pendulum models. There are surely many other examples of gradually more approximate models, but, as we have seen, it is not possible to generalize this situation to all scientific models. Models that cannot be deidealized are not able to improve their degree of approximation to the phenomena, whatever the way the concept of approximation be understood. If we take description, explanation, and prediction of the phenomena as the fundamental aims of science, we can determine in many cases that some models provide richer descriptions, better explanations, and more accurate predictions of those phenomena than other models. Sometimes, deidealized complex models are descriptively and predictably superior to simpler and more idealized models. On the other hand, if we were to endorse a strong realism according to which science aims at providing us a sequence of progressively better approximations to the truth concerning the things in themselves, we would not be able to determine that deidealized models approach that aim. We cannot compare our models to the world in itself nor have an idea of how a complete representation of the phenomena could be.

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For those reasons, we are unable to estimate how far from the absolute truth are the representations provided by our best models.20

4.8 Conclusion Let us revisit the three questions posed at the beginning of Sect. 7. The first question, concerning whether all scientific models can or cannot be deidealized, does not admit a general answer. We have clear examples of models in which deidealization is possible and has been effectively accomplished, and we have some examples of models that apparently cannot be deidealized. Up to now, there is no general argument to the effect that all models represent their targets holistically in such a way that they do not permit deidealization even in principle. We may suspect that very complex models are mostly not deidealizable but in many cases that may be the case simply because we do not know how to do it. As I have said, nothing follows about the impossibility of deidelization from the fact that some models are extremely complex or from the fact that some idealizations are not easily reversible. Consequently, the question of deidealization must be examined case by case in order to determine whether a given model can or cannot be decomposed into parts that can deidealized one by one. As to the second question, the answer is that sometimes we know how to deidealize a model and sometimes we do not know how to do it. For different reasons, deidealizing may be a complex and difficult process we are unable to address with the mathematical or practical tools at our disposal. It is not possible to ascertain a priori whether or not we will be able to know how to deidealize a given model in the future. Like science itself, the development of this specific knowledge is intrinsically unpredictable. As to the third question, even when the deidealization of a model is possible and practically feasible, it should be put into perspective. Deidealizing is not an end in itself but rather a means to other ends, such as improving the explanatory or predictive capacity of a model. Idealized models are built with a definite purpose and deidealization sometimes may frustrate that very purpose. For instance, if a model was built in order to achieve mathematical tractability, a mathematically intractable deidealization of that model would be overtly counterproductive. Simplified and approximate models are built mostly due to the complexity of the modeled phenomena and our limited epistemic capacities. These are constraints we cannot modify at will. Deidealizing a model will neither simplify the phenomena we face in our experience nor will enhance our epistemic capacities. What we do obtain from models is a limited and partial epistemic access to complex phenomena. Of course, our knowledge of the phenomena can be improved and, as a consequence, we are enabled to build better and richer models of those phenomena. 20 This point would need some more elaboration but it is not my purpose to discuss here the vexed questions of realism and truthlikeness.

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However, we will never obtain a model lacking any idealization. Models are, by definition, idealized human constructs, and all deidealized models are still idealizations to some degree. Moreover, the very process of deidealization often consists in introducing all kinds of new approximations, as the example of the pendulum shows. The idea of an exact and complete representation of the phenomena cannot be maintained even as a regulative ideal. The aim of deidealization is not to approach a perfect representation. In practice, a model is deidealized when we believe that this contributes to achieving the purpose for which it was built in the first place or when the model turns out to be inadequate to solve a new or more definite problem.

4.9 Afterword: Idealization without Representation I have conducted my analysis of deidealized models within the standard framework of the representationalist conception of models. In most of its formulations, this conception assumes -usually in an implicit way- that models aim at providing us with approximately true descriptions of the real-world phenomena. This stance looks uncomfortable because it seems to endorse a sort of naïve realism. It may be interpreted as if it were assumed that we are able to know the things in themselves and then compare them with our idealized models in order to determine that they do not represent them truthfully. If we drop those realist assumptions, we are not anymore in a position to assert that our models are false or that idealizations distort the reality. What we can do is to say is that models do not represent properly all we know about the phenomena, or even better, all we believe about them. We believe, for instance, that spacetime is continuous and we then say that a model that represents spacetime as discrete (at a given scale) is an idealization. However, we cannot say that such a model distorts the things as they really are because we have no way to know that. The model involves a distortion with respect to our accepted knowledge concerning spacetime or anything else. But the continuity of spacetime is just a hypothesis like any other empirical conjecture whose truth cannot be proven by any amount of evidence. This is commonplace in confirmation theory. The same can be said of models that represent water as a continuum and so on. Perhaps in the far future, we will be driven by the newly discovered evidence to accept that spacetime is discrete and that atoms and molecules do not exist after all. This would imply reinterpreting all the evidence we have collected concerning spacetime or atoms. This reinterpretation would be very drastic, but nothing in our present knowledge could prove that it is impossible. Idealizations then only pose an epistemological problem in the light of some realist assumptions about the world and the way our scientific constructs relate to reality. Idealized models are regarded as problematic because they do not give us undistorted, truthful, representations of the world. However, if we were to endorse a strict instrumentalist stance concerning scientific models, the traditional problem of idealization -the consequences of being a deliberate misrepresentation- vanishes. From this point of view, we could say that models do not represent anything at all

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concerning the real world.21 They are simply useful devices for gaining access and handling the phenomena as they appear to our experience. Idealizations are then only a convenient means to do it, not just for predicting the phenomena, but also to explore or explain them. Simplicity has been always the preferred epistemic virtue for instrumentalists. Models can be regarded as paradigmatic realizations of this virtue: to the extent that they simplify the complexity of the phenomena, they help us to reach one of the main aims of science. From this perspective, idealizing is almost always a virtue and deidealizing may be a vice, if it seriously compromises the simplicity of our models. Deidealizing a model can only be justified if it contributes to more expedient handling of the phenomena. It is pointless to say that idealizations are false or that they distort reality. Science is not in the business of searching the truth or of providing us with realistic or truthful representations of how things are in themselves. This general view of idealization can be endorsed, possibly with some amendments, by many antirealist conceptions regarding the aims of science, including fictionalism (at least in some of its versions) and artifactualism about models. Within an antirealist and non-representationalist framework, idealization is not a problem but rather a problem solver. Acnowledgments I am grateful to Tarja Knuuttila and Juan Redmond for their comments on the first draft of this chapter.

References Baker, G., & Blackburn, J. (2005). The pendulum: A case study in physics. Oxford: Oxford University Press. Batterman, R. (2002a). The devil in the details: Asymptotic reasoning in explanation, reduction, and emergence. New York: Oxford University Press. Batterman, R. (2002b). Asymptotics and the role of minimal models. The British Journal for the Philosophy of Science, 53(1), 21–38. Batterman, R. (2009). Idealization and modeling. Synthese, 169(3), 427–446. Batterman, R. (2010). On the explanatory role of mathematics in empirical science. The British Journal for the Philosophy of Science, 61(1), 1–25. Batterman, R., & Rice, C. (2014). Minimal model explanations. Philosophy of Science, 81(3), 349–376. Cartwright, N. (1989). Nature’s capacities and their measurement. Oxford: Clarendon Press. Cartwright, N. (1999). The dappled world: A study of the boundaries of science. Cambridge: Cambridge University Press. Frigg, R., & Nguyen, J. (2017). Models and representation. In L. Magnani & T. Bertolotti (Eds.), Springer handbook of model-based science (pp. 49–102). Cham: Springer. Frigg, R., & Nguyen, J. (2020). Modelling nature: An opinionated introduction to scientific representation. Cham: Springer. Gelfert, A. (2016). How to do science with models: A philosophical primer. Cham: Springer. Giere, R. (2006). Scientific perspectivism. Chicago: The University of Chicago Press. 21 Certainly,

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Alejandro Cassini is Professor of Philosophy and History of Science at the University of Buenos Aires and Senior Researcher at the Consejo Nacional de Investigaciones Científicas y Técnicas of Argentina. He served for twenty years as editor of the Revista Latinoamericana de Filosofía. He is the author of El juego de los principios: Una introducción al método axiomático Buenos Aires: A-Z Editora, Second edition: 2013. First edition: 2007). He is the editor (with Laura Skerk) of Presente y futuro de la filosofía (Buenos Aires: Ediciones de la Facultad de Filosofía y Letras de la Universidad de Buenos Aires, 2010).

Chapter 5

Scientific Representation as Ensemble-Plus-Standing-For: A Moderate Fictionalist Account José A. Díez

Abstract José A. Díez examines the reasons for claiming that models involve fictions. He opposes the claim that, in order to account for some key features of the practice of modeling in science, such as the existence of unsuccessful representations and also of successful yet inaccurate or idealized ones, it is necessary to accept fictional entities. In resisting such a view, he sketches an account of scientific modeling and argue that according to such account there is no need for strong factionalism, only a weak, unproblematic fictionalist component is needed. Keywords Representations · Inaccuracy · Fictions · Modeling

5.1 Introduction It has been argued that, in order to account for some key features of the practice of modeling in science, such as the existence of unsuccessful representations and also of successful yet inaccurate or idealized ones, it is necessary to accept fictional entities. The goal of this paper is to sketch an account of scientific modeling and argue that according to such account there is no need for strong fictionalism; only a weak, unproblematic fictionalist component is needed. According to the Ensemble-Plus-Standing-For (EPS) account, the practice of constructing a representational model consists in building ensembles and taking some elements/parts/features of the ensemble as standing for other entities. In order to develop a plausible account out of this idea, it is crucial, first, to neatly distinguish between conditions for preforming the representation and conditions for the accuracy Research for this work has been supported by the research project FFI2016-76799-P, Spanish Ministry of Science and Innovation. I want to thank Otávio Bueno, Roman Frigg, Carl Hoefer, Ulises Moulines, James Nguyen, José Luis Falguera, Stathis Psillos and Albert Solé for comments and criticisms to earlier versions of this paper. J. A. Díez (B) Department of Philosophy, University of Barcelona, Barcelona, Spain e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Cassini and J. Redmond (eds.), Models and Idealizations in Science, Logic, Epistemology, and the Unity of Science 50, https://doi.org/10.1007/978-3-030-65802-1_5

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of the representation; and second, to introduce the essentially pragmatic notion of logical-form-in-use (together with other more standard context-dependent features). This chapter proceeds as follows. In the first section I introduce the three main problems of scientific representation that the account aims to answer and clarify some terminology and the implicit parameters in representational sentences. In the second section I present the EPS account in its simple, non-fictionalist version, and make some necessary clarifications about some of its clauses. Finally, I discuss a specific problem that the simple version faces and introduce the moderate, unproblematic fictionalist component to fix it. Before starting, two important caveats are in order. On the one side, the account aims at being both enough monistic and enough substantive: the analysis aims to give necessary conditions that are “nearly sufficient”, i.e. as strong as possible (which implies that given two alternative analyses, the strongest is, ceteris paribus, the best). The goal is to exclude all cases of non-modeling, and include all paradigmatic case of modeling, and to do so providing substantive clauses that explicate the concept, not merely mentioning some minimalist platitudes. In doing so, we try to resist both pluralist and minimalist withdrawals that proliferate nowadays. On the other side, the account is restricted to the representational use of scientific models. Scientific models can serve other goals. They may be designed just to test whether certain empirical hypothesis or theory is coherent without any attempt to tell how reality actually is (e.g. Tobin 1970 ultra-Keynesian model). Or may have a merely predictive function (e.g. some computer simulations, Humphreys 2004; Weisberg 2013; Winsberg 2010), that is constructed to carry out relevant observational predictions in a given field but without aiming that non-observational components represent something actually present in the phenomenon (of course, merely predictive models are representational in a minimal sense, namely they represent the observable phenomena). Some models, that will play a crucial role below, are only partially merely predictive, that is together with elements that aim to correspond to observable predictable phenomena, they have some other components that play a role in the model and aim to correspond to something on-observable in the phenomena, but also include other components that though playing an essential inferential role do not aim to have any representational import (e.g. theBurridge-Knopoff 1967 block model of earthquakes, or the ancient Antikythera Mechanism for the Solar System, cf. Carman et al. 2012; Jones 2017). And there may be other non-representational uses as well, for instance, as heuristic tools such as Maxwell’s vortex fluid dynamical model of the causal relation of electricity and magnetism (Morrison 2015), or some of the uses of models as mediators (cf. e.g. Morgan and Morrison 1999), or as epistemic artifacts (Knuuttila 2011), or as explanatory fictions (Bokulich 2009), or as tools for theory construction (Hartmann 1995) might be (partially) non-representational as well (cf. Peschard 2011 for not-only-representational uses of models). We will then focus only on scientific representational models, but in a wide sense of ‘model’, that is referring to any scientific representans, including theories, engineering graphs, scientific images/drawings, geographical maps, etc. With regard non-scientific representations, although the account may have some application to some of them, it does

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not aim to apply to non-scientific representations in general and must then be taken as confined to scientific cases. The following are some examples of scientific modelings/representantia that, without aim of exhaustion, cover a variety of models one may find in scientific practice that our account aims at applying to: the double-helix, and the triple-helix models of DNA; the ideal pendulum model; the kinetic theory of gases and its billiard ball analogy; the phlogiston and oxygen theories of combustion; Ptolemy’s, Brahe’s and Copernicus’ models of the Solar System; Newton’s and Einstein’s theories of gravitation; the Lotka-Volterra predator-prey model; the above mentioned BurridgeKnopoff block model of earthquakes; the Bak-Sneppen model of species replacement; the computational model of the mind; Thomson’s, Rutherford’s and Bohr’s models of the atom; XIX century corpuscular and wave theories of light; the pipesreservoirs-fluid Philips-Newlyn hydraulic model of economic dynamics; the numerical representation of magnitudes in Representational Measurement Theories; the scale Mississippi River Basin model; the scale aircraft wind-tunnel simulation; the use of Drosophila as a model of general genetic phenomena; the experiments with mice for modeling human pain or reactions to drugs; the Mercator and Gall-Peters two dimensional maps of continental land. This is only a small sample of the vast plethora of models and representations in scientific practice. They cover a whole variety of scientific models/representations: material models; theoretical models; qualitative models; mathematical models; scale models; phenomenological models; analogical models; idealized models; organism models; in vivo and in vitro models; reductive models; etc. The goal of the EPS account is to resist the current pluralistic attitude and present a sufficiently unified and substantive account that applies to this plethora of scientific modeling explicating what they all have in common.

5.2 Settling the Analysis EPS aims to answer the following three key problems of scientific representation: (PD) Problem of directionality: In virtue of what does the direction of the representation run from model to target? (PE) Problem of existence: In virtue of what does the model represent its target? (PS) Problem of success or accuracy: In virtue of what, a model that represents a target does it successfully/adequately? There are other problems related to modeling that one may find in the literature (see e.g. Frigg and Nguyen 2017a for a comprehensive overview) but we will focus on these three. In particular, with regard the problem of ontology (What kind of entities are models?) EPS does not have a particular ontology to propose, the account just puts some constraints to possible ontologies.

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In answering these three problems, and specially PE and PS, it is going to be essential for the account to distinguish between conditions of performance or existence of the representation (“P-conditions” for short) and adequacy or success conditions (“S-conditions”). In analogy with speech acts (asserting, commanding, questioning, …), modeling may fail in two different senses. In one sense, one may fail in performing it. Mere intention does not suffice (contrary to Callender and Cohen 2006): one may have the intention to represent something else and fail (as one may have the intention of asserting, and not even asserting, e.g. because one did not utter a grammatical utterance). And, given that one has succeeded in performing a representation, then one may fail in a second sense, namely the representation does not successfully/adequately represent its target (as one may do an assertion and failing in not being true, or do a command and failing in not being obeyed). So, for the representation to be performed, certain conditions must be met (problems of directionality and existence above). And once the representation has been properly performed, additional conditions must be satisfied for the representation to be successful in the second sense (problem of success above). Models/representations, as speech acts, have certain goals constitutively associated to them, and the model succeeds in this second sense if its constitutive goal is satisfactorily reached. If one accepts this,1 then the analysis has to come in two subsequent steps. First, one has to provide an answer to PE, and then to PS (including as precondition that the performance conditions have already been satisfied). In order to avoid terminological confusions, let me clarify the use of some crucial expressions I will be using: – I will use ‘model-description’ to refer to the linguistic items (if any) used by the modeler to describe the model or to instruct its material construction (if the model is material): ink-words in papers and text-books, lists of instructions, written equations, etc. (I say “if any” because some models may come without physical signals/symbols but directly described-instructed “by the mind”). – I will use ‘model’ to refer to what the model description describes or instructs, which according to EPS, is basically an ensemble of entities together with a standing for relation. There is more to be said (see below) but by now it suffices to say that, for instance, in material models such as the physically constructed double helix DNA model, the ensemble is the complex physical entity constructed. Or in a mathematical model the ensemble (or one subpart of it) is a certain mathematical function with some properties such as being over the real numbers, being differentiable, etc. – I will use ‘target system’ to refer to the phenomenon in nature that the model aims to represent/account for. Here we have to take a decision. One may individuate the target as (allegedly) constituted by the entities the model attributes to it. If we 1 One

might not, for instance defending that no-representation simply is a limiting case of misrepresentation, that is that mere will suffices for representing and if one does it completely badly then we can talk of no-representation. If I understand them well, this is the stance taken by Chakravartty (2009) and Ducheyne (2008), for whom some degree of success is necessary for the existence of representation.

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do this, when a model attributes to the target an entity that actually does not exist (phlogiston, caloric, ether and other similar cases), then there is no target, the model is targetless (thus, strictly speaking, the model does not attribute the nonexistent entity “to the target”, since there is no target either). The main problem of this option is that this forces us to use what I take to be a counterintuitive talk: on the most common talk both phlogiston and oxygen models have the same target, they both represent (differently) combustion-calcination; the same in other cases (caloric and kinetic theories have the same target, they both represent heat phenomena; ether and electromagnetic models have the same target, they both represent light; etc.). I take this is the most common talk, which is worth to preserve and that the alternative option cannot naturally2 preserve. I will then use ‘target’ to refer to the previously identified and individuated natural phenomenon (by, say, features t1 , .., tn ), towards which the model is addressed;3 for instance, though phlogiston model attributes phlogiston to the target, and oxygen model attribute oxygen to the same target, neither phlogiston nor oxygen individuate the target (they may, if the model turns out to be correct, “be in the target”, but the target is not individuated by them). – Finally, I will use ‘model content’ to mean what the model “expresses”, the alleged state of affairs that according to the model exists in the world (for this it is crucial to bear in mind that not all the components of the models are intended to be projected to the world, more on this below). I do not use ‘refer’ here, for it is one important feature of this account that when the model fails (e.g. phlogiston) there is no entity/thing in the world, different from the model, to which ‘model content’ refers. I’ll discuss this in the last section. We have been talking as if the representational relation involved only two parameters, model and target. Nevertheless, as many have emphasized (prominently Giere 2004, 2010, and Van Fraassen 2008), this is a simplification: “M represents T ” makes implicit reference to other parameters. In my account, at least four other additional parameters: Subject S. Representations do not obtain in abstracto, they are the result of a scientific practice performed by particular individuals or communities. We use the variable S to refer to the relevant (individual or collective) subject that performs the representation, i.e. for the representator: “S uses M to represent T ”. Respects R. Given a subject, a model and a target, the assessment of the model as successful or adequate often depends on certain respects, the features/properties 2 One

might preserve this talk making the target be the phenomenon plus what “the true model” attributes to the world behind the phenomenon; the target would be the content of “the correct model”, whichever it is. I think this option is not “natural” since, given the uncertainty about truth, we would never know what the targets are, which I find counterintuitive. The only thing we could say with certainty is that the target is the phenomenon “together with whatever is behind it”, which I do not think is very illuminating, nor that it provides any benefit that my own, simpler talk does not. 3 It follows that as far as a target exists, its individuative features t , …, t also exist; how these 1 n features are themselves individuated (partially theoretically?, fully observationally?), is left here as an open issue.

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of the model that are taken as relevant for the representation. As the bidimensional continental maps example above exemplifies, the same model may be adequate taking into consideration some of its respects or properties but inadequate if we take into consideration others. The Mercator model is adequate with respect the shape, but not with respect the area dimension; while for the Gall-Peters the opposite is the case. It is then relative to some of its respects/properties that are taken into consideration, and not others, that a model is assessed: “S uses M to represent T in respects R (of M)”. Purposes P. Respects come with purposes. If the shape-respect of Mercator map is relevant for assessing it, it is so relative to certain purposes, in this case obtaining faithful information about a continent’s real shape; and analogously for the GallPeters map and the purpose of obtaining faithful information about the dimensions of continental land areas. And, as it happens in this case of bidimensional continental maps, it may very well be that there is no model adequate for all purposes: If a bidimensional map correctly represents continental shape, it does not correctly represent area dimensions; and the other way around. “S uses M to represent T in respects R for purposes P”. Context C. Finally, the context of use C is essential for the assessment. Representations are not all/nothing successful; success comes in degrees. Given the same subject, model, target, respects and purposes, a very stringent context may assess the model as unsuccessful if the degree or level of accuracy in which the relevant respects satisfy the purposes is below the demanding limit considered desirable by the context; while in other, less demanding context, the same model may be regarded as sufficiently adequate in the same respects for the same purposes. Context C is then essential in determining the degree of accuracy (and maybe other factors) for the assessment: “S uses M in C to represent T in respects R for purposes P”.

5.3 The EPS Account: A Simple, Non-Fictionalist Version With the above preparatory notions at hand, we can now introduce our EPS account. In this section we present a simplified version that elaborates the basic intuitions but omits the fictionalist component, and in the next section we will justify and introduce the fictionalist complexization needed. As we have emphasized, the problems of existence and success must be neatly distinguished, thus accordingly EPS comes in two steps. The first step accounts for the existence or performance of representation (EPS-P) in terms of conditions that must be met for an agent that intends to perform a representation to actually perform it. In this regard, as said above, mere intention does not suffice; intention suffices for directionality, but (as in speech acts) other conditions must be met for properly performing the act of representation. These conditions make use of the key notions of “ensemble”, “standing for” and “logical congruency constraint”. First, a model is an ensemble. An ensemble is simply a bunch of entities (individuals, properties, relations, …) articulated/structured in a specific manner. For

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instance, two individuals exemplifying two first order properties and related by a first order binary relation; or three first order properties related by a second order triadic relation. Notice that for a bunch of entities to be able to form an ensemble they must have the appropriate relative logical forms; two individuals and a ternary first order relation, or one individual with a second order monadic property, cannot form ensembles. Whether there is a realm of “simple” entities with a logical-form-in-se (as in the Tractatus) is an open question, but EPS does not depend on it since in practice we always face entities that very likely are not simples; it is in the act of representing that we take/use them as-simples and attribute a logical category to them, what I call a logical-form-in-use. A red circle may be taken as simple (e.g. as one element of the ensemble consisting in the London tube map), or it may be taken as an ensemble itself (consisting for instance in one individual instantiating two monadic properties; so taken this ensemble may represent e.g. that King Arthur is fat and honest). So, in representing, the subject takes some entities as simples, attribute certain logical category to each one and constructs an ensemble out of them. Second, the model is an ensemble plus a standing for relation. That is, an ensemble in which the subject takes some of its (taken-as-simple) parts (the respects R) as standing for other (usually different) entities. For the representation to be performed, the bunch of stood-for entities must be able to form an ensemble themselves, that is they must have the proper logical categories (also in-use) to be “ensembleable” to each other. This is easily satisfied if the entities in M stand for entities of their very same logical category. But this is too stringent. For instance, there is nothing that conceptually excludes that the ensemble consisting of an individual instantiating a first order monadic property represents a second order dyadic relation instantiating a third order monadic property. If we relax the same-logical-category condition for this more flexible constraint we obtain the following logical-congruency constraint: the respects in M must stand for entities that “preserve their logical congruency”. We cannot give here a precise definition of logical congruency, but to our present goals this informal characterization suffices; it excludes, for instance, that in an aimed model that is an ensemble formed by two individuals instantiating a binary first order relation, the two individuals stand for first order properties and the first order binary relation stands for a second order triadic relation. Such ensemble with such standing for relation violates the congruency constraint and therefore the aimed representation is not even performed. With these tools at hand we can now introduce the first step of our explication, the conditions for the performance of a representation: (EPS-P; simple version) S uses M to represent T in respects R with purposes P in context C: (i) (ii)

S has in C the intention-of-representing T (individuated by features t 1 , .., t n ) with M in respects R, for purposes P; in C, M is an ensemble of entities, those of which are respects in R are such that S intends that they stand for other entities satisfying the logical congruence constraint;

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(iii)

in C, all target individuating features t 1 , .., t n are stood for by respects R of M.

Some aspects of this characterization are worth commenting. First, representing “for the whole” is completely different from representing “for its parts”: the relation of M representing T is of a complete different nature than, and derivative from, the standing for relation of its parts/respects. Second, clause (i) explicates directionality: for the direction of the representation be from M to T the mere intention of S suffices.4 As the hyphenation connotes, this S-intention-to represent is a primitive concept, not analyzable in terms of “intending” and “represent” on pain of circularity. This intentionality suffices for directionality, but as announced it does not suffice for existence: subject’s intention, though necessary, is not sufficient for representation, two additional conditions must be met. Clause (ii) demands that M is taken by S in C as an ensemble (thus formed by taken-as-simple parts with some appropriate logical-forms-in-use) and that S intends that the relevant components (respects) of M stand for other entities respecting the logical congruency constraint.5 But here we have a complication. We know that many times S may intend that some part of M stands for something that actually does not exist, e.g. phlogiston, caloric, and the like. This problem is solved by talking of S intending that the respects stand for (existing) entities instead of simply saying that the respects must stand for entities. Standing for is factive, if x stands for y both x and y exist. Then the condition cannot be that respects stand for other things since many times some of them do not stand for anything. That is why (ii) talks, not of standing for, but of S’s intention that respects stand for other entities, and in cases like phlogiston this intention is present, though fails in the target lacking the intended entities (this failure, though, is not a failure in performing the representation, but a failure of adequacy). The final clause (iii) accounts for the fact that we cannot represent (not even wrongly) if the standing for relation does not appropriately connect the respects in M and the individuative elements of the target towards which, according to (i), S intends M to be addressed. Suppose that I intend-to-represent the killing of Caesar. For doing that, I construct on top of my table an ensemble made of two medium size pieces of wood and a little one of plastic, taken as individuals, and certain (first order) spatio-temporal relations. But then, I intend the standing for relation as follows: one of the wood pieces stands for Maradona, the other for the 1986 English goalkeeper, the plastic piece for the ball in the quarter-final, and the spatiotemporal relations on 4 Note that, for this intention to be present, the target T must exist, the target may lack many of the features attributed by the modeler, but not all. So, according to our use, one cannot intend to represent no-thing. A “representation” of say, Atlantis, who is acknowledged as non-existing, is not a representation in pour sense (unless what one wants to represent, the target, is intended to be a fictional entity as fictional, or an abstract conceptual entity). 5 Ensemble is a conceptually (not metaphysically) primitive notion, only introduced by examples; it is not explicated in other terms such as set-theoretical structure or the like. This avoids the problems of set-theoretical structuralism in representation (Frigg 2003), but, admittedly, at the cost of taking it as primitive (for a discussion of models as representational structures, cf. Morrison 2008 and Thomson-Jones 2011).

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the table for spatiotemporal relations in the Mexican stadium in Mexico City (this standing for relation satisfies the logical congruency constraint). I do not think that, given that I have intentionally addressed the model as targeting the killing of Caesar, we can say that I am nevertheless representing it, although very wrongly. If the objects that the representator intends to be stood-for do not belong to representator’s intentional target, the representator is “objectually” incoherent and, even if logically congruent, the representation is not successfully performed. In order to properly represent, our selection of a target and our intended (target-individuative) stood for entities must be coherent, the later must belong to the former. After the explication of the existence of representation, we are now in a position to give as a second step the analysis of the success or adequacy of a (n existing) representation: (EPS-S; simple version) S successfully uses M to represent T in respects R with purposes P in context C: i. ii. iii. iv.

S uses M to represent T in respects R with purposes P in context C (according to EPS-P); all entities that respects in R aim to stand for, exist; entities that respects in R stand for, behave to each other in T C-like their corresponding respects in M ; purposes P are achieved, and are so in virtue of (iii).

EPS-S combines the grain of truth of previous inherentist and functional accounts. I qualify as inherentist the accounts according to which the representation obtains in virtue of the inner natures/structures of model and target and of how they relate to each other. Although the proponents are unclear and many simply ignore the distinction between existence and success of the representation, these accounts have some plausibility only as accounts of success, not of performance. Inherentist accounts come in different versions, depending on the relation between M and T that explicates success, from similarity (e.g. Giere 1988, 2004, 2010; Weisberg 2013) to several kinds of morphisms or structure preservation (Suppes 1967, 1974, 1988; Sneed 1971; Balzer, Moulines and Sneed 1987; Mundy 1986; Swoyer 1991; Bueno 1997; Bueno, French and Ladyman 2002; Da Costa and French 2003; Bueno and French 2011; French 2014; Van Fraassen 1980, 2008; Contessa 2007). I cannot enter into the details here, but all these versions suffer from being either too weak (as similaritivism in general and some weak partial homomorphism versions) or too strong (as many morphism versions). In general, if the proposed relation is too weak, it does not exclude cases we want to exclude; and if it is too strong, it excludes cases we do not want to exclude. There is, though, a grain of truth in them, namely, that some communality-of-behavior relation must obtain in a given particular context for the representation being successful (although, against any particular inherentist proposal, no particular such relation is demanded in absolutely all contexts). The moral is that the context C specifies the particular communality-of-behavior relation relevant in C. This is what clause (iii) expresses.

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With regard functionalisms, the main ingredient in their analysis of scientific representation has to do with the function that the model performs with respect to the target, namely, some kind of information transference or surrogative reasoning (some representatives are Suárez 2004, 2010, 2015; Contessa 2007, 2011; Ducheyne 2012; and also, to some extent, some advocates of the representation-as account, e.g. Elgin 2010, Van Fraassen 2008, Frigg and Nguyen 2016, 2018; and some fictionalist accounts as well, such as Frigg’s 2010 relevant fictional information account). Although many authors do not distinguish here between existence and success either, one must distinguish them also here. It is plausible to regard the generation of surrogative reasoning as a condition for existence, while the correctness of the drawn inferences as a condition for success. EPS-P incorporates this functional condition of existence in the fact that the satisfaction of EPS-P conditions habilitates for transferring (correctly or wrongly) information from M to T. As for success, EPS-S(iv) is the clause that incorporates the crucial functional condition that the function/purpose is actually satisfied. Note, though, that this clause goes much further, or deeper, than regular functionalism, for it claims that the satisfaction of the purpose must be in virtue of there actually be a communality of behavior. Thus EPS-S does not combine structural and functional conditions by mere conjunction, it does it in a deeper manner, namely by showing that the satisfaction of the function is grounded in the communality of behavior.6 As for the other clauses, (i) is simply the precondition of performance, and (ii) demands that the respects in M that S intends to stand for entities in T, actually do so, thus that the intended stood for entities exist. This concludes the summarized presentation of EPS. Notice that once representational success is explicated, misrepresentation is immediately explicated as unsuccessful representation: S misrepresents T with M in respects R with purposes P in context C if and only if: (i) S uses M to represent T in respects R with purposes P in context C; and (ii) S does so unsuccessfully. We then have three different sources of misrepresentation. The representation may fail due to the non-existence of a postulated entity. Or all postulated entities (together to features t 1 , …, t n of T ) exist, but they do not behave to each other C-like the corresponding constituents of the model behave in the model. Or, finally (and admittedly unusual), the postulated entities exist, they behave C-like the constituents of the model in the model, but either (due to unexpected factors) the purposes are not reached, or they are reached but not in virtue of the communality of behavior. Finally, as many have emphasized (e.g. Elgin 1996, 2010, Thomson-Jones 2005, Godfrey-Smith 2006, Morrison 2008, Bokulich 2009, Cartwright 2010, Toon 2012, Frigg and Hartmann 2012), representations, even when successful, are almost always inaccurate. Inaccuracy is an essential feature of successful representations that any analysis must account for. Inexactness, distortion and idealization are essential to modeling: perfect spheres, infinitesimal distances, frictionless planes, point-masses, 6 Some functionalist, e.g. Suárez 2015, also claim that the function must be performed in every case

due to specific relation between M and T, but explicitly denies that this is part of the concept of representation.

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infinite particle collections, perfectly isolated populations, and the like. Models include these individuals and properties or relations, but the modeler does not aim that the parts of the target for which they stand have exactly these features; it is aimed only that they approximate them to a certain degree (determined by C): almost perfect spheres, very small distances, very small massive bodies, indefinitely large finite collections, almost isolated populations, etc. The analysis must do two things in this regard. First, to include an element that makes room for this essentially gradual approximative feature of successful models. Second, to show that there are formal tools for (re)constructing models that make models the kind of entities for which the degrees of satisfaction talk formally makes sense. With regard the former, as we advanced in Sect. 1 introducing the context of modeling C, in EPS-S one of the roles of the context is to determine the degree of accuracy required for the purposes in point. For instance, if we model parabolic shot, the degree of accuracy sufficient in one context (e.g. football match) may be insufficient in other, more demanding context (e.g. war bombing). In this respect, there is no substantive conceptual constraint applicable across all models; the only general, and almost empty conceptual constraint, as in our analysis, is that every context determines its relevant degree of representational accuracy. With regard the latter, EPS-S simply assumes in each case the formal tool that the philosopher considers the best for reconstructing idealized models, either topological (e.g. Balzer, Moulines and Sneed 1987), or analytic (e.g. Nowak 1980), or algebraic (e.g. Ducheyne 2012), or other. Notice that this inaccuracy as idealization/approximation is different from inaccuracy as abstraction, which in our account is explicated with the notion of respects, since abstracted features of the target are simply features for which there are not respects in the model.

5.4 EPS: The Complete, Moderate Fictionalist Version The issue of inexactness brings us to the feature that calls for the introduction of the moderate fictionalist element announced. Remember that respects of M are the components of the model that have a role in the representation. For instance, in the material double helix model of the DNA, shape properties (helicoidal, pentagonal, hexagonal, …), color of helix salient pieces, other straight pieces uniting the former ones, and other components, are elements of M that are respects in R, aimed to stand for other elements (individuals, properties, relations, …) in the target, the DNA; other components, such as the specific material constitution of the pieces (e.g. metallic instead of plastic) are not such respects since they are not aimed to play any role in the representational function at all. In most cases, as in this one, all Mrespects in R are aimed to stand for entities in the target. But this is not always the case. As we quickly mentioned in the introduction, some models are only partially representative, meaning that the modeler constructs the model using some parts of the model as relevant respects for obtaining the information within the model that she is willing to transfer to the target, but some of these relevant parts/respects of the model are not intended to correspond to anything in the target. In some occasions, respects

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in R, i.e. elements of M that are relevant for the informational function, are not aimed to correspond-to/stand-for anything in the target, as in the Burridge-Knopoff model for earthquakes or the astronomical Antikythera Mechanism mentioned above. In this later case, for instance, the model includes, relevantly for the calculations, a division of a circle in 360 sections of different magnitude each, with no intention to correspond each division to anything in the skies, this is a completely instrumental part of the device, yet, to repeat, essential for the calculations. For this kind of cases, the second condition EPS-P (ii) of the simple version, namely that all respects are aimed to stand for (satisfying the logical congruence constraint) entities in the target, is not satisfied, but nevertheless we still have a representation. This is also a kind of inexactness, though different from idealization and abstraction. Some could be tempted to argue that this is actually a case of idealization, but with zero or null degree of approximative correspondence. I do not think this is a plausible strategy, since the difference between zero degree of intended correspondence and no intended correspondence at all is merely verbal. In this kind of cases we simply have some respects with no intention by the modeler that they stand for something else in the target. We need then to weaken such condition including the possible presence of some non-representational respects, respects that are not really aimed to stand for something. How to weaken such condition, though, is not an easy issue. As we have mentioned, they are elements in M that, although not intended to correspond to anything, are nevertheless used in (in Hughes’ 1997, 2010 terminology) demonstrating the information we want to transfer to/interpret in the target. Then, they are relevant modeling elements, respects in our terminology, of M in the sense of being utilized in the demonstration, but not in the sense of being intended to correspond to something in T. How can they perform their role without being projected? The only manner I think this problem may be faced is going fictional in some way, and accepting the consequence that these models have something that makes their representational success partially mysterious. The model does not aim the non-projected element to stand for anything (not even approximately), nevertheless it proceeds as if it were projected. It does not stand for; it is not even intended that it stands for; the model just proceeds as if the element were aimed to be projected although it is explicit that it is not so aimed. In this sense its projection is just a pretense, a fiction.7 Admittedly, as we will discuss below, the representational success of the whole model, based on the modelized behavior of this fictional element with the other elements that are really intended to stand for things in T, is somehow mysterious. Let us introduce now the final account of existence including this fictionalist complexization: (EPS-P) S uses M to represent T in respects R with purposes P in context C:

7 To

avoid confusion, notice that this fictional as-if component has nothing to do with the representation-as mentioned in Elgin 1996, 2010, Van Fraassen 2008 or Frigg and Nguyen 2016.

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S has in C the intention-of-representing T (individuated by features t 1 , .., t n ) by M with respects R, for purposes P; in C, M is an ensemble of entities, those of which are respects in R, are such that S intends, actually or fictionally (at least one respect actually), that they stand for other entities in T satisfying the logical congruence constraint; in C, all target individuative features t 1 , …, t n of T are stood by M respects in R.

And the corresponding amendment for success: (EPS-S) S successfully uses M to represent T in respects R with purposes P in context C: (i) (ii) (iii) (iv)

S uses M to represent T in respects R with purposes P in context C; all entities that respects in R actually (not fictionally) aim to stand for, exist; entities stood for respects in R that S actually aims to stand for, behave to each other in T C-like their corresponding respects in M; purposes P are achieved, and are so in virtue of (iii) (respects, if any, that only fictionally aim to stand for, matter for demonstration in M but not for behavior alikeness in T ).

Admittedly, the fictionalist element introduced in EPF-S is somehow tricky. This element seems to undermine the proclaimed benefits of the syncretic clause (iv). If there are cases of bona fide representations in which some model respects are not aimed to stand for anything in the target, but are nevertheless essential for the “behavioral information” of M that is projected over T, and some of such models are intuitively successful, how can it be that the fulfillment of the purpose is due to the alikeness in behavior if there are respects, i.e. informationally relevant parts of M, that are not really aimed to correspond to anything in T ? How can there be similarity in behavior if for that respects there is nothing in T about which we can ask whether they behave alike such respects in M? Clause (iv) demands that the purposes are obtained in virtue of the behavior of the T-elements being C-like the behavior of the M-elements (deployed in gathered the information) that stand for them, but, how can the purposes be reached in virtue of a communality of behavior that presupposes a parts-correspondence that at least partially does not obtain? This in a nutshell is the problem that the presence of non-representational elements in representational models poses for EPS-S. The only coherent way of facing this problem is, as in EPS-S, to restrict the alikeness in behavior condition in (iii) to the non-fictionally aimed to stand for respects. Does this undermine the analysis? I do not think so. Admittedly, the representational success of the whole model in these cases, based on the modelized behavior, including the behavior of such fictional respects with the other elements that are really intended to stand for things in T, is somehow mysterious. The “in virtue of” part of clause EPS-S(iv) must be read, with some oddity, counterfactually, including this fictional reading of the non-representational respects of M; that is, as if such respects were also aimed to correspond to something in T. I do not think this is

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a specific deficiency of this analysis, but rather the analytical reconstruction of how things are in these quite unusual cases of successful models with some non-projected respects. Every analysis has to face these odd cases with (not merely idealized but completely) fictional components; and either you get silent on them but by getting also silent about what grounds success in general (Nguyen 2016), or you explicate the grounds of success and witness these rare cases as partially meeting the conditions only fictionally. I do not see a tertium datur, and I take that the latter is better than the former. Leaving aside this problem, that I claim partially representational models pose to all accounts, the fictionalist component introduced is relatively moderate an unproblematic. This account is fictionalist in including in the concept of representation the notion of fictionally intending that something stands for something else, which I take to be metaphysically innocuous. But, contrary to other, stronger fictionalist accounts (e.g. Godfrey-Smith 2009; Cartwright 2010; Contessa 2010; Frigg 2010; Toon 2012) this account does not take models as fictions (in any of the several senses involved in this literature), and accounts of inexactness and idealization without substantive factionalist machinery. It is in this sense that the account sketched here is only moderately fictional, and thus, other things equal, preferable to other strongest fictionalist accounts. In particular, there is no need of any fictional component with regard the content of models in cases of misrepresentation. When a model misrepresents (e.g. phlogiston, caloric, …), there is nothing concrete in the world, no concrete fact or state of affairs, that corresponds to the model (in all its M-respects and behavior). “What” then does the model express?, what entity its “content” is? According to EPS, no-thing, nothing at all; of course, as just said, no concrete state of affairs, but no-thing else either. In particular, nothing fictional either. According to one interpretation of fictions, fictional contents are abstracta. According to other, fictional contents are no-entities, but the imaginings that result in games of make-belief; that is, fictional texts exist, but what they express simply does not. According to EPS, there is no need to go either way. When the model represents but fails, there is no fiction in any interesting sense. What happens is simply that the ensemble plus the standing for do not have correspondence in the world, period. In a Tractarian mood, model’s content “is” in the ensemble-plus-standing for; when the model is correct, there is a concrete state of affairs, but when it is incorrect, there is (besides the target) no-thing else, neither concrete, nor fictional, abstract, or whatsoever.

5.5 Concluding Remarks To build a model consists in picking some “pieces” and assembling them together constructing a structured complex. The pieces/components are taken-as-simple, that is, even if complex themselves, its inner complexity does not matter for the modeling effects. In being ensembled/structured to each other, the pieces acquire a logical category in-use. This act of constructing an ensemble must come with two other actions, intentional in nature. First, the intention to use the constructed complex to represent

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a specific target phenomenon previously individuated by some detached features. Second, the (in general actual, but as we have seen, maybe for some pieces merely fictional) intention of taking the relevant pieces of the ensemble (its “respects”) as standing for pieces in the target (including some attributed to the target but not individuative of it). For succeeding in performing the representation, these two intentions must be logically and materially coherent: there must be logical coherence between the logical multiplicity/combinatoriality given to the pieces of the model and the pieces of the target, and there must be material coherency between the intended target and the intended stood for entities, which (the existing ones) must belong to the intended target. These conditions secure the existence of a representation of a specific target. For this representation to be (approximately) adequate/correct, additional conditions must be satisfied: the (not fictionally) intended stood for entities must exist, and behave to each other C-as (similarity of behavior determined by the context) its originals in the model; and the modeling purposes are achieved in virtue of such communality of behavior. The complete elaboration and defense of this account vis a vis its inherentist and functional competitors cannot be done here, but it is my claim that it fares better than them, and, though it leaves a strong role to the pragmatic context, EPS provides an enough monistic and substantive explication of the concept of scientific representation, being extensionally adequate and intensionally explicatory at the same time. And it does so without the cost of any strong fictionalist component, just with a weak, and unavoidable, innocuous fictional-intention element.

References Balzer, W., Moulines, C. Ulises, & Sneed, J. (1987). An architectonic for science: The structuralist program. Dordrecht: Reidel. Bokulich, A. (2009). Explanatory fictions. In M. Suárez (Ed.), Fictions in science: Philosophical essays on modeling and idealization (pp. 91–109). London: Routledge. Bueno, O. (1997). Empirical adequacy: A partial structure approach. Studies in History and Philosophy of Science, 28(4), 585–610. Bueno, O., & French, S. (2011). How theories represent. The British Journal for the Philosophy of Science, 62(4), 857–894. Bueno, O., French, S., & Ladyman, J. (2002). On representing the relationship between the mathematical and the empirical. Philosophy of Science, 69(3), 452–473. Burridge, R., & Knopoff, L. (1967). Model and theoretical seismicity. Bulletin of the Seismological Society of America, 57(3), 347–371. Callender, C., & Cohen, J. (2006). There is no special problem about scientific representation. Theoria, 21(1), 67–85. Carman, C., Thorndike, A., & Evans, J. (2012). On the pin-and-slot device of the Antikythera mechanism, with a new application to the superior planets. Journal for the History of Astronomy, 43(150), 93–116. Cartwright, N. (2010). Models: Parables vs fables. In R. Frigg & M. Hunter (Eds.), Beyond mimesis and convention: Representation in art and science (pp. 19–32). Berlin-New York: Springer. Chakravartty, A. (2009). Informational versus functional theories of scientific representation. Synthese, 172(2), 197–213.

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Contessa, G. (2007). Scientific representation, interpretation, and surrogative reasoning. Philosophy of Science, 74(1), 48–68. Contessa, G. (2010). Scientific models and fictional objects. Synthese, 172(2), 215–229. Contessa, G. (2011). Scientific models and representation. In S. French & J. Saatsi (Eds.), The continuum companion to the philosophy of science (pp. 120–137). London: Continuum Press. Da Costa, N., & French, S. (2003). Science and partial truth: A unitary approach to models and scientific reasoning. Oxford: Oxford University Press. Ducheyne, S. (2008). Towards an ontology of scientific models. Metaphysica, 9(1), 119–127. Ducheyne, S. (2012). Scientific representations as limiting cases. Erkenntnis, 76(1), 73–89. Elgin, C. (1996). Considered judgement. Princeton: Princeton University Press. Elgin, C. (2010). Telling instances. In R. Frigg & M. Hunter (Eds.), Beyond mimesis and convention: Representation in art and science (pp. 1–17). Berlin-New York: Springer. French, S. (2014). The structure of the world: Metaphysics and representation. Oxford: Oxford University Press. Frigg, R. (2003). Re-presenting scientific representation. Ph.D. Dissertation. London: London School of Economics and Political Science. Frigg, R. (2010). Models and fiction. Synthese, 172(2), 251–268. Frigg, R., & Hartmann, S. (2012). Models in science, Stanford Encyclopedia of Philosophy. https:// plato.stanford.edu/entries/models-science/. Frigg, R., & Nguyen, J. (2016). The fiction view of models reloaded. The Monist, 99(3), 225–242. Frigg, R., & Nguyen, J. (2017). Models and representation. In L. Magnani & T. Bertolotti (Eds.), Springer handbook of model-based science (pp. 49–102). Dordrecht-New York: Springer. Frigg, R., & Nguyen, J. (2018). The turn of the valve: Representing with material models. European Journal for the Philosophy of Science, 8(2), 224–225. Giere, R. (1988). Explaining science: A cognitive approach. Chicago: The University of Chicago Press. Giere, R. (2004). How models are used to represent reality. Philosophy of Science, 71(5), S742–S752. Giere, R. (2010). An agent-based conception of models and scientific representation. Synthese, 172(2), 269–281. Godfrey-Smith, P. (2006). The strategy of model-based science. Biology and Philosophy, 21(5), 725–740. Godfrey-Smith, P. (2009). Models and fictions in science. Philosophical Studies, 143(1), 101–116. Hartmann, S. (1995). Models as a tool for theory construction: Some strategies of preliminary physics. In W. Herfel, W. Krajevski, I. Niiniluoto, & R. Wojcicki (Eds.), Theories and models in scientific processes (pp. 49–67). Amsterdam: Rodopi. Hughes, R. I. G. (1997). Models and representation. Philosophy of Science, 64 (Proceedings):S325– S336. Hughes, R. I. G. (2010). The theoretical practices of physics: Philosophical essays. Oxford: Oxford University Press. Humphreys, P. (2004). Extending ourselves: Computational science, empiricism, and scientific method. Oxford: Oxford University Press. Jones, A. (2017). A portable cosmos: Revealing the Antikythera mechanism, scientific wonder of the ancient world. New York: Oxford University Press. Knuuttila, T. (2011). Modelling and representing: An artefactual approach to model-based representation. Studies in History and Philosophy of Science, 42(2), 262–271. Morgan, M., & Morrison, M. (Eds.). (1999). Models as mediators: Perspectives on natural and social science. Cambridge: Cambridge University Press. Morrison, M. (2008). Models as representational structures. In S. Hartmann, C. Hoefer, & L. Bovens (Eds.), Nancy Cartwright’s philosophy of science (pp. 67–90). New York: Routledge. Morrison, M. (2015). Reconstructing reality: Models, mathematics, and simulations. New York: Oxford University Press. Mundy, B. (1986). On the general theory of meaningful representation. Synthese, 67(3), 391–437.

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Nguyen, J. (2016). How models represent. Ph.D. Thesis. London: London School of Economics and Political Science. Nowak, L. (1980). The structure of idealization. Dordrecht: Springer. Peschard, I. (2011). Making sense of modeling: Beyond representation. European Journal for the Philosophy of Science, 1(3), 335–352. Sneed, J. (1971). The logical structure of mathematical physics. Dordrecht: Reidel. Suárez, M. (2004). An inferential conception of scientific representation. Philosophy of Science, 71(5), 767–779. Suárez, M. (2010). Scientific representation. Philosophy Compass, 5(1), 91–101. Suárez, M. (2015). Deflationary representation, inference, and practice. Studies in History and Philosophy of Science, 49(1), 36–47. Suppes, P. (1967). Set-theoretical structures in science. Stanford: Institute for Mathematical Studies in the Social Sciences. Suppes, P. (1974). The axiomatic method in the empirical sciences. In L. Henkin et alia (Eds.), Proceedings of the Tarski Symposium (pp. 465–479). Providence: American Mathematical Society. Suppes, P. (1988). Representation theory and the analysis of structure. Philosophia Naturalis, 25, 254–268. Swoyer, C. (1991). Structural representation and surrogative reasoning. Synthese, 87(3), 449–508. Thomson-Jones, M. (2005). Idealization and abstraction: A framework. In M. Thomson-Jones & N. Cartwright (Eds.), Idealization XII: Correcting the model-idealization and abstraction in the sciences (pp. 173–218). Amsterdam: Rodopi. Thomson-Jones, M. (2011). Structuralism about scientific representation. In A. Bokulich & P. Bokulich (Eds.), Scientific structuralism (pp. 119–141). Dordrecht: Springer. Tobin, J. (1970). Money and income: Post hoc ergo propter hoc? The Quarterly Journal of Economics, 84(2), 301–317. Toon, A. (2012). Models as make-believe: Imagination, fiction and scientific representation. Basingstoke: Palgrave Macmillan. Van Fraassen, B. (1980). The scientific image. Oxford: Clarendon Press. Van Fraassen, B. (2008). Scientific representation: Paradoxes of perspective. Oxford: Oxford University Press. Weisberg, M. (2013). Simulation and similarity: Using models to understand the world. New York: Oxford University Press. Winsberg, E. (2010). Science in the age of computer simulation. Chicago: The University of Chicago Press.

José A. Díez is Professor of Logic and Philosophy of Science at the University of Barcelona, and member of the LOGOS Research Group and of the Barcelona Institute of Analytic Philosophy. He is the author (with C. Ulises Moulines) of Fundamentos de filosofía de la ciencia (Barcelona: Ariel, Third edition: 2008. First edition: 1997), Iniciación a la lógica (Barcelona: Ariel, 2002) and (with Andrea Iacona) of A short philosophical guide to the fallacies of love (London: Bloomsbury, 2021).

Chapter 6

Seven Myths About the Fiction View of Models Roman Frigg and James Nguyen

Abstract Roman Frigg and James Nguyen present a detailed statement and defense of the fiction view of scientific models, according to which they are akin to the characters and places of literary fiction. They argue that while some of the criticisms this view has attracted raise legitimate points, others are myths. In this chapter, they first identify and then rebut the following seven myths: (1) that the fiction view regards products of science as falsehoods; (2) that the fiction view holds that models are data-free; (3) that the fiction view is antithetical to representation; (4) that the fiction view trivializes epistemology; (5) that the fiction view cannot account for the use of mathematics in the modeling; (6) that the fiction view misconstrues the function of models in the scientific process; and (7) that the fiction view stands on the wrong side of politics. As a result, they conclude that the fiction view of models, suitably understood (as an account of the ontology of models, rather than their function), remains a viable position. Keywords Fiction view of models · Imagination · Falsehood · Data · Representation

R. Frigg Department of Philosophy, London School of Economics and Political Science, Houghton Street, WC2A 2AE London, UK R. Frigg · J. Nguyen (B) Centre for Philosophy of Natural and Social Science, London School of Economics and Political Science, Houghton Street, WC2A 2AE London, UK e-mail: [email protected] J. Nguyen Institute of Philosophy, School of Advanced Study, University of London, Senate House, Malet Street, WC1E 7HU London, UK Department of Philosophy, University College London, Gower St, WC1E 6BT London, UK © Springer Nature Switzerland AG 2021 A. Cassini and J. Redmond (eds.), Models and Idealizations in Science, Logic, Epistemology, and the Unity of Science 50, https://doi.org/10.1007/978-3-030-65802-1_6

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6.1 Introduction What are scientific models? An answer to this question that has gained some popularity over the last decade is that models are akin to the characters and places of literary fiction. This is the core of the fiction view of models (“fiction view”, for short). The fiction view has attracted a number of criticisms. While some of these criticisms raise legitimate points, others are myths. They are based on misunderstandings, misrepresentations, or mischief. This paper identifies seven myths about the fiction view and explains where they go amiss. The seven myths we identify are (1) that the fiction view regards products of science as falsehoods; (2) that the fiction view holds that models are data-free; (3) that the fiction view is antithetical to representation; (4) that the fiction view trivializes epistemology; (5) that the fiction view cannot account for the use of mathematics in the modeling; (6) that the fiction view misconstrues the function of models in the scientific process; and (7) that the fiction view stands on the wrong side of politics. After a brief statement of the fiction view we discuss the seven myths one at a time. Every excursion into mythology faces an immediate problem. Some myths have found canonical expressions and their message is well documented. Others are the subject matter of oral traditions. They are passed on through word of mouth, with their content being in flux and difficult to pin down exactly. We relied on myths’ “scriptures” wherever possible, and we made an honest attempt to pin down oral narratives where no canonical versions were identifiable.

6.2 The Fiction View of Models The core idea of the fiction view is clearly stated in the following often-quoted passage by Peter Godfrey-Smith: I take at face value the fact that modelers often take themselves to be describing imaginary biological populations, imaginary neural networks, or imaginary economies. […] Although these imagined entities are puzzling, I suggest that at least much of the time they might be treated as similar to something that we are all familiar with, the imagined objects of literary fiction. Here I have in mind entities like Sherlock Holmes’ London, and Tolkien’s Middle Earth. […] the model systems of science often work similarly to these familiar fictions (2006, p. 735).

On this view, then, models are akin to the characters and places of literary fiction. To illustrate this idea, consider Fibonacci’s model of population growth. This model was one of the first mathematical models used to study population growth, and despite its simplicity it is still an important point of reference in population dynamics. Imagine you have a new-born pair of rabbits, one male and one female. Six months after birth they mate, and a further six months later a new male-female pair of rabbits is born. Now you want to know how this rabbit population evolves over time. To aid the mathematical description of the situation, you label the instants of time when

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rabbits mate and give birth by t1 , t2 , . . ., where t1 is the moment when you get the first pair of rabbits. Two consecutive instants are always separated by a six-months interval. You also let N (ti ) be the number of rabbit pairs at a certain instant of time ti . Now you assume that the pattern of the first birth continues: each pair of rabbits mate six months after birth, and a new male-female pair of rabbits is born to each pair six months after mating. You further assume that supplies of food and living space are unlimited, and that the rabbits are immortal. Under these assumptions, one can show that the number of rabbit pairs at a given time ti is the sum of the numbers of pairs at the previous two instants of time: N (ti ) = N (ti−1 ) + N (ti−2 ). Using this formula one quickly finds the rabbit pair numbers at all times: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …. These numbers are known as the Fibonacci numbers. What you have constructed is a model of the rabbit population. Your model world is one in which rabbits procreate at fixed instants of time, every female gives birth to exactly one male-female pair, rabbits are incestuous and only mate with their siblings, rabbits are immortal, and food and living space are unlimited. This is an imaginary population. Real rabbits are not like that, and their living environment doesn’t match the conditions of the model. Scientists of course know that. When they present this model, as Godfrey-Smith says, they take themselves to be describing an imaginary population. Or, in other words, they take themselves to be describing a fictional scenario. And the Fibonacci population model is no exception. Scientific discourse is rife with passages that describe fictional scenarios. Students of mechanics investigate the motion of perfectly spherical planets in otherwise empty space and economists study economies without money and transaction costs. This observation is the starting point of the fiction view: models, like Fibonacci’s model of a population, are like the places and objects of literary fiction. This idea can be developed in different ways, and a number of different versions of the fiction view have been formulated in recent years. Here is not the place to fully explicate and compare these different options.1 But laying out some basic distinctions now will prove useful for framing some of our discussions to come. First, we’re assuming an indirect view of scientific modeling throughout this paper. According to indirect views, a model description specifies a scientific model (understood as a fictional entity), just as a literary text specifies a fictional world. In turn this entity represents a physical target system.2 With this in place, one can ask about the ontological status of these models, construed as fictional entities. Here one can distinguish between realist and anti-realist fiction views. According to the former, models should be identified with objects that, in some sense, exist (e.g. abstractly in the actual world, or concretely in another possible world). According to the latter, models, still construed as fictions, should not be identified with existing objects and their fictional nature should be understood differently. A version of this anti-realist approach that has gained attention 1 For

a review of the various options see Chap. 6 of Frigg and Nguyen (2020). versions of the fiction view take the descriptions to be fictional direct descriptions of their targets. See, for instance, Levy (2015) and Toon (2012); for a discussion of their views see Frigg and Nguyen (2016). The distinction between the two versions of the fiction view is not too important for our current purposes.

2 Other

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in recent discussions of the fiction view of models appeals to Walton’s (1990) pretense theory in order to understand the nature of fictions. For a discussion of this view see Frigg (2010a) and Salis (2019).

6.3 First Myth: Fictions Are Falsehoods A frequent complaint is that regarding models as fictions misconstrues the epistemic standing of models because it relegates models to the barren land of falsehoods. Portides submits that to label something a fiction is to draw a contrast with something being a truth and therefore “we label X fictional in order to accentuate the fact that the claim made by X is in conflict with what we observe the state of the world to be” (2014, p. 76). For this reason we only classify X as a fiction “if we think that the truth valuation of the claim ‘X represents (an aspect of) the world’ is false”, and classifying models in this way, Portides argues, “obscures the epistemic role of models” (ibid.). Teller notes that “[t]he idea that science often purveys no more than fictional accounts is very misleading” because even though science has elements that are fictional, the presence of such elements “does not compromise the ways in which science provides broadly veridical accounts of the world” (2009, p. 235). Winsberg says that “[n]ot everything […] that diverges from reality, or from our best accounts of reality, is a fiction”, which is why “we ought to count as nonfictional many representations in science that fail to represent exactly” (2009, pp. 179–80). Among those nonfictional representations we find models like the “frictionless plane, the simple pendulum, and the point particle” (ibid.). The point that Teller and Winsberg make is that even though models often have fictional elements, it would be a mistake to count the whole model as fiction. In a similar fashion, Morrison argues that calling all models fictions is too coarse and a finer grained distinction should be introduced, namely “one that uses the notion of fictional representation to refer only to the kind of models that fall into the category occupied by, for example, Maxwell’s ether models” (2015, p. 90). She characterizes the ether model as one “that involves a concrete, physical representation but one that could never be instantiated by any physical system” (ibid., p. 85). Hence “[f]ictional models are deliberately constructed imaginary accounts whose physical similarity to the target system is sometimes rather obscure” (ibid., p. 90). Finally, the proposal for this book states that “fictionalism” is a position according to which “scientific models are just useful fictions that do not intend to provide truthlike descriptions of natural phenomena or to explain their causes”. Common to each of these criticisms is the characterisation of fictions as falsehoods, and hence interpreting the fiction view of models as a view that portrays models as falsehoods. First appearances notwithstanding, the fiction view is not committed this kind of nihilism. To see why this is so, we have to distinguish between two different notions of fiction and clarify which notion is at work in the fiction view of models.

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The two notions are what we call fiction as infidelity and fiction as imagination.3 Let us discuss each of these notions in turn. In the first usage, something is qualified as a “fiction” if it deviates from reality. The nature of this deviation depends on what is qualified as fiction. If we qualify a sentence (or proposition) as a fiction, the relevant kind of deviation is falsity: the sentence is a fiction if it is false. If we qualify an object as a fiction, the relevant kind of deviation is non-existence: the object is a fiction if it does not exist. We use “fiction” in this sense if we say “the Iraqi weapons of mass destruction were a fiction” to express that there are no, and never were, such weapons. In the second usage, “fiction” applies to a text and qualifies it as belonging to a particular genre, literary fiction, which is concerned with the narration of events and the portraiture of characters. Novels, stories, and plays are fictions in this sense. Rife prejudice notwithstanding, the defining feature of literary fiction is not falsity. It’s not the case that everything said in a novel is untrue: historical novels, for instance, contain correct factual information. Nor does every text containing false reports qualify as fiction: a wrong news report or a flawed documentary do not become fictions on account of their falsity—they remain what they are, namely wrong factual statements. What makes a text fictional is not its falsity (or a particular ratio of false to true claims), but the attitude that the reader is expected to adopt towards it, namely the attitude of imaginative engagement.4 Readers of a novel are invited to imagine the events and characters described. They are expressly not meant to take the sentences they read as reports of fact, neither true nor false. Someone who reads Tolstoy’s War and Peace as a report of fact and then accuses Tolstoy of misleading readers because there was no Pierre Bezukhov just doesn’t understand what a novel is. The two usages of “fiction” are neither incompatible nor mutually exclusive. In fact, some of the places and persons that appear in literary fiction are also fictions in the first sense because they do not exist. But compatibility is not identity. From the fact that something appears in a fiction of the second kind one cannot—and must not—automatically infer that it also is a fiction in the first sense. Pierre Bezukhov and Napoleon both appear in War and Peace, but only the former is a fiction in the first sense; it would be a grave error to infer that Napoleon does not exist because he appears in a work of fiction. Returning to the fiction view of models, the crucial point is that the notion of fiction involved in the fiction view is the second notion. When models are likened to fictions, this is taken to involve the claim that models prescribe certain things to be imagined while remaining noncommittal about whether or not the entities or processes in the model are also fictions in the first sense. Just as a novel can contain characters that exist and ones that don’t, and prescribe imaginings that are true and 3 The notion of fiction as infidelity is common sense and can found in most dictionaries. The online

version of Oxford Living Dictionaries, for instance, defines fiction as “something that is invented or untrue”. The idea that fiction is defined in terms of imagination is developed in Evans (1982) and Walton (1990). 4 Imagination can be propositional and need not amount to producing mental pictures. For a discussion of the notion of imagination with special focus on imagination in scientific modeling see Salis and Frigg (2020).

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ones that are false, models, understood as fictions, can feature existing and nonexisting entities alike, and they can ground true and false claims. An assessment of which of a model’s elements exist and which of its claims are true is a separate issue, one that is in no way prejudged by the fact that a model is a fiction. Different versions of the fiction view do this in different ways, and we will discuss one version in more detail later. But all versions of the fiction view of models share a commitment to the second sense of fiction and do not mean to brand models as falsities when they liken them to fictions. For this reason, as an anonymous referee points out, it might have been more apt to refer to this position as the “imagination view of models” rather than the “fiction view of models”. We agree that the “imagination view of models” would be a more informative label, and if one could turn back the wheel one could consider labelling the position in this way.5 However, for better or worse, the position has become known as the “the fiction view of models”, and it is critisized as such. For this reason, we stick to the (by now) conventional label “fiction view of models”. One might worry that this proclamation is in tension with the way in which the fiction view has been introduced and motivated earlier.6 In our example we introduced immortal rabbits living in an unlimited environment mating with their siblings and producing offspring according to a strict rule, and we noted that there are no such rabbits. If examples like these provide the motivation for the fiction view, then, despite invoking fiction as imagination rather than fiction as infidelity, isn’t it the case the latter notion is required to account for the use of fictions in modeling? Not quite. Thinking about model descriptions as describing objects that don’t exist doesn’t preclude them from, at least in part, being true of actual systems. In the context of model building and investigation, the question of whether model descriptions additionally describe actual target systems, or are related to actual target systems in some other way, is put aside. The motivation for doing so is to gain creative freedom. Scientists need to have the freedom to play with assumptions, consider different options, and ponder variations of hypotheses. In doing so they bracket the question whether the model-objects correspond to real-world objects for a moment to investigate a certain scenario that seems interesting enough to have a closer look. But this does not commit them to believing that the scenario is completely false when taken to be about a real-world target. In fact, what often motivates scientists to consider a certain scenario is that they think that the scenario has something to do with the target system that they are interested in. However, what that something is can only be assessed ex post facto, once all the details in the model are worked out. Fibonacci didn’t consider a population of immortal rabbits thinking “oh, it’s all false but it’s fun”. Fibonacci’s motivation to consider such a population was that he 5 This would be in line with Friend (2020, p. 103) who speaks of an “account of models as imagined

systems” and Frigg (2003, p. 87) who speaks of “models as imagined objects”. It would also be in line with Thomson-Jones (2020, p. 75) who notes that scientists “devote considerable time and energy to describing and imagining systems that cannot be found in the world around us”, and Thomasson (2020, p. 51) who sets herself the task of analyzing the core idea of regarding models as “imaginary” objects. 6 This point has been made by Martin Zach in his presentation at the workshop ‘Scientific Contents: Fictions or Abstract Objects?’ at University of Santiago de Compostela in January 2017.

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would eventually be able to learn something true about real rabbits from the model. But to be able to say what the true bits are, he first had to work out what happens in the model. The fiction view of models utilizes fiction as imagination to understand this practice, and this remains compatible with the idea that model descriptions are, at least in part, true (i.e. not fictional in the sense of infidelity) when applied to an actual system.7 The point is that this latter question is simply irrelevant with respect to the questions that the fiction (in the sense of imagination) view is attempting to address. This notion is not just a philosophers’ fancy. It is rooted in scientific practice, where, when a model is proposed, it is often unknown whether the entities in a model have real-world correspondents, and if they do what the nature the correspondence is. Consider the example of the Higgs boson. Elementary particle models that featured the Higgs boson were formulated in the 1960s. Much scientific work was dedicated to studying these models and understanding what was true and what was false in them by teasing consequences out of the model’s basic assumptions. Yet, this implied no commitment to either the existence or nonexistence of the Higgs boson. In fact, modelers remained expressly non-committal about this question and referred it to their experimental colleagues in CERN, where, after much work, a Higgs particle was found in 2012, and it was agreed to be “the” Higgs boson in 2013. Viewing models as fictions does not prejudge the matter either way. The question whether there is a Higgs boson stands outside the model in which it appears. Before moving on then, it is worth noting that there is a use of “fictionalism” in the philosophy of science that is more amenable to being interpreted in terms of infidelity. This occurs in the context of the debate about scientific realism. Scientific realists hold that mature scientific theories provide a (at least approximately) true account of the parts of the world that fall within its scope. Scientific anti-realists disagree and submit that we should only take claims about observables at face value and, depending on the kind of anti-realism one advocates, either remain agnostic about, or downright renounce commitment to, the theoretical claims of a scientific theory. Fine advocates such a position and calls it fictionalism: “Fictionalism” generally refers to a pragmatic, antirealist position in the debate over scientific realism. The use of a theory or concept can be reliable without the theory being true and without the entities mentioned actually existing. When truth (or existence) is lacking we are dealing with a fiction. Thus fictionalism is a corollary of instrumentalism, the view that what matters about a theory is its reliability in practice, adding to it the claim that science often employs useful fictions (1998, p. 667).

Being a fictionalist in this sense means being an anti-realist about a certain domain of discourse, and the domains that can be, and have been, given a fictionalist treatment range from morality to mathematics (for contemporary discussions see the contributions to Armour-Garb and Kroon (2020)). Regardless of the merits of this view in the context of the realism debate in the philosophy of science, it has not been 7 Usually,

at best parts of such descriptions are true of actual systems, but nothing rules out the existence of limiting cases where even the entire description can be true.

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offered as a systematic account of modeling, and we stress again that the fiction view of models is not a version of fictionalism in that sense.

6.4 Second Myth: Fictional Models Are Data-Free The second myth is a cousin of the first in that it portrays the fiction view as being committed to there being an unbridgeable gap between model and fact. The difference is that second myth is couched in terms of data rather than model-world correspondence. Kvasz notes that “when working with the model we often must supplement the data of the model description by some empirical observation. E.g. geophysicists used astronomical data of the Babylonians to fix the value of friction between the Earth’s kernel and its mantel”.8 He then claims that the fiction view cannot accommodate data because “[i]t is absurd, that Flaubert would need to ask experts to fix the size of Bovary’s foot, her hair length, and after filling this information into the novel, its plot would start to work. The quality of the plot of a work of fiction does not depend on empirical data. So models and fiction may be also in this respect dissimilar.” This is begging the question against the fiction view. As we have seen in the introductory example, mathematics can be part of what is imagined when dealing with a model (more on the use of mathematics below), and there is just no reason why data could not be part of an imaginary activity too. Fibonacci could have observed how rabbits breed and built empirical facts into the model, for instance by replacing the assumption that each rabbit pair produces exactly one pair of offspring by an empirical figure for the number of offspring. This would have complicated the arithmetical treatment of the model, but it would not have caused any problems for the philosophical analysis of it. Likewise, when Newton modeled the Sun-Earth system, his model system consisted of two imagined perfect spheres with homogenous mass distributions in empty space attracting each other gravitationally. This is a fictional system. But there is nothing stopping us from adding data to this model, for instance by inserting values for the masses of the Sun and the Earth, as well as for the gravitational constant. As far as the fictional status of the model system is concerned, there is no difference between imagining a perfect sphere tout court, or a perfect sphere with a mass of 5.972 × 1024 kg (the mass of the Earth). Furthermore, it is patently false that fiction and fact are antithetical and that no facts—or “data”—are ever inserted into fictions. Historical novels like War and Peace contain a plethora of historical facts, and authors who write such novels often carefully research the history on which they base their novel. The question concerns only which facts are inserted into the novel. Kvasz is of course right that facts about the size of Emma Bovary’s feet are not part of the novel. But that’s not because novels,

8 The

point has been made in a talk in Prague on 29 May 2018. The quotes are from the slides; italics are original; bold-face has been removed for typographical unity. The talk is available online at http://stream.flu.cas.cz/categories/representation-in-science.

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as matter of principle, resist facts; it is because her foot size is irrelevant to the plot.9 Imagine another author, Maubert, who works on a novel whose protagonists are a dancer, and orthopaedic surgeon, and a foot fetishist. Maubert is a social realist and sees it as his mission to document everyday lives. We can be sure that plenty of facts about feet will go into that novel!10

6.5 Third Myth: The Fiction View Is Antithetical to Representation Another myth concerning the fiction view of models is that it is antithetical to representation. To motivate her own proposal, Knuuttila rhetorically asks “if scientific models are considered as fictions rather than representations of real-world target systems, how are scientists supposed to gain knowledge by constructing and using them?” (2017, p. 2, emphasis added), thereby driving a wedge between fictions and representations. In the same vein, the proposal for this book says: Fictionalism and artifactualism remove the concept of representation from its central place in the philosophical analysis of scientific models. Regarding idealization, it remains to be seen how this concept could be accommodated by these accounts.

Artifactualists can speak for themselves. But in the context of the fiction view of models, the assumption seems to be that the view highlights the ontological status of models at the expense of according them representational content. As we have seen, the fiction view emphasizes thinking about model descriptions as describing and investigating the features of fictional systems. In some cases, this may be done at the expense of focusing on the target system, and the relationship between the things that are written down in scientific papers and textbooks and the parts of the world one might think they are trying to describe falls into oblivion. The focus is on the fiction rather than the actual world. Our discussion of the nature of fiction in Sect. 2 might be thought to lead to something like this view. If, as we suggested, the relevant notion of fiction involved here is that such descriptions are “fictional” in the sense that whether they are true or false when applied to a target system is irrelevant to their function, just as readers of literary fictions are expressly not meant to take the sentences they read as reports of fact, neither true nor false of the world we inhabit, 9 A parallel point is true of models, which are often poor in detail because certain details don’t matter

in a given context. It would, however, be mistaken to take models to be defective just because they do not contain certain details. Thanks for Martin Zach for pointing this out to us. 10 There is another objection to the fiction view lurking in the vicinity of this myth. Roughly speaking, it runs as follows: if works of literary fiction contain facts, these facts are not relevant to the aesthetic value of the work of fiction; they are not relevant to the value of the fiction qua fiction. Or more generally, the epistemic value of a work of fiction is not relevant to its aesthetic value. In contrast, whether a scientific model, understood as a work of fiction, contains or doesn’t contain the relevant facts is clearly relevant to the value of the scientific model. Therefore, scientific models and works of literary fictions should not be identified, because what makes them valuable qua model and qua work of fiction respectively, are not the same. We return to this objection in Sect. 6.

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then it looks like actual target systems have no space in the theory. Thus, the fiction view is antithetical to representation because, whilst it might provide an account of what models are, and moreover, whilst it might provide an account of what sorts of things are true in the model, it does so in a manner that is disconnected from the relevant systems in the world we typically understand models as representing. This objection fails to take into account the division of labour in philosophical discussions of scientific modeling. The fiction view of models is, first and foremost, an attempt to understand the ontology of scientific models. What is a frictionless plane? What is an ideal agent? What is a population of immortal animals? These things aren’t, at least not in any straightforward sense, homely concrete objects existing in our universe. But in order to understand what it is that these models tell us about actual slopes, customers and consumers, and animal populations, we first have to understand what models are and how they operate in a context of investigation. We then have to say how they represent their target systems. The objection, presumably, would insist that this can’t be done—and that is simply wrong. We don’t speak for everyone associated with the fiction view of models here, but our own preferred account of scientific modeling combines the fiction view with what we call the DEKI account of scientific representation. We understand scientific models to be fictional systems that in turn represent target systems in virtue of meeting four conditions: the model denotes the target system; the model exemplifies certain features; these features are converted by a key into a collection of other features; and these other features are imputed to the target system (2016, 2018). Logically speaking, the two accounts are independent of one another: one could subscribe to the fiction view and then offer an alternative account of how models represent, and one could adopt the DEKI account of scientific representation but combine it with another account of the ontology of scientific models (e.g. an account where models are construed as mathematical structures). The important point here is that the two accounts can be combined. So, at least on our approach to the question, the fiction view of models does not “remove the concept of representation from its central place in the philosophical analysis”; rather it is supposed to be supplemented with an account of representation to deliver a more encompassing account of scientific modeling. Whilst this goes some way to alleviating the worry that the fiction view is antithetical to representation, understood in this way, more specific worries can be targeted at the view. Most pertinently, one can now ask whether or not the fiction view is consistent with one’s preferred account of scientific representation. If it’s not, then although the view isn’t designed to downplay the importance of representation, it may nevertheless rule out scientific models from representing for ontological reasons. A worry of this sort is the following: in order for a scientific model to represent its target, we have to provide some way of making sense of ascribing physical features to models and of comparing these features with features of the target. Examples can illustrate what we mean by this. The imagined object sliding down the surface of a frictionless plane in the model experiences a certain degree of friction and is subject to less friction than the skier on the ski run; after a few years, the population of actual rabbits breeding in the back garden has a certain size and is smaller than the

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population of immortal rabbits in the model; the bob on the string in the idealized pendulum has a certain mass and weighs less than the weight at the bottom of the pendulum in the clock in the corner of the library; and so on. In order to make sense of such property attributions and comparisons, we have to be able to analyze the apparent ascription and comparison of model-target features.11 How to understand these ascriptions and comparisons will depend on the details of the fiction view in question. Fictional realists who think that fictional objects are concrete systems, existing in another possible world for example, presumably won’t have an issue with how to ascribe physical features to models, and presumably the comparisons will be phrased in terms of trans-world comparative statements. The objection is more pressing for fictional realists who think that these objects don’t have physical features, because they are actually existing abstract objects for example, since such abstract objects, presumably, are not subject to friction, and are not the sorts of things that have size or weight (Thomson-Jones 2010). Given this, it is also difficult to see how we can go about comparing them to their physical targets with respect to those properties. Fictional anti-realists, like those using the Waltonian framework face part of the worry: whilst the view allows us to accommodate what look like ascriptions of physical features by paraphrasing them as occurring in a game of make-believe, accommodating the comparisons is more problematic, given that there is no obvious game of make-believe associated with the comparisons. To illustrate: a claim like “the bob on the end of the pendulum weights x grams” might be true in the game of make-believe, but it remains disconnected from comparisons with the target because the weight at the end of the pendulum in the clock is not part of that game of make-believe. So it seems to remain unclear what is compared with what, and how. More generally then, if an account of scientific representation requires that we can provide an account of what looks like the ascription of physical features to scientific models, and an account of how these features are compared to features of their targets, and if one’s preferred fiction view of models cannot do this, then the view will be inconsistent with the idea that models, understood as fictions, can play a representational role. There are two ways of answering this objection. First, one can deny that an account of scientific representation requires an analysis of ascription and/or comparison of physical features. Second, one can attempt to accommodate them in one’s preferred fictional framework, where the details of such an accommodation will depend on the details of the framework in question. We here want to run the first line of argument and set aside the question of whether one can accommodate property ascriptions and comparisons. We note, however, the proponents of the fiction view using the Waltonian framework have number of options available to them if they want to pursue that second avenue (Salis 2016).

11 Another

objection might be that the fiction view cannot account for the idea that models denote their targets, which is a requirement of many accounts of representation. A failure of denotation is supposed to come about because in order for X to denote something X has to exist in some ontologically robust sense which fictions lack. We answer that objection in Salis et al. (2020).

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Why think that the way in which models represent requires ascribing and comparing the physical features of models with the physical features of their targets? Presumably this depends on the account of scientific representation that is adopted. A popular account invokes the notion of similarity to establish (at least in part) either representation or accurate representation,12 the basic idea being that a model represents its target in virtue of someone proposing that the two are similar in certain respects and to certain degrees. The model is accurate if the two are in fact so similar. In general, it is accepted that these sorts of similarities turn on the co-instantiation of certain physical features. So in order for the instances of modeling that we take to be accurate to come out as accurate on such an account, it better be the case that we can ascribe models the relevant physical features (population size, bob weight, and so on), and make model-target comparisons in terms of these features. Thus, a commitment to the similarity view of scientific representation engenders a commitment to understanding the sorts of ascriptions and comparisons offered above. And, so the objection goes, these are difficult to make sense of for, at least some, versions of the fiction view.13 However, as noted, the fiction view of models is by no means committed to any particular account of scientific representation.14 Our account of scientific modeling combines it with the DEKI account of representation. Recall that on this account a model exemplifies certain features, and a key is used to translate those features into other features to be imputed onto the target. If the target has those features, then the model is accurate with respect to them.15 Notice, then, that there is neither a demand that the model exemplifies physical features, nor that keys have to link features of the same kind. A key has to convert some feature P of a model into a feature Q, which is then imputed to the target. Such an act does not require that P is a physical feature. In fact, generating this claim does not require any explicit feature comparison at all. 12 For

more on the role of similarity in scientific representation see chapter 3 of Frigg and Nguyen (2020); see the rest of the book for more on scientific representation in general. 13 A related objection that has been directed at the fiction view is that it cannot accommodate “design models” (i.e. blueprints, plans, and so on), precisely because the fiction view requires comparing the features of models with the features of their targets. In the case of design models, there is no target system (at least at a certain stage in the modeling process), and therefore no target features with which to compare to the model’s features (Currie 2017). In response to this objection we note, again, that the fiction view concerns the ontology of models, not their representational content. As such, a model doesn’t have to represent in order to be considered a fictional object. Such models are a kind of targetless model, and we discuss them in Frigg and Nguyen (2020, see in particular Chaps. 8-9). 14 For a discussion of how the fiction could be combined with different accounts of representation see Frigg and Nguyen (2016). 15 The notion of a key was also invoked by one of us in a sketched precursor to the DEKI account (Frigg 2010b). As stated there, keys work by taking “facts” about the model to claims about the target. Toon (2012, p. 58) and Levy (2015, pp. 789–90) objected that according to the fiction view there are no facts about models, thus there is nothing for the key to apply to. This objection relies on an overly stringent understanding of “model-fact”: there’s nothing to prevent keys being applied to “facts” which are only fictionally true in the game of make believe associated with the model. Indeed, articulating what is fictionally true in a model is one of the main tasks for the fiction view of models.

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As previously noted, the DEKI account can be combined with any of the different fiction views of models. Combining it with a realist view that takes fictions to be concrete possible objects would lead to the view the model can exemplify physical features, and that these physical features can be acted on by a key that links physical features with physical features. Combining it with a realist view that takes fictions to be abstract actual objects, the model can exemplify the sorts of features that such abstract objects have, and these can be acted on by a key linking them to physical features of targets. Combining it with an anti-realist view of the Waltonian stripe, the model can be said to exemplify certain physical (or non-physical) features in the relevant game of make-believe, and then these features can be linked, via a key, to the features of target systems. In none of these cases do we have to compare the model-features with features of targets; the key is simply a way of taking us from a statement of the form “the model has feature P” to one of the form “the target has (physical) feature Q”. It makes no difference to the function of the key if the former sentence is embedded in a game of make-believe, and the latter is not. So, as we have seen, the fiction view of models is not antithetical to representation; in the first instance because it’s not designed to be an account of how models represent and in the second, because there are routes available to combine the view with such accounts. These routes can involve embracing the challenge of making sense of ascribing physical features to fictional entities, and comparing them with the physical features of their targets, or they can simply deny that such comparisons need to be made to make sense of scientific representation. Either way, combining the fiction view with an account of scientific representation is not the impossible task that the myth takes it to be.

6.6 Fourth Myth: Fiction Trivializes Epistemology Another objection that may be directed at the fiction view of models stems from an existing literature regarding the cognitive, or epistemic, value of art.16 There the debate concerns whether or not works of art, including works of fiction, have epistemic value and the connection that this has to aesthetic value. “Cognitivists” are committed to both (i) the idea that art, at least in part, has epistemic value because we learn from it, and (ii) that art’s epistemic function positively contributes to its aesthetic value.17 “Anti-cognitivists” deny one, or both, of these claims. These considerations can be brought to bear in the context of the fiction view of models via an (anticognitivist) argument that runs as follows: works of fiction have no, or at best little, 16 We

use the phrases “cognitive value” and “epistemic value” interchangeably. we will see, it’s important to distinguish between these two aspects of cognitivism in the context of the fiction view of models, but it is also important in the context of analyzing the value of art itself. Both Gaut (2003) and Thomson-Jones (2005) characterize the debate about art in this way. Gibson (2008) prefers a different characterization according to which the question is whether the artwork itself, qua artwork, contains cognitive content. The difference is not important for our current project. 17 As

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epistemic value; scientific models have epistemic value; therefore scientific models are not works of fiction.18 Obviously we don’t deny that scientific models have epistemic value: in fact we take it that they are bearers of such value par excellence. As such we have to address the first premise; that works of fiction lack such value. This leads to two questions: first, is it actually the case that works of fiction lack epistemic value, and second if they do, does this stem from their fictional nature (rather than some other property that some works of fiction happen to have more conventionally) in such a way that this carries over to the fiction view of models? Notice that a positive answer to the first question doesn’t entail a positive answer to the second: even if works of fiction, now understood to exclude scientific models, have little to no epistemic value, this need not necessarily arise from their fictional nature (or at least the aspects of their fictional nature that the fiction view of models exploits); and as such, even if it turns out that the anti-cognitivist’s arguments against the epistemic value of literary works of fiction are sound, this does not necessarily entail that they carry over to the fiction view of models. As it happens, we think that the anti-cognitivists arguments fail even in the case of literary works of fiction, which we will address first. But we’ll also comment on a plausible way in which one could be an anti-cognitivist about literary fiction whilst still adopting the fiction view of models. In order to address the question of whether works of fiction have epistemic value it is crucial to delineate the kind of epistemic value that is in dispute, at least in the context of defending the fiction view of models from anti-cognitivist arguments. The idea that fiction, and art more generally, has “epistemic value” can be explicated in multiple different ways, and thus, there are many different ways in which this can be denied. So it’s useful to disentangle the different kinds of knowledge that fiction might be said to provide. First and foremost, in the current context, is the idea that fiction provides propositional knowledge concerning the actual world. This is to be contrasted with other kinds of knowledge that fiction might plausibly provide: philosophical knowledge, i.e. knowledge of concepts; knowledge of possibilities; practical knowledge, i.e. knowledge of how to do certain things; knowledge of the significance of events; phenomenological knowledge, i.e. knowledge of what it feels like to be in a certain situation; and knowledge of values (see Gaut 2003, pp. 437–90 and the references therein). This is not to say that there is a clear division between these sorts of knowledge, and the fiction view of models does not rule out the idea that models can also provide knowledge beyond propositional knowledge of the actual world (for example, it’s very plausible that many models provide us with knowledge of possibilities, sometimes called “how-possible” models (Bokulich 2014) and models in engineering can provide us with practical knowledge, and so 18 Something similar to this argument is considered by G. Currie (2016, p. 304). Note, however, that

he does accept that we can learn from fictions in some sense, albeit in a sense that doesn’t support the analogy between literary fiction and scientific modeling. The problem with locating this myth, as Currie points out, is that there is relatively little agreement about the extent to which fictions hold cognitive or epistemic value. As such, those who think they don’t will take the fiction view of models to be a non-starter (Portides 2014).

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on). But for our current purposes we can put these sorts of questions aside and focus on whether works of fiction provide us with knowledge of the actual world. In the first instance, it’s difficult to deny that works of fiction provide us with knowledge of this kind. We learn much about the academic environments in the UK and USA in the 1970s from reading Changing Places and much about the geography of Dublin at the turn of the century from Ulysses. In light of the above discussion, there are two ways for the anti-cognitivist to respond. They could either accept that we do acquire knowledge from fiction but deny that this contributes to their aesthetic value (denying (ii) above, something we return to later), or they could argue that all that we learn from fiction is, in some sense, banal, or a collection of truisms. This latter approach is argued by Stolnitz (1992). He invites us to consider what we learn from Pride and Prejudice. Presumably, given that we’re interested in what we learn about the actual world from the fiction, the cognitive value of the work is not directly concerned with the details of the novel itself. We’re not concerned with learning about the events in the fiction alone; we’re concerned with how the fiction can help us gain knowledge about actual systems in the world. As such he argues that the cognitivists won’t: settle for other truths, of which there are a great many, about Elizabeth Bennet and Mr Darcy [but rather t]hey will settle for nothing less than psychological truths about people in the great world, truths universal, more or less […] Hence [what we learn is that]: Stubborn pride and ignorant prejudice keep attractive people apart (p. 193).

His point, then, is that in moving from the world of the fiction to learning something about the world itself, we have had to abstract from the details. In his own words, if we want to specify what we learn about the actual world from the novel we: abandon the setting of the novel in order to arrive at psychological truth. Yet in abandoning Hertfordshire in Regency England, we give up the manners and morals that influenced the sayings and doings of the hero and heroine [and thus] we abandon their individuality in all of its complexity and depth. My statement of the psychological truth to be gained from the novel [that stubborn pride and ignorant prejudice keep attractive people apart] is pitifully meagre by contrast. Necessarily, since the psychologies of Miss Bennet and Mr Darcy are fleshed out and specified within the fiction only. Once we divest ourselves of the diverse, singular forces at work in its psychological field, as we must, in getting from the fiction to the truth, the latter must seem, and is, distressingly impoverished. Can this be all there is? From one of the world’s great novels? (p. 194).

Elgin calls this the argument from banality (2017, p. 245). Since the purpose of scientific models is to give us knowledge, and ideally precise and specific knowledge, they cannot, so the argument goes, be works of fiction, or even be like works of fiction.19 The question, then, is whether Stolnitz is right that all we can learn about the actual world from works of fiction are banal truisms. It’s far from clear whether this is the 19 G. Currie gestures at a similar worry when he notes that “[w]e have no more than the vague suggestion that fictions sometimes shed light on aspects of human thought, feeling, decision, and action” (2016, p. 304). We take it that the below discussion concerning whether what we learn from fiction is “trivial” or “banal” carry over to whether what we learn “sheds light” on phenomena in the world. A position similar to G. Currie’s is also advocated by Portides (2014).

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case. Even in his own example, why should we think that it is banal that pride and prejudice keep attractive people apart? Those inexperienced in the trials and tribulations of dating and relationships often don’t realize that if they swallowed their pride a little more often, their relationships would go much smoother. (In fact, one could argue that it’s precisely because of the fact that we are so familiar with Pride and Prejudice that Stolnitz’s example of a psychological truth might appear uninformative; perhaps it’s not that the novel reflects something already widely known, perhaps it’s widely known because of the cultural familiarity with the novel.)20 Other examples of fiction providing us with non-trivial knowledge abound. As previously mentioned, Changing Places teaches us about the difference between American and British academic cultures in the 1970s; Harper Lee’s To Kill a Mockingbird provides detailed insights into the racial inequality and its dealing with rape; Harriet Stowe’s Uncle Tom’s Cabin taught its readers much about slavery in the USA; and George Orwell’s Animal Farm provides a fine-grained account of the phoney pretention of communism. And in the latter case, it, along with the likes of Doctor Zhivago, being banned in the USSR suggests that there is something non-trivial about what we learn about Soviet Communism and the Russian Revolution from reading such works of fiction (we are not claiming that all banned books are banned because of what we learn from them; plausibly some are banned for other reasons; but Animal Farm and Doctor Zhivago were banned for what they say about communism rather than for, say, being obscene). Similarly, the Catholic church’s Index Librorum Prohibitorum was not restricted to scientific works; it included Milton’s Paradise Lost and Voltaire’s Candide, amongst others, again suggesting that works of fiction can contain the sort of cognitive content that certain institutions would prefer were not widely accessible.21 There is, however, a grain of truth in Stolnitz’s argument. In order to learn actual psychological truths about from Pride and Prejudice we abstract away from the details of the fictional novel. Whilst we think that Stolnitz is incorrect to conclude that the result of such a process leads to triviality, he is correct that what works of fiction tell us about the world is, often at least, less precise than the details of the fictional world. But this is by no means unique to novels. Many elements of science, including, but not limited to, models don’t represent the world in its full messy detail. Laws of nature only apply ceteris paribus (Cartwright 1983), and scientific 20 Kvasz offers the further objection that scientific models should be distinguished from works of fictions because the former, but not the latter, provide novel knowledge about their targets. Again, it is unclear to us why one should think that by working through the implications of a fictional novel one would not learn anything new. 21 In order to handle these sorts of examples in the context of discussing the cognitive, or epistemic, value of art, the anti-cognitivist can accept that we do learn from fiction, but argue that the fact that we do so is irrelevant to the works’ aesthetic value (this may seem plausible with respect to learning about Dublin’s geography from Ulysses, but we’re unsure whether it’s correct with respect to the other examples discussed). However, here it serves to distinguish between the debate in the philosophy of art and the fiction view of models. In the case of the latter it’s irrelevant as to whether or not what we learn contributes to the works’ aesthetic value. So once the anti-cognitivist’s target is the connection between learning from fiction and its aesthetic value, rather than whether or not we learn from fiction at all, the fiction view of models is no longer threatened by their claims.

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models are also abstractions in the sense of not representing everything in a target system (Thomson-Jones 2005). This is particularly clear in the case of so-called “toy models”, where one of us has argued that this process of abstracting from the details of the model is required in order to understand what they tell us about their target systems, and that it’s this process that explains why such models are so prevalent in science (Nguyen 2020). The claim is that the representational content of these models is also relatively abstract in the sense of applying broadly to a number of target systems and at a relatively high level of generality, such as “mild preferences tend to lead to social segregation” (Schelling’s model), or “asymmetrical information tends to lead to Pareto-inefficiency” (Akerlof’s model). One might argue that these propositions are “banal” or “trivial”, and there we have no knock-down argument against this claim because there is no clear criterion for when a truth is “banal” or “trivial”. We note, however, that both models contributed to their authors being awarded Nobel Memorial Prizes in Economics, which would suggest that the scientific community took a less bleak view of these insights than the “banality argument” would suggest we should. It seems to us that the works of fiction (and scientific models) in question do teach us important truths about the world, but for the sake of argument, let’s suppose, with the anti-cognitivist, that we’re wrong. Does this vindicate the argument given at the beginning of this section regarding the idea that the fiction view of models must fail because fictions, in virtue of their fictional nature, have no epistemic, or cognitive value? Here we see again a disconnect between the anti-cognitivist arguments against the cognitive value of art and attempts to turn those arguments against the fiction view of models. The former arguments take as data observations about what we learn from actual works of fiction; they don’t turn on the ontological status of such fictions. Since the fiction view of models is only committed to the abstract mechanism of fiction (as outlined previously), defenders of the view can plausibly argue that the implementation of that mechanism in the context of scientific models is, in an important sense, different from how it has been implemented in the cases of fictional novels. This would allow them to accept (at least for the sake of argument) that literary fictions lack cognitive value, without thereby undermining the fiction view of models itself. It’s worth briefly sketching the sorts of differences that one could appeal to in order to justify such a position.22 First, one could argue that, in general at least, literary fictions don’t have a fully specified target system, and it’s this that tells against the idea that they have cognitive value. But as we saw in the previous section, there’s nothing about the nature of fiction itself that prevents a fiction playing a representational role, so even if this were true about literary fiction, it has no bearing on the fiction view of models.23 Second, one could argue that the ways in which literary 22 See

Frigg and Nguyen (2017, pp. 56–58) for further elaboration.

23 We want to further note here that we think many literary fictions do have target systems—Animal

Farm is quite clearly targeted at Soviet Communism and further examples are not difficult to find; Erich Maria Remarque’s All Quiet on the Western Front and Kurt Vonnegut’s Slaughterhouse-Five

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fictions are interpreted as representing their targets is much less constrained than the way in which scientific models are so interpreted. Drawing on Friend (2017a) for an example, following the publication of Jean Rhys’s Wide Sargasso Sea, adopting a postcolonial interpretation of Jane Eyre yields very different insights from the behaviour of Bertha Antoinetta Mason about Jamaica’s colonial past.24 More generally, considering alternative interpretations of literary fictions is a practice to be celebrated. But in the case of science, if one interprets a scientific model in a nonstandard way, it often suggests that the model hasn’t been understood. So flexibility of interpretation counts against the claim that works of fiction have cognitive value. This argument seems problematic for two reasons. First, even if we assume that it is correct that in general that works of literature have more flexibility with respect to interpretation than scientific models,25 it remains unclear to us why this flexibility should count against the cognitive value of fiction. That we can reinterpret Jane Eyre doesn’t make the work cognitively inert. Second, even if this was the case, the point doesn’t carry over to the fiction view of models because scientific fictions are, usually at least, much more constrained. Finally then, one could consider the differences with respect to style in literary and scientific fictions. In the case of literature, the way in which a work is written seems to have clear ramifications concerning what, if anything, we learn from it. But, as G. Currie notes: ‘[m]odels are not dependent for their value in learning on any particular formulation’ (2016, p. 305). So this difference could play a role in distinguishing between the cognitive value of literary and scientific fictions. Again it’s not obvious how this tells against the fiction view—that models are (supposedly) independent of their formulation, or at least less dependant upon their formulation than fictions, is no reason not to think of them as fictions. And again, the difference seems more of a matter of degree than kind. As noted by, e.g., Vorms (2011, 2012), the ways in which scientific models are presented can make a difference to their epistemic value (consider, for example the importance of the coordinate system used in presenting a mechanical problem!). To summarize then, it doesn’t seem like anti-cognitivist arguments in the philosophy of art put the fiction view of models in any particular danger. First, the arguments can only be brought to bear if it’s agreed that fictions don’t have epistemic, or cognitive value. And as we have seen, there is no widespread agreement on this issue. Second, even if one were to grant that works of fiction don’t have such value, it still remains the case that a defender of the fiction view of models can utilize differences (even if they are matters of degree rather than of kind) between literary and scientific are passionate denunciations of the horrors of the First and Second World Wars (respectively)— and as discussed above at least some scientific models lack targets. So this way of distinguishing between literary and scientific fictions seems implausible. 24 Friend (2017a) argues in favour of pluralism about interpretation—which she construes in terms of what is true in the fiction—but her arguments carry over to what the resulting fictions tell us about the world. 25 While this may be the case in general, there are cases from the history of science where considering an alternative interpretation of a model has increased its cognitive value in a way to be celebrated. Consider, for instance, Bohr’s interpretation of a celestial two-body model in terms of atomic structure, or Goodwin’s interpretation of an ecological predation model in terms of economic firms.

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fictions in order to explain why the latter hold cognitive value despite the fact that they are on a part with the former with respect to their ontological status.

6.7 Fifth Myth: The Fiction View Is Antithetical to Mathematisation The main competitor to the fiction view of models is the view that models should be thought of as mathematical structures. From this it is tempting to infer that the fiction view is in tension with the idea that scientific models involve mathematics. And given the prevalence of mathematics in many (although, we submit, not all) modeling contexts, if this were the case then the fiction view would face a serious objection. This brings us to our fifth myth: that the fiction view of models is antithetical to mathematisation. Busting this myth requires explaining how mathematics enters the picture on the fiction view.26 Answering this objection requires understanding how model descriptions are related to the models they describe. Again, the exact details of this will depend on the details of the particular version of the fiction view in question. The descriptions can be taken to describe concrete possible objects (or collections of concrete possible objects) in the same way in which as statements about other possible worlds describe those worlds (Contessa 2010). The descriptions can be taken to describe abstract actual objects, or indeed be taken to bring those objects into existence, in the sense of creating abstract artifacts (Thomasson 1999). Or they can be taken to be (partly) constitutive of the fictional entity, understood in the Waltonian sense (by serving to specify the primary truths in the relevant game of make-believe). The point, though, is that all (indirect) fiction views have a role for model descriptions describing, in some sense, the fictional entities in question. And the crucial thing to note in this context is that these model descriptions often include statements in the language of mathematics.27 The question then, is how do model descriptions, understood as being at least in part mathematical, relate to the mathematical features of the models they describe. Again, this will depend on the details of the particular fiction view. When they’re seen as describing possible concrete objects, the relationship between a mathematical description and such an object will be understood in the same way as we understand the relationship between a mathematical description and an actual concrete 26 This objection has been put to us in personal conversation, but we have not been able to locate it in print. However, given that the main competitor to the fiction view, at least in terms of the ontology of scientific models, is the structuralist view, understanding how mathematics enters the picture on the fiction view is crucial for understanding what’s at stake in the debate. Structuralism about models comes in two versions. The traditional version, associated with Suppes (2002) submits that models are structures, while a contemporary alternative, associated with Bueno and French (2018), regards structures as a meta-level representation of models. 27 Even in cases where they are not explicitly mathematical, one might still want to apply mathematical tools to them, we discuss these sorts of cases in Frigg and Nguyen (2016).

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object. Presumably, whatever is going on when we describe actual rabbit populations in mathematical language is also what’s going on when we describe possible rabbit populations in mathematical language. For a discussion of a number of such suggestions see, for instance, Shapiro (1983). If models are understood as abstract actual objects, the question is trickier: in what sense can we describe such objects mathematically? To the best of our knowledge, there hasn’t been much explicit discussion of this sort of position in the literature. However, what’s important for our current purposes is that this is a question of metaphysics: presumably how you go about thinking about these sorts of objects will inform how they can be described mathematically. But notice that thinking about models as abstract actual objects isn’t antithetical to mathematising them, it just requires that some story be told, and it’s not obvious that there is anything blocking such a telling. In the case of Waltonain anti-realism, mathematics enters into games of makebelieve in two places. First, mathematical model descriptions can serve to specify primary truths in the game of make-believe associated with a model. When we specify Fibonacci’s model of population growth we use the equation N (ti ) = N (ti−1 ) + N (ti−2 ). This makes it fictionally true in the model that the population size at a time t i is the sum of the population size at the two previous time steps. Second, recall that games of make-believe are also associated with rules of generation (for more on the details of the Waltonian framework in the context of the fiction view see Frigg (2010a) and Salis (2019)). These rules act to generate secondary truths, which, when combined with the primary truths, provide the collection of propositions that are fictionally true in the model. The nature of these rules of generation can vary across different games. One that is in operation in many literary fictions is the so-called “Reality Principle” (Walton 1990). According to this rule, the truths of the games of make-believe in which it is in operation are what would be true in the actual world, were the primary truths of the game true.28 In the context of considering how mathematics enters the picture in the pretense version of the fiction view of models, it’s important to note that the rules of generation often (perhaps always) consist of the rules of logic and mathematics. So far from being antithetical, mathematics fits seamlessly into this version of the fiction view. What are we to make of the comparison between models and literary fictions in light of this observation? Kvasz argues that: “even though we have the means to go beyond the explicitly stated content by using certain rules of inference, the rules used in fiction are very different from those used in scientific modelling”.29 This objection is hard to pin down because we’re not told how the rules are different, and without knowing what the alleged difference is, it is difficult to assess how damaging the difference really is. We suppose that Kvasz alludes to the fact that rules of generation in literary fiction are “informal” while rules of generation in modeling are mostly 28 Friend

(2017b) offers a slightly different rule, the “Reality Assumption”, according to which everything which is actually true is fictionally true, just so long as it isn’t explicitly ruled out by the primary truths. 29 Again, the quote is from the slides of his talk, cf. footnote 6.

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mathematical. If this is the relevant difference, then it is not a problem for the fiction view. First, even though many of the rules in literary fiction are informal, they are still (usually) constrained by logic and mathematics (when Yossarian calculates bomb trajectories in Catch-22, these trajectories can enter into all sorts of mathematical relationships). That rules of generation in fiction include non-mathematical rules doesn’t mean they don’t also contain mathematical ones. Second, even if it were the case that the rules used in games of make-believe associated with scientific models were more explicitly mathematical than the rules used in games associated with literary fictions, this does nothing to undermine the (Waltonian version of) the fiction view. As argued previously, the view is an attempt to understand the ontology of scientific models. If, in order to do this, we are lead to understanding their function differently to the function of literary fiction, this is not an objection to the account. In fact, given that the Waltonian framework involves rules of generation, but allows these rules to vary across different games, we can accommodate the differences between scientific models and literary fictions by considering how their rules differ. But, again, the fact that the rules are different does not mean that the ontology story offered by the account cannot get off the ground. So, again, although there might be differences concerning how mathematics enters the picture on different versions of the fiction view of models, the idea that the view somehow stands in tension with the role of mathematics in modeling is mistaken. Each version of the view can accommodate scientific models being described in the language of mathematics, and having, in some sense, mathematical features.

6.8 Sixth Myth: Fiction Misconstrues the Function of Models A related objection is that the fiction view misidentifies the aims of models. Giere deems it “inappropriate” to “regard scientific models as works of fiction” even though they are ontologically on par. The reason for this is “their differing function in practice” (Giere 2009, p. 249).30 Giere identifies three functional differences (ibid., pp. 249–252). First, while fictions are the product of a single author’s individual endeavours, scientific models are the result of a public effort because scientists discuss their creations with their colleagues and subject them to public scrutiny. Second, there is a clear distinction between fiction and non-fiction books, and even when a book classified as non-fiction is found to contain false claims, it is not reclassified as fiction. Third, unlike works of fiction, whose prime purpose is to entertain, scientific models are representations of certain aspects of the world. In a similar vein, Magnani dismisses the fiction view for misconstruing the role of models in the process of scientific discovery. On his account, the role of models is to be “weapons” in what he calls “epistemic warfare”, a point of view “which sees scientific enterprise as a complicated struggle for rational knowledge in which it is crucial to distinguish 30 Similar

arguments are made by Liu (2014) and Portides (2014).

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epistemic (for example scientific models) from non-epistemic (for example fictions, falsities, propaganda, etc.) weapons” (2012, p. 2; cf. 2020). Fictions, so the argument goes, are not involved in any such epistemic warfare. Neither of these objections is on target. As regards Giere’s, it’s not part of the fiction view to regard “scientific models as works of fiction” in some vague and unqualified sense, much less to claim that literary fictions and scientific models perform the same function. Proponents of the fiction view are careful to specify the respects in which models and fictions are taken to be alike, and none of the aspects Giere mentions are on anybody’s list. Furthermore, even if they were, they wouldn’t drive Giere’s point home. First, whether a fiction is the product of an individual or a collective effort has no impact on its status as a fiction; a collectively produced fiction is just a different kind of fiction. Even if War and Peace (to take Giere’s own example) had been written in a collective effort by all established Russian writers of Tolstoy’s time, it would still be a fiction. Vice versa, even if Newton had never discussed his model of the Solar System with anybody before publishing it, it would still be science. The history of production is immaterial to the status of a work. Second, as noted previously, falsity is not a defining feature of fiction. We agree with Giere that there is a clear distinction between texts of fiction and non-fiction, but we deny that this distinction is defined by truth or falsity; it is the attitude that we are supposed to adopt towards the text’s content that makes the difference. Third, proponents of the fiction view agree that it is one of the prime functions of models to represent, and they go to great length to explain how models do this. We have discussed this issue in Sect. 5. Magnani’s criticism is also based on an understanding of fiction as falsity, which supposedly implies that fictions can play no epistemic role. We repeat that fiction is not defined through falsity and that models, even if understood as fictions in one of the qualified senses discussed in this chapter, can play epistemic roles.

6.9 Seventh Myth: The Fiction View Stands on the Wrong Side of Politics Finally, Giere (2009, p. 257) complains that the fiction view plays into the hands of irrationalists.31 Creationists and other science sceptics will find great comfort, if not powerful rhetorical ammunition, in the fact that philosophers of science say that scientists produce fiction. This, so the argument goes, will be seen as a justification of the view that religious dogma is on par with, or even superior to, scientific knowledge. Hence the fiction view of models undermines the authority of science and fosters the cause of those who wish to replace science with religious or other unscientific worldviews. Needless to say, we agree that irrationalists deserve no support. In order not to misidentify the problem it is important to point out that Giere’s claim is not that the view itself—or its proponents—support creationism; his worry is that the view can be 31 This

discussion of Giere’s objections is based on Sect. 4 in Frigg (2010c).

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misused if it falls into the wrong hands. True, but almost anything can. What follows from this is not that the fiction view itself should be abandoned. What follows is that some care is needed when communicating the view to non-specialist audiences. As long as the fiction view presented carefully and with the necessary qualifications, it is no more dangerous than other ideas, which, when taken out of context, can be put to uses that would (probably) send shivers down the spines of their progenitors (think, for instance, of the use of Darwinism to justify eugenics).

6.10 Conclusion We’ve attempted to vanquish seven myths about the fiction view: (1) it does not regard the products of science as falsehoods; (2) it allows that data can be part of models; (3) properly understood (as an account of the ontology of models, rather than their function) it is perfectly compatible with the fact that models represent; (4) it allows that we can learn important truths about the world from models; (5) it allows for this to involve mathematics; (6) it doesn’t misconstrue the function of scientific models in practice; and finally (7) it is no more politically problematic than many other philosophical and scientific ideas. Our hope then, is to have cleared away some important misconceptions about the view, thereby making it plain what the view does and does not entail about scientific modeling. This, of course, is not to say that the view faces no difficulties, or is obviously true. But if it’s going to be attacked or criticized, or if it’s going to be compared to other accounts of the ontology of scientific models, then hopefully such discussions can proceed without repeating any of the misconceptions we have discussed here. Acknowledgements We would like to thank Martin Zach and an anonymous referee for comments on an earlier draft, and Joe Roussos and Fiora Salis for helpful discussions when producing the manuscript. James Nguyen recognizes the support of the Jeffrey Rubinoff Sculpture Park for support during the preparation of this chapter. Thanks also to Alejandro Cassini and Juan Redmond for inviting us to contribute to this project.

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Salis, F. (2019). The new fiction view of models. The British Journal for the Philosophy of Science Online First. https://doi.org/10.1093/bjps/axz015. Salis, F., & Frigg, R. (2020). Capturing the scientific imagination. In P. Godfrey-Smith & A. Levy (Eds.), The scientific imagination: Philosophical and psychological perspectives (pp. 17–50). New York: Oxford University Press. Salis, F., Frigg, R., & Nguyen, J. (2020). Models and denotation. In C. Martínez-Vidal & J. L. Falguera (Eds.), Abstract objects: For and against (pp. 197–219). Cham: Springer. Shapiro, S. (1983). Mathematics and reality. Philosophy of Science, 50(4), 523–548. Stolnitz, J. (1992). On the cognitive triviality of art. The British Journal of Aesthetics, 32(3), 191– 200. Suppes, P. (2002). Representation and invariance of scientific structures. Stanford: CSLI Publications. Teller, P. (2009). Fictions, fictionalization, and truth in science. In M. Suárez (Ed.), Fictions in science: Philosophical essays on modeling and idealization (pp. 235–247). New York: Routledge. Thomasson, A. L. (1999). Fiction and metaphysics. New York: Cambridge University Press. Thomasson, A. L. (2020). If models were fictions, then what would they be? In A. Levy & P. Godfrey-Smith (Eds.), The scientific imagination: Philosophical and psychological perspectives (pp. 51–74). New York: Oxford University Press. Thomson-Jones, K. (2005). Inseparable insight: Reconciling formalism and cognitivism in aesthetics. Journal of Aesthetics and Art Criticism, 63(4), 375–384. Thomson-Jones, M. (2005). Idealization and abstraction: A framework. In M. Thomson-Jones & N. Cartwright (Eds.), Idealization XII: Correcting the model-idealization and abstraction in the sciences (pp. 173–218). Amsterdam: Rodopi. Thomson-Jones, M. (2010). Missing systems and face value practice. Synthese, 17(2), 283–299. Thomson-Jones, M. (2020). Realism about missing systems. In A. Levy & P. Godfrey-Smith (Eds.), The scientific imagination: Philosophical and psychological perspectives (pp. 75–101). New York: Oxford University Press. Toon, A. (2012). Models as make-believe: Imagination, fiction and scientific representation. Basingstoke: Palgrave Macmillan. Vorms, M. (2011). Representing with imaginary models: Formats matter. Studies in History and Philosophy of Science, 42(2), 287–295. Vorms, M. (2012). Formats of representation in scientific theorising. In P. Humphreys & C. Imbert (Eds.), Models, simulations, and representations (pp. 250–273). New York: Routledge. Walton, K. L. (1990). Mimesis as make-believe: On the foundations of the representational arts. Cambridge, MA: Harvard University Press. Winsberg, E. (2009). A function for fictions: Expanding the scope of science. In M. Suárez (Ed.), Fictions in science: Philosophical essays on modeling and idealization (pp. 179–191). New York: Routledge.

Roman Frigg is Professor of Philosophy at the London School of Economics. He was Associate Editor of The British Journal for the Philosophy of Science and member of the steering committee of the European Philosophy of Science Association. He has developed the fiction view of models and the DEKI account of scientific representation. He is the author (with James Nguyen) of Modelling nature: An opinionated introduction to scientific representation (Cham: Springer, 2020). James Nguyen is the Jacobsen Fellow at the Department of Philosophy, University College London and the Institute of Philosophy, University of London. He is also a Research Associate at the Centre for Philosophy of Natural and Social Science, London School of Economics and Political Science. He is the author (with Roman Frigg) of Modelling nature: An opinionated introduction to scientific representation (Cham: Springer, 2020).

Chapter 7

Bridging the Gap: The Artifactual View Meets the Fiction View of Models Fiora Salis

Abstract Fiora Salis compares the fictional and the artifactual views of models. She argues that both accounts contain several deep insights concerning the nature of scientific models but they also face some difficult challenges. She then puts forward an account of the ontology of models intended to incorporate the benefits of both views avoiding their main difficulties. Her key idea is that models are human-made artifacts that are akin to literary works of fiction. In this view, models are complex objects that are constituted by a model description and the model content generated within a game of make-believe. As per the fiction view, model descriptions are construed as props in a game of make-believe, where props are concrete objects that prescribe certain imaginings. As per the artifactual view, model descriptions are construed as concrete representational tools that enable and constrain a scientist’s cognitive processes and provide intersubjective epistemic access to their imaginings. Keywords Artifactualism · Fictionalism · Model descriptions

7.1 Introduction What are scientific models? It is common to distinguish between two main types of models, material and theoretical. Material models are ordinary objects that serve as representations of physical systems—or target systems. Among these are, for example, Kendrew’s plasticine model of myoglobin (Frigg and Nguyen 2016), the hydraulic model of the San Francisco Bay (Weisberg 2013), the Newlyn-Phillips hydraulic model of the economy (Morgan 2012) and model organisms of different sorts (Leonelli and Ankeny 2013). Theoretical models do not exist as ordinary objects, yet they also serve as representations of physical systems. Among these are Newton’s model of the Solar System (Frigg 2010a), the Lotka-Volterra model of predator-prey interaction (Weisberg 2013), Goldbeter’s model of circadian rhythm (Bechtel and Abrahamsen 2010) and Turing’s model of the embryo (Levy 2015). F. Salis (B) Department of Philosophy, University of York, Heslington YO10 5DD, York, UK e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Cassini and J. Redmond (eds.), Models and Idealizations in Science, Logic, Epistemology, and the Unity of Science 50, https://doi.org/10.1007/978-3-030-65802-1_7

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Theoretical models typically involve linguistic and mathematical descriptions—or model descriptions—that specify a simplified surrogate system—or model system— to be studied for the ultimate purpose of learning about reality. Material models are concrete, physical systems that do not raise any ontological concerns beyond those related to the nature of physical objects. Theoretical models, however, do raise serious concerns. What are they exactly? In this chapter I will answer this question by integrating two major approaches, the fiction view and the artifactual view of models, into a new account of models as human-made artifacts that are akin to literary works of fiction. Call this new proposal the integrated fiction view of models. The contemporary English noun ‘fiction’ originates from the Latin verb ‘fingere’, which has several meanings. Among these, two are relevant for our present discussion: fingere as imagining, supposing and hypothesising, and fingere as forming, shaping and creating. These different meanings correspond to two different notions of fiction. The former corresponds to the notion of fiction as a construct of the imagination and the object of an imaginative response. The latter corresponds to the notion of fiction as a human-made object or artifact. The fiction view of models is inspired by the literature on fiction in aesthetics—including Walton’s (1990) theory of fiction as make-believe—that assumes the first notion of fiction as imagination. The artifactual view of models is akin to the artifactual view of fiction in metaphysics—including Thomasson’s (1999) artifactual theory of fiction—that assumes the second notion of fiction as a human-made artifact. As I will argue, however, the two notions can be naturally integrated to produce a better theory that avoids certain important challenges faced by the fiction view and the artifactual view of models. According to the fiction view, theoretical models are akin to the imaginary objects (people, places, events) of literary fictions. Godfrey-Smith originally emphasized that modelers often think of themselves as describing ‘imaginary biological populations, imaginary neural networks, or imaginary economies’ which ‘might be treated as similar to […] the imagined objects of literary fiction’ (2006, 735). Indeed, modelers do think and talk about model systems as if they were concrete objects. This is what Deena Skolnick-Weisberg calls the folk ontology of models (cited in Godfrey-Smith 2006, 735), which is integral to what Thomson-Jones (2010) calls the face-value practice of modeling. Model descriptions involve the attribution of properties that only concrete objects can have, yet there are no objects instantiating those properties. Scientists, of course, know this. They merely imagine that there are some concrete systems having such and such properties. The face-value practice is a claim about scientific cognition as a social phenomenon that cannot be reduced to a mere psychological account of how individual modelers think in their own idiosyncratic and subjective way. Godfrey-Smith emphasizes that model-based science ‘has sociological and formal features, as well as psychological ones’ (2006, 728) and emphasizes that he is not interested in providing an account of the psychological mechanisms

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underlying model-based reasoning.1 His focus is on the social practice of modelbased science, and hence on the ways in which scientists think and talk about models as members of specific scientific communities. Godfrey-Smith identifies a special strategy of model-based science that is composed of two steps. First, modelers present a model description that specifies an imaginary system to be studied for the purpose of learning about reality. Second, they translate what they have learned about the imaginary system into knowledge of reality via model-world comparisons that are based on a relation of resemblance. On this approach, modelers ‘gain understanding of a complex real-world system via an understanding of simpler, hypothetical system that resembles it in relevant respects’ (2006, 726). This hypothetical system is different from reality and it is furthermore interpreted as a representation of reality. For this reason, the strategy of model-based science is described as a strategy of indirect representation.2 This idea led to a variety of accounts approaching theoretical models as fictions. Despite their many differences, standard proposals appeal to Walton’s (1990) theory of fiction as a game of make-believe (Barberousse and Ludwig 2009; Frigg 2010a, b; Frigg and Nguyen 2016, 2017; Levy 2015; Toon 2012; Salis 2016, 2019 ). Makebelieve is a social activity involving imaginings that are highly constrained by props and principles of generation.3 Props are concrete objects that prescribe to imagine certain things. Principles of generation are rules of inference that are understood as being in force in a particular game. Together, the props and the principles of generation determine, enable and constrain the imaginings of those playing the game. These imaginings are the fictional truths—or f -truths—of the game. They are the propositions that are true in the game of make-believe and hence fictional. In modeling, model descriptions are the props that prescribe to imagine that some particular system—the model system—is so and so. Typically, they are concrete linguistic descriptions and mathematical formulas (such as marks on a paper or on a computer screen). The model system is an imaginary object that is manipulated and developed in the imagination by drawing certain inferences according to the principles of generation. On the fiction view, models are the imaginary systems specified by model descriptions and these imaginary systems are the vehicles of representation of targets. Understanding how modeling works requires focusing on how these imaginary systems are constructed and manipulated. And this, in turn, requires focusing on the cognitive aspects of modeling, namely on the imaginative abilities of modelers as participants in games of make-believe. The fiction view of models is often developed together with an antirealist approach to imaginary systems according to which there are no such systems. This generates two main ontological problems. First, if models are identified with model systems and model systems do not exist then models do not exist. Second, if model systems 1 See the contrast he explicitly draws between his approach and the psychological account of model-

based reasoning advanced by Nersessian (1999). Weisberg (2007) for a similar idea that does not appeal to the analogy with fiction. 3 See Salis and Frigg (2020) for a thorough analysis of make-believe in the context of the scientific imagination. 2 See

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are the vehicles of representation of targets and there are no model systems then models cannot represent. Furthermore, Knuuttila (2017)argues that the fiction view faces three main coordination problems generated by the separation between model descriptions and imaginary systems. First, the fiction view cannot explain how imaginary systems relate to the descriptions that specify them. Second, it cannot explain how different scientists can have imaginings that are about the same imaginary system. Third, it cannot explain how imaginary systems relate their targets in terms of resemblance. To avoid these difficulties, Knuuttila puts forward the artifactual view of models as an alternative according to which model descriptions are ‘irreducible parts of any representational vehicle, such as a model’ (2017, 12). On this view, scientific models are autonomous objects that can be conceived of as human-made artifacts endowed with particular functions in a given context (Boumans 2012; Currie 2017; Knuuttila 2009, 2011, 2017; Knuuttila and Loettgers 2017; Knuuttila and Voutilanen 2003). Models involve model descriptions as representational tools that are among their essential components. Representational tools are concrete, material objects that can be perceived by an agent and have informational content. In order to understand this content, an agent must master the rules of interpretation that are in force in the particular episode of modeling. These rules depend upon the format of the representational tools (which pertain to some symbol system with its own objective syntactic and semantic rules) and on the agent’s cognitive abilities and purposes in context.4 Knuuttila explicitly criticizes the fiction view because of its identification of model systems as the vehicles of representation of targets and because of the separation of model systems from model descriptions. She emphasizes that the fiction view is ‘bound to approach modeling as a primarily mental activity, and the activity of using external representational means as that of merely (partially) describing the mental content of scientists’ (2017, 3). The artifactual view, instead, focuses on the external representational tools that enable and extend scientific reasoning. While Knuuttila recognizes the commonalities between models and fictions, she argues that the focus should be on the fictional uses of models rather than considering models as fictions. The implication of this move is that ‘the artifactual account renders visible how the characteristics of the various representational tools used in model construction enable and shape scientific imagination, also delimiting what can be thought of’ (2017, 3). That is, representational tools enable and at the same time constrain what can be imagined. For this reason, they should be at the centre of an account of what models are. Knuuttila submits that the artifactual view is independent from the framework of make-believe and that imagination in modeling does not play any crucial role. This disregard for the role of imagination raises four challenges for the artifactual view. First, the view cannot explain how models are built and developed coherently with the face-value practice of modeling. Second, it cannot account for the attribution of

4 Vorms (2011)

emphasizes the crucial role of the format of a representational tool for the inferences that can be drawn from it.

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concrete properties to model systems. Third, it cannot clarify the notion of representation involved in the identification of model descriptions as representational tools. Fourth, it cannot explain how scientists engage in model-world comparisons. In this chapter, I will present a theory of theoretical models that integrates the key insights of the fiction view with those of the artifactual view and overcomes the specific challenges that each of them faces. On this new proposal, models are akin to fictional stories, not fictional objects. They are complex objects constituted by model descriptions and the model propositional content. Model descriptions are concrete representational tools that function as props in a game of make-believe. Together with the principles of generation, they determine the model content. Thus, on this new integrated fiction view, both model descriptions and model content are essential components of models.5 In what follows I will critically assess Frigg and Nguyen’s (2016) most recent version of the fiction view of models (Sect. 7.2) and Knuuttila’s (2017) recent version of the artifactual view (Sect. 7.3). Then, I will present the integrated fiction view of models (Sect. 7.4).

7.2 The Fiction View of Models What are models according to the fiction view? The word ‘model’ is ambiguous between different uses. Some upholders of the fiction view identify models with model systems and, coherently with the strategy of indirect representation, argue that model systems are the vehicles of representation of targets (Godfrey-Smith 2006; Frigg 2010a, b). Since model systems are akin to the imaginary entities of fictional stories, the approach raises the usual metaphysical controversies surrounding fictional entities.6 On the one hand, realists about model systems argue that they are abstract created entities (Giere 1988, 2009; Knuuttila 2009; Thomasson 2020; Thomson-Jones 2020). On the other hand, antirealists about model systems argue that there are no such entities (Frigg 2010a, b; Frigg and Nguyen 2016, 2017). Both approaches face the problem of how to make sense of the concrete character of model systems. Abstract objects cannot have the sort of concrete properties that modelers attribute to them (Hughes 1997; Salis 2016). And non-existent objects cannot have any properties (Salis 2016). Standard solutions have been offered in terms of imagination and games of make-believe. Scientists merely imagine that some (abstract or non-existent) object is so and so. These imaginings can be assessed as correct (or incorrect) if they conform (or do not conform) to the rules of the particular game of make-believe they are playing. To avoid the metaphysical controversies mentioned above, Toon (2012) and Levy (2015) reject model systems and identify models with model descriptions. On this 5 See

Salis (2019) for a similar development of the analogy between models and literary works of fiction that does not integrate the original insight of the artifactual view of models. 6 See Salis (2013a) for a review of these controversies.

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view, model descriptions prescribe imaginings that are directly about real systems as having properties and features that they do not really have. The main problem with this proposal is that it cannot account for the indirect strategy of model-based science and for the typical surrogative reasoning that is enabled by models (Frigg and Nguyen 2016; Knuuttila 2017).7 It is widely agreed that one of the pivotal features of modeling is the fact that models facilitate reasoning that proceeds on the basis of surrogate systems (Barberousse and Ludwig 2009; Frigg and Nguyen 2016, 2017; GodfreySmith 2006; Knuuttila 2017; Mäki 2011; Suárez 2009; Swoyer 1991; Weisberg 2007, 2013). Scientists present model descriptions that specify model systems as the objects of study. It is in virtue of manipulating, investigating and developing these surrogate systems in the imagination that they can formulate hypotheses about reality. In their recent contribution to this debate, Frigg and Nguyen (2016) endorse the analogy between theoretical models and fictions coherently with the strategy of indirect representation. In the case of physical models, the vehicle of representation is a concrete system constructed and manipulated in reality; in the case of theoretical models, the vehicle of representation is an imaginary system constructed and manipulated in the imagination. Frigg and Nguyen’s ultimate goal is to offer a pragmatic theory of models as representations according to which models are objects that an agent can endow with a representational function. For this purpose, they put forward the DEKI account of representation, named after its four elements: denotation, exemplification, keying up and imputation. Denotation is a two-place relation between a vehicle of representation—what they call an O-object (where ‘O’ stands for the specification of the particular kind of thing the object is)—and the object it represents (the target). Exemplification is a mode of reference involving properties that an object instantiates, that are epistemically accessible and that are made relevant by context. Exemplification is selective because not all properties that an object instantiates are contextually relevant and epistemically accessible. Imputation is usually a matter of stipulation. An agent imputes certain properties exemplified by the model to the target. However, model and target are two very different things. Thus, an interpretation key—such as resemblance—is needed to associate properties exemplified by the model with properties of the target. DEKI builds on Goodman (1976) and Elgin’s (2004, 2010) theory of pictorial representation, which relies on the notion of representation-as. Representation-as involves a vehicle X representing a target system Y as a Z. Material models such as Kendrew’s plasticine model fit well with this scheme: a plasticine rope (X) represents myoglobin (Y ) as a folded chain of amino acids (Z). The plasticine rope has many different properties and it can be described in many different ways (as a plasticine rope or as an object made of calcium, salt and petrol). It becomes an O-object when it is described in a particular way (say as a plasticine rope). The O-object becomes a Z-representation when O-properties are interpreted as Z-properties. Obviously, a plasticine rope cannot instantiate the property of being a folded chain of amino acids. Thus, imagination plays an important role already in this concrete case, where 7 See Salis (2019) for a thorough criticism of the direct fiction view of models that identifies further

specific problems.

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interpretation requires that one imagine the plasticine rope to be a folded chain of amino acids. Imagination plays a more complex role in the case of theoretical models. These models present a serious challenge to DEKI because they do not involve any physical object X that represents a target Y as a Z. Inspired by Walton’s (1990) theory of fiction as a game of make-believe, Frigg and Nguyen suggest that theoretical models involve a model description D—the prop—prescribing to imagine that so and so is the case and thereby generating an imaginary system that is akin to the fictional characters of literature. This fictional system, they argue, is the vehicle of representation X—the O-object—that can be further interpreted as a Z. For example, Newton’s model of the Sun-Earth system generates an imaginary two-body system (X) that represents the Sun-Earth system (Y ) as the Sun and the Earth being attracted to each other by the force of gravity (Z). Thus, imagination plays a role both in the specification of X as the imaginary system and in the interpretation of this system as a Z. In the game of make-believe for Newton’s model, the imaginary two-body system is composed of two homogeneous perfect spheres, one large and one small, attracted to each other with a certain force proportional to 1/r 2 . This is then interpreted as a Z by further imagining that the large sphere is the Sun and the small sphere is the Earth and that they are attracted to each other by gravity.8 On this view, all scientific models, be they physical or theoretical, are Zrepresentations. In this sense, DEKI offers a uniform account of models that construes them as complex objects composed of a representational vehicle X together with an interpretation I, orM = X, I . Interpretation is a bijective function that maps Oproperties onto Z-properties (properties of the two-body system onto properties of the Sun-Earth system). X is the vehicle of representation, which can be physical (the plasticine rope) or fictional (the two-body system). In both cases, imagination is needed for interpretation. In the case of theoretical models, it is also needed for the specification of the vehicle of representation. The DEKI account of theoretical models faces five problems. First, DEKI is neutral with respect to the ontology of fictional entities, but Frigg and Nguyen express their preference for an antirealist interpretation according to which there are no such entities. This, however, raises one important ontological problem for theoretical models. If models are complex objects constituted by a representational vehicle X and an interpretation I, but for theoretical models X does not exist, then the only component of a theoretical model MT is the interpretation I, namelyMT = _, I .9 That is, there is a gap in the place where X should be (the O-object is fictional). One could endorse some form of realism about fictional objects and hold that X is some 8 According

to Frigg and Nguyen, these two steps correspond to two parts of the model description D, namely DX (where ‘X’ stands in for the vehicle X) and DI (where ‘I’ stands for the interpretation that transforms X into a Z-representation). They notice, however, that DX and DI are not always separated as in Newton’s model. In some cases, the imaginary system has the relevant properties and the interpretation is one of simple identity. For example, in Fibonacci’s model of population growth DX specifies a rabbit population and the model is a rabbit-population-representation. 9 An anonymous referee further notices that M = _, I  looks like a structure without domain, T which is not a structure at all.

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sort of abstract entity. However, the motivation for endorsing realism about fictional entities should be independent of the particular issue at stake, or the position would be legitimately deemed ad hoc. A second problem for DEKI concerns the representational relation between imaginary systems and reality. Salis (2019) emphasizes that if imaginary systems are the vehicles of representation and if representation involves a denotation condition, then theoretical models cannot represent. Denotation, as stated above, is a two-place relation between a vehicle of representation and the object it represents. According to Frigg and Nguyen, however, the vehicle of representation is X, which does not exist. This entails that there cannot be any denotation relation between models and targets. Again, one could assume realism about fictional entities, but this should be justified independently of this particular problem. Knuuttila identifies three coordination issues concerning the fiction view of models, which apply also to DEKI. On her view, ‘The problem is how the imaginings of scientists, which are mental phenomena, and as such not intersubjectively available, can be coordinated with other dimensions of modeling practice’ (2017, 9). This problem effectively boils down to the problem of intersubjective epistemic access to models as imaginary systems. The first of these coordination issues concerns the relation between imaginary systems and reality. As stated in the Introduction, according to Godfrey-Smith’s original version of the fiction view, this is to be explained in terms of a relation of resemblance. How are imaginary systems compared to target systems? Since he voices his preference for a deflationary ontology of imaginary objects according to which they do not exist, they cannot have the sort of properties that they purportedly share with their targets. Frigg and Nguyen (2016) emphasize that DEKI does not face the problem because it explains the way in which scientists learn about reality through modeling in terms of the notions of imputation and keying-up. Modelworld comparisons do not play any fundamental role in their account. Nevertheless, modelers do engage in model-world comparisons for different purposes and these are an important part of the modeling practice. Frigg and Nguyen (2016, 239) refer to Salis (2016)for possible solutions that would be fully compatible with DEKI. On her account, model-world comparisons (and other sorts of comparisons including those between models) require an analysis in terms of imagination and make-believe. The resemblance relation between model systems and reality thus comes down to a sort of imagined resemblance (or imagined similarity) that is construed and constrained by the rules of an unofficial game of make-believe involving ad hoc rules stipulated by scientists. The second coordination issue concerns the relation between imaginary systems and model descriptions. On the fiction view, theoretical models are imaginary systems that are specified by model descriptions. The DEKI account of representation fully endorses this interpretation and construes the vehicles of representation in theoretical models as fictional. Knuuttila, however, argues that model descriptions have a fundamental epistemic role that is neglected by the fiction view. Model descriptions enable scientists to intersubjectively access imaginary systems that would otherwise remain epistemically inaccessible to the scientific collective. To emphasize her

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point, she notices that changes in model description provide epistemic access to distinct model systems. The example she mentions is provided by Weisberg and Reisman’s (2008)reconstruction of the Lotka-Volterra model of predation. The original model treats predation as a population-level phenomenon investigated through two famous differential equations. A more recent individual-based model studies the same phenomenon at an individual-level. This model involves a set of state variables for each individual within the predator and prey populations, and a set of assumptions about how individuals behave and interact over time. In this case, the same phenomenon is investigated computationally rather than analytically. Knuuttila posits that the upholder of the fiction view ‘would presumably assume that the two different mathematical representations describe the same imagined system, as one of the benefits of the fictional approach is precisely to maintain the identity of the model system under different descriptions’ (2017, 9). But I doubt it. In this particular case, the imaginary systems described by the two models are clearly different because they (fictionally) instantiate different properties. For example, the individual-based model describes an imaginary system where individual predators move randomly, while the original Lotka-Volterra model describes an imaginary system involving no individuals moving in any way. In fact, the upholder of the fiction view would agree with Knuuttila in holding that different model descriptions (of the kind specified in the example) give us access to different imaginary systems. This may be a good reason to include model descriptions as essential components of what models are. In fact, this is what I will do in Sect. 7.4. But for the moment, let us focus on the last issue. The third coordination issue relates to the previous one and concerns scientists’ intersubjective identification of the same imaginary system. The worry, anticipated by Weisberg (2013), is that imaginings are private mental states that can vary from scientist to scientist. So, how can we be sure that different scientists imaginatively engaging with the same model effectively imagine the same fictional system? Upholders of the fiction view argue that scientists’ imaginative engagement with models is constrained by the rules of the relevant game of make-believe. As Frigg puts it, ‘[a]s long as the rules are respected, everybody involved in the game has the same imaginings’ (2010a, 264). Knuuttila, however, submits that the rules themselves are in fact made intersubjectively available to scientists by the particular representational tools that specify imaginary systems. In the language of Walton’s theory, the props—the model descriptions—prescribe to imagine that so and so is the case. Without these, scientists could not construct and intersubjectively investigate the same imaginary system. Furthermore, modelers themselves recognize the importance of model descriptions, especially of different mathematical representations that enable the specification of different properties of model systems (as illustrated in the previous paragraph). Thus, the focus should be on model descriptions as the representational tools that enable and constrain the imaginings of scientists rather than on the imaginary systems themselves. The standard fiction view of theoretical models identifies models with imaginary systems created through the games of make-believe played by scientists. This generates the first two problems discussed above. Furthermore, it distinguishes between

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model descriptions and the imaginary systems they specify. This generates the three coordination issues identified by Knuuttila (2017). In response to the last three of these problems, Knuuttila puts forward the artifactual view as an alternative that under reasonable adjustments could integrate the original insight of the fiction view (model descriptions specify imaginary systems) without posing the same problems. As I will argue in the next section, the artifactual view advances some important ideas about the nature of models and how they represent reality. However, Knuuttila explicitly argues that the view is independent of the make-believe framework and does not need to rely on the role of imagination to explain what models are and how scientists deploy them. This raises certain important challenges that will need to be answered through the development of a better proposal that integrates the key insights of both views.

7.3 The Artifactual View of Models Upholders of the artifactual view construe models as concrete human-made artifacts that function as epistemic tools and as objects of enquiry. As concrete human-made artifacts, models require human agency for their creation and also for their endowment with some particular function. As epistemic tools, they enable scientists to learn about reality when they are used as instruments of investigation and interpretation of selected aspects of the world in view of some particular epistemic purpose. As objects of enquiry, they facilitate and constrain surrogative reasoning through their materiality. In this section, I will explore these ideas through paradigmatic development of this proposal put forward by Knuuttila (2017), which she explicitly presents as an alternative to the fiction view of models discussed above. Knuuttila offers a uniform account of both physical and theoretical models as concrete human-made artifacts that are endowed by an agent with a specific function in a given context. In both cases, modelers create and manipulate concrete entities, which can be three-dimensional objects—such as a plasticine rope supported by wooden rods in Kendrew’s model of myoglobin—or linguistic and mathematical marks on paper—such as linguistic descriptions and mathematical equations in Newton’s model of the Sun-Earth system. On this view, all models (be they theoretical or material) have both abstract and concrete dimensions. These dimensions are provided by their representational modes and their representational media. For example, Kendrew’s plasticine model involves a certain 3D-configuration (representational mode) that inhabits a plasticine rope (a representational medium). Newton’s model involves linguistic and mathematical configurations (representational media) that can inhabit different representational media, including ink on paper or electrical signals generating certain configurations on a computer screen. These objects, like all artifacts, can be used for a variety of purposes. Kendrew’s plasticine model can be used as a representation of the structure of myoglobin (in the Cambridge Laboratory for Molecular Biology) or as an historical item to observe and study for educational

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purposes (in the Science Museum in London). And Newton’s model of the SunEarth system can be used to calculate the orbit of the Earth around the Sun (in an astronomical observatory) or as an example of an application of classical Newtonian mechanics (in a physics textbook). Knuuttila emphasizes that scientists construct models by using external representational tools that are the material embodiment of their ideas and thoughts. This material embodiment is an irreducible part of the model that plays two main roles. First, it constrains and enables a scientist’s cognitive processes (of reasoning, memorisation, computation and demonstration). Second, it provides a vehicle of communication within the scientific community. As she puts it: ‘Scientists do not read the minds of each other, and neither are they able to process even modestly complicated relations or interactions between different components without making use of external representational scaffolding’ (2017, 12). Concerning the first role, the materiality of models constrains and at the same time enables possible interpretations and uses, including surrogative reasoning. To function in this way, models require an agent to create them with a particular purpose, which furthermore explains what features of models are judged as relevant. These constraints and affordances reveal models as intersubjectively available objects of investigation that can function also as knowledge objects. We can learn about Newton’s model by exploring and manipulating its linguistic and mathematical assumptions, and this is enabled and constrained by its materiality, that is by the representational means (linguistic description and mathematical formulae) that constitute it. For example, Newton’s model describes a two-body gravitational system composed of two perfect spheres interacting with each other. The model enables us to calculate the orbit of the Earth around the Sun when applied to the Sun-Earth system. Concerning the second role, the materiality of models enables communication among scientists by providing intersubjective epistemic access to their cognitive processes. The three coordination problems discussed in the previous section concern different aspects of the same issue: how can scientists intersubjectively access a model’s imaginary system? Knuuttila submits that the representational means of models give scientists intersubjective access to the model systems that are built and manipulated in surrogative reasoning. However, she rejects the idea that model systems are imagined, non-existent entities that are created through the imaginings of scientists in games of make-believe. Inspired by Thomasson’s (1999) account of fictional characters as abstract artifacts created by authors through the activity of storytelling, Knuuttila thinks of model systems as abstract artifacts created by scientists through the activity of modeling. Fictional characters depend on fictional stories for their existence. For example, the Anna Karenina character can exist only as long as there is some copy of the homonymous book by Lev Tolstoy or some other material rendering of the story (film, theatre piece, oral communication, etc.). Analogously, fictional systems ontologically depend on material representational tools such as scientific papers, textbooks, oral presentations and communications. Knuuttila claims that, contrary to the fiction

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view of models, ‘the artifactual theory does not need to invoke pretension’ (2017, 16). This, however, generates the following four problems. First, the artifactual view cannot explain how scientists build and manipulate models. Knuuttila explains the practice of model building in terms of the purposeful selection of the representational means that are more apt to deliver the particular end that modelers have in view. While this is certainly crucial to an appropriate explanation of model building, it misses the important fact that any object needs a relevant interpretation to be used as a representation of anything. Mathematical equations and linguistic descriptions are not models unless interpreted in a certain way. A physicist could study the equation for Newton’s law of gravitation (Fg = Gm 1 m 2 ) without any reference to the Sun-Earth system. Any particular inscription r2 of the equation on a piece of paper, a whiteboard or a computer screen is concrete, but it is not a representation of any particular system unless it is used under some interpretation for some purpose. For example, the equation can be used in Newton’s model of the Sun-Earth system if Fg is interpreted as standing for the force of gravity, m 1 and m 2 for the masses of the idealized Sun and Earth respectively, r for the distance between the two, and G for the gravitational constant. In this particular case, applying the equation requires more than a simple mapping of the symbols to the objects they stand for. It requires a number of idealizing assumptions. Among these, for example, are the assumptions that the Sun and the Earth are perfect spheres, that they have a homogenous mass distribution, that gravity is the only force relevant to the Earth’s motion and that the Earth does not have any gravitational interaction with anything else. These idealized assumptions specify an imaginary system that differs in relevant respects from the model target. Explaining how this particular system is built and manipulated requires an appeal to imagination and the framework of make-believe as a social imaginative activity with a normative and objective character. Second, the artifactual view cannot explain how scientists can attribute concrete properties to model systems. As noticed above, abstract objects cannot have the sort of concrete properties that model systems are supposed to instantiate. They can only have such properties within the games of make-believe played by scientists. Thomasson (2003) recognizes the problem in the context of literature and suggests that a Waltonian account of fictional discourse in terms of pretense and imagination contributes a plausible explanation of the similar ways of thinking and talking about fictional characters in stories. For example, we say that Frankenstein’s monster was a creation of Dr Frankenstein, which is true in the fiction without being true. In reality, Frankenstein’s monster was a creation of Mary Shelley. Thomasson argues that the latter claim is true (Frankenstein’s monster was created by Mary Shelley), while the former claim (Frankenstein’s monster was a creation of Dr Frankenstein) is only fictionally true—or true in the game of make-believe for Shelley’s novel. A similar distinction is necessary to explain scientists’ ways of thinking and talking about fictional systems. For example, consider our ways of thinking and talking about the fictional system specified by Newton’s model. We say that the two-body gravitational system is composed of two homogeneous perfect spheres, one large and one small, attracted to each other with a certain force proportional to 1/r 2 . This is true in the model, or fictionally true, without being true. In reality, on Knuuttila’s realist

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construal, the two-body gravitational system is an abstract object that cannot have any concrete properties. Perfect spheres cannot be large or small and they cannot be attracted to each other with any force. The representational means that are irreducible components of the model involve the attribution of concrete properties to abstract objects that cannot instantiate them. It is therefore necessary to appeal to imagination and make-believe to make sense of this attribution in modeling. Third, the artifactual view cannot explain an important distinction between two types of representational relations within modeling. Originally, Frigg (2010b, 100) noticed that “there are two very different relations that are commonly called ‘representation’ and a conflation between the two is the root of some of the problems that (allegedly) beset scientific representation”. The first relation is between model descriptions and model systems. The second relation is between models and their targets. The latter is usually explained in terms of denotation. The former, however, is quite distinct in that model descriptions do not stand in any denotation relation with model systems. Explaining this notion of representation requires reliance on the resources provided by the framework of make-believe.10 Fourth, the artifactual view cannot explain model-world comparisons. As stated above, modelers do engage in these comparisons. Yet, they are difficult to explain without appealing to some sort of imagination and pretense. In the case of DEKI, that was due to the inexistence of a model system having the sort of properties that can be shared with a real system. In the case of Knuuttila’s artifactual view, the problem originates from the fact that abstract objects (the model systems that she interprets coherently with Thomasson’s (1999) ontology of fictional entities) cannot have any of the concrete properties that modelers supposedly consider when comparing them with real world systems. Here I also refer the reader to Salis (2016), who describes different solutions involving the notion of imagination and make-believe in the case of abstract artifacts. The artifactual view of models advances several important ideas. In particular, it recognizes the crucial role of the materiality of models in enabling and constraining a scientist’s cognitive processes and in providing the vehicle of communication within the scientific community. However, the view faces also a number of problems generated by its failure to recognize the relevance of the scientific imagination for model building and development, for the attribution of properties to model systems, for the interpretation of the distinct types of representation relations involved in modeling and for the explanation of model-world comparisons. In the next section I will explain how these problems can be overcome within a novel approach to models as fictions that integrates the original insights of the artifactual view of models and of the standard fiction view.

10 An anonymous referee notices that this distinction holds only if one upholds an indirect view of representation. As mentioned above, the direct view of representation does not fit well with the face-value practice of modeling. In particular, it cannot account for the indirect strategy of modelbased science identified by Godfrey-Smith (2006) and for the typical surrogative reasoning that is enabled by models. Knuuttila (2017) explicitly recognizes the latter point as especially problematic.

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7.4 The Integrated Fiction View of Models So, what are scientific models? Here I propose a new account that preserves the two main insights of the fiction view and of the artifactual view. Coherently with the fiction view, I hold that model descriptions specify imaginary systems that are akin to the characters of literary fictions. Coherently with the artifactual view, I hold that models are concrete human-made artifacts that can be endowed with distinct functions in different contexts. Knuuttila rejects the notion of fiction as imagination in favour of the notion of fiction as a human-made artifact. But in fact, the two notions are not incompatible and the view that I will put forward naturally reconciles them on the basis of a new analogy between models and fictions that recognizes the fundamental role of imagination. The integrated fiction view builds on the analogy between theoretical models and literary works of fiction. Literary works of fiction are human-made artifacts created through the author’s activity of storytelling. Fictional storytelling relies on imagination both in its making and in its reception. Neither the author of fiction nor her audience believes the content of a fictional story, they merely imagine it. Similarly, theoretical models are human-made artifacts created through a scientist’s activity of modeling. As recognized by the face value practice, modelers use model descriptions to specify model systems having properties that only concrete objects could have. Yet there are no such objects. Scientists know this. They use their imagination to build and manipulate surrogate systems for different purposes in different contexts. The standard fiction view was originally motivated by the face value practice of modeling. However, this view separates model descriptions from the imaginary systems they specify and it identifies models with the latter. In Sect. 7.2, I argued that this view faces five main problems. The first two emerge from Frigg and Nguyen’s (2016) antirealism about imaginary systems, which implies that theoretical models do not have any representational vehicle and that they cannot stand in a denotation relation with their targets. The other three (originally identified by Knuuttila (2017)) boil down to the problem of intersubjective epistemic access to models as imaginary systems, namely how imaginary systems relate to model descriptions, how different scientists can have imaginings that are about the same imaginary system and how imaginary systems relate to reality based on a relation of resemblance. To avoid these difficulties, Knuuttila puts forward the artifactual view of models as an alternative according to which model descriptions are irreducible constituents of models as representational vehicles. I agree with her conclusion that model descriptions must be included in a characterisation of what models are. However, Knuuttila makes the further claim that pretense and imagination are dispensable in an account of models as artifacts. This is where I disagree. In Sect. 7.3, I argued that failure to recognize the crucial role of imagination in modeling generates four problems concerning how scientists build and develop models, how they attribute properties to model systems, what representation relations are involved in modeling, and how to explain model-world comparisons. Solving these problems requires a different proposal that recognizes the essential role of imagination.

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Like other upholders of the fiction view, I also build my proposal on Walton’s (1990) theory of fiction as a game of make-believe. But I do this on the basis of a different analogy between theoretical models and literary works of fiction. Literary works of fiction are syntactic-semantic entities wherein certain syntactic symbols function as props that prescribe to imagine certain propositions. These propositions together with those generated via the principles of generation are the f -truths that constitute the story’s content. Analogously, theoretical models involve a model description, which is composed of certain linguistic and mathematical symbols functioning as props that prescribe to imagine certain propositions. These propositions are the primary f -truths of the model from which further implied f -truths can be inferred via the principles of generation. Together, the model’s primary and implied f -truths constitute the model’s content. Model descriptions prescribe imaginings that seem to be about some particular fictional systems. Imaginings, however, do not commit to the existence of any fictional entities. Coherently with Walton’s analysis, the integrated fiction view assumes antirealism about fictional entities, including model systems.11 On the integrated fiction view, a model is a complex object constituted by model description and model content. The model is analogous to a literary work of fiction, the model description is analogous to the text of a story and the model content is analogous to the content of a story. A literary work of fiction is a human-made artifact that is created through the imaginative activities of an author (or a group of authors). It involves certain symbolic devices (usually linguistic) that inhabit (or ontologically depend on) concrete representational media (typically, ink on paper). The symbols express a certain content (usually, but not exclusively, propositional). Models, like stories, are human-made artifacts created through an agent’s imagination. They involve model descriptions, which are symbolic devices (linguistic and mathematical) that rely on certain concrete representational media. They are indispensable constituents that enable and constrain the model content (propositional and mathematical) together with the principles of generation in force in the game of make-believe. Now we can see how this view helps solving the problems encountered by the standard fiction view and the artifactual view of models. By putting imagination at the centre of modeling, the integrated fiction view can explain the four problems faced by the artifactual view. Scientists build models by selecting the model descriptions that best serve the specific purposes they have in mind in a particular context. Model descriptions are mere symbols that need an interpretation to be used for any particular purpose. This interpretation is enabled by what Kind and Kung (2016) call ‘transcendent uses of imagination’, i.e. uses that divert from reality in some respects and to certain degrees for the purpose, in this case, of building and manipulating a surrogate system in the imagination. Model descriptions (through their concrete manifestations in representational media) are the concrete symbols that function like props prescribing certain imaginings. These imaginings constitute the model’s idealized linguistic and mathematical assumptions that specify an imaginary system. 11 For an explanation of the aboutness of our thought and discourse about fictional entities coherent with fictional antirealism see Salis (2013b, 2019) and Friend (2011, 2014).

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Scientists manipulate the imaginary system by manipulating the concrete interpreted symbols from which further implied fictional truths can be inferred via principles of generation. Different principles of generation may be needed in specific domains of scientific enquiry, including general theoretical principles and mathematical principles. It is a virtue of the framework of make-believe that it can accommodate a variety of principles (Salis and Frigg 2020). For example, the linguistic and mathematical descriptions Newton used when building the model of the Sun-Earth system are the props that prescribe imagining certain fictional propositions. The model description prescribes imagining that the force acting between the Sun and the Earth is gravity and that its magnitude is given by Newton’s law of gravity. Applying this law requires making a number of idealizing assumptions that specify an imaginary system wherein two perfect spheres attracted to each other are further imagined as being the Sun and the Earth attracted by gravity. To develop the model, Newton inserted his force law in his equation of motion to obtain the differential equation describing the Earth’s trajectory. By solving this equation under the relevant interpretation, we find that, in the model, the Earth moves on an elliptic orbit around the Sun. Property attribution can be also explained within this framework in terms of the model content. When looking at the face value practice, scientists seem to attribute concrete properties to imaginary systems. When scientists do this, they produce claims that involve the predication of such properties within a game of make-believe. So, they can say things like ‘the small sphere moves on an elliptic orbit around the large sphere’ or ‘two perfect spheres are attracted to each other by gravity’. Claims like these are only true in the model—or f -true—and do not require the postulation of any exotic entities to assess their conditions of correctness. The integrated fiction view provides also the theoretical resources to explain the representational relation between model descriptions and model systems. Model descriptions prescribe imagining that certain objects are so and so. This notion corresponds to what Frigg (2010a) calls ‘p-representation’, where ‘p’ stands in for prop. Walton’s theory of make-believe is neutral with respect to the ontology of fictional objects, which is why even Thomasson (2003) recognizes that in story telling the author of fiction engages in a game of make-believe. A similar move is available, and in fact even necessary, to the upholder of the artifactual view of models. Model descriptions are the props that, through their prescriptions to imagine, enable, and constrain an agent’s imaginings together with the principles of generation. Now we can also see how the integrated fiction view can solve the first two problems faced by the fiction view of models and in particular by the analysis proposed by DEKI. The integrated fiction view offers a characterisation of models as complex objects constituted by model descriptions and model content. The integrated fiction view identifies models with these complex objects, and these are the representational vehicles standing in a denotation relation with targets. Since both model descriptions and model content exist, models exist and they can therefore stand in such a relation. The integrated fiction view can also account for the three coordination problems identified by Knuuttila. The view is fully compatible with the analysis of

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model-world comparisons advanced by Salis (2016) in terms of games of makebelieve. The problem of the relation between model descriptions and model systems can be addressed by recognizing that the properties that model systems have are specified in the model content, which is partially constituted by the imaginings prescribed by model descriptions together with the principles of generation. Thus, model descriptions do not fully determine the properties of the imaginary system, but they contribute in an essential way to its constitution in the imagination and provide the material scaffolding that constraints and enables surrogative reasoning. Thus, construing model descriptions as props in a game of make-believe delivers an explanation of how they constrain and enable a scientist’s imagination. To build and manipulate a model for a specific purpose, scientists must stipulate certain prescriptions to imagine and then conform to them to derive the relevant outcomes from the initial assumptions and the principles of generation. The inclusion of model descriptions as essential constituents of models takes also care of the problem of communication among scientists thinking and talking about the same imaginary system. Intersubjective identification of the same imaginary system is enabled by reference to the same model content which is partially determined by model descriptions stipulating the relevant imaginings. Individual scientists may engage in idiosyncratic imaginings that may determine different imaginary objects. However, model descriptions (together with the principles of generation) impose the constraints that determine what one should imagine and thereby exclude any imaginings that do not conform. In this way, the integrated fiction view has unified the key insights of the standard fiction view and the artifactual view in a way that avoids the main challenges faced by each view. This new account combines the notions of fiction as imagination and fiction as a human-made artifact by building on the analogy between models and literary works of fiction. Literary works of fiction are syntactic-semantic artifacts that are created through the imaginative activities of authors. Analogously, theoretical models are complex objects constituted by model descriptions and their content, which are created through the imaginative activities of scientists for specific purposes in a given context. Models are intersubjectively available tools of enquiry and objects of knowledge that crucially rely on imagination and the social activity of make-believe for their construction and manipulation in particular scientific communities.

References Barberousse, A., & Ludwig, P. (2009). Models as fictions. In M. Suárez (Ed.), Fictions in science: Philosophical essays in modeling and idealizations (pp. 56–73). New York: Routledge. Bechtel, W., & Abrahamsen, A. (2010). Dynamic mechanistic explanation: Computational modeling of circadian rhythms as an exemplar for cognitive science. Studies in History and Philosophy of Science, 41(3), 321–333. Boumans, M. (2012). Mathematics as quasi-matter to build models as instruments. In D. Dieks, W. J. Gonzalez, S. Hartmann, M. Stöltzner, & M. Weber (Eds.), Probabilities, laws, and structures (pp. 307–318). Dordrecht: Springer.

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Currie, A. (2017). From models-as-fictions to models-as-tools. Ergo, 4(27), 759–781. Elgin, C. Z. (2004). True enough. Philosophical Issues, 14(1), 113–131. Elgin, C. Z. (2010). Telling instances. In R. Frigg & M. Hunter (Eds.), Beyond mimesis and convention: Representation in art and science (pp. 1–18). Berlin and New York: Springer. Friend, S. (2011). The great beetle debate. Philosophical Studies, 153(2), 183–211. Friend, S. (2014). Notions of nothing. In M. García-Carpintero & G. Martí (Eds.), Empty representations (pp. 308–332). Oxford: Oxford University Press. Frigg, R. (2010a). Models and fiction. Synthese, 172(2), 251–268. Frigg, R. (2010b). Fiction and scientific representation. In R. Frigg & M. Hunter (Eds.), Beyond mimesis and convention: Representation in art and science (pp. 97–138). New York: Springer. Frigg, R., & Nguyen, J. (2016). The fiction view of models reloaded. The Monist, 99(3), 225–242. Frigg, R., & Nguyen, J. (2017). Models and representation. In L. Magnani & T. Bertolotti (Eds.), Springer handbook of model-based science (pp. 73–126). Berlin: Springer. Giere, R. N. (1988). Explaining science: A cognitive approach. Chicago: The University of Chicago Press. Giere, R. N. (2009). Why scientific models should not be regarded as works of fiction. In M. Suárez (Ed.), Fictions in science: Philosophical essays on modeling and idealization (pp. 248–258). New York: Routledge. Godfrey-Smith, P. (2006). The strategy of model-based science. Biology and Philosophy, 21(5), 725–740. Goodman, N. (1976). Languages of art. Indianapolis and Cambridge: Hackett. Hughes, R. I. G. (1997). Models and representation. Philosophy of Science (Proceedings), 64, S325–S336. Kind, A., & Kung, P. (2016). Introduction. In A. Kind & P. Kung (Eds.), Knowledge through imagination (pp. 2–38). Oxford: Oxford University Press. Knuuttila, T. (2009). Representation, idealization, and fiction in economics: From the assumptions issue to the epistemology of modelling. In M. Suárez (Ed.), Fictions in science: Philosophical essays on modeling and idealization (pp. 205–231). New York: Routledge. Knuuttila, T. (2011). Modelling and representing: An artefactual approach to model-based representation. Studies in History and Philosophy of Science Part A, 42(2), 262–271. Knuuttila, T. (2017). Imagination extended and embedded: Artifactual versus fictional accounts of models. Special Issue of Synthese on Modeling and Representation. https://doi.org/10.1007/s11 229-017-1545-2. Knuuttila, T., & Loettgers, A. (2013). Basic science through engineering: Synthetic modeling and the idea of biology-inspired engineering. Studies in History and Philosophy of Biological and Biomedical Sciences, 44(2), 158–169. Knuuttila, T., & Voutilainen, A. (2003). A parser as an epistemic artifact: A material view on models. Philosophy of Science, 70(5), 1484–1495. Leonelli, S., & Ankeny, R. (2013). What makes a model organism? Endeavour, 37(4), 209–212. Levy, A. (2015). Modeling without models. Philosophical Studies, 172(3), 781–798. Mäki, U. (2011). Models and the locus of their truth. Synthese, 180(1), 47–63. Morgan, M. S. (2012). The world in the model: How economists work and think. Cambridge: Cambridge University Press. Nersessian, N. (1999). Model-based reasoning in conceptual change. In L. Magnani, N. Nersessian, & P. Thagard (Eds.), Model-based reasoning in scientific discovery (pp. 5–22). New York: Kluwer/Plenum. Salis, F. (2013a). Fictional entities. Online Companion to Problems in Analytic Philosophy. Retrieved from https://repositorio.ul.pt/bitstream/10451/10860/1/Fictional%20Entities3_Salis% 2c%20Fiora_Companion2013.pdf. Salis, F. (2013). Fictional names and the problem of intersubjective identification. Dialectica, 67(3), 283–301. Salis, F. (2016). The nature of model-world comparisons. The Monist, 99(3), 243–259.

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Salis, F. (2019). The new fiction view of models. British Journal for the Philosophy of Science. https://doi.org/10.1093/bjps/axz015. Salis, F., & Frigg, R. (2020). Capturing the scientific imagination. In P. Godfrey-Smith & A. Levy (Eds.), The scientific imagination: Philosophical and psychological perspectives (pp. 18–50). New York: Oxford University Press. Suárez, M. (2009). Fictions in scientific practice. In M. Suárez (Ed.), Fictions in science: Philosophical essays in modeling and idealizations (pp. 3–11). New York: Routledge. Swoyer, C. (1991). Structural representation and surrogative reasoning. Synthese, 87(3), 449–508. Thomasson, A. L. (1999). Fiction and metaphysics. Cambridge, MA: Cambridge University Press. Thomasson, A. L. (2003). Speaking of fictional characters. Dialectica, 52(2), 205–223. Thomasson, A. L. (2020). If models were fictions, then what would they be? In P. Godfrey-Smith & A. Levy (Eds.), The scientific imagination: Philosophical and psychological perspectives (pp. 52– 74). Oxford: Oxford University Press. Thomson-Jones, M. (2010). Missing systems and the face value practice. Synthese, 172(2), 283–299. Thomson-Jones, M. (2020). Realism about missing systems. In P. Godfrey-Smith & A. Levy (Eds.), The scientific imagination: Philosophical and psychological perspectives (pp. 76–101). Oxford: Oxford University Press. Toon, A. (2012). Models as make-believe: Imagination, fiction and scientific representation. Chippenham: Palgrave Macmillan. Vorms, M. (2011). Representing with imaginary models: Formats matter. Studies in History and Philosophy of Science Part A, 42(2), 287–295. Walton, K. L. (1990). Mimesis as make-believe: On the foundations of the representational arts. Cambridge, MA: Harvard University Press. Weisberg, M. (2007). Who is a modeler? The British Journal for the Philosophy of Science, 58(2), 207–233. Weisberg, M. (2013). Simulation and similarity: Using models to understand the world. New York: Oxford University Press. Weisberg, M., & Reisman, K. (2008). The robust Volterra principle. Philosophy of Science, 75(1), 106–131.

Fiora Salis is Associate Lecturer in Philosophy at the University of York and Associate Researcher at Centre for Natural and Social Science at the London School of Economics. She has held positions at the University of Lisbon and the London School of Economics, and visiting appointments at the Institute of Philosophy of the University of London and the University of Geneva. Her main research focus is on the nature and varieties of imagination, and on its distinct uses in artistic fictions, scientific models, and thought experiments. She also works on fiction and the semantics and pragmatics of fictional discourse, fictional names, mental files, episodic memory and imagery.

Chapter 8

Models as Hypostatizations: The Case of Supervaluationism in Semantics Manuel García-Carpintero

Abstract Manuel García Carpintero defends a form of antirealism for the explicit talk and thought both about fictional entities and scientific models: a version of StephenYablo’s figuralist brand of factionalism. He argues that, in contrast with pretense-theoretic fictionalist proposals, on his view, utterances in those discourses are straightforward assertions with straightforward truth-conditions, involving a particular kind of metaphors or figurative manner. But given that the relevant metaphors are all but “dead”, this might suggest that the view is after all realist, committed to referents of some sort for singular terms in the relevant discourses. He revisits these issues from the perspective of the more recent work on them and applies his view to recent debates in semantics on the role and adequacy of supervaluationist models of indeterminacy. Keywords Fictional entities · Antirealism · Semantics · Supervaluationism

8.1 Introduction In previous work García-Carpintero (2010a) I extended a fictionalist account of apparent reference to fictional characters in fictional discourse (García-Carpintero 2019a) to an equally fictionalist account of scientific models. Here I will return to it, by considering recent developments and relying on more recent work on fiction (GarcíaCarpintero 2013a, 2019b, c), learning from fiction (García-Carpintero 2019d) and fictional reference (García-Carpintero 2018, 2019a, e). Also, I will illustrate the Financial support was provided by the DGI, Spanish Government, research projects FFI2016-80588R and FFI2016-81858-REDC, and the award ICREA Academia 2018 funded by the Generalitat de Catalunya. A version of this paper was presented at the Workshop on Imagination and Fiction in Scientific Modelling, University of York, 2019; thanks to participants for their comments, in particular to the organizers Mary Leng and Fiora Salis; thanks also to Ali Abasnezhad and Marta Campdelacreu, and to Michael Maudsley for the grammatical revision. M. García-Carpintero (B) University of Barcelona, Barcelona, Spain e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Cassini and J. Redmond (eds.), Models and Idealizations in Science, Logic, Epistemology, and the Unity of Science 50, https://doi.org/10.1007/978-3-030-65802-1_8

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view by applying it to recent debates in semantics on the role and adequacy of supervaluationist models of indeterminacy. Salis (2019) offers desiderata for philosophical accounts of models that I find plausible, in addition to a view to deal with them rather close to mine. As she points out— following Godfrey-Smith (2006) and others—thinking of models as fictional objects holds the promise of illuminating the intuitive impression that model-descriptions characterize concrete, actual systems, in spite of the fact that, straightforwardly taken, they fail to do so; for this is analogous to what happens with fictional works like War and Peace. However, as Currie (2016) and Yablo (2020) note, the promise is tainted by popular realist views on the ontology of fictional characters (Thomasson 2015, 2020). Like Friend (2020), and along the lines of Yablo (2020), I argue that the issue of realism is a red herring when it comes to these matters, and that accounts of fictional discourse help to illuminate philosophical concerns about scientific models. I will illustrate my points with the case of supervaluationist models of vagueness in semantics. In the next section I outline the account of fiction and fictional reference on which I will rely. Section 3 moves on to the case of models and the illustration from semantics. I wrap up in Sect. 4, by taking up Salis’s desiderata and summarizing relative to them the proposal on Sense, pretense, and reference models on the basis of the previous discussion.

8.2 Realism and Irrealism About Fictional Characters In this section I will outline the background views about fiction. I will briefly rehearse first a set of convenient distinctions. Let us assume that an assertion is what is done by default by means of declarative sentences: “[i]n natural language, the default use of declarative sentences is to make assertions” (Williamson 1996, 258).1 It is a feature of assertions that we evaluate them as correct or otherwise depending on whether they are true. Let us thus consider three sorts of prima facie assertoric uses made with declaratives in discourses involving fictions: (1) (2) (3)

When Gregor Samsa woke, he found himself transformed into a gigantic vermin. According to Metamorphosis, when Gregor Samsa woke, he found himself transformed into a gigantic vermin. Gregor Samsa is a fictional character.

Consider first an utterance of (1) by Kafka, as part of the longer utterance of the full discourse which, with a (good, in this case) measure of idealization, we can think constitutes the act of putting forward Metamorphosis for us to enjoy. These uses of fictional declarative discourse, which I will call textual,2 are distinguished by the fact 1 Cf.

García-Carpintero (forthcoming-a) for elaboration and defense. borrow this and the other two related labels from Bonomi (2008); I find the package particularly apt. Thomasson (2003, 207) has similar distinctions. 2I

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that they are not intuitively on a par with straightforwardly truth-evaluable discourse. Intuitively, ‘Gregor Samsa’ fails to refer in them to anything; given this, an assertion of (1) would intuitively fail to be true and would therefore be incorrect. However, we do not intuitively find it plausible to criticize Kafka on this account (cf. Van Inwagen 1977, 301). The other two kinds differ in so far as they lack this feature. There is, firstly, the use of sentences such as (1) to report on what goes on in a fiction, that is, the character of the fictional world it presents, its plot. I will call these plot-reporting uses paratextual; according to Lewis (1978) and others, they are simply elliptic for intuitively equivalent ascriptions of propositional content like (2), which on such grounds I will also count as paratextual. Readers of Metamorphosis would count (1) in such a use as straightforwardly, actually true, as they would (2), and reject the results of substituting ‘rat’ for ‘vermin’ in them. Finally, I will call the uses of sentences such as (3) metatextual; they are also intuitively truth-evaluable vis-à-vis actuality but not content-reporting, in that they are not prima facie equivalent to content-reporting ascriptions like (2). Kripke (2013, based on talks originally delivered in 1973) argues that a proper account of metatextual uses requires interpreting names like ‘Gregor Samsa’ in them as referring to fictional entities. In a similar vein, Van Inwagen (1977) provides an influential Quinean argument for realism about fictional entities, arguing that it allows for a straightforward explanation of the validity of arguments involving apparent reference to and quantification over them in metatextual discourse.3 Ficta could then be taken to be concrete Meinongian non-existent entities (Priest 2011), concrete non-actual possibilia (Lewis 1978), or (as both Kripke and Van Inwagen recommend) abstract existing entities of various sorts, Platonic abstracta like Wolterstorff’s (1980) or Currie’s (1990) roles, or rather created artifacts, as in Salmon (1998), Thomasson (1999, 2003, 2020) or Schiffer (2003).4 Artifactualists think of fictional characters as having an ontological status analogous to that of the fictional works in which they occur (Thomasson 1999, 143, 2003, 220, 2020, 71; Salmon 1998, 78–9). As I will explain in a moment, my brand of fictionalism about fictional characters claims only that we do not need to take referential expressions in discourses of any of the three kinds to really refer to them in order to understand how they work. However, I do not have any ontological scruples about ficta when they are understood on the artifactualist proposal, along the lines I will suggest. Moreover, to assume them greatly facilitates presenting my own view. Perhaps then there is no substantive difference between the fictionalism I hold and the artifactualist view.5 Let me elaborate on how I understand that proposal. 3 Cf.

however Kroon (2015) for a serious challenge. and Voltolini (2016) offer helpful discussion and further references. 5 Like Thomasson (2015, 262) (specially so given my ultimate fictionalism), I am not much disturbed by Brock’s (2010) main criticism of created fictional characters concerning the particular circumstances of their creation. Everett and Schroeder’s (2015) alternative proposal that they are spatially discontinuous concrete “ideas for fictional characters” is insightful. I cannot go here into the reasons why I think the “social construct” account I favor is more apt, nor address the intuitions that they (ibid., 284–5) marshal against it. 4 Kroon

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Fictional works result in my view (García-Carpintero 2013a, 2019b, c) from the communicative acts of fiction-makers; they are social constructs, abstract created artifacts with norm-regulated functions.6 They have a complex structure, grounded on that of the vehicles that express them; they are in part composed of singular representations (García-Carpintero forthcoming-b). It is these singular representations that I will (roughly) take fictional characters to be on the artifactualist account I will use as a foil: on the proposal, terms like ‘Gregor Samsa’ in metatextual uses have as semantic value a singular representation associated with that name, which is a constituent of Kafka’s Metamorphosis.7 (3) makes a true assertion of this. These fictional entities could then be invoked to account for textual and paratextual uses. The contextualist views defended by Predelli (1997), Recanati (2000, 213–226), Reimer (2005) and Voltolini (2006) develop the idea. The context in which ‘The battle happened here.’ is uttered might require us to evaluate the assertion not with respect to the place where the utterance occurs but another, contextually provided location. On those views, the context of textual uses of (1) similarly leads us to evaluate their truth not at the actual world, but at a counterfactual or imaginary one, “the” world of the fiction—actually, a plurality thereof if this is theoretically explicated by means of standard possible worlds ideology. Predelli (1997) only considers examples involving real names, but he then (Predelli 2002) extends the view to cases involving fictional names, arguing that they refer to ficta—actual abstract created existents.8 I will provide, for later use, a few more details on how I suggest to think of the semantic values of prima facie empty names on the realist proposals just outlined— on which, of course, the prima facie impression of vacuousness is wrong: textual, paratextual and metatextual uses of declarative sentences using such names are all in fact used to make truth-apt claims, some of them true. In my work on reference, I have been promoting a version of a view that it is now standard in current semantics (cf. García-Carpintero 2018, forthcoming-b). On this view, referential expressions like indexicals and proper names carry presuppositions of acquaintance, or familiarity. This is frequently cashed out by assuming that contexts include discourse referents, which we may think of as shareable singular representations that may well not pick out anything.9 For proper names, the relevant discourse referents are crucially defined 6 There

is no difference in these respects with other communicative acts; they also generate (when they do not misfire) social constructs of that kind, cf. García-Carpintero forthcoming-a. 7 Cf. MacDonald (1954, 177): “Characters, together with their settings and situations, are parts of a story”. 8 For reasons I cannot go into here (García-Carpintero 2019e), if we hold this view of textual and paratextual discourse, we should take both expressions like ‘Pierre Bezhukov’ in War and Peace which do not pick out any actual person, and those like ‘Napoleon’ which do, as equally having the associated representations as semantic values. Although I cannot elaborate this point here, it plays a very important role in the account of models in general and semantic models below. 9 Instead of characterizing the singular representations the proposal takes fictional characters to be in terms of discourse referents we could invoke mental files, insofar as we think of them as public and normatively characterized; cf. Orlando (2017), Terrone (2018). What about expressions of plural reference, like ‘the Hobbits’, or ‘the Dwarves’ (Kroon 2015)? I assume these could be handled in a related way, given an adequate semantic account; cf. Moltmann (2016) for discussion of how such an account should look like.

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by naming practices (distinct ones for the ‘David’ that picks out Lewis and the one that picks out Hume); typically existing ones, but in some cases practices created with the very discourse including the name. For indexicals, they might be constituted by perceptual information, or information available in previous discourse to which the expression is anaphorically linked. Fictional realism (and the fictionalist view that uses it as a convenient presentational device that I endorse) holds that all this carries over to textual discourse. The singular representations that it takes to be the semantic value of referential expressions are thus to be individuated by such discourse referents.10 There is a tradition that associates some descriptions with entities of the kind we are positing, roles (Rothschild 2007; Glavaniˇcová 2017; Stokke 2020) like the president of the USA or the mayor. Such roles can be understood as sets of the properties by means of which they identify their occupiers or, for purposes of formal modeling, as Carnapian individual concepts picking referents out relative to worlds, to the extent that we think of them as merely partial functions. If we model the fictional world by means of standard possible worlds, the role that we are taking as the semantic value of ‘Gregor Samsa’ will pick out different individuals in different such worlds. There is a well-known wrinkle in this proposal. While the entities that realists posit may well instantiate the properties predicated of them in metatextual uses like (3), this is not so clear for the two other uses. Such entities are not easily taken to be the sort of thing capable of waking or going to sleep, for these capacities appear to require having causal powers which abstract objects, created or Platonic, appear to lack. A standard way to deal with this distinguishes between two types of predication: having and holding. The subject-predicate combination in (1) does not mean that the semantic value assigned to the subject-term truly instantiates (has) the property expressed by the predicate, but merely that the former represents something to which the latter is ascribed in its encompassing fiction (holds). This helps with a point that Everett (2013, 163–178) emphasizes, that is, that there are many mixed cases such as (4) below: (4)

At the start of Metamorphosis, Gregor Samsa—an emotional alter ego created by Kafka for that novel—finds himself transformed into a gigantic vermin.

Following Everett and Schroeder (2015, 286–8) and Recanati (2018), we explain such mixed cases in that they involve a form of independently well-attested metonymyinduced, “regular” polysemy, as when we straightforwardly apply ‘lion’ and ‘ferocious’ to a lion-representation that literally, primarily is not a lion, like a sculpture 10 This semantic proposal for referential expressions in textual and paratextual discourse is an elaboration of Frege’s view that referential expressions shift their semantic values in intentional contexts to what in extensional contexts are their senses; Sainsbury’s (2018) “display” account of attitude ascription is an alternative. If paratextual uses of referential expressions occur (implicitly or explicitly) in intensional contexts, as on Lewis’s (1978) view, the parallel is immediate for them. Textual uses would also straightforwardly fit the bill if they were also elliptical for some operatorinvolving analogue of (2), as Devitt (1981, 172) and Orlando (2017) defend. This is objectionable, however, as Bertolet (1984)and Predelli (1997) pointed out; the proposal in the main text obtains essentially the same result without positing implicit operators.

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of one; for we also naturally find similarly mixed cases there. Thus, a sculptor can say of one of her creations: (5)

That lion is the best sculpture I’ve made this month; it is as ferocious as the one we saw yesterday at the zoo.

Contextualism similarly takes the inserted metatextual claim in (4) to involve straightforward, having predication, while the one in the main clause is rather of the holding variety: we are just saying of the relevant Samsa-representation that it represents someone to which, in the work, the predicate applies—the way the statue is metonymically said in (5) to represent a ferocious lion. Now, although this is a consistent and helpful account, I do not really think it is a good idea to regard textual uses as assertions, to be evaluated as literally true or untrue, except not at the actual world but at “the” world of the fiction (see Walton 1990, 41– 2; Everett 2013). Even some realists about fictional characters share the view that textual and paratextual discourse should not be taken as straightforwardly assertoric, but rather as involving pretense, cf. Thomasson (2003, 210–4, 2020, 56–7). On some views, the apparent assertion conveyed by (1) in textual and paratextual uses is just an Austinian locutionary act, a mere “act of speech”, as when I utter ironically ‘you are very considerate’: the assertion that the addressee is very considerate is merely pretend, I am not committing myself to it—in fact I assertorically commit myself to the opposite. My own view on textual and paratextual uses endorses Currie’s (1990) view that, as in the irony case, pretense is not all that there is to them. In irony, there is also an assertoric act. In fiction, there is also a speech act of its own, with specific force and content (fiction-making, as Currie calls it), cf. García-Carpintero (2019b, d). They are pretend assertions, but also alternative acts to be evaluated with respect to norms other than truth vis-à-vis the character of “the” fictional world they represent. Moreover, I have explained in detail why we certainly do not need to assume that referential expressions in such uses like indexicals or proper names refer to fictional characters on the artifactualist construal, or to anything at all; all we need is a richer presuppositional semantics for them than Millians assume, along the lines of what is offered by empirically adequate current semantics (García-Carpintero 2018, 2019a).11 I extend this irrealist view of textual and paratextual uses to metatextual discourse. I hold a Yablo’s (2001) version of figuralist brand of fictionalism, on which the semantic referential apparatus (de jure directly referential expressions such as names and indexicals, quantifiers generalizing over the positions they occupy, expressions for identity) is used metaphorically in the likes of (3), deploying the figure of speech 11 In

Thomasson’s (2003) terms, the relevant pretense in textual and paratextual uses is not de re (vis-à-vis artifactualist fictional characters) but de dicto. However, the presuppositional account of singular reference I uphold, unlike Thomasson’s construal of the relevant contents merely as existential generalizations, conveniently explains the impressions of singularity for such uses that writers like Friend (2011) rightly emphasize (García-Carpintero 2010b, 2018). Maier (2017) offers a DRT implementation of ideas very close to mine, except for his quasi-Meinongian account of metatextual discourse, for which I favor the Yablonian proposal below.

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called hypostatization (García-Carpintero 2010a). It is a rather dead, conventionalized kind of metaphor; so in contrast to pretense-theoretic fictionalist proposals, on this view utterances in metatextual discourse are straightforward assertions with truth-conditions.12 This might suggest that the view is after all realist, committed to referents of some sort for singular terms in metatextual discourse, but I do not think so. One could follow Brock (2002) and claim that the literal content apparently involving commitment to fictional entities is in fact along the lines of (2): one about what is true according to a pretense—the pretense that some realist theory is true. Or—like Yablo (2001) himself—one could follow Walton (1993) in thinking that this applies in general to metaphors, which are a “prop-oriented” form of make-believe put forward with the aim of asserting a metaphorical content non-committal to fictional entities, through the process that Richard (2000) calls “piggybacking”.13 My own preferred line, however, follows Yablo’s (2014) recent development of his views,14 articulating the notion that the truth of metatextual sentences including fictional names and their generalizations does not really commit us to the existence of fictional characters. For this is merely pretend-presupposed and, when we look at what they are really about (the truth-makers for the claims we make with them), we do not find the referents they appear to pick out.15 Yablo explains subject-matter, or what claims are about, in terms of the semantic notion of an answer to a question. In current semantics, questions like ‘Is Trump US president in 2021?’ are analyzed as sets of their possible answers, hence sets of sets of truth-making “worlds”, actually partitions of the space of worlds in the mathematical sense—in our case, the set including the set in all whose worlds Trump is president in 2021, and the one including all those in which he is not. In the relevant cell of the partition giving what (3) is about, we perhaps find fictional worlds in which the role that is the semantic value of ‘Gregor Samsa’ does pick out an object, together with the actual world in which it doesn’t pick 12 The pretense involved is not pragmatic but semantic in Armour-Garb’s and Woodbridge’s (2015)

classification, if I understand them correctly; cf. García-Carpintero (2019a). mother tells her child “the cowboy should now wash his hands for dinner”. She is making an utterance that would be true-in-the-pretense if certain conditions obtained (mother and child are playing a game of cowboys and Indians, with specific principles of generation), with the intention of asserting such conditions (i.e., that the boy dressed as a cowboy now has certain obligations). Cf. also Evans (1982, 363–4). 14 Yablo (2020) provides a helpful outline, applied to the case of models we will be discussing below. Hoek (2018) offers a precise, neat variation on Yablo’s ideas. 15 Cf. Cameron (2012) and von Solodkoff (2019) for similar views. Their proposals, however, raise a serious concern: in general, we do not want to say that entities are fictional just because they are grounded on more fundamental entities. Social constructs (which is what products of speech acts are, on my view) nonetheless exist as social kinds, even if they should be grounded on more fundamental entities (Schaffer 2017). Why should it be different with fictional characters? After all, on the realist proposal made above they have exactly the same ontological status as social constructs like speech acts. The figuralist account helps here: even if conventionally standardized, apparent references to fictional characters in metatextual discourse are just metaphors (hypostatizations), and, as such, we do not need to assume that they are actual successful references to characterize the contents of claims made by means of them. 13 A

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out anything. But in all of them there is a fiction like Metamorphosis referentially deploying ‘Gregor Samsa’. What we thus find corresponding to the name in truthmakers for (3) are the “ideas for fictional characters” of Everett and Schroeder (2015), or representations thereof.16 My preferred story thus shares significant similarities with artifactualist views. We end up interpreting (2), (3), and (4) as making genuine assertions, whose truth is grounded on the pretenses thereof in textual uses of (1). Why not then endorse the realist proposal outlined above? My reasons are essentially Yablo’s (García-Carpintero 2019a) : it is not needed on the view just outlined, and there are good reasons to reject it.17 If we take it seriously that fictional names refer to ficta, we have to find reasons to choose among the different candidates that realists offer, some of which we mentioned above; but there is no rational way of taking a stance on such matters. For all we can tell, the relevant empirical and theoretical considerations do not select just one of them. The situation is analogous to the hermeneutics of fictions, when there are several alternative interpretations consistent with all data—as in Blade Runner (on whether or not Deckard is a replicant) or The Turn of the Screw (on whether or not the ghosts are figments of the governess’s imagination). The fictionalist attitude allows us to ignore the issue in good faith.18 We’ll come back to these indeterminacy concerns, which are at the heart of my illustration of a fictionalist view of models.

8.3 Models as Fictions: An Illustration from Semantics After Tarski, logicians applied the set-theoretical notion of model to characterize the semantics of formal languages and to define their logical properties.19 When such models are used to explain logical properties like validity, given ordinary conceptions thereof, it is natural to think that they play the role of their scientific counterparts as idealizations that help us to better understand a target domain—in this case that on which the ordinary conception of logical validity is defined, the representational acts of inferring, believing or asserting and the propositions we represent in them.20 16 Hoek (2018, Sect. 4) discusses related cases in detail. Crimmins (1998), Sainsbury (2011), Howell

(2015) and Manning (2015, 297–301) defend similar views. 17 As Everett (2013, 143) neatly puts it: “I do not mean to deny that in some cases the entities invoked

by certain fictional realists, who then go on to identify these entities with fictional characters, genuinely exist. My complaint is simply that, in these cases, the relevant entities are not fictional characters; the identification made is wrong”. Cf. also Paganini (2020). 18 If we adopt (as I think we should) Williamson’s (2018, ch. 10) view that philosophy resorts to models in the same spirit as sciences, we may think of the imaginary referents posited for referential expressions in metatextual discourse in Maier’s (2017) formal account as fictions exactly along the lines suggested in the next section for semantic models. Williamson in fact illustrates his suggestions with semantic models for intensional discourse. 19 Cf. Hodges (1985–1986) for historical details. 20 Cf. García-Carpintero (1993, 123, 128), Glanzberg (forthcoming). More on propositions momentarily.

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Montague’s (1974, 222) vision and work extended those techniques to the semantics of natural languages. Research pursuing the development of his program now constitutes a discipline of its own, a most recent scientific offshoot of philosophy. It is thus as apt as it in the case of logic to think of models posited in contemporary semantics to meet its explanatory aims as their scientific namesakes.21 Central explanatory tasks for semantic theories of natural languages include: accounting for facts about systematicity and productivity in understanding, communication and acquisition; explaining judgments about entailments, truth-value or correctness relative to particular situations.22 These facts are supposed to be empirically revealed in part by the speaker’s intuitions concerning the representational acts performed by uttering the relevant sentences. It has been traditionally taken for granted that the meanings thus revealed include a propositional component that might be common to acts of different kinds, like commands, promises or assertions. This is still the standard assumption among semanticists. It has been questioned by Hanks (2015) and others, but, as I have argued (García-Carpintero forthcoming-c), their arguments are not compelling. That paper also provides reasons for a minimalist stance towards propositions. On this view, propositions lack any structure of their own; adopting a Stalnakerian picture, they are just properties of circumstances of evaluation (Richard 2013). What are such circumstances? For many they are complete and consistent possible worlds, for Lewisians centered possible worlds. I will think of them as “smaller” than full possible worlds, as in Situation Semantics (Kratzer 2017) or in Truthmaker Semantics (Fine 2017).23 Models deployed in semantics are standardly taken to be classical: they ascribe to predicates extensions and anti-extensions exhaustive of the domain with respect to every circumstance of evaluation. This fits badly with the semantically significant intuitive data manifesting the omnipresence of vagueness in natural language and thought—the existence of borderline cases for predicates like ‘bald’, and their Sorites-proneness. In support of classical logic, Williamson (1994) has argued that such data should be accounted for epistemically: predicates in natural language have classical intensions; it is just that competent speakers are afflicted by an unescapable form of ignorance regarding their precise profiles. I am far from alone in being deeply skeptical of this view. My reasons are essentially of a metasemantic nature (cf. Weatherson (2003a), Heck (2004), Leitgeb (forthcoming, Sect. 1). While in the case of predicates like ‘being water’ there are good reasons to endorse the realist, “correspondence-truth”, externalist intuitions unveiled by Burge, Kripke and Putnam—which entail that such predicates may truly apply 21 Cf.

Yalcin (2018), Williamson (2018, ch. 10). I do not share Yalcin’s Chomskian skepticism regarding the intuitive adequacy of the contents ascribed to utterances (sentences-in-context) by semantic theories, but I won’t go into this here, cf. García-Carpintero (forthcoming-a). 22 Cf. Yalcin (2014, 18–23) for more details on those explanatory goals. Systematicity concerns the fact that speakers who, say, competently understand ‘John loves Mary’ can equally understand ‘Mary loves John’; productivity, the fact that competent understanding is in principle unbounded: ‘the son of Mary swims’, ‘the son of the son of Mary swims’ … . 23 “Smaller” not just spatiotemporally (say, limited to events in our light cone), but also at the level of detail (say, domain of individuals, relevant features, etc.) at which events are specified.

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or fail to apply in spite of inescapable ignorance and error on the part of competent speakers –, they fail to extend to, say, ‘being sort of dirty’ or ‘being kind of cute’. In cases like ‘water’ there are convincing explanations (i.e., a decent metasemantics) for how the expressions came to acquire their externalist meanings, but Williamson’s clever proposals fail to fit the bill for the ones just mentioned. In contrast, there are good alternative semantics for which an adequate metasemantics is in the offing. The ones I favor deploy the supervaluationist techniques devised by Van Fraassen (1966) and further developed by Fine (1975) to account for vagueness, along the lines of the semantic indecision interpretation suggested by Lewis (1993). In a nutshell, the idea is to associate predicates (and thereby sentences and nouns) with a plurality of classical models as opposed to just one, which constitute admissible precisifications, or ways of making them precise enough to have classical extensions compatible with their core meanings. The true extension of a predicate is then the intersection of all those it receives in each admissible precisification.24 Now, there are two different ways of understanding the supervaluationism-cumindecision view, SI. I will put it in terms of Lewis’s (1975) convenient ideology of languages and language. The former are abstract entities, consisting of (recursive, compositional) assignments of putative meanings to expressions; the latter are those abstract entities made concrete enough by their being used by actual populations. To be sure, supervaluationist precisifications are languages in the first sense. But on the first, more traditional understanding of SI, they are not languages in the second sense; none of them really is the natural language spoken by an actual population. In leaving borderline cases undecided, actual languages have, as it were, a shortage of meaning (Williamson 1994, 142): precisifications repair the deficit. On a second, plurivaluationist interpretation recently promoted by Caie (2018) and Sud (2020), all admissible precisifications are actually used: vagueness shows that natural languages have an affluence of meanings.25 As indicated above, the intuitions of speakers that are significant for metasemantic purposes discern propositions in the meanings of the representational acts (literally) made with utterances of sentences of natural languages. In these terms, the plurivaluationist take contends that, in speaking literally, language users assert many different precise propositions. In previous work (García-Carpintero 2007, 2008, 2010a, b, c) I have in effect argued that, although there is something to the plurivaluationist 24 Leitgeb (forthcoming) provides an interesting alternative that surrenders less of classical logic than standard supervaluationism. The idea, in a nutshell, is to have the semantic theory specifying truth-conditions for the object language relatively to an arbitrarily chosen admissible precisification. For reasons indicated in a moment, I prefer a complex account of the sort outlined below that heavily relies on supervaluationist notions. But I could make the points below for which I appeal to plurivaluationist ideas by relying instead on Leitgeb’s account. 25 Fine (1975, 282f.) describes these two possibilities with a nice metaphor: “Ambiguity is like the super-imposition of several pictures, vaguenessVagueness like an unfinished picture, with marginal notes for completion. One can say that a super-imposed picture is realistic if each of its disentanglements are; and one can say that an unfinished picture is realistic if each of its completions are. But even if disentanglements and completions match one for one, how we see the pictures will be quite different”. Vagueness here is a deficit, ambiguity (as Fine designates the first option, which manifests I think his own preference for the other) over-abundance.

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proposal, it doesn’t offer the whole story on the semantic effects of indeterminacy.26 On an alternative pluralist view, although the propositions literally asserted with vague sentences might be understood for some purposes as precise, along the plurivaluationist picture, we also need to assume that, more fundamentally, we assert in the very same cases propositions that are themselves vague. This more complex story is required to properly defend SI from what I take to be the two most serious arguments against it: first, Williamson’s (1994, 187–98; Andjelkovic and Williamson 2000) claim that, in Wright’s (2004, 88) terms, SI has to “surrender the T-scheme”; second, Schiffer’s (1998, forthcoming) argument based on ascriptions (particularly, de re ascriptions) of what is said with vague utterances. The first argument is based on the following consideration. Take a truth-bearer consisting of a vague predication of a borderline case. SI declares it neither true nor false, so the left-hand side of the relevant instance of the T-schema should be (determinately) false. On SI, however, the right-hand side is neither true nor false. Both sides thus differ in truth-value, and hence the whole biconditional should be false, or unassertable. In reply I have argued that, assuming Williamson’s (1994, 162–4; Andjelkovic and Williamson 2000, 216) articulation of the T-scheme as conditional on what truth-bearers say, there is a well-motivated notion of what is said, or propositional content, on which indeterminate truth-bearers say or assert different fully precise contents, given which Williamson’s argument can be resisted (García-Carpintero 2007);27 this assumes the plurivaluationist interpretation. There is another notion of what is said on which they do not (determinately) say any of those precise propositions but a vague one instead, on which the relevant instances of the T-schema should be rejected; but there is no serious philosophical worry about it. The details are complicated and we do not need to go into them any further here. The standard, simpler way of dealing with this first argument against SI on the traditional “meaning deficit” interpretation eschews vague propositions, sticking to the idea that it is just indeterminate which precise propositions modeled by the admissible precisifications of a truth-bearer are asserted in an utterance thereof (Keefe 2010). I do not think this offers a good reply to the argument (Rohrs 2017, 2190– 4), nor that this strategy properly answers Schiffer’s argument, particularly when it comes to the ascription of de re contents (Merlo 2017, 2646–51). Heck (2004, 126) rightly appreciates that these issues require a good account of the relation between precisifications and the propositional contents asserted by vague utterances, but he 26 As usual in discussing these matters, I am ignoring here the additional complexity created by higher-order vagueness—the fact that theoretical notions that we use in our metalanguage like borderline case or admissible precisification are themselves vague; that, as Williamson insists, we are doomed to conduct our investigations of these matters in a language that exhibits the very phenomena we are theorizing about. Heck (2004, 123) warns against making the case for higherorder vagueness too easily, but I think it can be convincingly made anyway.) In my view, Caie’s (2018, 128–32) and Sud’s (2020, Sects. 5–6) arguments to stick just to the plurivaluationist view underestimate the problems posed to their proposals by the vagueness of our theoretical discourse itself (Rohrs 2017, 2194–7), and they overlook the resources of the pluralist view I will outline; but I cannot develop these points here. 27 López de Sa (2009) and Iacona (2010) offer a similar view; Fine (2007) a critical appraisal.

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(ibid., 115) overlooks the considerations I am about to rehearse for taking such contents to be themselves vague. Once more, debates about this second argument are too complex to be properly examined here. I will limit myself to outlining my view, showing how the fictionalist view of supervaluationist models helps to alleviate prima facie difficulties. A fuller defense would require a global appraisal of the views, and for it a level of detail that I cannot indulge here. Schiffer’s argument concerns the ascription of de re, singular contents. Imagine that Alex said ‘the money is buried there’; a singular proposition has been expressed, concerning a particular demonstrated location. On the deflationary conception of propositions outlined above, singular propositions are properties of truth-making situations, constitutively individuated by reference to a particular entity. Consider thus a de dicto (i.e., as specific as possible about the ascribed content) ascription regarding what Alex said: ‘Alex said that the money is buried there’. This is also a de re ascription, because for our ascription to be as faithful as possible to the content that Alex expressed, we need to refer ourselves to the particular location that Alex meant; hence our ascription entails that there is a location such that Alex said that the money is buried there. Now, which one is that location? There only appear to be precise locations, precise positions in physical space; but there are plenty of them that might be the one that Alex meant, and there appear to be as little metasemantic material to fix upon a specific one as in other cases. This is a particular case of the “problem of the many” (Weatherson 2016). It arises for any ascription of contents expressed by means of referentially indeterminate expressions like ‘there’ above, or ‘Kilimanjaro’: to the extent that ‘there’ is supposed to refer to precise locations, there is a plurality of candidate referents for it; the same applies to ‘Kilimanjaro’, if it is meant to pick out precise atom-constituted mountains—just consider an atom in a candidate boundary for the mountain, and two aggregates respectively including and excluding it. I will focus on locations, because the best-known version of the proposal I will make, which posits singular propositions individuated by vague objects (Lowe 1995; Tye 1996; Sattig 2013; Korman 2015; Jones 2015), doesn’t easily get a grip in such cases; for we cannot naturally distinguish in the location case something like a mountain or a cloud from, say, the aggregates of particles constituting it. The standard supervaluationist treatment of the problem posits different precisifications, different models in which different precise locations are assigned as semantic values to the constant translating ‘there’ (Weatherson 2003b; Keefe 2010). Alex’s utterance would be true iff the money is in any of them, while ‘there’ doesn’t determinately refer to any of them, and the ascription about what was said is true to the extent that the singular term used in it is somehow coordinated with the one Alex used, so that they are jointly precisified. Schiffer argues that, whatever the plausibility of this view regarding Alex’s utterance, it fails for the ascription, because, intuitively, Alex “didn’t say” any such singular proposition. The standard supervaluationist response (Keefe 2010) is to accept the intuition only if it assumes some theoretical vocabulary (“Alex didn’t determinately say …”), which in its turn should be understood as subject to supervaluationist treatment. This, however, raises serious concerns (García-Carpintero 2010c; Merlo 2017, 2646–51), which my proposal circumvents.

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Remember that the favored realism about fictional characters discussed in the previous section posits roles as semantic values for prima facie empty fictional names. There is no reason not to extend such potential semantic values to all singular terms, including the use of ‘there’ in our example or ‘Kilimanjaro’; the theoretical resources employed in the case of fictional names extend in fact to all.28 Thus we get a particular role as semantic value for the envisaged utterance of ‘there’ in the previous example. This is a vague entity, because the relevant role constitutively has the job of picking out (precise) locations, and there are many different ones it can correctly pick out.29 Barnes and Williams (2011) make a good case that supervaluationist techniques can also be used to articulate an intelligible version of the view that vagueness is in some cases ontic, with precisifications representing the alternative state of affairs that might constitute the facts—the way things actually are.30 Does this amount to abandoning SI as a general account of vagueness? I have argued that not really (García-Carpintero 2010c); because the entities we are now classifying as vague (roles, and the singular propositions individuated by means of them) are themselves representational in nature. Be that as it may, once again the concern is mitigated by the Yablonian fictionalist take on roles outlined in the previous section.

8.4 Conclusion: Models and Fictions I am now in a position to sum up the fictionalist account of models based on the illustration that the supervaluationist view offers. No matter whether we opt for the meaning-deficit or the meaning-abundance interpretation of precisifications, or whether we complement our theoretical account by ascribing to sentences vague propositions in addition to the precise propositions determined by the precisifications, these play a fundamental role in the explanation. Precisifications are models in the logical sense: interpretations of expressions corresponding to those in the natural languages we are studying. I have suggested to understand them also as models in the scientific sense, providing idealized versions of the languages speakers use, and to adopt a fictionalist view of them. Their targets are the facts about systematicity and productivity in understanding, communication and acquisition in language-use 28 This creates a systematic ambiguity, of course not unlike the one generated by Fregean “reference shifting” accounts, which must be dealt with along the lines of extant proposals for that case; cf. Orlando (2017) for discussion. 29 García-Carpintero (2020) discusses how this ontic indeterminacy arises as almost a matter of course for the realist view of fictional characters. An additional virtue of thus ascribing vague propositions to sentences including vague expressions is that we thus alleviate the concerns about SI raised by Fodor and Lepore (1996). 30 García-Carpintero (2013b) embraces this model for future contingents, deploying a non-standard “thin red line” form of supervaluationism on which truth is bivalent and non-equivalent admissible precisifications account for worldly indeterminacy at a time. Abasnezhad and Hosseini (2014) develop this model in precise detail for the case of the referential indeterminacy of ‘there’ or ‘Kilimanjaro’ in the examples above.

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mentioned above as empirical explananda for semantic theories. Given one or another interpretation, precisifications explain the facts under the fiction that speakers deploy the model languages. Let me conclude by locating the illustrated view in the current debate, as announced at the outset, relative to Salis’s (2019) three desiderata. The present fictionalist view of models is closer to direct views like those by Toon (2010a, b) and Levy (2015), than to indirect ones like Godfrey-Smith’s (2009), Frigg’s (2010a, b), or Frigg’s and Nguyen’s (2016, 2018). It is particularly close to Levy’s, which also uses Yablo’s (2014) semantic account. On direct views, the asserted content of claims prima facie referring to model-systems (precisifications, in our illustration) in fact concerns model targets (in our illustration, the relevant situations of language use). Model-descriptions descriptively characterize in the first place (essentially by reference to themselves as representational vehicles, see below) model systems, to be understood as scenarios that need not include the actual one; they can also, however, have as “parts” (Yablo 2014) actually satisfied scenarios concerning target systems, in the way discussed in the second section. Salis’s identification problem goes as follows: “what objects are models, and how do we identify them?” On the view outlined, models are primarily contents (“model systems”), the scenarios involving use of the precisified languages that we are meant to imagine. As in Salis’s and Frigg’s and Nguyen’s (2016, 2018) proposal, they essentially involve elements of (more or less precisely individuated) representational vehicles, “model-descriptions”. This is so in this case because logical models already are ascriptions of semantic values to expressions. More in general, the fictionalist view I outlined in the second section grounds the “references” of singular terms pretending to refer to fictional entities on the very referential expressions. Salis’s (2019)epistemic problem concerns “how models represent in ways that facilitate knowledge of reality”. I have not discussed this issue here. On the present proposal, the question is equivalent to ask how we can learn from fictions such as War and Peace. In recent work (García-Carpintero 2019d) I have defended an indirect speech act model to account for this, given the goals assumed in the modeling practice for the projects at stake (see Currie 2020, ch. 9 for a related view). Thus developed, the fictionalist view is close to inferentialist accounts like Suárez’s (2010). Finally, Salis identifies the problem of truth-conditions: “Only concrete objects can have the sort of properties that model-systems and target-systems supposedly share. But model-systems are not concrete. On the fiction view, they are merely imaginary systems. So, how can they share any properties with any physical systems, and what would ground the truth-conditions of model-world comparisons?” On the view outlined, the actually instantiated scenarios that are targets for explanations by means of models are “parts” (Yablo 2014) or “exculpatures” (Hoek 2018) of the fictional scenarios (model systems) that model-descriptions invite us to imagine; such scenarios, on the proposal to tackle the epistemic problem just summarized, are indirectly conveyed, given the nature of the scientific modeling practice. Remember, however, that I have no objection on ontological grounds to the abstract artifacts

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envisaged by realists such as, for the present case, Thomasson (2020) or ThomsonJones (2020); it is just that we have no need for them in order to account for theoretical claims like, in our case, those made by referring to precisifications.31 As indicated at the outset, Currie (2016) offers well-grounded skeptical remarks on the adequacy of the “fiction model” of model-based science. I nonetheless think that, as presented here, it provides conceptual relief for the ontological and epistemological puzzles the practice raises identified by Salis.

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31 Frigg and Nguyen (2016, 235) raise an interesting objection involving cases of theoretical models

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Manuel García-Carpintero is Professor of Philosophy at the University of Barcelona. He is the author of Las palabras, las ideas y las cosas. Una presentación de la filosofía del lenguaje (Barcelona: Ariel, 1996) and Relatar lo ocurrido como invención. Una introducción a la filosofía de la ficción contemporánea (Madrid: Cátedra, 2016). He is the editor (with Josep Macià) of Twodimensional semantics (Oxford: Oxford University Press, 2006); (with Max Kölbel) of Relative truth (Oxford: Oxford University Press, 2008) and The Bloomsbury companion to the philosophy of language (London: Continuum, 2012); (with Genoveva Martí) of Empty representations: Reference and non-existence (Oxford: Oxford University Press, 2014); (with Stephan Torre) of About oneself (Oxford: Oxford University Press, 2016); and (with Alessandro Capone and Alessandra Falzone) of Indirect reports and pragmatics in the world languages (Cham: Springer, 2018).

Chapter 9

Structural Representation and the Ontology of Models Otávio Bueno

Abstract This chapter introduces a structural account of representation through partial structures and examines its ontological commitments. It is pointed out that structural approaches to scientific representation emphasize the crucial role played by structures in representing salient features of the world. It is common to present and, in some cases, even to reify such structures as abstract entities, in particular as set-theoretic constructs. Against this view, it is argued that no such reification is called for and that several strategies can be articulated to avoid commitment to set-theoretic structures in the defense of structural representation. In the end, one can favor the latter without being platonist about the former. Keywords Representation · Partial structures · Models · Ontology

9.1 Introduction A central feature of scientific practice is to advance representations of relevant parts of the world. Whether researchers are interested in determining the chemical composition of ribosomes, displaying the molecular structure of albumin or modeling the likely trajectory of a hurricane, representational devices are a ubiquitous feature of their activity. Representations allow for the display and articulation of relevant data about the target and to draw inferences about it from available information, however partial such information may be (see Suárez 2004; Hughes 2010; Bueno and Colyvan 2011; Bueno and French 2018). The display, extraction, and transmission of information via inferential devices are among the key roles played by a representation. As an illustration, properly designed maps display geographical and topographical data about a region, and offer an effective means of extraction of information about the location and relative distances of landmarks, which, in turn, allows map users to draw inferences about their own position in the environment, thus enabling them to navigate successfully a O. Bueno (B) Department of Philosophy, University of Miami, Coral Gables, FL 33124, USA e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Cassini and J. Redmond (eds.), Models and Idealizations in Science, Logic, Epistemology, and the Unity of Science 50, https://doi.org/10.1007/978-3-030-65802-1_9

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place they have never been to. In this way, the three roles of representation are often thoroughly integrated. But how are representations achieved? In the case of science, models are the most common vehicles of such representations (see Da Costa and French 2003; Van Fraassen 2008; Frigg and Nguyen 2020). It is by using models that significant portions of scientific information can be displayed, extracted, and transmitted. As a result, models and scientific representation go hand in hand (Hughes 2010, chapter 5). If models are crucial for scientific representation, does their ontology impact on the way in which representations are achieved? In particular, in the case structural approaches to scientific representation (which emphasize the crucial role played by structural considerations, such as various morphisms connecting different models, in the representation of phenomena), the issue arises as to whether one is forced to decide on a particular ontology of models in order to adopt a structural approach to representation. Central to structural approaches is the role played by structures in the representation of salient features of the world. It is usual to have such structures presented and, in some cases, even reified, as abstract entities, in particular as set-theoretic constructs. Is this demanded by structural representation or is it just a contingent, convenient, but ultimately dispensable, feature of the way in which the approach is articulated? In this paper, I argue that no such reification is called for. I examine several strategies that can be advanced to avoid commitment to set-theoretic structures in a defense of structural representation. In the end, one can favor such representation without being platonist about sets.

9.2 Structural Representation: Central Features 9.2.1 Partial Structures To be concrete, I will consider one structural approach to scientific representation: the one in terms of partial structures, which is typically formulated in terms of set theory. (For a critical examination of various views of scientific representation, and a defense of a different approach, see Frigg and Nguyen 2020). The partial structures approach relies on three main concepts: partial relation, partial structure and partial truth (see Mikenberg et al. 1986; Da Costa and French 2003). One of the main motivations for introducing this proposal derives from the need for supplying a formal framework in which the openness and incompleteness of the information that is dealt with in scientific practice can be accommodated. This is accomplished, first, by extending the usual notion of structure, in order to accommodate the partialness of information we have about a certain field of inquiry (introducing then the notion of a partial structure). Second, the Tarskian characterization of the concept of truth is generalized for partial contexts, which then leads to the introduction of the corresponding concept of partial truth.

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The first step, then, to characterize partial structures is to formulate a suitable concept of a partial relation. In order to investigate a certain field of inquiry  (say, quantum particles), researchers formulate a conceptual framework that helps them systematize and interpret the information they obtain about . This field can be represented by a set D of objects (which includes observable objects, such as configurations in a Wilson chamber and spectral lines, and unobservable objects, such as quarks). D is studied by the examination of the relations that hold among its elements. However, it often happens that, given a relation R defined over D, we do not know whether all objects of D (or n-tuples thereof) are related by R, or we need to ignore some of the relations that are known to hold among objects of D, in order to study other relations about it in a tractable way. This is part of the incompleteness and partiality of our information about , and is formally accommodated by the concept of a partial relation. The latter can be characterized as follows. Let D be a non-empty set. An n-place partial relation R over D is a triple R1 , R2 , R3 , where R1 , R2 , and R3 are mutually disjoint sets, with R1 ∪ R2 ∪ R3 = Dn , and such that: R1 is the set of n-tuples that (we know that) belong to R; R2 is the set of n-tuples that (we know that) do not belong to R, and R3 is the set of n-tuples for which it is not known (or, for reasons of simplification, it is ignored that it is known) whether they belong or not to R. (Note that if R3 is empty, R is a usual n-place relation that can be identified with R1 .) But in order to accommodate the information about the field under study, a concept of structure is needed. The following characterization, spelled out in terms of partial relations and based on the standard concept of structure, offers a concept that is broad enough to accommodate the partiality usually found in scientific practice. A partial structure A is an ordered pair D, Rii ∈ I , where D is a non-empty set, and (Ri )i ∈ I is a family of partial relations defined over D. (The partiality of partial relations and structures is due to the incompleteness of our knowledge about the field under investigation. With additional information, a partial relation can become a full relation. Thus, the partialness examined here is not ontological, but epistemic.) We have now defined two of the three basic concepts of the partial structures approach. In order to spell out the last one (partial truth), we will need an auxiliary notion. The idea here is to use the resources supplied by Tarski’s definition of truth. But since the latter is only defined for full structures, we have to introduce an intermediary notion of structure to link partial to full structures. This is the first role of those structures that extend a partial structure A into a full, total structure (which are called A-normal structures). Their second role is model-theoretic, namely to put forward an interpretation of a given language and to characterize semantic notions. Let A = D, Ri i ∈ I be a partial structure. We say that the structure B = D , R i i ∈ I is an A-normal structure if (i) D = D , (ii) every constant of the language in question is interpreted by the same object both in A and in B, and (iii) R i extends the corresponding relation Ri (in the sense that, each R i , supposed of arity n, is defined for all n-tuples of elements of D ). Note that, although each R i is defined for all n-tuples over D , it holds for some of them (the R i1 -component of R i ), and it doesn’t hold for others (the R i2 -component).

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As a result, given a partial structure A, there are several A-normal structures. Suppose that, for a given n-place partial relation Ri , we don’t know whether Ri a1 …an holds or not. One of the ways of extending Ri into a full R i relation is to look for information to establish that it does hold; another way is to look for contrary information. Both are prima facie possible ways of extending the partiality of Ri. But the same indeterminacy may be found with other objects of the domain, distinct from a1 , …, an (for instance, does Ri b1 …bn hold?), and with other relations distinct from Ri (for example, is Rj b1 …bn the case, with j = i?). In this sense, there are too many possible extensions of the partial relations that constitute A. Therefore, we need to provide constraints to restrict the acceptable extensions of A. In order to do that, we need first to formulate a further auxiliary notion (see Mikenberg et al. 1986). A pragmatic structure is a partial structure to which a third component has been added: a set of accepted sentences P, which represents the accepted information about the structure’s domain. (Depending on the interpretation of science that is adopted, different kinds of sentences are to be introduced in P: realists will typically include laws and theories, whereas empiricists will add mainly certain regularities and observational statements about the domain in question.) A pragmatic structure is then a triple A = D, Ri , Pi ∈ I , where D is a non-empty set, (Ri )i ∈ I is a family of partial relations defined over D, and P is a set of accepted sentences. The idea is that P introduces constraints on the ways that a partial structure can be extended (the sentences of P hold in the A-normal extensions of the partial structure A). Our problem is: given a pragmatic structure A, what are the necessary and sufficient conditions for the existence of A-normal structures? Here is one of these conditions (Mikenberg et al. 1986). Let A = D, Ri , Pi ∈ I be a pragmatic structure. For each partial relation Ri , we construct a set M i of atomic sentences and negations of atomic sentences, such that the former corresponds to the n-tuples that satisfy Ri , and the latter to those n-tuples that do not satisfy Ri . Let M be ∪ i ∈ I M i . Therefore, a pragmatic structure A admits an A-normal structure if and only if the set M ∪ P is consistent. Assuming that such conditions are met, we can now formulate the concept of partial truth. A sentence α is partially true in a pragmatic structure A = D, Ri , Pi ∈ I if there is an A-normal structure B = D , R i i ∈ I such that α is true in B (in the Tarskian sense). If α is not partially true in A, we say that α is partially false in A. Moreover, we say that a sentence α is partially true if there is a pragmatic structure A and a corresponding A-normal structure B such that α is true in B (according to Tarski’s account). Otherwise, α is partially false. The idea, intuitively speaking, is that a partially true sentence α does not describe, in a thorough way, everything that it is concerned with, but only an aspect of it: the one that is delimited by the relevant partial structure A. After all, there are several different ways in which A can be extended to a full structure, and in some of these extensions α may not be true. Thus, the concept of partial truth is strictly weaker than truth: although every true sentence is (trivially) partially true, a partially true sentence may not be true (since it may well be false in certain extensions of A). To illustrate the use of partial truth, let us consider an example. As is well known, Newtonian mechanics is appropriate to explain the behavior of bodies under certain

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conditions (say, bodies that, roughly speaking, have a low velocity with respect to the speed of light, that are not subject to strong gravitational fields etc.). But with the formulation of special relativity, we know that if these conditions are not satisfied, Newtonian mechanics is false. In this sense, these conditions specify a family of partial relations, which delimit the context in which Newtonian theory holds. Although Newtonian mechanics is not true (and we know under what conditions it is false), it is partially true; that is, it is true in a given context, determined by a pragmatic structure and a corresponding A-normal one (see Da Costa and French 2003). But what are the relations between the various partial structures articulated in a given domain? It is in terms of such relations that scientific representation is ultimately implemented. Since we are dealing with partial structures, a second-level of partiality emerges: we can only establish partial relations between the (partial) structures at our disposal. This means that the usual requirement of introducing an isomorphism between theoretical and empirical structures (see Van Fraassen 1980, p. 64) can hardly be met. After all, researchers typically lack full information about what they study. Thus, relations weaker than full isomorphism (and full homomorphism) need to be introduced (see French and Ladyman 1997, 1999; Bueno 1997; Bueno and French 2018). In terms of the partial structures approach, however, appropriate characterizations of partial isomorphism and partial homomorphism can be offered (see French and Ladyman 1999; Bueno 1997; Bueno et al. 2002). And given that these notions are more open-ended than the standard ones, they accommodate better the partiality of structures found in scientific practice. Let S = D, Ri i ∈ I and S  = D , R i i ∈ I be partial structures. So, each Ri is a partial relation of the form R1 , R2 , R3 , and each R i a partial relation of the form R 1 , R 2 , R 3 . (For simplicity, I’ll take the partial relations in the definitions that follow to be two-place relations. The definitions, of course, hold for any n-place relations.) We say that a partial function (that is, a function that is not defined for every object in its domain) f : D → D is a partial isomorphism between S and S  if (i) f is bijective, and (ii) for every x and y ∈ D, R1 xy ↔ R 1 f (x)f (y) and R2 xy ↔ R 2 f (x)f (y). So, when R3 and R 3 are empty (that is, when we are considering total structures), we have the standard notion of isomorphism. Moreover, we say that a partial function f : D → D is a partial homomorphism from S to S  if for every x and every y in D, R1 xy → R 1 f (x)f (y) and R2 xy → R 2 f (x)f (y). Again, if R3 and R 3 are empty, we obtain the standard notion of homomorphism as a particular case. There are two crucial differences between partial isomorphism and partial homomorphism. First, a partial homomorphism does not require that the domains D and D of the partial structures under study have the same cardinality. Second, a partial homomorphism does not map the relation R i into a corresponding relation Ri . Clearly a partial homomorphism establishes a much less strict relationship between partial structures. Partial isomorphism and partial homomorphism offer mappings among partial structures that are less tight than their corresponding full counterparts (isomorphism

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and homomorphism). Partial mappings, as transformations that connect different partial models that may be used in scientific practice, allow for the transferring of information from one domain into another, even when the information in question is incomplete. After all, if a sentence is partially true in a given partial structure S, it will also be partially true in any partial structure that is partially isomorphic to S (see Bueno 2000). It is also worth noting that the partial structures approach offers a straightforward way of accommodating Mary Hesse’s [1970] tripartite account of analogy in terms of positive, negative and neutral analogies (see Da Costa and French 2003, pp. 47–52). R1 relations correspond to the positive analogies: those that are known to carry over from the source to the target. R2 relations, in turn, correspond to negative analogies: those whose target are known not to carry over from the source. Finally, R3 relations correspond to neutral analogies: those for which it is unknown whether the source carries over or not to the target. The third component is crucial since it allows one to explore the openness and heuristic fruitfulness of analogies in scientific representation, while being fully grounded on the various partial mappings that hold among the relations for which enough information is available.

9.2.2 Partial Structures and Structural Representation: A Challenge Given the considerations above, partial mappings can be used as mechanisms of representation in scientific practice. It is in virtue of the fact that certain structures that characterize a given phenomenon bear significant relations with the latter (in the sense that there is a partial mapping between the two) that these structures can be used to represent the relevant features of the phenomenon under study. Of course, which features are relevant is a pragmatic matter, largely dependent on the context under consideration. (Further details about the approach can be found, e.g., in Da Costa and French 2003; Bueno et al. 2002, Bueno 1997; Bueno and French 2018.) Structural representation emphasizes the need for an intentional component in scientific representation. One needs to choose the relevant structures that are going to be used as representational devices. To represent some phenomena as a Newtonian system is to select models of Newtonian physics as the appropriate representational devices. But to implement such a representation also requires establishing suitable (partial) mappings between the models and the relevant phenomena. Note that the mappings are ultimately established (in many cases, after embedding models into other models) between models and the phenomena rather than between models and representations of the phenomena. We are, after all, trying to explain the phenomena rather than their representations. It may be objected that the very notions of isomorphism and homomorphism (or their corresponding partial versions) require that both the models in question and the related phenomena be represented set-theoretically, given that such morphisms are

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mappings over sets. The formulations of partial isomorphism and homomorphism presented above seem to support this point. Thus, before any representation in terms of suitable mappings can get off the ground, one needs first to have the phenomena represented within set theory (see Frigg and Nguyen 2020). In response, it is important to resist reifying the notions of (partial) isomorphism and homomorphism and their characterizations. There is a perfectly reasonable sense of (partial) isomorphism that connects abstract objects, or even concrete entities, directly to objects in the world. When setting up the table for your dinner guests, you want to make sure that there is an isomorphism between those who will have dinner and the number of plates on the table. Very clearly, there is an isomorphism between these objects; the objects themselves rather than their set-theoretic representation. The situation is not fundamentally different in the sciences; although, of course, the parameters there are far more complex. But eventually, after layers of models, one eventually aims to accommodate real phenomena. A related concern emerges from the fact that not only the various (full or partial) morphisms have been formulated set-theoretically, but the entire partial structures approach has been characterized in set theory. Thus, on the partial structures approach, and the point can easily be made with regard to other structural approaches to scientific representation (such as, for instance, those advanced in Van Fraassen 1980, 2008), the models used in scientific representation are themselves set-theoretic in nature. As a result, or so the argument goes, one needs to be a platonist about mathematics (one needs to be committed to the existence of mathematical objects, in particular, of sets) to be able to adopt structural representation. The difficulty, it seems, is especially telling in the case of partial structures, given the thorough set-theoretic way in which they have been formulated. It may seem that for those who are already committed to platonism, this objection may not seem to be problematic. The platonist already takes the existence of mathematical objects, including sets, as a given. The fact that this commitment shows up also in the context of scientific representation need not be surprising or upsetting. Two problems, however, need to be addressed, especially by platonists: (a)

(b)

Intrinsic formulations: It is unclear why the representation of physical (concrete) phenomena should require the use of abstract (causally inert and non-spatiotemporally locatable) objects. In principle, more intrinsic accounts (Field 2016, pp. P4–P7, 44–50), formulated in terms of the very objects that are being represented, rather than in terms of extrinsic entities, should be articulated, without the need for invoking extraneous objects. In the case of scientific representation, given that the phenomena under study are ultimately concrete, the use of abstract entities in their investigation takes the representations away from something intrinsic. Presumably, even platonists should be able to recognize the advantage of more intrinsic formulations: they are not explanatorily artificial or ill-motivated. Epistemological concern: An account is also needed of how one can have knowledge of, or, at least, form reliable beliefs about, abstract objects if they are used in scientific representation, so that one can properly integrate one’s beliefs

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about the physical phenomena with the representational devices invoked to accommodate them. Despite the variety of platonist approaches that have been developed, such as standard, object-oriented platonism (Gödel 1964/1983, and Quine 1960), mathematical structuralism (Shapiro 1997; Resnik, 1997), full-blooded platonism (Balaguer 1998), or neo-Fregeanism (Hale and Wright 2001), the need to provide an account of mathematical knowledge, or, at least, of the reliability of mathematical beliefs, still remains the weakest aspect of platonist views (see Bueno 2011). So, whether one is a platonist or not, the use of set-theoretic devices in scientific representation poses a significant difficulty. The remainder of this paper is designed to address it.

9.3 Representation Without Set-Theoretic Structures: The Ontology of Models Set Free Scientific representation should not require commitment to the existence of abstract entities, including sets. After all, the phenomena under study are typically concrete. (Pure mathematics, although part of the sciences, broadly understood, is, of course, an exception, given that the justification of mathematical results does not usually depend on experience.) Given the kind of phenomena under investigation in the sciences, it is sensible and desirable to search for more intrinsic accounts of scientific representation, since one thereby avoids the introduction of unnecessary, extraneous entities that are not part of the phenomena at stake. In this way, by avoiding the commitment to sets, one can in principle undermine the concern resulting from the lack of intrinsic formulations in scientific representation. Moreover, if no commitment to sets is involved in an account of structural representation, the epistemological problems of explaining how knowledge of these objects is possible and how such knowledge is invoked to account for scientific representation do not emerge. As a result, difficulties resulting from the epistemological front can be resisted. Four strategies can be advanced to avoid commitment to sets in structural representation: (a) one can adopt a modal-structural interpretation of set theory; (b) one can reconstruct the relevant mathematics in terms of second-order logic; (c) one can resist the need for a metaphysical interpretation of set theory, or (d) one can use ontologically neutral quantifiers when quantifying over sets. These strategies have in common the fact that they all allow for quantification over sets, but articulate different ways of avoiding commitment to their existence. As will become clear, some of them are more successful than others, but each suggests a path that, at least in principle, would offer resources to implement a structural approach to scientific representation without requiring platonism either about set theory or about the relevant models. In what follows, I critically examine each strategy, and indicate, as appropriate, some of the concerns and limitations they face.

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9.3.1 Modalizing Set-Theoretical Structures: A Modal-Structural Interpretation One way of avoiding commitment to sets in scientific representation is by considering only the possibility of certain structures rather than the existence of mathematical objects, including sets, in the formulation of the relevant theories. Mathematical theories are often articulated in such a way that what matters is not the existence or the nature of the objects that are posited, but the relations among them. Consider, for instance, arithmetic. Rather than the stipulation of an infinitude of numbers with a particular nature, the central feature of the theory is that there enough of them in a certain order. What they ultimately are does not really matter. There is not more to the number 1 than the facts that it is the unique successor of 0 and the only natural number to precede 2. Crucial here are the relations among these objects, which are understood just as placeholders, rather than things with fundamental natures. This is the key structuralist insight about arithmetic and mathematics more generally (see Shapiro 1997; Resnik 1997). The focus lies on the overall structure, the family of relations among certain objects, rather than on individual objects themselves and their nature. The structuralist insight can be extended further so that there is no commitment even to actual structures. This is articulated by a modal-structural interpretation of mathematics (Hellman 1989). Only the possibility of the relevant structures needs to be entertained. In the case of arithmetic, consider, for example, the statement S to the effect that there are infinitely many prime numbers. It can be formulated, in a modal second-order language, in terms of two kinds of statements: (a) (b)

If there were structures satisfying the axioms of Peano Arithmetic, it would be true in those structures that S. It is possible that there are structures satisfying the axioms of Peano Arithmetic.

Taken together, the two statements provide a translation into a modal language of statement S that, prima facie, seems to be committed to the existence of abstract entities (natural numbers). But, in contrast with S, the modal translation only considers the possibility of the relevant structures rather than the actuality of the mathematical objects in question. Statement (a) indicates that S would hold in the relevant structures if there were such structures (but never asserts that there are). Statement (b) indicates that the structures are possible (but never asserts that they exist). This is important to prevent that the negation of S also holds in the relevant structures, which would be the case if the structures in question were not possible. Throughout, possibility is taken by the modal-structural interpretation as primitive, and it should never be characterized in non-modal ways, such as in terms of possible worlds or abstract entities (otherwise, one would simply reintroduce, by the back door, the kinds of entities that were to be avoided in the first place). A concern can be raised to structuralist accounts of mathematics (including the modal-structural interpretation) regarding the identity conditions for numbers across different structures: is the natural number 2 the same object as the real number 2?

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Structuralists respond to this question differently. One possible answer challenges that there is a fact of the matter to resolve the issue: given that the properties of the relevant numbers are tied to the structures they are related to (namely, the natural number structure and the real number structure), there is nothing that could settle the issue in a principled way if these two structures are the only ones being considered (see Resnik 1997). Alternatively, given that the properties of the numbers depend on the relevant structures, and since the natural number structure and the real number structure exhibit each number with different properties, the numbers themselves are ultimately different (see Shapiro 1997). Of course, assuming that both natural- and real-number structures are embedded into a single larger structure, one can easily identify the number 2 of each structure for simplicity. This is done by traditional object-oriented platonists, who assume, implicitly, a single underlying universe. Interestingly, this option is also open to all forms of structuralism (including the modal-structural one), and invokes a larger structure in which the two other structures are embedded to identify pragmatically the corresponding numbers. If set theory itself could be formulated in a modal second-order logic (for a discussion, see Hellman 1989, pp. 53–93), one could in principle formulate the relevant mathematical resources for structural representation in a way that would avoid ontological commitment to sets: only the possibility of the relevant structures would be involved. As it turns out, however, it is unclear how far this strategy actually goes. It is arguably more successful for narrow mathematical theories, such as arithmetic and analysis, which are important to scientific representation, as a number of scientific theories relies on them for their formulation, and their modal-structural characterization, albeit somewhat awkward, can in principle be implemented. Set theory, however, is a different matter. To be formulated, models of set theory typically require inaccessible cardinals, and the sheer possibility of such cardinals is not enough to generate the relevant models. One needs to postulate the existence of various inaccessible cardinals to obtain the models in question, and this requires a commitment not just to their possibility, but to their actuality. Thus, as things stand, the modal-structural approach does not seem to offer a fully developed strategy to dispense with sets, although it seems more successful in accommodating narrower mathematical theories that are also used in scientific representation (for further discussion, see Bueno 2013).

9.3.2 Partial Structures and Second-Order Logic A second, albeit at best partial, approach to avoid commitment to sets in structural representation is to use second-order logic to characterize some of the structures and at least the relevant mappings among such structures. Although not all mathematical structures can be formulated in second-order logic alone, a number of them can. The scope is similar, to a certain extent, to the successful portion of the modal-structure interpretation, but without requiring the translation of the relevant statements into a

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modal second-order language. In fact, a surprising amount of mathematics can be formulated in systems of second-order arithmetic alone, including arithmetic and analysis (see Simpson 2009). In this respect, large portions of the mathematics that is used in scientific practice and in scientific representation can be formulated just in terms of second-order logic, independently of set theory. The issue arises as to whether second-order logic is indeed logic or just “set theory in sheep’s clothing” (Quine 1986, p. 66). Part of the concern is that second-order logic allows for the formulation of substantial mathematical statements, such as statements that have a model if, and only if, the Continuum Hypothesis holds (Väänänen 2020). And, the argument goes, if second-order logic has such an expressive power, it is nothing more than a disguised (form of) set theory. It is not clear, however, that this argument goes through. The fact that secondorder logic is able to express set-theoretic content just indicates that it has substantial expressive power. But there is a difference between expressiveness and ontological commitment. The Continuum Hypothesis can also be expressed in English. But the idea that a natural language, just because of the fact that it allows for the expression of certain situations, is thereby committed to the existence of the relevant objects or state-of-affairs does not make much sense. The existence of ghosts, phlogiston, witches, and hobgoblins can be similarly expressed in English, but clearly this provides no reason for anyone to believe that these things exist. Precisely the same point holds for second-order logic and its expressive resources. If such logic is able to express some set-theoretic content, this is simply a reflection of second-order logic’s expressive power rather than an argument for its commitment to the existence of sets. Ontological commitment, pace Quinean considerations, requires more than just the expression of a certain situation. One would need to have independent reasons to believe that what is expressed, when properly interpreted, is also true. But this is an entirely separate matter from the expressive power of the logic in question: it is a matter of what exists, and that is not something that can be settled by logic alone. It is, therefore, crucial not to conflate existence and expressive power, as the objection above does. (For additional defenses of second-order logic, see Shapiro 1991; Bueno 2010; a helpful survey of philosophical issues about second-order logic can also be found in Väänänen 2020.) Furthermore, various morphisms that are central to structural representation can be formulated in second-order logic alone. For instance, in a second-order language, the statement to the effect that that there is a one-to-one correspondence between F and G can be straightforwardly expressed as follows (for further discussion, see Boolos 1998): ∃R[∀x∀y∀z(Rx y&Rx z → y = z) &∀x∀y∀z(Rx z&Ryz → x = y) & ∀x(F x → ∃y(Gy&Rx y)) &∀y(Gy → ∃x(F x&Rx y))]. In this way, the use of second-order logic allows for the expression of crucial components of structural representation. This includes significant portions of the

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mathematics that are required for the sciences as well as the relevant mappings among the structures in question. No commitment to set theory is needed for that.

9.3.3 Set Theory and Metaphysical Interpretations A third strategy in support of structural representation vis-à-vis set theory consists in stepping back and considering precisely the nature of the difficulty at hand. Strictly speaking, the concern is not with the commitment to the existence of sets per se, but with a certain understanding of their nature, according to which sets are abstract entities: causally inert, non-spatiotemporally located objects. It is the abstract nature of sets that makes intrinsic formulations of structural representation so hard to come by, given the concreteness of the empirical phenomena that are being represented. It is the abstractness of sets that makes one’s knowledge of, or one’s reliable beliefs about, them so difficult to secure. However, if one resists taking sets as abstract entities, the two problems that plagued structural accounts of scientific representation become far more manageable. But how can it be claimed that sets are not abstract? Interestingly, the proposed strategy does not make any such claim. It highlights that settling the issue regarding the ontological nature of sets (namely, whether they are abstract or not) is not needed to make sense of the use of sets in scientific representation. In fact, there is nothing in scientific (or mathematical) practice that requires settling such issue. One can simply leave it open. What matters to the practice are the particular mathematical results that can be obtained or formulated using set theory: theorems about a variety of entities and relations among them, involving, for instance, topological and Hilbert spaces, differential equations, complex numbers, or probability functions. Whether any of these objects is abstract or not turns out to be an additional, metaphysical question about them that the practice itself does not resolve, nor does it have to. On this strategy, one should resist the need to provide a metaphysical gloss on mathematical practice, including on set-theoretic structures used in scientific representation. The axioms of set theory, such as those of Zermelo-Fraenkel set theory with the axiom of choice (ZFC; see, for instance, Jech 2003, p. 3), do not specify anything about the metaphysical nature of sets. In particular, sets are not characterized as being abstract entities, nor are they characterized as being concrete. The issue is simply not addressed. Consider, for instance, the extensionality axiom, according to which if sets x and y have the same elements, then x = y. (The converse implication holds as a matter of logic alone.) Nothing in this axiom determines the nature of such sets. One can, of course, interpret the sets in question as being abstract. This is not ruled out by the axiom, but nor is it required by it either. The same considerations apply to all the other axioms of ZFC. Consider, for example, the following ones: The axiom of pairing states that, given sets a and b, there is a set {a, b} containing exactly a and b. The axiom of union specifies that for any set x, there is the union of all elements of x. The power set axiom, in turn, conveys that, for any set x, there is the set of all subsets of x. Once again, nothing in

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these three axioms requires sets to be abstract. They can be, of course, but need not. The issue is just left open. Perhaps the situation is different with the axiom schema of separation, according to which if P is a property (with a parameter p), for any set x and parameter p, there is a set that contains all members of x that have the property P. If properties are themselves characterized as abstract entities, it seems that this axiom would require the introduction of abstract entities. But just as sets need not be interpreted as being abstract, properties need not either. As a result, the axiom schema of separation does not force one to take sets to be abstract entities. How about the axiom of infinity, according to which there is an infinite set? If the concrete world is finite, there would not be concrete entities to satisfy such axiom. This is correct, but this can also be interpreted as a potentially infinite set rather than an actually infinite one, thus leaving it open the matter of the abstract nature of sets. Not even the axiom of choice, according to which every family of nonempty sets has a choice function (that is, a function that selects a member of each of the nonempty sets), requires sets to be abstract. One may insist that functions are abstract entities, and thus the axiom of choice requires sets to be so as well. But just as sets need not be interpreted as being abstract, functions need not either. In this way, it is possible to make sense of the axiom of choice without taking sets themselves to be abstract entities. Similar points can be made regarding the remaining ZFC axioms (replacement and regularity). Given these remarks, to assert that sets are abstract is to add a particular metaphysical interpretation to the set-theoretic framework provided by ZFC, an interpretation that takes sets to be objects that are causally inert and non-spatiotemporally locatable. To assert that sets are concrete, that they are located wherever their members are located, is similarly to add a metaphysical interpretation to set theory. One that is similarly not required. Given ZFC’s silence on the metaphysical nature of sets, the proposed strategy insists that one can adopt set theory to formulate the relevant models and structures using the very terms in which the scientific theories in question are characterized; one can also adopt set theory to express the mappings involved in the structural representation of the phenomena. But in doing so, one need not be committed to the metaphysical interpretation of sets as abstract entities. This issue is left entirely open. As a result, the use of sets becomes simply a convenience to express the structures under consideration rather than a reification procedure that settles the nature of the objects involved. Note that there is no tension here with the search for intrinsic formulations since, on this approach, sets are just expressive devices and do not characterize the nature of the phenomena under study (their own nature, as noted, is left open). It is similarly not mysterious how one can have knowledge of, or form reliable beliefs about, sets: whatever follows from the axioms of set theory provides the relevant knowledge. Of course, as with most fields of inquiry, a number of issues will remain unknown, given the inherent limitations of the information about such fields (and, in this case, in light of limitative results about the objects in question due to Gödel’s incompleteness theorems).

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9.3.4 Ontologically Neutral Quantification The fourth and final strategy to avoid ontological commitment to sets in structural representation consists in noting that sheer quantification over certain objects does not entail (or require) the existence of the objects over which one quantifies. This includes quantification over indispensable entities, those without which the relevant theories cannot be effectively formulated nor can the relevant explanations be successfully implemented (Colyvan 2001). After all, one can, and regular does, quantify over entities that do not exist. Quantification over fictional characters provide a clear illustration. Even granting that our literary practices require quantification over Hamlet, Anna Karenina, and many other fictional characters, that is, granting that quantification over them may be indispensable to explain and accommodate central features of our understanding and engagement with the relevant works, these considerations alone provide no reason to believe in the existence of fictional characters. It would be a mistake to look for Anna Karenina in Russia (or anywhere else in the world). Despite that, she bears relations with other people and objects in the world. Even though she does not exist, it is still correct to claim that some people are more faithful to their partners than Anna Karenina was to her husband. The same point goes through in the case of the sciences. The states of a quantum system can be represented in terms of suitable vectors in a Hilbert space. Does this entail that such vectors exist? It need not (as physicists often acknowledge), since Hilbert space vectors are only part of the mathematical formalism used to formulate quantum mechanics. The commitments of the theory are about its physical content rather than the underlying mathematical setting. Despite the fact that typically one cannot even express (non-relativist) quantum mechanics without quantifying over and referring to vectors in a Hilbert space (given the standard formulations of the theory), no such vectors are taken to exist, despite the theory’s impressive empirical success. This illustrates that one can, and regularly does, quantify over things that do not (or need not) exist. In this respect, scientific practice seems at odds with the indispensability argument, which requires commitment to all entities that are indispensable to our best theories of the world. Such practice is far more discriminative than the argument suggests, since it usually restricts commitment only to those entities for which there is good, independent evidence. (Colyvan 2001 provides a defense of the indispensability argument; Maddy 1997 offers a searching critique.) These considerations motivate an important distinction: quantifying over some objects is not the same as being committed to their existence. It, thus, makes sense to distinguish quantification from commitment to existence (see Azzouni 2004; Bueno 2005). One may quantify over the entire domain (universal quantification) or over a portion of the domain (existential quantification) without thereby assuming (or requiring) the existence of the objects in the domain, even if such quantification is indispensable, that is, it cannot be eliminated or dispensed with. To indicate the existence of the relevant objects, an existence predicate needs to be introduced, which

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indicates the conditions under which the objects in question are taken to exist. This is an ontologically neutral account of quantification. Interestingly, it is unclear that one can provide uncontentious necessary criteria for existence. It is easier, and often less controversial, to advance sufficient conditions. Few would question the sufficient condition to the effect that if an object is causally active and located in spacetime, it exists. However, if such condition is also taken to be necessary, that is, if an object exists, then it is spatiotemporally locatable and causally active, platonists will immediately complain that the condition clearly begs the question against their view. The complaint is warranted. Sufficient conditions for something to be taken to exist are less troublesome. The distinction between quantification and commitment to existence is also supported by considerations from mathematical practice. A mathematician, discussing classical set theory, may claim that: “There are sets that are too big to exist”. If quantification is characterized as being ontologically committing, the statement is turned into a contradiction: there exist sets that do not exist. But there is a perfectly uncontentious interpretation of the statement in which it is not contradictory. If one keeps quantification and ontological commitment apart, and resists giving the existential quantifier an ontologically loaded reading, what is stated is only that, among the sets, some (such as the Russell set) do not exist. In other words, some sets do not exist or, in symbols, ∃x (Sx ∧ ¬Ex), in which ‘S’ is the predicate ‘is a set’ and ‘E’ is the existence predicate. Of course, different views will characterize the existence predicate differently. Some will specify it in terms of ontological independence (Azzouni 2004); others in terms of observability, verifiability, or causal accessibility. However this is ultimately decided (if it can be decided at all), it is not a matter of logic, but ultimately depends on what there is. This is as it should be, given that what exists or not should not be a matter of logic. In this way, in light of the various strategies just outlined, one can allow for quantification over sets in structural representation without commitment to the existence of such sets. Structural representation, formulated in terms of partial structures, can be articulated without taking a platonist stance on the ontology of models.

9.4 Conclusion If the considerations above are near the mark, structural representation can be maintained without set-theoretic reification. It is possible to use set theory in an ontologically neutral way, for instance, by formulating the relevant structures in a modal second-order language (at least in the context of mathematical theories that can be properly modalized in this way), or directly in second-order logic (again at least regarding the mathematical theories that can be so formulated), or by resisting the urge to interpret the relevant structures as platonic entities, or, finally, by invoking ontologically neutral quantifiers. In none of these strategies is there a commitment to the existence of sets, even though sets are freely quantified over throughout.

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The considerations above should also make clear that there is no need for those who favor a structural approach to representation to identify models (as vehicles of representation) with set-theoretic structures. After all, scientific representation involves a variety of different sources, such as, iconic models, maps, mock-ups, and prototypes, none of which are sets (see Da Costa and French 2003). As a result, structural representation need not be tied to set-theoretic structures. It is also worth noting that structural approaches are also able to accommodate the representation of the behavior of their targets even in cases in which such behavior may seem not to have any structural similarity with the targets in question. Consider, for instance, the Newlyn-Phillips machine (Frigg and Nguyen 2020), which arguably bears no structural similarity with the target it aims to represent: an economy is not, after all, a series of interconnected water tanks; although there may be some similarity with the target’s behavior (Weisberg 2013), namely, the flow of water through tanks and of money in an economy. Interestingly, even in the case of modeling behavior, there are structural similarities between the machine and the flow of money, given that both involve a given amount of fluid/money that goes cyclically through various channels, whether they are tanks/chambers or types of economic expenditures, with the money flowing from the treasury to education or health services. Without such structural connections, made explicit via suitable (partial) morphisms, it would be unclear how the representation could ever be successfully implemented or assessed. To sum up, this work’s central idea is that structural representation does not require a platonist interpretation of the models that are invoked to implement the representing activity. Scientific representation can be, in this way, set free. Acknowledgements My thanks go to an anonymous reviewer and especially to Alejandro Cassini for extremely helpful comments on an earlier version of this work. They led to significant improvements.

References Azzouni, J. (2004). Deflating existential consequence: A case for nominalism. New York: Oxford University Press. Balaguer, M. (1998). Platonism and anti-platonism in mathematics. New York: Oxford University Press. Boolos, G. (1998). Logic, logic, and logic. Cambridge, MA: Harvard University Press. Bueno, O. (1997). Empirical adequacy: A partial structures approach. Studies in History and Philosophy of Science, 28(4), 585–610. Bueno, O. (2000). Empiricism, mathematical change and scientific change. Studies in History and Philosophy of Science, 31(2), 269–296. Bueno, O. (2005). Dirac and the dispensability of mathematics. Studies in History and Philosophy of Modern Physics, 36(3), 465–490. Bueno, O. (2010). A defense of second-order logic. Axiomathes, 20(2), 365–383. Bueno, O. (2011). Logical and mathematical knowledge. In S. Bernecker & D. Pritchard (Eds.), Routledge companion to epistemology (pp. 358–368). London: Routledge.

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Bueno, O. (2013). Nominalism in the philosophy of mathematics. Stanford Encyclopedia of Philosophy (Fall 2013 edition). https://plato.stanford.edu/archives/fall2013/entries/nominalismmathematics/. Bueno, O., & Colyvan, M. (2011). An inferential conception of the application of mathematics. Noûs, 45(2), 345–374. Bueno, O., & French, S. (2018). Applying mathematics: Immersion, inference, interpretation. Oxford: Oxford University Press. Bueno, O., French, S., & Ladyman, J. (2002). On representing the relationship between the mathematical and the empirical. Philosophy of Science, 69(3), 497–518. Colyvan, M. (2001). The indispensability of mathematics. New York: Oxford University Press. Da Costa, N., & French, S. (2003). Science and partial truth: A unitary approach to models and scientific reasoning. New York: Oxford University Press. Field, H. (2016). Science without numbers: A defense of nominalism. Second edition. Oxford: Oxford University Press. French, S., & Ladyman, J. (1997). Superconductivity and structures: Revisiting the London account. Studies in History and Philosophy of Modern Physics, 28(3), 363–393. French, S., & Ladyman, J. (1999). Reinflating the semantic approach. International Studies in the Philosophy of Science, 13(2), 103–121. Frigg, R., & Nguyen, J. (2020). Modelling nature: An opinionated introduction to scientific representation. Cham: Springer. Gödel, K. (1964/1983). What is Cantor’s continuum problem? In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics: Selected readings. Second edition, (pp. 470–485). Cambridge: Cambridge University Press. Hale, B., & Wright, C. (2001). The reason’s proper study. Oxford: Oxford University Press. Hellman, G. (1989). Mathematics without numbers: Towards a modal-structural interpretation. Oxford: Clarendon Press. Hesse, M. (1970). Models and analogies in science. Oxford: Oxford University Press. Hughes, R. I. G. (2010). The theoretical practices of physics: Philosophical essays. Oxford: Oxford University Press. Jech, T. (2003). Set theory. Third, revised and expanded edition. Berlin: Springer. Maddy, P. (1997). Naturalism in mathematics. Oxford: Clarendon Press. Mikenberg, I., Da Costa, N., & Chuaqui, R. (1986). Pragmatic truth and approximation to truth. Journal of Symbolic Logic, 51(1), 201–221. Quine, W. V. O. (1960). Word and object. Cambridge, MA: The MIT Press. Quine, W. V. O. (1986). Philosophy of logic. Second edition. Cambridge, MA: Harvard University Press. Resnik, M. (1997). Mathematics as a science of patterns. Oxford: Clarendon Press. Shapiro, S. (1991). Foundations without foundationalism: A case for second-order logic. Oxford: Clarendon Press. Shapiro, S. (1997). Philosophy of mathematics: Structure and ontology. New York: Oxford University Press. Simpson, S. (2009). Subsystems of second order arithmetic. Second edition. Cambridge: Cambridge University Press. Suárez, M. (2004). An inferential conception of scientific representation. Philosophy of Science, 71(5), 767–779. Väänänen, J. (2020). Second-order and higher-order logic. Stanford Encyclopedia of Philosophy (Fall 2020 Edition). https://plato.stanford.edu/archives/fall2020/entries/logic-higher-order/. Van Fraassen, B. C. (1980). The scientific image. Oxford: Clarendon Press. Van Fraassen, B. C. (2008). Scientific representation: Paradoxes of perspective. Oxford: Clarendon Press. Weisberg, M. (2013). Simulation and similarity: Using models to understand the world. New York: Oxford University Press.

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Otávio Bueno is Professor of Philosophy and Cooper Senior Scholar in Arts and Sciences at the University of Miami. He has defended a structural account of representation, and is the author (with Steven French) of Applying mathematics: Immersion, inference, interpretation (Oxford: Oxford University Press, 2018). He is also the editor (with Ruey-Lin Chen and Melinda Fagan) of Individuation, processes, and scientific practices (New York: Oxford University Press, 2018), and (with George Derby, Steven French and Dean Rickles) of Thinking about science, reflecting on art: Bringing aesthetics and philosophy of science together (London: Routledge, 2018), and (with Øystein Linnebo) of New waves in philosophy of mathematics (Hampshire: Palgrave MacMillan, 2009). He is Editor-in-Chief of Synthese.

Chapter 10

Representation and Surrogate Reasoning: A Proposal from Dialogical Pragmatism Juan Redmond

Abstract Juan Redmond presents an inferential conception of scientific representation based on the ludic perspective of dialogical pragmatism. He conceives of his proposal as an answer to the question “How models are used to represent the world?” Consequently, he lines up with the host of pragmatic approaches that stress the importance of the notion of use and users for representing and modeling. The main point of his proposal is that users or agents use models to represent their targets doing inferences concerning those targets on the basis of reasoning dialogically about the models. Keywords Inferentialism · Surrogative Reasoning · Pragmatics · Dialogical Logic

10.1 Introduction In this article, we embrace the approaches that criticize the assumption that representation is a dyadic relation of correspondence between the representative vehicle (the model) and its target. For this reason, we take distance from any notion that resends a certain type of “mirroring” engaged in concepts such as isomorphism, similarity, likeness, or resemblance or, in general, between the properties of the vehicle of representation and its target (Suárez 2004). However, we intend to maintain the idea that a model represents its target system. For this reason, in this work, we line up with the host of pragmatic approaches to representation that stress the importance of the notion of use and users for representation (Suárez 2004; Giere 2004; Bailer-Jones 2003; Frigg 2003). Indeed, we defend that the intentional participation of the users is what guides and determines the representation function that the models fulfill in the scientific practice and, therefore, it allows to explain accurately the dynamics of the modeling process. In that sense, concerning the function of representation that is at the core of the process of modeling, following Giere (2004), our proposal is one answer to the question “how models are used to represent the world”. J. Redmond (B) Condell 391, 2520000 Viña del Mar, Chile e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Cassini and J. Redmond (eds.), Models and Idealizations in Science, Logic, Epistemology, and the Unity of Science 50, https://doi.org/10.1007/978-3-030-65802-1_10

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Thus, the main point compromised in our proposal is that users or agents use models to represent their target doing inferences regarding the target (T ) on the basis of reasoning about the model (M). And we sustain in this article (following Clerbout 2014) that this intentional practice of modeling through representation by inferring, could be better understood as an interactive and dynamic process that requires an adequate theoretical structure. And that the theoretical perspective of the dialogical pragmatism could contribute significantly in to this last point. Some pragmatic approaches (Suárez 2004) have defended the importance of the level of competence and information of the users in the act of inferring. Thus, for an agent to draw inferences regarding T on the basis of reasoning about M, requires pragmatic skills that depend on the aim and context of the particular inquiry. We will sustain that these conditions are supplied by the dialogical approach to inferences where users are integrated elements of the inferential structure itself. In other words, we think that conditions of “level of competence” or “adequacy” required for users are integrated or incorporated as part of the ludic structure of the game-theoretical semantics of dialogical approach to inferences. It is exactly, indeed, what is required for the Proponent and the Opponent, the users (players) in the dialogical structure. Using dialogical logic as the basis for inference, users turn out to be part of the underlined logic for surrogate reasoning. In that way, as Suárez affirms (2004: 773), the skills thought for the users become the pragmatic “skills” of a logic used by an agent.

10.2 Representation and Surrogate Reasoning: General Insights We believe that any attempt to clarify in an acceptable way how we gain knowledge through the practice of modeling in science, should state how the reasoning made in a model leads to statements about that portion of reality (the target) pointed by the model. Indeed, we deliver or convey affirmations from the model to the target system. The question is on what foundation we deliver those affirmations from one to the other. The answer depends on the way in which each philosopher or scientist think of the relationship between the model and the target system. In generalized literature, this relationship is mostly known as “representation.” That is, depending on the way in which each theory understands representation, it will be the way in which it explains on what foundation it is possible to deliver affirmations from the model to the target system. Although the different approaches differ in the way they understand what it is to represent (isomorphism, similarity, etc.), there seems to be a common feature in all of them: the model inferentially fulfills the function of a substitute. The latter is the point of view of Swoyer (1991: 449) who defined this function as surrogate reasoning. What kind of surrogate function accomplish the model? Well, initially we will consider that a model—as a scientific representation—should allow us

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to generate hypotheses about their target systems. We would then have those approaches in which the notion of surrogate reasoning derives from the definition of representation, and other approaches—such as the one to which we endorse in this article-, which defines representation as a specific mode of surrogate reasoning. The latter are known in generalized literature as inferential perspectives of representation (Hughes 1997, 2010; Suárez 2004, 2010). While the former establishes a correspondence between the model and its target that can be described as formally or materially bijective (mirroring), the latter stresses this correspondence inferentially. The widespread acceptance of surrogate reasoning as a condition of adequacy for all models as scientific representations makes it possible that a definition centered on it leaves no approach outside. That is, in the first case there would be a one-to-one correspondence between the parts of each one: when it is said that M represents T, a formal or material commitment is established with T. Even when scientists are not sure if T exists. For example, an architect is committed to the scale ratio of the distances between the parts of the model of a bridge and the bridge itself (the target system), even if the bridge has not yet been built; the same in the past for a physicist committed to the isomorphism of the “planetary” model of the atom. In general, perspectives with these commitments are called “representationalist”; while those that put the weight on the other functions that a model fulfills are called “non-representationalist.”

10.3 What is to Represent? We have then that the modeling practice is based on the relationship established between the model and the target system. We call this relationship representation and each approach—to be well defined- [following Frigg and Nguyen 2017], according to their foundations, fill the blank in: S is a scientific/epistemic representation of T iff ___

This formula is understood as the Scientific/Epistemic Representation Problem (SR-Problem or ER-Problem for short). The difference between Scientific and Epistemic refers to what Callender and Cohen (2006: 68–69) point out as the “demarcation problem” (Popper) but for representations: “scientific representations” for those who demarcate scientific from non-scientific representations; “epistemic representations” for those who consider irrelevant that distinction—following a suggestion of Contessa’s (2007) for broadening the scope of the investigation. It depends on this definition whether the perspective is representationalist or not. For our article we retain that representationalist perspectives consider surrogate reasoning as a by-product while the non-representationalist perspective we defend here considers surrogate reasoning as the core of the definition. Other issues that should answer a perspective on modeling, to be well defined, are the following:

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• The Representational Demarcation Problem (the question of how scientific representations differ from other kinds of representations). • The Problem of Style (what styles are there and how can they be characterized?). • To formulate Standards of Accuracy (how do we identify what constitutes an accurate representation?). • The Problem of Ontology (what are the kind of objects that serve as representations?). In addition to these issues, as Frigg and & Nguyen (2017) points out, we also have “conditions of adequacy” that must be completed satisfactorily. These are: Surrogate Reasoning, Possibility of Misrepresentation (which should be possible to distinguish from a non-representation); Targetless Models; Requirement of Directionality (From M to T but not vice versa); Applicability of Mathematics. • Surrogative Reasoning (scientific representations allow us to generate hypotheses about their target systems). • Possibility of Misrepresentation (if S does not accurately represent T, then it is a misrepresentation but not a non-representation). • Targetless Models (what are we to make of scientific representations that lack targets?). • Requirement of Directionality (scientific representations are about their targets, but targets are not about their representations). • Applicability of Mathematics (how does the mathematical apparatus used in some scientific representations latch onto the physical world). Our dialogical point of view of the process of modeling will offer an answer to the Scientific/Epistemic Representation Problem. This answer, as we signalize above, is aligned with pragmatic and inferentialist approaches (mainly following Suárez 2004). But at the same time, we believe that another of the relevant contributions of the dialogical approach is that it makes evident the importance of considering one more condition of adequacy among those cited above: the applicability of logic. In the same context of the problems and questions about the applicability of Mathematics to a physical system (Pincock 2012), it is possible to ask about the adequacy of the logic we use to obtain conclusions that go from M to the empirical world (hypothesis proposal). More details below.

10.4 A Ludic-Dialogical Approach to Representation: Interaction and Applicability of Logic As noted above, we defend here the ludic-dialogical approach as a proper frame for addressing notions of representation and modeling in science. Even for those proposals that are not inferentialist. For those that are, there is a complementary advantage that is explained in (ii) below. In effect, our claim is that the ludic-dialogical

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approach is an optimal “frame” to capture the whole process of representation and modeling in science and the main contributions are the following: i.

ii.

The ludic-dialogical approach allows to formally reflect the interaction between the model and the target system during the modeling process. In effect, due to its interactive conformation (pragmatic approach elaborated from the notions of use and agency), the dialogical frame allows to capture the fluidity of the exchanges between the model and the target system as a formal dialog. Particularly to give an account of the implicit interaction of a process where from M we generate hypotheses on TS. The ludic-dialogical approach offers the possibility of elaborating different rules of inference that responds to the criterion of applicability of logic as a condition of adequacy. We will point out one of the problems of not taking into account this criterion: we refer to the targetless condition. In effect, if the logic underlying surrogate reasoning is committed ontologically, as is the case with classical logic (Frege-Russell-Quine tradition), it will prevent certain accuracy conditions from being met, as is the case, for example, with targetless. That logic will lead to a predictable failure. But the interesting thing is that the failure is not due to a problem of the standard of adequacy and the possibility of misrepresentation. This is a case where we apply the wrong frame because we didn’t take into account the applicability of logic.

Both contributions, from our point of view, allow us to believe that dialogic is an optimal frame to capture with greater fidelity the insights of the inferential approach to scientific representation. Indeed, when choosing an inferential approach, as Knuuttila affirms (2005: 36), we are not contesting representation itself but rather the way it is conventionally understood. We think that the pragmatic point of view of the dialogical approach could give new insights into the notion of representation. To reach this goal we sustain too that it is not enough to get away from the notion of denotation or reference refusing all kinds of mirroring approaches. Our belief is that it is necessary to use an appropriate relation of inference for surrogate reasoning. In other words, if—as Suárez says (2004 and 2010)-, the representation cannot be reduced to the mere reference or denotation of an object by another, we should base the surrogate reasoning in a semantics not engaged with those notions as is the case of the dialogical frame. In that way we propose dialogic as a general frame for capturing the dynamics present in the practice of representing and modeling. Based on these notions we will formulate a dialogical conception of representation and modeling. Let’s now first introduce the general insights of dialogic.

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10.5 Interaction and Language Games in Logic: Origins of Another Paradigm Within the mathematical logic of the twentieth century, a set of techniques, concepts and results emerged and constituted a sort of paradigm in which the idea of logical inference is a particular case of the interaction between the participants of a critical dialogue. As already remarked in the works of Per Martin-Löf (1996), the philosophical vocabulary often presents the following ambiguity: the same term designates both an action and the content or result of such action. This is the case, among others, of “reasoning” and “proposition". Johan van Benthem points out (1991:159) that this ambivalence, which oscillates between a “static” pole (the content) and a “dynamic” pole (the action), confirms the different representations of what the task of logic should be. For the tradition of mathematical logic that reaches its zenith in Frege’s perspective, logic is the study of a structure composed of propositions (independent objects, see Satz an Sich of Bolzano) and of relations between these propositions (the relation of “logical consequence” is the most important). But from the thirties, a new current thinks that the theory of meaning and content of thought (static tradition) must be accompanied by the theory of the act of thinking or giving meaning (dynamic point of view). We can consider the intuitionism of L.E.J. Brouwer as the starting point of this tradition. The propositional structure that is the object of the static tradition, is defined semantically as a Boolean structure, where propositions are considered as truth values and logical constants as operators on those values. Syntactically, as an algebra of pure signs on which we operate via calculation rules. The existence of such structures is considered a mathematical fact, and its suitability to give the rules of reasoning as evidence. Therefore, in this perspective, in the words of van Benthem: In other words, the emphasis lies on "that" or "whether" certain statements are true about a situation, not so much on "how" they come to be seen as true. To some, this ‘declarative’ bias, as opposed to a ‘procedural’ one, is even a laudable hallmark of logical approaches as such. But, in recent years, there has been a growing tendency in logical and linguistic research to move dynamic considerations of cognitive action to the fore, trying to do justice to the undeniable fact that human cognitive competence consists in procedural facility just as much as communion with eternal truths. (van Benthem 1991: 159)

Focusing on “how” statements come to be seen as true has important consequences, both philosophical and technical.1 It is here, precisely, that intuitionist logic comes into play insofar as it is the first attempt to develop these consequences. In fact, there are at least two principles that are considered valid for classical logic but are presented as problematic for those who pretend to consider the mode of apprehension of the truth of a statement by a subject of knowledge: the first is double negation, the second is the law of excluded middle. 1 See

Rahman/McConaughey/Klev/Clerbout (2018) developed a thorough approach to logic where games of giving and asking questions are extended to games of How.

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The first is the nucleus of a crucial mode of inference in mathematics: reasoning by the absurd. To deduce A from its double negation, according to the intuitionists, generates problems that directly concern the existential quantifier: we can show by the absurd the existence of mathematical entities without the need to exhibit them or build them. The latter, for intuitionists, challenges the significance of the quantifier. If our interest is the mode of apprehension of the truth of a statement, it seems more reasonable to demand that the condition of recognition of the truth of an existential affirmation be the capacity to determine a particular value for a quantified variable, in such a way that the statement of the correctly instantiated formula is true. With respect to the law of excluded middle, the argument that proves its validity hides a subtlety unacceptable to intuitionists: the demonstration of the principal disjunction is carried out without either of the two members of the disjunction being proven. In this sense the intuitionists argue that it would be reasonable to perform this test by demonstrating at least one member of the disjunction (as the function of a disjunction is defined in the theory of demonstration). As Dummett (1977) points out, if we do not want to consider a theory of truth independently of a theory of the mode of recognition of that truth, the excluded middle is unacceptable since it forces us to consider in a demonstration the existence of demonstration that we do not possess. For all this, the logician who decides to take into account the recognition of truth, in the form of a theory of the construction of demonstrations or an epistemology of the means of verification, is led without delay to modify his conception of the laws of logic, which gives rise to non-classical logics. However, the development of intuitionist logic encounters a major difficulty of semantic order. For classical propositional logic is provided a notion of semantics developed from the works of Alfred Tarski (1983): theory of models. This theory assumes of the notion of truth via the notion of reference: from a function of interpretation of individual terms and predicates, it is possible to make explicit the value of truth of a statement relative to the structure. But the big problem here is that the Tarskian definition of models presupposes the validity of the third excluded. Therefore, intuitionist logic emerges as a pure calculation without semantics. In this sense, the dialogic logic developed by Paul Lorenzen (1960) and Paul Lorenzen and Kuno Lorenz (1978) arises directly from the intention of giving intuitionist logic a semantics of its own. In general, we have two traditions that claim to implement the notion of language games in logic. On the one hand, the logic of Lorenzen and Lorenz was born directly from the intention to give intuitionist logic a semantics of its own. On the other hand, the semantics of Hintikka games (the GTS = games semantics), which shares the game-theoretical tenets of dialogical logic for logical constants; but turns to standard model theory when we reach the level of elementary statements. At this level standard truth-functional formal semantics comes into play. More recently Rahman and his team of Lille, in order to develop dialogues with “content” they enriched the dialogical framework with fully interpreted languages (as implemented within Per Martin-Löf’s Constructive Type Theory). They call it “Immanent Reasoning” (Rahman/McConaughey/Klev/Clerbout (2018)). One of the chief ideas animating Immanent Reasoning is that the origin of concepts is rooted not

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only in games of giving and asking for reasons (games involving Why-questions), but also in games that include moves establishing how it is that the reasons brought forward accomplish their explicative task. Thus, Immanent Reasoning games are dialogical games of Why and How.

10.5.1 Dialogical Logic2 The dialogic approach focuses on the procedural dimension of the demonstration in order to give a semantics to the statements. Indeed, what is at issue here is to know to what extent it is possible that the notion of demonstration, which is normally absent from current linguistic practices, can give semantics to statements. And it is, in fact, in the notion of dialogue that Lorenzen and Lorenz (1978) find the concept that allows them to explain the meaning of logical constants, keeping intact the current linguistic intuitions and stressing the importance of the procedural and epistemic dimension of the notion of demonstration. Dialogues are mathematically defined language games that establish the interface between the concrete linguistic activity and the formal notion of demonstration. Two interlocutors (Proponent and Opponent) exchange movements that are concretely linguistic acts. The Proponent enunciates a thesis, the thesis of the dialogue, and undertakes to defend it by responding to all the opponent’s criticisms. The allowed criticisms are defined in terms of the structure of the statements affirmed in the dialogue. For example, if a player affirms conjunction A and B, at the same time he gives the opponent the possibility to choose one of the two and to demand that he affirm it. The very notion of asserting is defined by the context of critical interaction: asserting means committing oneself to providing justification to a critical interlocutor. But in dialogues there is no general theory of justification but only insofar as they are logically complex statements that find their justification from simple statements. In turn, simple statements are justified in reciprocal action with the critical interlocutor. That is, as the rule exhorts, the Proponent may consider an elementary statement justified, if and only if the Opponent has granted that justification. This rule confirms the formality of the dialogues: the Proponent wins without presupposing justifications for any particular statement. A formal dialog (Keiff 2012) follows two kinds of rules: rules of particles and structural rules according to the following extension of the first-order language: FO[τ], as the result of enriching a first-order language over vocabulary τ with the following metalogical symbols: i. ii. iii. 2 For

two force symbols, ? and !; the symbols 1, 2, ∀x/c, ∃x (where x and c stand, respectively, for any variable and any constant of the vocabulary τ); two labels, O and P (standing for the players, Opponent and Proponent, respectively). a more detail presentation see the Appendix at the end of this chapter.

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Particle Rules

The particle rules constitute the local semantics of a logic, for they determine the dialogical meaning of each logical constant. They abstractly describe the way a formula of a given main connective may be objected to (challenge), and how to answer the objection (defense). Particle rules make no reference to the context of argumentation in which the rule is applied. But they are intrinsically dependent on the notion of dialogue, since they describe sequences of language acts. An example for the conjunction:

10.5.1.2

Structural Rules

In the same way as particle rules describe the local meaning of the logical constants, structural rules determine their global semantics leading the general organization of the dialogues. The structural rules are meant to organize the application of the particle rules in such a way that the set of moves resulting from the application of the rules to an initial formula (called the thesis) yields a dialogue that can be seen as a valid argument for the thesis. A particular development of both of these rules will be given below for a Targetless Dialogical Logic.

10.6 The Dialogical take of Representation We propose the following answer for the S/E representation problem: 1. 2.

M is a model for T iff M represents T. M represents T iff the following conditions are met: a.

b.

The representational force of M points to T: there is an interactive relationship established between M and T (a dialogue scheme Δ) that allows competent agents to defend statements in M (winning strategies) as standing affirmations of a (real or possible) target system. There is a set Δ1 (dialogues) such that their winning strategies for a set of n thesis on M (Dn) are considered by competent agents as winning strategies for the set of m thesis on T (Dm): Dm ⊂ Dn.

The set of dialogues and winning strategies are the inferences draw by competent and informed agents on M and regarding T. In other words, the set of n thesis on M

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which are considered at the same time by competent agents as theses on T, are the hypotheses that should be tested.

10.6.1 Representational Force of the Model Following Mauricio Suárez, we consider the “representational force of the model” as the capacity of the model to lead a competent user to a consideration of the target (2004: 768). Thus, if force is a relational and contextual property determined mostly by the intentional use of agents, our proposal—the dialogical perspective— , contributes positively to this point by preserving the relational (interactive) and contextual features of the process of modeling (principally surrogate reasoning), through a semantic based on agents and the notion of use. The model, through the intentional use of competent agents, must provide with specific information regarding their target. That’s why it is not an arbitrarily chosen sign. More informative and less informative is at stake here. In that way our proposal rejoins the idea of Suárez (2015: 43) which is nicely presented in his example of the graph (a Model) representing a bridge (the target system): “The force of a representation is essentially linked to a practice of interpreting features of the graph as standing for features of a (possible, but not actual at that time) bridge.” We recover this idea by saying that the Model allows -in a dialogue- to defend statements in M as standing affirmations of a (real or not real) target system. Thus, our starting point is that there is an interactive relationship established between M and T (a dialogue scheme Δ) that allows competent agents to defend statements in M (winning strategies) as standing affirmations of a (real or possible) target system.3

From a dialogical point of view, T and M assume the roles of the Opponent and Proponent, respectively. And the thesis affirmed on ‘0’ is an affirmation in M that will be defended from attributes (properties and conditions) assigned in T (in “C”). Thus, the interaction between M and T is presented as an interactive game. Let’s see a more detailed example:

3 “M

773).

allows competent and informed agents to draw specific inferences regarding T ” (Suárez, 2004,

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M stars the dialog on ‘0’ with the claim BT v CT (hypothesis about T ). The Opponent (T ) challenge that claim on ‘1’ and ‘3’. But the Proponent (the Model) could defend it successfully on the basis of information provided on ‘C’ by the Opponent (T ) him/herself. The fact that M could defend successfully his/her claim using the information (data) conceded by T, makes it possible to state that it is an interactive relationship (a dialogue) where an agent defends statements in M as standing affirmations of a (real or possible) target system. To present it in this way we presupposed that the relationship is established in a propositional language (without quantification).

10.7 Final Remarks Our proposal fits the Epistemic/Scientific Representation Problem from an interactive and inferential point of view bringing the notions of ‘force’ and ‘correctly drawing inferences’ according to a specific dialogical logic. In this way, we provide a neat explanation of the possibility of misrepresentation regarding the two forms in which this argument comes (following Suárez 2004: 775): inaccuracy and mistargeting. Part (i) of the dialogical conception takes care of mistargeting by explicitly bringing in the notion of force into the definition of representation. Part (ii) of this conception accounts for inaccuracy since it demands that we correctly draw inferences from the source about the target, without any logical or metaphysical commitment (“true”, ontological or wherever). In this sense, if M does not accurately represent TS, then it is a misrepresentation but not a non-representation. In our account there is no emphasis on one-to-one model-target correspondence. That’s why we do agree that inference should not engage mirroring. But if we don’t pay attention to the logic we use to infer, the latter could be a failed attempt. In this sense, we don’t believe that it is possible that the source could be non-isomorphic or dissimilar to the target and still licenses true conclusions. If we are dealing with the notion of ‘true’ of classical logic and obtaining conclusions, for example, we should accept that every singular term identified in the model (as representation of the target system) should depict the corresponding object in the target system. For example,

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if we affirm that k1 is a particle that moves in a Hilbert space of 10 dimensions, on which logical semantics do we justify that this affirmation is “true”? Classical logic does not help much in this respect since from the above statement we deduce that it is also “true” that there is something moving in Hilbert’s space of ten dimensions (Ak1 → ∃ xAx). And the existential quantifier ‘∃’, for the true or false, leaves no doubt here about the ontological commitment of k1 . Any attempt to clarify ‘true’ in other sense must be done by choosing a specific ‘semantic framework’. Our choice was the approach of dialogical pragmatism. However, inside the dialogical frame it is necessary still to elaborate the rules of inference that allow a free logic (a task for a future article). We also have that, according to the representational force of the model (2a), targetless models are dealt with successfully because the dialogical approach of representation does not require the existence of a target. That’s why our proposal is not compatible with the Representational Demarcation Problem and is ontologically non-committal because anything that has an internal structure that allows an agent to draw inferences can be a representation. The condition (2a) also explains the directionality of representation: to defend statements in M (winning strategies) as standing affirmations of the target system not entail defending statements in the target system as standing affirmations of the model. To answer the Problem of Style, our claim is that M representing T involves the claim that there is an interactive relationship (style of the representation) established between M and T (a dialogue scheme Δ). Finally, once again, we stress the importance of the applicability of logic as a condition of adequacy. We believe that the ludic-dialogical approach responds to this requirement in an appropriate manner. Especially if the approach is inferentialist, but from this same ludic-dialogical approach, the requirements of the other perspectives can be met. Thus, the aim of this paper was to propose an inferential approach to modeling in science based on the ludic perspective of dialogical pragmatism. With this objective we proposed a dialogical-inferential definition of the notion of representation by filling the gap on the ER/SR-Problem and thus giving an interactive approach to the modeling process. The latter was developed from the conviction that adequacy of logic should be taking into account between the conditions for any theory of scientific representation, especially in the frame of the practice of modeling in science. Focusing on the adequacy of logic and surrogate reasoning allows us to propose an inferential and no-representationalist conception of modeling in science.

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Appendix: Standard Dialogical Logic4 Let L be a first-order language built as usual upon the propositional connectives, the quantifiers, a denumerable set of individual variables, a denumerable set of individual constants and a denumerable set of predicate symbols (each with a fixed arity). We extend the language L with two labels O and P, standing for the players of the game, and the question mark ‘?’. When the identity of the player does not matter, we use variables X or Y (with X=Y). A move is an expression of the form ‘X-e’, where e is either a formula ϕ of L or the form ‘?[ϕ1 , . . . , ϕn ]’. We now present the rules of dialogical games. There are two distinct kinds of rules named particle (or local) rules and structural rules. We start with the particle rules.

4 The

following brief presentation of standard dialogical logic is due to Nicolas Clerbout. The main original papers on dialogical logic are collected in Lorenzen/Lorenz (1978). For a historical overview see Lorenz (2001). Other papers have been collected more recently in Lorenz (2008, 2010a, b). A detailed account of recent developments since, say, Rahman (1993) and Felscher (1994), can be found in Rahman/Keiff (2005) and Keiff (2009). For the underlying metalogic see Clerbout (2013a,b). For a textbook presentation: Redmond/Fontaine (2011) and Rückert (2011a). For the key role of dialogic in regaining the link between dialectics and logic, see Rahman/Keiff (2010). Keiff (2004a, b) study Modal Dialogical Logic. Fiutek et al. (2010) study the dialogical approach to belief revision. Redmond (2010) studied Dialogic and fiction. Clerbout/Gorisse/Rahman (2011) studied Jain Logic in the dialogical framework. Popek (2012, p. 223–244) develops a dialogical reconstruction of medieval obligationes. See also Magnier (2013)—on dynamic epistemic logic and legal reasoning in a dialogical framework.

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In this table, the ai s are individual constants and ϕ(ai /x) denotes the formula obtained by replacing every occurrence of x in ϕ by ai . When a move consists in a question of the form ‘?[ϕ1 ,…,ϕn ]’, the other player chooses one formula among ϕ1 ,…, ϕn and plays it. We can thus distinguish between conjunction and disjunction on the one hand, and universal and existential quantification on the other hand, in terms of which player has a choice. In the cases of conjunction and universal quantification, the challenger chooses which formula he asks for. Conversely, in the cases of disjunction and existential quantification, the defender is the one who can choose between various formulas. Notice that there is no defense in the particle rule for negation. Particle rules provide an abstract description of how the game can proceed locally: they specify the way a formula can be challenged and defended according to its main logical constant. In this way we say that these rules govern the local level of meaning. Strictly speaking, the expressions occurring in the table above are not actual moves because they feature formulas schemata and the players are not specified. Moreover, these rules are indifferent to any particular situation that might occur during the game. For these reasons we say that the description provided by the particle rules is abstract. The words “challenge” and “defense” are convenient to name certain moves according to their relationship with other moves. Such relationships can be precisely defined in the following way. Let  be a sequence of moves. The function p assigns a position to each move in , starting with 0. The function F  assigns a pair [m,Z] to certain moves N in , where m denotes a position smaller than p (N) and Z is either C or D, standing respectively for “challenge” and “defense”. That is, the function F  keeps track of the relations of challenge and defense as they are given by the particle rules. A play (or dialogue) is a legal sequence of moves, i.e., a sequence of moves which observes the game rules. The rules of the second kind that we mentioned, the structural rules, give the precise conditions under which a given sentence is a play. The dialogical game for ϕ, written D(ϕ), is the set of all plays with ϕ as the thesis (see the Starting rule below). The structural rules are the following: SR0 (Starting rule) Let ϕ be a complex formula of L. For every π ∈ D(ϕ) we have: • pπ (P − ϕ) = 0, • pπ (O-n: = i) = 1,

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• pπ (P-m: = j) = 2. In other words, any play π in D(ϕ) starts with P-ϕ. We call ϕ the thesis of the play and of the dialogical game. After that, the Opponent and the Proponent successively choose a positive integer called repetition rank. The role of these integers is to ensure that every play ends after finitely many moves, in a way specified by the next structural rule.

SR1 (Classical Game-Playing Rule) • Let π ∈ D(ϕ). For every M in π with pπ (M) > 2 we have Fπ (M) = [m’,Z] with m’ < pπ (M) and Z∈{C,D} • Let r be the repetition rank of player X and π ∈ D(ϕ) such that – the last member of π is a Y move, – M 0 is a Y move of position m0 in π , – M 1 ,…,M n are X moves in π such that Fπ (M 1 ) = … = Fπ (M n ) = [m0 ,Z]. Consider the sequence5 π ’ = π ∗ N where N is an X move such that Fπ ’ (N) = [m0 ,Z]. We have π ’∈ D(ϕ) only if n < r. The first part of the rule states that every move after the choice of repetition ranks is either a challenge or a defense. The second part ensures the finiteness of plays by setting the player’s repetition rank as the maximum number of times he can challenge or defend against a given move of the other player. SR2 (Formal rule) Let ψ be an elementary sentence, N be the move P- ψ and M be the move O-ψ. A sequence π of moves is a play only if we have: if N∈ π then M∈ π and Pπ (M) 100 existing. The reason that such atoms are not observed naturally is that their nuclei undergo spontaneous fusion, and are unstable.” According to Yuri Oganessian et al. (2006: 044,602–1) the first attempt to synthesize 118 X was made by the Polish physicist Robert Smolanczuk. Indeed, after recognizing the synthesis of elements 110 to 112 by Sigurd Hofmann and collaborators in Darmstadt, Smolanczuk (1999: 2638) concludes that the synthesis of heavier nuclei might be possible if adequate bombarding energy was available. For him the most optimistic synthesis was that of isotope 293 118 with excitation energy of 13.31 MeV. Well, since 2002 the Oganessian team carried out experiments, reactions between atoms of Californium and Calcium, and Curium and Calcium, to try to synthesize isotopes of element 118, using Curium and Californium as targets, with the result that “The properties of the nuclei produced in these experiments corroborated the assignment of the first observed event to the decay of 294 118.” (Oganessian et al. 2006: 044602–2). And in the conclusion of the paper they claim: “A new element with atomic number 118 was synthesized for the time in the 248 Cf + 48 Ca reaction.” That is, “As a whole, the results of the experiments agree with the predictions of theoretical models concerning the properties of the superheavy nuclei in the vicinity of closed nuclear shells.” (Oganessian et al. 2006: 044602–1). In November 2016 the IUPAC named Oganesson the new synthesized element and gave it the chemical symbol Og. That is, 118 X became finally 118 Og, and it is

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the last chemical element of the periodic table. Kurt Kolasinski (2017) presents the complete periodic table, which already includes Og. Since the discovery of Og, researches have been carried out to discover its atomic and molecular properties. This is the case, for example, of Clinto S. Nash (2005: 3499), for whom “Element 112 is likely to be similar in most respects to Hg, element 114 is likely to be significantly less active than Pb although not inert, and element 118 seems nearly certain to be far more active than Rn, or perhaps even element 114 or element 112.” And Paul Jerabek and co-authors (2017: 4), who recognize that the isotope 294 of 118 Og was produced in a heavy fusion reaction of a Calcium beam and a Californium target, attribute to Og the electronic configuration indicated above, with which Og completes row 7 of the periodic table, and conclude that superheavy atoms, like Og are “qualitativey different from the lighter congeners”. The prediction of the existence of element 118 has been confirmed and consolidated experimentally.

11.5 Conclusion From what has been presented in the preceding sections, it can be inferred that explanation and prediction are capacities of the theories/models of science that go hand in hand. As a matter of fact, a model capable of explaining the Periodic Table of the elements is also capable of predicting the existence of new elements; and the prediction of supernovae is unthinkable without previously having an explanatory supernova theoretical model. If models are understood as tools, or devices, for prediction and (theoretical) explanation, then it seems clear that they do not need to be compared with realworld objects, nor does the problem of coordination between model descriptions and imagined entities appear, nor the problem of coordinating the imaginings of different scientists that supposedly assumes the fictional approach. (Cf. Knuuttila 2017: Sect. 2.3). But the artifactual approach, that considers models as objects, would seem more appropriate since, adapting the main idea of this perspective (Cf. Knuuttila 2017: 3) to my own approach, models are hypothetical constructs intentionally proposed in science for some purposes, in particular those of prediction and theoretical explanation. The difference with the artifactual account is that this approach works with a very general model concept that is materialized in different representational modes and media, while, from my own perspective, a theoretical model of physics is a physical–mathematical construct designed to facilitate calculation; a calculation that, of course, can also be done using computational means, but that does not require material embodiment, except perhaps for the purposes of illustration or to make the model intuitive. Thus, for instance, the interior model of stars of the main sequence is a set of five differential equations, five gradients of pressure, mass, luminosity and temperature, whose combination in specific circumstances allows the calculation and comparison of stars with each other and with the observational data. I agree with

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Pierre Duhem (1969: 117) that “the hypotheses of physics are mere mathematical contrivances devised for the purpose of saving the phenomena.” In conclusion, representational modes and media are only relevant insofar as any theoretical model of physics has to be designed or presented in a way accessible to public knowledge, especially that of experts. In this work I have labelled the theoretical models of physics as constructs, tools, instruments or devices for the purposes of prediction and explanation. Adding the tag of artifacts can enrich the number of metaphorical adjectives for the use of theoretical models; in the understanding that it is only taken as an additional term without a privileged use with respect to the others.

References Babor, J., & Ibarz, J. (1968). Química general moderna. Barcelona: Editorial Marin. Bransden, B., & Joachain, C. (1983). Physics of atoms and molecules. Harlow: Addison Wesley Longman. Duhem, P. (1969). To save the phenomena: An essay on the idea of physical theory from Plato to Galileo. Chicago: The University of Chicago Press. Original edition: Essai sur la notion de théorie physique de Platon à Galilée. Paris: Hermann, 1908. Hund, F. (1978). Geschichte der physikalischen Begriffe. Teil 2: Die Wege zum heutigen Naturbild. Mannheim: Bibliographisches Institut, Band 544. Jerabek, P. et al. (2017). Electron and nucleon localization functions of oganesson: Approaching the Thomas-Fermi limit. arXiv:1707.08710v3 [nucl-th] 6 December 2017. Knuuttila, T. (2017). Imagination extended and embedded: Artifactual versus fictional accounts of models. Synthese Online First. https://doi.org/10.1007/s11229-017-1545-2. Kolasinski, K. (2017). Physical chemistry: How chemistry works. Chichester: Wiley. Misner, C., Thorne, K., & Wheeler, J. (1973). Gravitation. New York: Freeman and Co. Nash, C. (2005). Atomic and molecular properties of elements 112, 114, and 118. Journal Physical Chemistry A, 109(15), 3493–3500. Oganessian, Y., et al. (2006). Synthesis of the isotopes of elements 118 and 116 in the 246 Cf and 245 Cm+48 Ca fusion reactions. Physical Review C, 74, 044602. Ohnaka, K., et al. (2017). Vigorous atmospheric motions in the red supergiant star Antares. Nature, 548(7667), 310–312. Ostlie, D., & Carroll, B. (1996). Modern stellar astrophysics. Reading, MA: Addison-Wesley. Rivadulla, A. (2003). Revoluciones en física. Madrid: Trotta. Rivadulla, A. (2005). Theoretical explanations in mathematical physics. In G. Boniolo (Ed.), The role of mathematics in physical sciences (pp. 161–178). Dordrecht: Springer. Rivadulla, A. (2006). Theoretical models and theories in physics: A rejoinder to Karl Popper’s picture of science. In I. Jarvie (Ed.), Karl Popper: A centenary assessment (Vol. III, pp. 85–96). Aldershot: Ashgate. Rivadulla, A. (2016). Models, representation and incompatibility: A contribution to the epistemological debate on the philosophy of physics. In J. Redmond (Ed.), Epistemology, logic and the impact of interaction (pp. 521–532). Cham: Springer. Rivadulla, A. (2017). Archaeological researches on Popper’s philosophy of science: Lights and shadows. Ápeiron. Estudios De Filosofía, 6, 115–130. Rivadulla, A. (2019). Causal explanations: Are they possible in physics? In M. Matthews (Ed.), Mario Bunge: A centenary festschrift (pp. 303–328). Cham: Springer. Smolanczuk, R. (1999). Production mechanism of superheavy nuclei in cold fusion reactions. Physical Review C, 59(5), 2634–2639.

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Van Fraassen, B. (2008). Scientific representation: Paradoxes of perspective. Oxford: Clarendon Press. Yasuda, N., et al. (2019). The hyper suprime-cam SSP transient survey in COSMOS: Overview. Publications of the Astronomical Society of Japan. https://doi.org/10.1093/pasj/psz050.

Andrés Rivadulla is Professor of Philosophy (retired) at the Complutense University of Madrid. His research interests are the general philosophy of science, the history and philosophy of probability and theoretical statistics, the history and philosophy of physics and the epistemology and the methodology of science. He is the author of Filosofía actual de la ciencia (Madrid: Tecnos, 1986); Probabilidad e inferencia científica (Barcelona: Anthropos, 1991); Revoluciones en física (Madrid: Trotta, 2003); Éxito, razón y cambio en física. Un enfoque instrumental en teoría de la ciencia (Madrid: Trotta, 2004); and Meta, método y mito en ciencia (Madrid: Trotta, 2015).

Chapter 12

Commented Bibliography on Models and Idealizations Alejandro Cassini

Abstract This chapter provides a classified and commented bibliography of printed books on the philosophy of scientific modeling and related issues, such as representation, idealization, computer simulation, and others. It is intended as a guide for further readings concerning the main topics of the preceding chapters. Keywords Models · Idealizations · Fictions · Computer simulations Scientific models have been one of the favorite issues for philosophers of science in the last two decades. A complete, or almost complete, bibliography would be too long to be compiled here. Besides that, in the Internet era, an extensive printed bibliography has a relative utility; it cannot be complete and it will become out of date in a few years. For those reasons, we have decided to offer a selective bibliographical list, which is restricted to printed books, both monographs and collections of articles. There are too many articles to be listed, even being very selective. Extensive references to philosophical articles can be found in Bailer-Jones (2009), concerning older publications up to 2007, and in Magnani and Bertolotti (2017), and Frigg and Nguyen (2020), which include more recent works. We have selected here monographic books and collections of articles that deal specifically with scientific models, scientific representation, computer simulations, idealization, and scientific fiction, most of them written from a philosophical point of view. We have included a commentary on the listed bibliography in which the cited works are classified. The selection is limited to empirical sciences; we do not include, for instance, model theory in logic or mathematical fictionalism. It is also limited to philosophical literature, and for that reason, it does not include purely scientific works on modeling in the different sciences. We have not included textbooks, handbooks, or companions on the general philosophy of science, which usually contain chapters or short articles on most of the topics this book deals with. Many books on general topics in the philosophy of science, say, the realism-antirealism debate, such as Rowbottom A. Cassini (B) Department of Philosophy, University of Buenos Aires, Puán 480, 1406 Buenos Aires, Argentina e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Cassini and J. Redmond (eds.), Models and Idealizations in Science, Logic, Epistemology, and the Unity of Science 50, https://doi.org/10.1007/978-3-030-65802-1_12

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(2019), discuss the issue in terms of models, besides theories. We have generally not selected those books. General works on scientific models: Morgan and Morrison (1999) can be regarded as the work that originates a good deal of the present discussion on scientific modeling. Cartwright (1999) is another influential work, which is commented on and discussed in Hartmann et al. (2008). Hughes (2010) includes some influential papers on models and representation, which were originally written in the 1990s. There are a few comprehensive or general books on the topic: Harré (2004), Toon (2012), Gelfert (2016), Gerlee and Lundh (2016), and Downes (2021) provide good introductions from different standpoints. Da Costa and French (2003), Weisberg (2013), and Morrison (2015) include more advanced discussions. De Charadevian and Hopwood (2004), and Sterrett (2006) are about material and scale models. Boumans (2005), and Morgan (2012) deal with models in economics, Fox Keller (2002) with models in biology, Klein (2002) with models in chemistry, and Winsberg (2018) with climate models. Bailer-Jones (2009) is mainly about the history of the philosophy of modeling. Ippoliti et al. (2016) includes some general articles on scientific models. Historical origins: McGuinness (1974) includes Boltzmann’s 1902 seminal article on scientific models. Duhem (1906) is a classic, pioneering work in the philosophy of science with a negative assessment of the use of models in physics. Campbell (1920) contains an early account of analogical models and theories. There was not very much discussion of models in empirical sciences by the philosophers of science until the 1950s. Some older books that deal with the general philosophy of science, or with special topics, that include some discussion of scientific models are Braithwaite (1953), Hesse (1954), (1961), Harré (1960), Nagel (1961), and Hempel (1965). Early contributions: Freudenthal (1961) was probably the first book entirely devoted to the philosophy of scientific models. Black (1962) is a collection of articles on different topics that includes an influential account of models and metaphors. Hesse (1963) was the first monographic book on models and analogies in science. Achinstein (1968) was another early account of models. Bunge (1973) pioneered the idea of models as mediators between theories and phenomena. Wartofsky (1979) is an interesting collection of papers on models and scientific representation. Hacking (1983), and Mayo (1996) contain accounts of models of data and experiments. Galison (1997) includes an account of the use of simulations in experimentation. Herfel et al. (1995) contains several papers on models. Aronson et al. (1995) includes a discussion of models in physics. Suppes (1969), (1993), (2002) collect most of his contributions to the philosophy of scientific models since the 1960s. Wimsatt (2007) collects several influential articles. The model-theoretic view of scientific theories: The so-called semantic conception of theories conceives of empirical theories as collections of related models. It originated in early works by Suppes (1957), (1969). Endorsers of this view do not understand the concept of the model in the same way. The structuralist version of this view was proposed by Sneed (1971), explained by Stegmüller (1973), and developed in detail by Balzer et al. (1987). Other versions of the semantic view are those by Van Fraassen (1980), (1989), Giere (1988), (1997), (1999), (2006), Suppe (1977),

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(1989), and Da Costa and French (2003). Krause and Arenhart (2017) compares the semantic and non-semantic approaches to theories. Hughes (1989), and Healey (2017) apply the semantic view to quantum mechanics; Thomson (1989), and Lloyd (1994) apply it to biological theories. Scientific representation: The concept of representation was much discussed in the fields of the philosophy of language, the philosophy of mind, and aesthetics before entering into the field of the philosophy of science. Goodman (1968) is an account of pictorial representation that has been influential on many philosophers of science. Wartofsky (1979) is a neglected work that includes some articles on scientific representation. Walton (1990) is an extensive work on literary fictional characters that have been used as a source of inspiration for several accounts of scientific models. Lynch and Woolgar (1990) deals with the practice of representing within different disciplines from a sociological point of view, whereas Copmans et al. (2014) is an updated sequel of that work. Boniolo (2007) is a monograph on the concept of representation and its history that examines scientific models. Giere (2006), Van Fraassen (2008), Pincock (2012), and French (2014) are more recent works devoted, among other issues, to scientific representation. Frigg and Hunter (2010) is a collection of papers on scientific and artistic representation. Bokulich and Bokulich (2011) includes an account of structural representation. Bueno and French (2018) and Beni (2019) deal with structural representation in more detail. Frigg and Nguyen (2020) is an up to date survey of the different philosophical theories of scientific representation. Model-based reasoning: Reasoning with models is a pervasive practice in all sciences. Many different aspects of model-based reasoning are dealt with in Magnani et al. (1999), Magnani and Nersessian (2002), and Magnani (2014). Magnani and Bertolotti (2017) is an encyclopedic compilation on every aspect of the field. Nersessian (2008) is also relevant to the topic. Models, analogies, and metaphors: The use of models in science has been related in different ways to the use of analogies and metaphors. The pioneering works on the scientific use of metaphors are Black (1962) and Hesse (1963). Leatherdale (1974) relates scientific models with analogies and metaphors. Ortony (1993) is a classic collection of articles in the philosophy of metaphor, which includes a section on science. Hallyn (2000) is another useful collection of papers. Bartha (2010) provides a detailed study of analogical reasoning. Idealization: Despite its importance for scientific modeling, the concept of idealization has not been analyzed in detail by philosophers of science until recent times. Nowak (1979) is a pioneering work, the first monography on the topic. Potochnik (2017) is a monography on scientific idealization that includes a discussion of models. Wheeler (2018) is devoted almost exclusively to physical laws. Shanks (1998) and Nowakowa and Nowak (2000) are collection of papers on idealizations in different sciences. Thomson-Jones and Cartwright (2005) is a valuable collection of papers on the same topic. Cartwright (1989) deals with abstraction and idealization. Niiniluoto (1999), Sklar (2000), and Weisberg (2013) include discussions of idealizations in science, physics, and model building, respectively. Batterman (2002) and Strevens (2008) deal with the explanatory roles of idealizations. Borbone and Brzechczyn

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(2016) is a collection of articles dealing with idealization in the natural and the social sciences. Lenhard and Carrier (2017) contains some articles on mathematical idealizations. Sprenger and Hartmann (2019) deals with the application of the Bayesian methods to highly idealized models. Fictionalism: Vaihinger (1927) was the pioneering work in scientific fictionalism. In recent years, Vaihinger’s philosophy has been vindicated by many philosophers, as Appiah (2017) shows. There has been an explosion of works on fictionalism in different areas of philosophical inquiry, although the very concept of fiction has been understood in different ways. Suárez (2009), and Woods (2010) are two useful collections of articles on scientific fictions. Levy and Godfrey-Smith (2020) and ArmourGarb and Kroon (2020) are more recent collections on a variety of philosophical fictionalisms that include several articles on scientific models. Thomason (1999), Sainsbury (2010), Armour-Garb and Woodbridge (2015), and García-Carpintero (2016) are general works on the philosophy of fiction, which provide different approaches to the topic, although they are not primarily devoted to fictions in science. Kroon et al. (2019) is a comprehensive introduction to fictionalism in metaphysics with some references to scientific models. Woods (2018) deals with the logic of fictions and examines formal models. Falguera and Martínez-Vidal (2020) includes several articles on the fiction view of models. Computer simulations: The use of simulation models has become a standard practice in all sciences. For that reason, the study of computer simulations has many implications for the study of scientific modeling. The first philosophical account of computer simulations in science was Humphreys (2004). More recent general accounts include Winsberg (2010), Resch et al. (2017), Durán (2018), Lenhard (2019), and Varenne (2019). There are some relevant treatments of the topic in Radder (2003), Lenhard et al. (2006), Humphreys and Imbert (2012), Tolk (2013), Weisberg (2013), Morrison (2015), Lenhard and Carrier (2017), and Winsberg (2018), although some of these books are not devoted specifically to computer simulations. Magnani and Bertolotti (2017) is a large collection of detailed review articles concerning the state of the art in modeling and simulation. Oberkampf and Roy (2010) is a useful treatise written by computer scientists. Beisbart and Saam (2019) contains many articles on different philosophical issues concerning simulations written by philosophers and computer scientists. Acknowledgements I am grateful to Juan Manuel Durán for his advice concerning the literature on computer simulations and to Tarja Knuuttila and Roman Frigg for providing some references on models and idealizations.

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Alejandro Cassini is Professor of Philosophy and History of Science at the University of Buenos Aires and Senior Researcher at the Consejo Nacional de Investigaciones Científicas y Técnicas of Argentina. He served for twenty years as editor of the Revista Latinoamericana de Filosofía. He is the author of El juego de los principios: Una introducción al método axiomático (Buenos Aires: A-Z Editora, Second edition, 2013. First edition: 2007). He is the editor (with Laura Skerk) of Presente y futuro de la filosofía (Buenos Aires: Ediciones de la Facultad de Filosofía y Letras de la Universidad de Buenos Aires, 2010).

Name Index

A Abasnezhad, Ali, 179, 191 Abrahamsen, Adele, 159 Achinstein, Peter, 250 Alexandrova, Anna, 52 Andjelkovic, Miroslava, 189 Ankeny, Rachel, 159 Appiah, Kwame Anthony, 252 Armour-Garb, Bradley, 139, 185, 252 Aronson, Jerrold, 250 Azzouni, Jody, 212, 213

B Babor, Joseph, 241 Bailer-Jones, Daniela, 7, 12, 14, 78, 217, 249, 250 Baker, Gregory, 98 Baker, Richard, 60, 61 Balaguer, Mark, 206 Balzer, Wolfgang, 10, 123, 125, 250 Barberousse, Anouk, 161, 164 Barnes, Elizabeth, 191 Bartha, Paul, 251 Batterman, Robert, 52, 54, 55, 58, 104, 105, 251 Bechtel, William, 159 Beisbart, Claus, 252 Beni, Majid, 251 Benthem, Johan van, 222 Bernstein, Julius, 60 Bertolet, Rod, 183 Bertolotti, Tommaso, 249, 251, 252 Blackburn, James, 98 Black, Max, 250, 251 Boesch, Brandon, 72

Bogen, Jim, 65 Bohr, Niels, 4, 8, 150, 236, 238, 243 Bokulich, Alisa, 116, 124, 146, 251 Bokulich, Peter, 251 Bolinska, Agnes, 71 Boltzmann, Ludwig, 7, 250 Boniolo, Giovanni, 251 Bonomi, Andrea, 180 Boolos, George, 209 Borbone, Giacomo, 251 Borges, Jorge Luis, 76 Boumans, Marcel, 162, 250 Bovens, Luc, 250 Boyle, Robert, 241 Braithwaite, Richard, 7, 8, 250 Bransden, Brian, 243, 245 Brock, Stuart, 181, 185, 252 Brzechczyn, Krzysztof, 251 Bueno, Otávio, 24, 25, 115, 123, 151, 199, 203, 204, 206, 208, 209, 212, 251 Bunge, Mario, 13, 250 Burridge, R., 116, 117, 126

C Caie, Michael, 188, 189 Callender, Craig, 19, 118, 219 Campbell, Norman, 2, 79, 250 Carnap, Rudolf, 3, 5–7 Carrier, Martin, 252 Carrillo, Natalia, 51, 53, 59, 62 Carroll, Bradley, 240 Cartwright, Nancy, 13, 54, 66, 72, 90, 124, 128, 148, 250, 251 Cassini, Alejandro, 1, 18, 36, 87, 155, 214, 249

© Springer Nature Switzerland AG 2021 A. Cassini and J. Redmond (eds.), Models and Idealizations in Science, Logic, Epistemology, and the Unity of Science 50, https://doi.org/10.1007/978-3-030-65802-1

259

260 Chakravartty, Anjan, 118 Chang, C., 9 Clerbout, Nicolas, 218, 222, 223, 229, 233 Cohen, Jonathan, 19, 118, 219 Cole, Kenneth, 60, 61 Colyvan, Mark, 199, 212 Contessa, Gabriele, 72, 123, 124, 128, 151, 219 Cook, Norman, 36 Copmans, Catelijne, 251 Craver, Carl, 53, 64, 65 Currie, Adrian, 144, 162, 180, 181, 184, 192, 193 Currie, Gregory, 150

D Da Costa, Newton, 24, 123, 200, 202–204, 214, 250, 251 Devitt, Michael, 183 Díez, José A., 115 Dirac, Paul, 4 Dretske, Fred, 72–75, 80 Ducheyne, Steffen, 118, 124, 125 Duhem, Pierre, 2, 7, 247, 250 Dummett, Michael, 223 Durán, Juan Manuel, 252

E Einstein, Albert, 3, 5, 6, 45, 117, 239 Elgin, Catherine Z., 124, 126, 147, 164 El Hady, Ahmed, 65 Evans, Gareth, 137, 185 Evans, James, 116 Everett, Anthony, 181, 183, 184, 186

F Falguera, José Luis, 252 Faraday, Michael, 242 Field, Hartry, 37, 205 Fine, Arthur, 39 Fine, Kit, 187–189 Floridi, Luciano, 84 Fodor, Jerry, 191 Fontaine, Matthieu, 229 Fox Keller, Evelyn, 250 French, Steven, 24, 25, 123, 199, 200, 203, 204, 214, 250, 251 Freudenthal, Hans, 250 Friend, Stacie, 40, 138, 150, 152, 173, 180, 184

Name Index Frigg, Roman, 18, 20, 39–42, 46, 53, 89, 115, 117, 122, 124, 126, 128, 133, 135– 138, 143, 144, 149, 151, 152, 154, 159, 161, 163–167, 171, 172, 174, 192, 193, 200, 205, 214, 217, 219, 220, 249, 251, 252

G Galison, Peter, 250 García-Carpintero, Manuel, 179, 182, 184– 186, 188–192, 252 Gaut, Berys, 145, 146 Gehring, Petra, 252 Gelfert, Axel, 58, 90, 250 Gerlee, Philip, 14, 15, 250 Gibson, John, 145 Giere, Ronald, 10, 11, 18, 19, 26–29, 36, 39, 45, 78, 95, 100, 119, 123, 153, 154, 163, 217, 251 Glanzberg, Michael, 186 Glavaniˇcová, Daniela, 183 Gödel, Kurt, 206, 211 Godfrey-Smith, Peter, 40, 55, 90, 95, 96, 124, 128, 134, 135, 160, 161, 163, 164, 166, 171, 180, 192, 252 Goodman, Nelson, 164, 251 Gorisse, Marie-Hélène, 229

H Hacking, Ian, 13, 43, 250 Hale, Bob, 206 Hallyn, Fernand, 251 Hanks, Peter, 187 Harré, Rom, 250 Hartmann, Stephan, 53, 116, 124, 250, 252 Heck, Richard G. Jnr., 187, 189 Heidegger, Martin, 4 Heimburg, Thomas, 63, 65 Heisenberg, Werner, 4 Hellman, Geoffrey, 207, 208 Hempel, Carl Gustav, 3, 238, 250 Herfel, William, 250 Hesse, Mary, 12, 25, 79, 204, 250, 251 Hilbert, David, 3–5 Hodges, Wilfrid, 186 Hodgkin, Alan, 53, 58–68 Hoefer, Carl, 115, 250 Hoek, Daniel, 185, 186, 192 Hopwood, Nick, 250 Hosseini, Davood, 191 Howell, Robert, 186

Name Index Hughes, R. I. G., 39, 126, 199, 200, 219, 250, 251 Humphreys, Paul, 116, 252 Hund, Friedrich, 242 Hunter, Matthew, 251 Huxley, Andrew, 53, 58–68

I Iacona, Andrea, 189 Ibarz, José, 241 Imbert, Cyrille, 252 Iwasa, Kunihiko, 63

J Jackson, Andrew, 63, 65 Jebeile, Julie, 90 Jech, Thomas, 210 Jerabek, Paul, 246 Joachain, Charles, 243, 245 Jones, Alexander, 116 Jones, Nicholas, 190

K Kaminski, Andreas, 252 Keefe, Rosanna, 189, 190 Keiff, Laurent, 224, 229 Keisler, H. Jerome, 9 Kind, Amy, 173 Klein, Ursula, 250 Knopoff, L., 116, 117, 126 Knuuttila, Tarja, 43, 51, 52, 55, 56, 68, 106, 111, 141, 162–164, 166–172, 174, 221, 246, 252 Kolasinski, Kurt, 246 Korman, Daniel Z., 190 Krajewski, Wladyslaw, 250 Kratzer, Angelika, 187 Kripke, Saul, 181 Kroon, Frederick, 139, 181, 182, 252 Kung, Peter, 173

L Ladyman, James, 123, 203, 204 Lavoisier, Antoine, 242 Laymon, Ronald, 36, 88, 95, 100 Leatherdale, W. H., 251 Leitgeb, Hannes, 187, 188 Lenhard, Johannes, 252 Leonelli, Sabina, 159 Lepore, Ernest, 191

261 Levy, Arnon, 40, 42, 53, 58, 64, 65, 90, 135, 144, 159, 161, 163, 192, 193, 252 Lewis, David, 181, 183, 188 Liu, Chuang, 153 Lloyd, Elizabeth, 251 Loettgers, Andrea, 52, 162 López de Sa, Dan, 189 Lorenzen, Paul, 223, 224, 229 Lorenz, Kuno, 223, 224, 229 Lowe, E. J., 190 Lowenhaupt, Benjamin, 65 Ludwig, Pascal, 161, 164 Lundh, Torbjörn, 14, 15 Lynch, Michael, 251 M MacKay, David, 78 Maddy, Penelope, 212 Magnani, Lorenzo, 153, 154, 249, 251, 252 Maier, Emar, 184, 186 Mäki, Uskali, 54, 66, 78 Manning, Luke, 186 Martin-Löf, Per, 222, 223 Maxwell, James Clerk, 7, 16, 83, 116, 136 Mayo, Deborah, 250 McGuiness, Brian, 250 McMullin, Ernan, 54, 88, 91, 95 Mendeleev, Dmitri, 241, 242, 245 Merlo, Giovanni, 189, 190 Meyer, Lothar, 241 Mikenberg, Irene, 200, 202 Millikan, Robert, 242 Misner, Charles, 238 Moltmann, Friederike, 182 Montague, Richard, 187 Morgan, Mary, 12, 13, 26, 52, 55, 68, 72, 106, 116, 159, 250 Morrison, Margaret, 12, 13, 36, 72, 90, 95, 97, 100, 116, 122, 124, 136, 250, 252 Moulines, C. Ulises, 10, 115, 123, 125, 250 Mundy, Brent, 123 N Nagel, Ernest, 8, 250 Nash, Clinton, 246 Nelson, Robert, 98, 99 Nernst, Walther, 60, 61, 66 Nersessian, Nancy, 161, 251 Newton, Isaac, 3, 4, 16, 34, 92, 117, 140, 154, 159, 165, 168–170, 174, 239 Nguyen, James, 18, 20, 39–42, 46, 89, 115, 117, 124, 126, 128, 133, 135, 143,

262 144, 149, 151, 159, 161, 163–166, 172, 192, 193, 200, 205, 214, 219, 220, 249, 251 Niiniluoto, Ilkka, 90, 95, 250, 251 Nola, Robert, 90 Novakowa, Izabella, 251 Nowak, Leszek, 125, 251

O Oberkampf, William, 252 Oganessian, Yuri, 245 Ohnaka, Keiichi, 236 Olsson, M. G., 98, 99 Orlando, Eleonora, 182, 183, 191 Ortony, Andrew, 251 Ostlie, Dale, 240

P Paganini, Elisa, 186 Pero, Francesca, 78, 79 Perrin, Jean Baptiste, 242 Peschard, Isabelle, 116 Piccinini, Gualtiero, 84 Pierce, John, 73 Pincock, Christopher, 220, 251 Portides, Demetris, 136, 146, 147, 153 Potochnik, Angela, 96, 107, 251 Predelli, Stefano, 182, 183 Priest, Graham, 181

Q Quine, Willard V. O., 206, 209

R Radder, Hans, 252 Rahman, Shahid, 222, 223, 229 Recanati, François, 182, 183 Redmond, Juan, 1, 38, 111, 155, 217, 229 Reichenbach, Hans, 3, 6 Reimer, Marga, 182 Reisman, Kenneth, 167 Resch, Michael, 252 Resnik, Michael, 206–208 Rice, Collin, 54, 55, 58, 68, 104, 105 Richard, Mark, 185, 187 Rivadulla, Andrés, 235, 237–239, 242 Rohrs, Benjamin, 189 Rothschild, Daniel, 183 Rowbottom, Darrell, 97, 249 Roy, Christopher, 252

Name Index Rutherford, Ernest, 15, 236, 243 S Saam, Nicole, 252 Sainsbury, Mark, 183, 186, 252 Salis, Fiora, 136, 143, 152, 155, 159, 161, 163, 166, 171, 174, 175, 180, 192 Salmon, Nathan, 181 Sattig, Thomas, 190 Scarantino, Andrea, 84 Schaffer, Jonathan, 185 Schaffner, Kenneth, 65 Schiffer, Stephen, 181, 189 Schroeder, Timothy, 181, 183, 186 Shanks, Niall, 251 Shannon, Claude, 72, 73, 76, 83 Shapiro, Stewart, 9, 152, 206–209 Simpson, Stephen, 209 Sklar, Lawrence, 55, 90, 95, 105, 251 Smolanczuk, Robert, 245 Sneed, Joseph, 10, 123, 125, 250 Sprenger, Jan, 252 Stegmüller, Wolfgang, 10, 250 Sterrett, Susan, 250 Stokke, Andreas, 183 Stolnitz, Jerome, 147, 148 Strevens, Michael, 54, 95, 96, 104, 251 Styer, Daniel, 105 Suárez, Mauricio, 19, 20, 22, 29–32, 71, 72, 78, 79, 124, 192, 199, 217–221, 226, 227, 252 Sud, Rohan, 188, 189 Suppe, Frederick, 6, 10, 44, 250 Suppes, Patrick, 10, 21, 25, 77, 123, 250 Swoyer, Christopher, 30, 123, 164, 218 T Takashima, Shiro, 63, 66 Tarski, Alfred, 8, 223 Tasaki, Ichiji, 63, 65 Teller, Paul, 20, 36, 39, 67, 78, 95, 96, 100, 136 Tent, Katrin, 21 Terrone, Enrico, 182 Thomasson, Amie L., 37, 40, 138, 151, 160, 163, 169, 171, 180, 181, 184, 193 Thomson-Jones, Katherine, 145 Thomson-Jones, Martin, 72, 78, 90, 95, 122, 124, 138, 143, 145, 149, 160, 163, 193, 251 Thomson, Joseph John, 242 Thomson, Paul, 251

Name Index Thorndike, Alan, 116 Thorne, Kip, 238 Tobin, James, 116 Tolk, Andreas, 252 Tolstoy, Lev, 154, 169 Toon, Adam, 26, 41, 42, 90, 92, 124, 128, 135, 144, 161, 163, 192, 250 Tye, Michael, 190 V Väänänen, Jouko, 209 Vaihinger, Hans, 37–39, 100, 252 Van Fraassen, Bas, 10, 77, 119, 123, 124, 126, 188, 200, 203, 205, 238, 250, 251 Van Inwagen, Peter, 181 Varenne, Franck, 252 Vertesi, Janet, 251 Voltaire, 148 Voltolini, Alberto, 181, 182 Von Neumann, John, 4 Von Solodkoff, Tatjana, 185 Vorms, Marion, 150 Voutilainen, Atro, 162 W Walton, Kendal, 41, 42, 136, 137, 152, 160, 161, 165, 167, 173, 174, 184, 185, 251

263 Wartofsky, Marx, 250, 251 Way, Eileen, 250 Weatherson, Brian, 187, 190 Weber, Marcel, 64 Weisberg, Michael, 18, 29, 36, 53, 54, 95, 101, 104, 116, 123, 159, 164, 167, 214, 250–252 Wheeler, Billy, 90, 251 Wheeler, John, 238 Williams, J. Robert, 191 Williamson, Timothy, 180, 186, 187, 189 Wimsatt, William, 250 Winsberg, Eric, 45, 106, 116, 136, 250, 252 Wojcicki, Richard, 250 Wolterstorff, Nicholas, 181 Woodbridge, James, 185, 252 Woods, John, 38, 252 Woolgar, Steve, 251 Wright, Crispin, 189, 206

Y Yablo, Stephen, 180, 184, 185, 192 Yalcin, Seth, 187 Yasuda, Naoki, 240

Z Ziegler, Martin, 21

Subject Index

A Abstract entities, 9, 11, 22, 25, 28, 40, 44, 166, 188, 200, 205–207, 210, 211 Abstraction, 15, 18, 33, 35, 45, 53, 64, 65, 72, 76, 78, 79, 82, 83, 89–93, 97, 125, 126, 149, 251 Accuracy, 28, 34, 45, 55, 57, 91, 93, 96, 100, 102, 107, 115, 117, 120, 125, 220, 221 Adequacy, 20, 24, 45, 118, 122, 123, 180, 187, 193, 218–221, 228 Agents, 13, 17–23, 25, 27–32, 42, 120, 142, 162, 164, 168, 169, 173, 174, 218, 225, 226, 228 Analogy, 2, 7, 18, 40–42, 43, 52, 54, 59, 61– 62, 65, 72, 75–79, 83, 117–118, 164, 172–173, 175, 204 -negative, 25, 42, 204 -neutral, 25, 26, 204 -positive, 25, 26, 40, 42, 204 Antirealism, 38, 172, 173 Approximation, 33, 35, 72, 83, 89, 91, 94, 97–100, 104, 105, 108, 110, 125 Arithmetics, 207–209 Artifacts, 14, 40, 43, 44, 56, 60, 67, 181, 192, 237, 247 Artifactualism, 43, 44, 111, 141 Atomic nuclei, 40, 236 Atoms, 8, 17, 26, 37, 39, 92, 110, 117, 190, 219, 236, 242–246 Axiomatics, 3, 5, 6, 8, 10, 11 Axiom of choice, 211 Axioms, 4–7, 10–12, 207, 210, 211

B Bohr’s atomic model, 8, 15, 40

C Coherence, 129 Communication channel, 72–78, 80, 83 Communication theory, 72, 74 Completeness, 12, 64, 74, 82, 83, 101, 233 Complexity, 15, 36, 75, 78, 109, 111, 128, 147, 189 Computer simulations, 14, 15, 17, 39, 44–46, 88, 91, 102, 116, 249, 252 Concepts, 1, 2, 7–10, 12, 14, 16–21, 27, 29–33, 37, 38, 41, 52, 53, 55, 68, 72, 87–90, 108, 116, 122, 128, 129, 139, 141, 142, 146, 183, 200–202, 217, 222–224, 236–238, 241, 246, 250–252 Concrete entities, 168, 205, 211 Conditions of adequacy, 220 Content, 6, 7, 83, 119, 128, 134, 141, 148, 149, 152, 154, 162, 163, 172–175, 181, 184, 185, 187, 189, 190, 192, 209, 212, 222, 223 Contextualism, 184 Continuum hypothesis, 209 Correlations, 81, 83

D Data, 13, 45, 61, 77, 88, 99, 140, 149, 155, 186, 187, 199, 238, 246 Deidealization, 35, 87–89, 95–110 Denotation, 31, 39, 164, 166, 171, 172, 174, 221

© Springer Nature Switzerland AG 2021 A. Cassini and J. Redmond (eds.), Models and Idealizations in Science, Logic, Epistemology, and the Unity of Science 50, https://doi.org/10.1007/978-3-030-65802-1

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266 Description, 5, 33, 35, 36, 38, 39, 41, 42, 77, 80, 83, 89, 95, 100, 108, 110, 134, 136, 160, 162, 167–170, 174, 175, 183, 192, 230, 237 Directionality, 20, 22, 27, 31, 117, 118, 120, 122, 220, 228 Discourse, 135, 139, 179, 180–183, 189 -metatextual, 181, 184–186 -Paratextual, 182–184 -textual, 182–184 Distortion, 33–35, 52, 55, 57, 58, 62, 65, 67, 78, 89–93, 96, 97, 105, 110, 124 Distortion, 33–35, 52, 55, 57, 58, 62, 65, 67, 124 Domain, 2, 8, 9, 13, 15, 17, 21, 23, 36, 43, 45, 88, 90, 93, 94, 96, 101, 102, 139, 174, 186, 187, 202, 203, 212 E Electron, 6, 17, 39, 59, 240, 242–245 Elementary particles, 16, 39, 139 Elements, 9, 11, 14, 21, 23, 24, 26, 27, 29, 39, 58, 62, 75, 108, 115, 116, 121, 122, 125–127, 129, 136, 138, 148, 164, 192, 201, 210, 218, 237, 240–242, 244–246 Embedding, 9, 21, 77, 204 Engineering, 60–62, 116, 146 Epistemic access, 15, 16, 40, 43, 56, 87, 96, 109, 166, 167, 169, 172 Epistemology, 43, 134, 145, 223 Equivocation, 72–75, 78, 80, 82, 83 Ether, 7, 16, 38, 119 Expediency, 34, 88, 100, 108 Experiments, 15, 44, 59, 60, 62, 63, 66, 88, 117, 242, 245, 250 Explanation, 33, 45, 52, 54, 64, 65, 77, 78, 88, 89, 96, 104, 105, 108, 170, 171, 175, 181, 188, 191, 192, 212, 227, 236–241, 244, 246, 247 F Falsehood, 58, 134, 136, 155 Fictional characters, 37, 40, 42, 165, 169, 170, 180–182, 184–186, 191, 212, 251 Fictional discourse, 38, 170, 179, 180 Fictional entities, 39–41, 115, 122, 135, 145, 151, 163, 165, 166, 171, 173, 181, 182, 185, 192 Fictionalism, 37–41, 111, 115, 136, 139– 141, 159, 181, 184, 249, 252

Subject Index -mathematical, 38, 249 Fictional places, 37 Fictions, 37–43, 100, 115, 116, 128, 134– 141, 143, 145, 149, 150, 151, 152, 154, 160–162, 164, 171, 172, 180, 186, 191–192, 237, 249, 252 -full, 37, 100 -literary, 2, 37, 40–42, 44, 134, 135, 137, 141, 146, 149, 150, 152–154, 160, 163, 172, 173, 175, 236, 251 -scientific, 40, 100, 149–151, 249, 252 -semi, 37–39, 100 Fiction view of models, 40–43, 134, 136– 146, 149–154, 160, 161, 163, 166, 168, 170, 174, 252 Force, 30, 34, 37, 60–62, 92, 93, 98, 119, 147, 161, 162, 165, 170, 173, 174, 184, 223–228, 238–240, 243 Functions, 6, 8, 9, 13–16, 21, 25, 27, 42, 45, 54, 63, 65, 77, 80–82, 87, 116, 118, 124, 125, 134, 141, 145, 153– 155, 162–165, 168, 169, 172, 173, 182, 183, 203, 210, 211, 217–219, 223, 230, 232

G Games of make-believe, 152, 153, 161, 163, 167, 169, 170, 175 Gravitation, 4, 34, 92, 93, 117, 170, 238

H Higgs boson, 139 Hodgkin and Huxley model, 51, 53, 58–68 Homomorphism, 9, 10, 21, 22, 24, 123, 203–205

I Idealization, 1, 2, 12, 17, 33–36, 39, 40, 43, 45, 51–59, 61–68, 71, 72, 78, 79, 82, 83, 87–91, 94–97, 99–107, 109–111, 124–126, 128, 141, 180, 186, 249, 251, 252 -Aristotelian, 91 -Galilean, 53, 54, 88, 91 Imaginary entities, 37, 41, 42, 163 Imagination, 34, 37, 40–42, 137–139, 160– 166, 168, 170–173, 175, 179, 186 Inaccuracy, 124, 125, 227 Incompleteness, 25, 33, 89, 101, 200, 201, 211 Indexicals, 182–184

Subject Index Individuals, 42, 52, 64, 119–122, 125, 153, 154, 160, 167, 175, 183, 187, 207, 223, 229, 230 Inexactness, 124–126, 128 Inference, 21, 30–32, 57, 72, 76, 124, 199, 218, 221–223, 227 Inferentialism, 29–32 Information, 2, 9, 15, 58, 71–84, 120, 124– 127, 137, 140, 149, 183, 199–204, 211, 218, 226, 232 Instrumentalism, 39, 43, 139 Intentionality, 122 Interpretation, 5, 7, 8, 53, 62–65, 77, 79, 128, 150, 162, 164–166, 168–171, 173, 174, 186, 188, 189, 191, 192, 201, 202, 206–208, 210, 211, 213, 214, 223, 237 Irrealism, 180 Isomorphism, 9, 10, 19, 21, 22, 24, 25, 31, 203–205, 217–219

K Kinetic theory of gases, 72, 77, 79–81, 117

L Language, 180, 186–189, 191, 192 Laws of nature, 33, 90, 148 Logic, 8, 10, 38, 152, 153, 187, 188, 206, 208–210, 213, 216–218, 220–222, 224, 225, 227, 228 -classical, 38, 187, 188, 221, 222, 227, 228 -dialogical, 218, 223–225, 227 -first-order, 10 -non-classical, 28, 223 -second-order, 206, 208, 209, 213 Lotka-Volterra’s prey-predator model, 16, 117, 159, 167

M Mapping, 21, 170, 203–205, 208, 210, 211 Maps, 10, 14, 17, 18, 28, 91, 96, 97, 116, 117, 120, 165, 199, 214 Mathematics, 4, 5, 9, 91, 107, 134, 139, 140, 151–153, 155, 205–207, 209, 210, 220, 223, 240 Maxwell’s ether models, 16, 136 Meaning, 3–6, 9, 11, 13, 14, 16–18, 33, 89, 90, 125, 160, 187–189, 222, 224, 225, 230

267 Mechanics, 3, 4, 34, 39, 77, 93, 100, 102, 103, 105, 107, 169, 202, 203, 212, 238, 243, 244, 251 -celestial, 34, 93, 103, 107 -Newtonian, 93, 100, 169, 202, 238, 240 -quantum, 3, 4, 39, 77, 212, 238, 243, 244, 251 Metaphysics, 152, 160, 252 Misrepresentation, 19, 22, 23, 27, 28, 31, 55, 57, 58, 66, 67, 94, 101, 110, 124, 128, 134, 220, 221, 227 Modeling, 13, 33, 35, 39, 44–46, 52–58, 66– 68, 89, 96, 101, 105, 124, 126, 128, 129, 162, 166, 183, 192, 199, 214, 217–221, 226, 228, 249–252 Models, passim -as imaginary systems, 166, 172 -as mediators, 13, 42, 116, 250 -atomic, 40, 236, 242, 243 -deidealized, 33, 35, 36, 90, 92, 95, 96, 99–102, 104, 107, 108, 110 -descriptions, 78, 82, 91, 118, 135, 138– 141, 151, 152, 160–168, 171–175, 246 -exploratory, 13 -functions of, 46, 153, 155 -idealized, 33–36, 39, 45, 88, 90, 92, 94–96, 98–100, 102–110, 117, 125, 252 -material, 40, 117, 118, 159, 160, 164, 250 -mathematical, 10, 21, 22, 25, 44, 61, 91, 94, 99, 108, 117, 118, 134, 152 -mechanical, 7, 16 -molecular, 92 -non-representational, 17 -nuclear, 236, 237 -of data, 250 -planetary, 34, 35, 39, 40, 219 -representational, 17, 20, 39, 52, 89, 95, 115, 127, 128 -systems, 41, 134, 160–164, 166, 167, 169–175 -targetless, 119, 220, 228 -theoretical, 28, 35, 40, 44, 45, 57, 77, 78, 88, 90–92, 94–97, 99, 117, 160, 161, 163–168, 172, 173, 175, 193, 236, 237, 239–241, 243–247 -use of, 7, 8, 12, 13, 34, 88, 236, 237, 247, 250–252 Model system, 41, 42, 134, 140, 160–164, 166, 167, 169–175, 192 Model theory, 8, 223, 249 Molecular biology, 168

268 Molecules, 19, 26, 79–82, 92, 103, 105, 110 Morphisms, 19, 21–23, 25, 123, 200, 204, 205, 209, 214

N Names, 181–186, 191 Nervous impulse, 51, 53, 60, 64, 68 Newlyn-Phillips hydraulic machine, 26, 40 Noise, 72–75, 77, 78, 80–83, 104 Non-existence, 124, 137 Non-existing entities, 138 Numbers, 21, 25, 29, 31, 37, 38, 40, 52, 61, 74, 76, 80, 81, 83, 91, 95, 103, 105, 118, 134, 135, 140, 143, 149, 152, 170, 171, 174, 205, 207, 208, 210, 211, 231, 232, 236–238, 240, 243–245, 247

O Observation, 76, 135, 140, 149, 152, 235, 240, 243 Observational terms, 5, 6 Ontological commitment, 208, 209, 212, 213, 228 Ontology, 15, 16, 28, 41, 42, 44, 46, 92, 117, 142, 153, 155, 160, 165, 166, 171, 174, 180, 200, 213, 220 Order, 4, 14, 17, 64, 100, 103, 105, 116, 142, 144, 181, 193, 207, 223, 241

P Parameters, 35, 77, 90, 91, 98, 99, 101, 102, 104, 106, 116, 119, 205, 211 Partial isomorphism, 9, 24, 203, 205 Partial structures, 24, 25, 200–205, 208, 213 Pendulum, 40, 78, 87, 97–99, 100–104, 106– 108, 110, 117, 136, 143 -compound, 97–99, 102, 107, 108 -ideal, 78, 108, 117 -physical, 99–103, 106, 108 -simple, 97, 99, 102, 136 Periodic table, 237, 241, 242, 244, 246 Phenomena, 10, 11, 13–19, 22, 25, 26, 28, 33, 35, 36, 39, 41–45, 52, 54–58, 76, 82, 87–91, 94–96, 99–102, 104–111, 116, 117, 119, 136, 166, 189, 200, 204–206, 210, 211, 238, 239, 247, 250 Planets, 8, 19, 34, 35, 92–94, 104, 135, 238 Platonism, 205, 206 Possibilia, 181

Subject Index Pragmatics, 2, 15, 16, 20, 27, 39, 42, 43, 45, 76, 94, 100, 102, 116, 129, 139, 164, 185, 202–204, 217, 218, 220, 221 Pragmatism, 39, 43, 218, 228 Prediction, 30, 33, 42, 45, 77, 78, 88, 89, 93, 95–97, 99, 102, 108, 116, 236, 237, 245–247 Pretense, 126, 184–186 Principles, 3, 13, 41, 161, 173–174, 185, 222 -of generation, 41, 161, 173–174, 185 Problems, 2–4, 14–16, 19, 20, 22, 23, 27– 29, 31, 36–40, 42–45, 56, 58, 66, 88, 90, 99–104, 106, 107, 110, 111, 116– 120, 122, 126–128, 134, 140, 150, 154, 161, 163–167, 169–172, 174, 175, 189, 190, 192, 202, 205, 206, 210, 219–221, 223, 225, 246 Problem solving, 45, 91, 172, 173 Properties, 6, 12, 15, 18, 19, 22, 24–33, 35, 38, 42, 55, 73–75, 78–82, 89, 90, 92, 94, 96, 101, 118, 120, 121, 125, 143, 146, 160, 163–167, 170–172, 174, 175, 183, 186, 187, 190, 192, 208, 211, 217, 226, 237, 241, 242, 245, 246 Propositions, 11, 41, 137, 149, 152, 161, 173, 174, 186–191, 222 Purpose, 13, 17, 19, 20, 23, 26–29, 32–34, 36, 42–45, 52, 56, 72, 73, 75–79, 87– 92, 94, 96–98, 100, 102–104, 107, 109, 110, 120, 124–127, 129, 135, 147, 152, 153, 160–162, 164, 166, 168–170, 172, 173, 175, 183, 188, 189, 237, 238, 246, 247 Q Quantification, 181, 206, 212, 213, 230 Quantities, 65, 73–75, 99, 102, 105 Quantum mechanics, 3, 4, 39, 77, 212, 238, 243, 244, 251 R Real, 13, 17, 18, 24, 26, 29, 37–39, 44, 55, 78, 79, 92, 95, 97–99, 101, 108, 111, 118, 120, 135, 138, 139, 182, 205, 207, 208, 225, 226, 237, 239, 241, 245 Realism, 33, 37, 89, 95, 97, 100, 108–111, 139, 165, 166, 180, 181, 183, 191 Reality, 15, 37, 41, 55, 96, 110, 111, 116, 136, 137, 160, 161, 164, 166, 168, 170, 172, 173, 192, 218

Subject Index Realtivity -special theory, 3, 4 Reference, 2, 9, 12, 20, 37, 119, 134, 164, 170, 175, 179–182, 184, 190–192, 221, 223, 237, 249, 252 Relations, 9–10, 16, 19, 21, 23, 25, 28, 32, 44, 54, 59, 65, 68, 83, 120, 122–123, 125, 169, 171–173, 201, 203–205, 207, 210, 212, 222, 230 -asymmetric, 22, 29 -equivalence, 10, 19, 22, 25 -intransitive, 19, 22 -irreflexive, 18, 22 -partial, 24, 200–203 -reflexive, 19, 22, 27 -transitive, 22 Relativity, 1, 3, 4 -general theory, 1, 3, 4 -special theory, 1, 3, 4 Representation, 1–2, 16–31, 33, 36, 39, 41– 46, 52–56, 58, 65, 67, 76, 87–89, 95–87, 100–101, 107–108, 110, 115– 126, 128–129, 133–134, 136, 141– 145, 161–166, 168, 170–172, 174, 182–184, 199–209, 203–206, 207– 214, 217–221, 225–228, 237, 249– 251 -DEKI account, 42, 142, 144, 164, 166 -direct, 21, 42 -indirect, 41, 161, 163, 164 -inferential view of, 30, 32 -pictorial, 164, 251 -similarity view of, 2, 31, 144 -structural view of, 22 -visual, 39, 95 Representationalism, 43, 88 Rules, 2, 5–8, 108, 152–153, 162, 166, 221–222, 225, 228–230 -of correspondence, 2, 5–8 -of generation, 152, 153 -of inference, 152, 161, 221, 228 Rutherford’s atomic model, 117

S Schelling’s segregation model, 149 Semantics, 2, 6, 8, 10–12, 21, 38, 162, 180, 182, 184–188, 201, 218, 221, 223–226, 228, 233, 250, 251 Sets, 5, 9–11, 21, 24–25, 32, 40, 77, 81, 183, 185, 200–201, 205–214 -set theory, 6, 21, 25, 200, 205, 206, 208–211, 213

269 Similarity, 2, 10, 19–21, 25–29, 31–32, 123, 127, 129, 136, 144, 166, 214, 217– 218 -formal, 20, 29 -informal, 2, 10, 20, 21, 25–29, 31 Simplicity, 5, 18, 21, 24, 61, 62, 95, 100, 102, 108, 111, 134, 203, 208 Simplification, 15, 33, 35, 53, 66, 89, 96, 119, 201 Solar System, 8, 16, 34, 35, 92–94, 103, 104, 116 Stellar models, 236 Structuralism, 2, 21, 24–25, 151, 122, 206, 208, 238 -modal, 208 Structures, 3, 4, 7–10, 12, 15, 16, 20–26, 30, 32, 33, 35, 39, 42, 55, 81, 87, 89, 92, 94, 122, 123, 142, 150, 151, 165, 168, 182, 187, 199–205, 207, 208, 210, 211, 213, 214, 218, 222–224, 228, 237, 238, 240, 245 Supernovae, 236, 237, 239, 240, 246 Supervaluations, 188, 191 Surrogative reasoning, 30–32, 124, 164, 168, 169, 175, 220 Systems, 5, 17, 26, 27, 40, 41, 42, 52, 54, 55, 58, 72, 82, 87, 92, 93, 94, 96, 105, 138, 139, 141, 142, 143, 147, 159, 160, 161, 162, 163, 164-167, 169, 170, 172-175, 180, 192, 209, 220 -imaginary, 40, 161, 162, 164–167, 169, 170, 172–175, 192 -real, 26, 27, 42, 164, 171 T Target, 18–20, 22–32, 36, 39, 41, 42, 44, 53, 65, 66, 68, 71, 72, 75, 76, 78, 89, 90, 96, 105, 106, 109, 117–120, 122–129, 135, 142–145, 148, 150, 154, 161, 162, 164–166, 170–172, 174, 186, 191–193, 199, 204, 214, 217–220, 226–228, 245, 246 Target system, 52, 55–58, 68, 72, 78, 118, 135, 136, 138, 141, 142, 145, 149, 159, 164, 166, 192, 217–221, 225– 228 Theoretical terms, 5, 6, 37 Theories, 1-14, 15, 17, 210-21, 35, 37, 43-45, 77, 92, 95, 101, 108, 116-117, 119, 139, 202, 207-208, 211-213, 238-239, 241, 246, 250-251 -classical view of, 7, 8, 12 -empirical, 2, 5–7, 10–12, 250

270 -formal, 5, 6, 9, 12 -physical, 3–8, 38 -semantic view of, 10, 12 Thermodynamic limit, 105 Thomson’s atomic model, 15, 117 Three-body model, 34, 93, 104 Truth, 37, 41, 43, 100, 108, 110, 111, 123, 136, 139, 147–149, 154, 182, 184– 186, 191, 200–202, 222, 223, 241 Truthlikeness, 108, 109 Two-body model, 34, 35, 93, 94

V Vagueness, 10, 27, 180, 187–189, 191 Values, 35, 74, 80-82, 91, 95, 102-104, 140, 146, 182, 190-190, 243 -epistemic, 53, 56, 63, 141, 145, 146, 149, 150 -semantic, 182, 190–192

Subject Index W Watson-Crick’s DNA model, 16 World, 13, 17, 18, 33, 36, 37, 39, 41-44, 52-55, 58, 62, 63, 65, 67, 87, 94, 95, 97, 100, 101, 108, 110, 111, 119, 128, 135, 136, 138-142, 146-149, 152, 153, 155, 161, 163, 166, 168, 171, 172, 175, 184-185, 192, 199, 200, 205, 211, 212, 217, 220, 246 -actual world, 141, 146, 147, 152, 182, 184 -fictional worlds, 135, 148, 181–185 -possible worlds, 135, 143, 151, 182, 183, 187, 207, 207

Z Zermelo-Fraenkel set theory, 210