This paper concerns Carnap’s early contributions to formal semantics in his work on general axiomatics between 1928 and 1936. Its main focus is on whether he held a variable domain conception of models. I argue that interpreting Carnap’s account in terms of a fixed domain approach fails to describe his premodern understanding of formal models. By drawing attention to the second part of Carnap’s unpublished manuscript Untersuchungen zur allgemeinen Axiomatik, an alternative interpretation of the notions ‘model’, ‘model extension’ and ‘submodel’ (...) in his theory of axiomatics is presented. Specifically, it is shown that Carnap’s early model theory is based on a convention to simulate domain variation that is not identical but logically comparable to the modern account. (shrink)

Structuralism, the view that mathematics is the science of structures, can be characterized as a philosophical response to a general structural turn in modern mathematics. Structuralists aim to understand the ontological, epistemological, and semantical implications of this structural approach in mathematics. Theories of structuralism began to develop following the publication of Paul Benacerraf’s paper ‘What numbers could not be’ in 1965. These theories include non-eliminative approaches, formulated in a background ontology of sui generis structures, such as Stewart Shapiro’s ante rem (...) structuralism and Michael Resnik’s pattern structuralism. In contrast, there are also eliminativist accounts of structuralism, such as Geoffrey Hellman’s modal structuralism, which avoids sui generis structures. These research projects have guided a more systematic focus on philosophical topics related to mathematical structuralism, including the identity criteria for objects in structures, dependence relations between objects and structures, and also, more recently, structural abstraction principles. Parallel to these developments are approaches that describe mathematical structure in category-theoretic terms. Category-theoretic approaches have been further developed using tools from homotopy type theory. Here we find a strong relationship between mathematical structuralism and the univalent foundations project, an approach to the foundations of mathematics based on higher category theory. (shrink)

This edited volume explores the previously underacknowledged 'pre-history' of mathematical structuralism, showing that structuralism has deep roots in the history of modern mathematics. The contributors explore this history along two distinct but interconnected dimensions. First, they reconsider the methodological contributions of major figures in the history of mathematics. Second, they re-examine a range of philosophical reflections from mathematically-inclinded philosophers like Russell, Carnap, and Quine, whose work led to profound conclusions about logical, epistemological, and metaphysic.

Informally, structural properties of mathematical objects are usually characterized in one of two ways: either as properties expressible purely in terms of the primitive relations of mathematical theories, or as the properties that hold of all structurally similar mathematical objects. We present two formal explications corresponding to these two informal characterizations of structural properties. Based on this, we discuss the relation between the two explications. As will be shown, the two characterizations do not determine the same class of mathematical properties. (...) From this observation we draw some philosophical conclusions about the possibility of a ‘correct’ analysis of structural properties. (shrink)

Øystein Linnebo and Richard Pettigrew have recently developed a version of non-eliminative mathematical structuralism based on Fregean abstraction principles. They argue that their theory of abstract structures proves a consistent version of the structuralist thesis that positions in abstract structures only have structural properties. They do this by defining a subset of the properties of positions in structures, so-called fundamental properties, and argue that all fundamental properties of positions are structural. In this article, we argue that the structuralist thesis, even (...) when restricted to fundamental properties, does not follow from the theory of structures that Linnebo and Pettigrew have developed. To make their account work, we propose a formal framework in terms of Kripke models that makes structural abstraction precise. The formal framework allows us to articulate a revised definition of fundamental properties, understood as intensional properties. Based on this revised definition, we show that the restricted version of the structuralist thesis holds. 1Introduction 2The Structuralist Thesis 3LP-Structuralism 4Purity 5A Formal Framework for Structural Abstraction 6Fundamental Relations 7The Structuralist Thesis Vindicated 8Conclusion. (shrink)

The article investigates one of the key contributions to modern structural mathematics, namely Hilbert’sFoundations of Geometry and its mathematical roots in nineteenth-century projective geometry. A central innovation of Hilbert’s book was to provide semantically minded independence proofs for various fragments of Euclidean geometry, thereby contributing to the development of the model-theoretic point of view in logical theory. Though it is generally acknowledged that the development of model theory is intimately bound up with innovations in 19th century geometry, so far, little (...) has been said about how exactly model-theoretic concepts grew out of methodological investigations within projective geometry. This article is supposed to fill this lacuna and investigates this geometrical prehistory of modern model theory, eventually leading up to Hilbert’sFoundations. (shrink)

The paper investigates Ernst Cassirer’s structuralist account of geometrical knowledge developed in his Substanzbegriff und Funktionsbegriff. The aim here is twofold. First, to give a closer study of several developments in projective geometry that form the direct background for Cassirer’s philosophical remarks on geometrical concept formation. Specifically, the paper will survey different attempts to justify the principle of duality in projective geometry as well as Felix Klein’s generalization of the use of geometrical transformations in his Erlangen program. The second aim (...) is to analyze the specific character of Cassirer’s geometrical structuralism formulated in 1910 as well as in subsequent writings. As will be argued, his account of modern geometry is best described as a “methodological structuralism”, that is, as a view mainly concerned with the role of structural methods in modern mathematical practice. (shrink)

Rudolf Carnap’s mature work on the logical reconstruction of scientific theories consists of two components. The first is the elimination of the theoretical vocabulary of a theory in terms of its Ramsification. The second is the reintroduction of the theoretical terms through explicit definitions in a language containing an epsilon operator. This paper investigates Carnap’s epsilon-reconstruction of theories in the context of pure mathematics. The main objective here is twofold: first, to specify the epsilon logic underlying his suggested definition of (...) theoretical terms and a suitable choice semantics for it. Second, to analyze whether Carnap’s approach is compatible with a structuralist conception of mathematics. (shrink)

Øystein Linnebo and Richard Pettigrew have recently developed a version of non-eliminative mathematical structuralism based on Fregean abstraction principles. They argue that their theory of abstract structures proves a consistent version of the structuralist thesis that positions in abstract structures only have structural properties. They do this by defining a subset of the properties of positions in structures, so-called fundamental properties, and argue that all fundamental properties of positions are structural. In this paper, we argue that the structuralist thesis, even (...) when restricted to fundamental properties, does not follow from the theory of structures that Linnebo and Pettigrew have developed. To make their account work, we propose a formal framework in terms of Kripke models that makes structural abstraction precise. The formal framework allows us to articulate a revised definition of fundamental properties, understood as intensional properties. Based on this revised definition, we show that the restricted version of the structuralist thesis holds. (shrink)

This paper provides a historically sensitive discussion of Carnaps theory will be assessed with respect to two interpretive issues. The first concerns his mathematical sources, that is, the mathematical axioms on which his extremal axioms were based. The second concerns Carnapcompleteness of the modelss different attempts to explicate the extremal properties of a theory and puts his results in context with related metamathematical research at the time.

In historical discussions of twentieth-century logic, it is typically assumed that model theory emerged within the tradition that adopted first-order logic as the standard framework. Work within the type-theoretic tradition, in the style ofPrincipia Mathematica, tends to be downplayed or ignored in this connection. Indeed, the shift from type theory to first-order logic is sometimes seen as involving a radical break that first made possible the rise of modern model theory. While comparing several early attempts to develop the semantics of (...) axiomatic theories in the 1930s, by two proponents of the type-theoretic tradition and two proponents of the first-order tradition, we argue that, instead, the move from type theory to first-order logic is better understood as a gradual transformation, and further, that the contributions to semantics made in the type-theoretic tradition should be seen as central to the evolution of model theory. (shrink)

In Untersuchungen zur allgemeinen Axiomatik and Abriss der Logistik, Carnap attempted to formulate the metatheory of axiomatic theories within a single, fully interpreted type-theoretic framework and to investigate a number of meta-logical notions in it, such as those of model, consequence, consistency, completeness, and decidability. These attempts were largely unsuccessful, also in his own considered judgment. A detailed assessment of Carnap’s attempt shows, nevertheless, that his approach is much less confused and hopeless than it has often been made out to (...) be. By providing such a reassessment, the paper contributes to a reevaluation of Carnap’s contributions to the development of modern logic. (shrink)

In historical discussions of twentieth-century logic, it is typically assumed that model theory emerged within the tradition that adopted first-order logic as the standard framework. Work within the type-theoretic tradition, in the style of Principia Mathematica, tends to be downplayed or ignored in this connection. Indeed, the shift from type theory to first-order logic is sometimes seen as involving a radical break that first made possible the rise of modern model theory. While comparing several early attempts to develop the semantics (...) of axiomatic theories in the 1930s, by two proponents of the type-theoretic tradition and two proponents of the first-order tradition, we argue that, instead, the move from type theory to first-order logic is better understood as a gradual transformation, and further, that the contributions to semantics made in the type-theoretic tradition should be seen as central to the evolution of model theory. (shrink)

This paper discusses Carnap’s attempts in the late 1920s to provide a formal reconstruction of modern axiomatics.1 One interpretive theme addressed in recent scholarly literature concerns Carnap’s underlying logicism in his philosophy of mathematics from that time, more specifically, his attempt to “reconcile” the logicist approach of reducing mathematics to logic with the formal axiomatic method. For instance, Awodey & Carus characterize Carnap’s manuscript Untersuchungen zur allgemeinen Axiomatik from 1928 as a “large-scale project to reconcile axiomatic definitions with logicism, and (...) transform implicit into explicit definitions.” It is argued that Carnap’s central idea was to balance a Fregean foundational stance with the modern model-theoretic viewpoint introduced in Hilbert’s Grundlagen der Geometrie ). It was also shown in recent literature that Carnap’s attempt to provide a logicist reconstruction of axiomatics is limited in several ways.2 No closer attention, however, has so far been dedicated to some of the details of his proposed reconciliation. (shrink)

This paper discusses Carnap’s attempts in the late 1920s to provide a formal reconstruction of modern axiomatics.1 One interpretive theme addressed in recent scholarly literature concerns Carnap’s underlying logicism in his philosophy of mathematics from that time, more specifically, his attempt to “reconcile” the logicist approach of reducing mathematics to logic with the formal axiomatic method. For instance, Awodey & Carus characterize Carnap’s manuscript Untersuchungen zur allgemeinen Axiomatik from 1928 as a “large-scale project to reconcile axiomatic definitions with logicism, and (...) transform implicit into explicit definitions.” It is argued that Carnap’s central idea was to balance a Fregean foundational stance with the modern model-theoretic viewpoint introduced in Hilbert’s Grundlagen der Geometrie ). It was also shown in recent literature that Carnap’s attempt to provide a logicist reconstruction of axiomatics is limited in several ways.2 No closer attention, however, has so far been dedicated to some of the details of his proposed reconciliation. (shrink)

The paper outlines a novel version of mathematical structuralism related to invariants. The main objective here is twofold: first, to present a formal theory of structures based on the structuralist methodology underlying work with invariants. Second, to show that the resulting framework allows one to model several typical operations in modern mathematical practice: the comparison of invariants in terms of their distinctive power, the bundling of incomparable invariants to increase their collective strength, as well as a heuristic principle related to (...) the search for new invariants. (shrink)

In Untersuchungen zur allgemeinen Axiomatik and Abriss der Logistik, Carnap attempted to formulate the metatheory of axiomatic theories within a single, fully interpreted type-theoretic framework and to investigate a number of meta-logical notions in it, such as those of model, consequence, consistency, completeness, and decidability. These attempts were largely unsuccessful, also in his own considered judgment. A detailed assessment of Carnap’s attempt shows, nevertheless, that his approach is much less confused and hopeless than it has often been made out to (...) be. By providing such a reassessment, the paper contributes to a reevaluation of Carnap’s contributions to the development of modern logic. (shrink)

A central topic in the logic of science concerns the proper semantic analysis of theoretical sentences, that is sentences containing theoretical terms. In this paper, we present a novel choice-semantical account of theoretical truth based on the epsilon-term definition of theoretical terms. Specifically, we develop two ways of specifying the truth conditions of theoretical statements in a choice functional semantics, each giving rise to a corresponding logic of such statements. In order to investigate the inferential strength of these logical systems, (...) we provide a translation of each truth definition into a modal definition of theoretical truth. Based on this, we show that the stronger notion of choice-semantical truth captures more adequately our informal semantic understanding of scientific statements. (shrink)

The paper surveys different notions of implicit definition. In particular, we offer an examination of a kind of definition commonly used in formal axiomatics, which in general terms is understood as providing a definition of the primitive terminology of an axiomatic theory. We argue that such “structural definitions” can be semantically understood in two different ways, namely as specifications of the meaning of the primitive terms of a theory and as definitions of higher-order mathematical concepts or structures. We analyze these (...) two conceptions of structural definition both in the history of modern axiomatics and in contemporary philosophical debates. Based on that, we give a systematic assessment of the underlying semantics of these two ways of understanding the definiens of such definitions, by considering alternative model-theoretic and inferential accounts of meaning. (shrink)

Mathematical structuralism can be understood as a theory of mathematical ontology, of the objects that mathematics is about. It can also be understood as a theory of the semantics for mathematical discourse, of how and to what mathematical terms refer. In this paper we propose an epistemological interpretation of mathematical structuralism. According to this interpretation, the main epistemological claim is that mathematical knowledge is purely structural in character; mathematical statements contain purely structural information. To make this more precise, we invoke (...) a notion that is central to mathematical epistemology, the notion of (informal) proof. Appealing to the notion of proof, an epistemological version of the structuralist thesis can be formulated as: Every mathematical statement that is provable expresses purely structural information. We introduce a bi-modal framework that formalizes the notions of structural information and informal provability in order to draw connections between them and confirm that the epistemological structuralist thesis holds. (shrink)

The turn of the last century was a key transitional period for the development of symbolic logic and scientific philosophy. The Peano school, the editorial board of the Revue de Métaphysique et de Morale, and the members of the Vienna Circle are generally mentioned as champions of this transformation of the role of logic in mathematics and in the sciences. The articles contained in this volume aim to contribute to a richer historical and philosophical understanding of these groups and research (...) areas in Italy, France and Austria. Specifically, the contributions focus on the following topics: a detailed investigation of the relation between structuralism and modern mathematics; different notions of definition and interpretation at the turn of last century; a closer understanding of the relation between the Vienna Circle, the Peano School and French philosophy in the first half of the twentieth century. (shrink)

This book provides a collection of chapters on the development of scientific philosophy and symbolic logic in the early twentieth century. The turn of the last century was a key transitional period for the development of symbolic logic and scientific philosophy. The Peano school, the editorial board of the Revue de Métaphysique et de Morale, and the members of the Vienna Circle are generally mentioned as champions of this transformation of the role of logic in mathematics and in the sciences. (...) The scholarship contained provides a rich historical and philosophical understanding of these groups and research areas. Specifically, the contributions focus on a detailed investigation of the relation between structuralism and modern mathematics. In addition, this book provides a closer understanding of the relation between symbolic logic and previous traditions such as syllogistics. This volume also informs the reader on the relation between logic, the history and didactics in the Peano School. This edition appeals to students and researchers working in the history of philosophy and of logic, philosophy of science, as well as to researchers on the Vienna Circle and the Peano School. (shrink)

In this paper, we aim to explore connections between a Carnapian semantics of theoretical terms and an eliminative structuralist approach in the philosophy of mathematics. Specifically, we will interpret the language of Peano arithmetic by applying the modal semantics of theoretical terms introduced in Andreas (Synthese 174(3):367–383, 2010). We will thereby show that the application to Peano arithmetic yields a formal semantics of universal structuralism, i.e., the view that ordinary mathematical statements in arithmetic express general claims about all admissible interpretations (...) of the Peano axioms. Moreover, we compare this application with the modal structuralism by Hellman (Mathematics without numbers: towards a modal-structural interpretation. Oxford University Press: Oxford, 1989), arguing that it provides us with an easier epistemology of statements in arithmetic. (shrink)

As Paul Feyerabend once remarked, philosophy of science is a subject with a great past. Let me for the moment leave aside his disillusioned impression that it had only a sad present and no future and concentrate on its past. It is surprising indeed that much has been published on the history of science in the last few decades, while only very few efforts have been made to give an overall description of the history of philosophy of science. That of (...) course presupposes a defi nition or at least a rough idea of the subject. And along with that goes an answer to the question when it started and what has been part of it during its development. Some seem to think that philosophy of science already began with Aristotle’s Analytica Posteriora, while others would be inclined to have it start more than 2000 years later, let’s say with the Vienna Circle. (shrink)

_Erich Reck* * and Georg Schiemer.** ** The Prehistory of Mathematical Structuralism. _Oxford University Press, 2020. Pp. 454. ISBN: 978-0-19-064122-1 ; 978-0-19-064123-8. doi: 10.1093/oso/9780190641221.001.0001.

Idealist philosopher Georg Hegel defied the traditional epistemological distinction of objective from subjective and developed his own dialectical alternative. Remarkable for its breadth and profundity, this work combines aspects of psychology, logic, moral philosophy, and history to form a comprehensive view that encompasses all forms of civilization. Its three divisions consist of the subjective mind (dealing with anthropology and psychology), the objective mind (concerning philosophical issues of law and morals), and the absolute mind (covering fine arts, religion, and philosophy). (...) Wide-ranging in its influence, this survey of the evolution of consciousness is essential reading for all students of philosophy and history. (shrink)

The description for this book, Desert, will be forthcoming.

TRANSLATORS PREFACE THE PRESENT TRANSLATION OF GEORG SIMMEL'S Schopen- hauer und Nietzsche: Ein Vortragszyklus (1907), ...

The ‘extended mind’ thesis asserts that cognitive processes are not bound by the skull or even skin of biological individuals, but actively incorporate environmental structures such as symbols, tools, artifacts, media, cultural practices, norms, groups, or institutions. By distributing cognition across space, time, and people in canny ways, we circumvent or overcome the biological limitations of our brains. Human beings are creative, albeit opportunistic experts in cognitive ‘self-transcendence.’ This entry surveys discussions of EM in philosophy of mind and cognitive science (...) which point to this overarching theme. (shrink)

What drives much of the current philosophical interest in the idea of group cognition is its appeal to the manifestation of psychological properties—understood broadly to include states, processes, and dispositions—that are in some important yet elusive sense emergent with respect to the minds of individual group members. Our goal in this paper is to address a set of related, conditional questions: If human mentality is real yet emergent in a modest metaphysical sense only, then: (i) What would it mean for (...) a group to have emergent cognitive states? (ii) Is this even a metaphysically coherent view? (iii) Relative to which notion of emergence do we have reason to believe that certain groups in fact have emergent cognitive states? We shall argue that evidence from a wide variety of social science domains makes it plausible that there are group cognitive states and processes no less metaphysically emergent than human cognitive (and other special science) states and processes. (shrink)

Georges Sorel's Study on Vico is a revelatory document of the depths and stakes of French social thought at the end of the 19th century. What brought Sorel to the 18th century Neapolitan theorist of history? Acute awareness of the limitations of Marxist thought in his day, a profound concern with the material underpinnings of language, law, and culture, and the imperative to understand the possibilities of revolutionary change. We find here a different Sorel, one who speaks in surprising ways (...) to the 21st century. The translation is accompanied by an introduction and by a set of notes which situate the text both in Sorel's overall intellectual trajectory and in the fin de siècle debates from which it emerged. (shrink)

Many epistemologists have been interested in justification because of its presumed close relationship to knowledge. This relationship is intended to be ...

Darwin the writer -- Learning to see : Darwin's prophetic apprenticeship on the Beagle voyage -- The prose of On the origin of species -- Surprise and paradox : Darwin's artful legacy -- Darwinian mind and Wildean paradox -- Hardy's Woodlanders and the Darwinian grotesque -- Coda : the comic Darwin.

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be (...) preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Stolen Legacy by George G.M. James refutes the Euro-centric myth that the origin of Western philosophy is Greek. First published in 1954, this book was seminal in leading to a radical reappraisal of a philosophical system long thought to be of European origin. It is an essential work in the syllabus for the study of Western philosophy.

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be (...) preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

How do we evaluate the power and utility of language when it has been made to articulate falsehoods in certain totalitarian regimes or has been charged with vulgarity and imprecision in a mass-consumer democracy? How will language react to the increasingly urgent claims of more exact speech such as mathematics and symbolic notation? These are some of the questions Steiner addresses in this elegantly written book, first published in 1967 to international acclaim.

Benjamin Franklin famously wrote that “the good [that] men do separately is small compared with what they may do collectively” (Isaacson 2004). The ability to join with others in groups to accomplish goals collectively that would hopelessly overwhelm the time, energy, and resources of individuals is indeed one of the greatest assets of our species. In the history of humankind, groups have been among the greatest workers, builders, producers, protectors, entertainers, explorers, discoverers, planners, problem-solvers, and decision-makers. During the late 19th (...) and early 20th century, many social scientists employed the notorious “group mind” idiom to express the sensible idea that groups can function as the seats of cognition, intelligence, and agency in their own right (Allport 1968; Wilson 2004). In their quest to stress (rightly) that group phenomena are something “over and above” the sum of individual contributions, a fondness for vitalist metaphors led them to believe (wrongly) that genuine group cognition must be the result of tapping into individualistically inaccessible, “holistic” forces. Today, inspired in part by historically unparalleled forms of mass collaboration enabled by the internet, it has once again become popular to speak of collective intelligence, group agency, or even the emergence of a “global brain” (cf. the wiki-edited MIT Handbook of Collective Intelligence). -/- In this chapter, I review some contemporary developments of the idea of group cognition, defined broadly as the collaborative performance of cognitive tasks such as remembering, problem-solving, decision-making, or verbal creativity for the purpose of producing a group-level outcome. My discussion serves a two-fold purpose. First, by discussing how the idea of group cognition can be operationalized, I seek to show that we can retain some central theoretical insights of the “group mind” thesis without succumbing to its eccentric metaphysical overtones. Second, by providing a useful array of generalizable taxonomic resources, I hope to foster greater degrees of mutual awareness among insufficiently integrated areas of research on group performance. (shrink)

In the light of two unpublished letters from Carnap to Kuhn, this essay examines the relationship between Kuhn's The Structure of Scientific Revolutions and Carnap's philosophical views. Contrary to the common wisdom that Kuhn's book refuted logical empiricism, it argues that Carnap's views of revolutionary scientific change are rather similar to those detailed by Kuhn. This serves both to explain Carnap's appreciation of The Structure of Scientific Revolutions and to suggest that logical empiricism, insofar as that program rested on Carnap's (...) shoulders, was not substantially upstaged by Kuhn's book. (shrink)

Consent theory -- The principle of fair play -- Multiple principle theory -- Limits of political obligation.