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Georg Schiemer
University of Vienna
  1. What Are Structural Properties?†.Johannes Korbmacher & Georg Schiemer - 2018 - Philosophia Mathematica 26 (3):295-323.
    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. (...)
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  2.  1
    The Pre-History of Mathematical Structuralism.Erich H. Reck & Georg Schiemer (eds.) - 2020 - Oxford, England and New York, NY, USA: Oxford University Press.
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  3.  36
    Hilbert, Duality, and the Geometrical Roots of Model Theory.Günther Eder & Georg Schiemer - 2018 - Review of Symbolic Logic 11 (1):48-86.
    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 (...)
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  4.  18
    The Structuralist Thesis Reconsidered.Georg Schiemer & John Wigglesworth - 2019 - British Journal for the Philosophy of Science 70 (4):1201-1226.
    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 (...)
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  5. Carnap’s Early Semantics.Georg Schiemer - 2013 - Erkenntnis 78 (3):487-522.
    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’ (...)
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  6.  27
    Cassirer and the Structural Turn in Modern Geometry.Georg Schiemer - 2018 - Journal for the History of Analytical Philosophy 6 (3).
    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 (...)
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  7.  51
    The Epsilon-Reconstruction of Theories and Scientific Structuralism.Georg Schiemer & Norbert Gratzl - 2016 - Erkenntnis 81 (2):407-432.
    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 (...)
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  8.  23
    The Structuralist Thesis Reconsidered.Georg Schiemer & John Wigglesworth - 2017 - British Journal for the Philosophy of Science:axy004.
    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 (...)
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  9.  38
    Carnap on Extremal Axioms, "Completeness of the Models," and Categoricity.Georg Schiemer - 2012 - Review of Symbolic Logic 5 (4):613-641.
    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.
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  10.  55
    Logic in the 1930s: Type Theory and Model Theory.Georg Schiemer & Erich H. Reck - 2013 - Bulletin of Symbolic Logic 19 (4):433-472.
    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 (...)
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  11.  63
    Carnap’s Early Metatheory: Scope and Limits.Georg Schiemer, Richard Zach & Erich Reck - 2017 - Synthese 194 (1):33-65.
    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 (...)
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  12. Invariants and Mathematical Structuralism.Georg Schiemer - 2014 - Philosophia Mathematica 22 (1):70-107.
    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 (...)
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  13.  32
    Carnap on Logic and Rationality.Georg Schiemer - 2017 - Synthese 194 (1):1-14.
    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 (...)
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  14.  16
    Carnap's Untersuchungen: Logicism, Formal Axiomatics, and Metatheory.Georg Schiemer - 2012 - In R. Creath (ed.), Vienna Circle Institute Yearbook. Springer Verlag. pp. 13--36.
    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 (...)
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  15.  18
    Logic in the 1930s: Type Theory and Model Theory.Georg Schiemer & Erich H. Reck - 2013 - Bulletin of Symbolic Logic 19 (4):433-472.
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  16.  13
    Carnap’s Untersuchungen: Logicism, Formal Axiomatics, and Metatheory.Georg Schiemer - 2012 - Vienna Circle Institute Yearbook 16:13-36.
    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 (...)
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  17.  19
    A Choice-Semantical Approach to Theoretical Truth.Holger Andreas & Georg Schiemer - 2016 - Studies in History and Philosophy of Science Part A 58:1-8.
    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, (...)
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  18.  16
    Introduction to Special Issue: Foundations of Mathematical Structuralism.Georg Schiemer & John Wigglesworth - 2020 - Philosophia Mathematica 28 (3):291-295.
    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 (...)
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  19.  15
    What are Implicit Definitions?Eduardo N. Giovannini & Georg Schiemer - 2021 - Erkenntnis 86 (6):1661-1691.
    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 (...)
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  20.  19
    Fraenkel's Axiom of Restriction: Axiom Choice, Intended Models and Categoricity.Georg Schiemer - 2010 - In Benedikt L.öwe & Thomas Müller (eds.), PhiMSAMP. Philosophy of Mathematics: Sociological Aspects and Mathematical Practice. College Publications. pp. 307{340.
  21.  4
    Modal Structuralism with Theoretical Terms.Holger Andreas & Georg Schiemer - forthcoming - Erkenntnis:1-25.
    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 :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 (...)
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  22.  18
    Reviews. [REVIEW]Hans-Joachim Dahms, Markus Lammenranta, Juha Manninen & Georg Schiemer - 2011 - Vienna Circle Institute Yearbook 15:337-349.
    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 (...)
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  23.  18
    Logicism and Ramsification: William Demopoulos: Logicism and its Philosophical Legacy. Cambridge: Cambridge University Press, 2013, Xii+272pp, $99.00 HB. [REVIEW]Georg Schiemer - 2014 - Metascience 23 (2):255-261.
    This excellent book presents a collection of eleven articles, all but one of which were written by William Demopoulos over the period of the last 19 years. The book comprises eight published articles, some of which have appeared only recently, as well as three new articles. The thematic scope of the topics investigated here is broad and ranges from Frege’s original logicist program outlined in his Grundlagen der Arithmetik (Frege 1884) to Carnap’s mature work on the logical reconstruction of scientific (...)
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