Carnap’s early metatheory: scope and limits

Schiemer, Georg; Zach, Richard; Reck, Erich H.

doi:10.1007/s11229-015-0877-z

Cited by 5 publications

(6 citation statements)

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“…44 But while Chasles clearly saw that Poncelet was wrong in believing that the justi ication of duality requires polar theory, Chasles still agrees 42 See e.g. (Badesa et al 2009), (Awodey & Reck 2002), (Schiemer et al 2017), and (Schiemer & Reck 2013) for more on the development of various metatheoretical concepts during that period. 43 The signi icance of transfer principles for the formation of metatheoretical conceptions has been emphasized in the context of a discussion of Frege's views on metatheory in (Tappenden 2005), an article that was a major inspiration for the current paper.…”

Section: Duality Transfer and The Harbingers Of Early Metatheorymentioning

confidence: 99%

Projective Duality and the Rise of Modern Logic

Eder¹

2021

Bull. symb. log

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The symmetries between points and lines in planar projective geometry and between points and planes in solid projective geometry are striking features of these geometries that were extensively discussed during the nineteenth century under the labels "duality" or "reciprocity". The aims of this article are, irst, to provide a systematic analysis of duality from a modern point of view, and, second, based on this, to give a historical overview of how discussions about duality evolved during the nineteenth century. Speci ically, we want to see in which ways geometer's preoccupation with duality was shaped by developments that lead to modern logic towards the end of the nineteenth century, and how these developments in turn might have been in luenced by re lections on duality. This is a ``preproof'' accepted article for The Bulletin of Symbolic Logic. This version may be subject to change during the production process.

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Section: Duality Transfer and The Harbingers Of Early Metatheorymentioning

confidence: 99%

Projective Duality and the Rise of Modern Logic

Eder¹

2021

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“…In his Untersuchungen, he mentioned, apart from the notions of "monomorphicity", "non-forkability" and "decidability", a fourth concept of completeness which is associated with the "extremal axioms" (cf. Schiemer et al [53], pp. [40][41].…”

Section: The Uniqueness Of the Model As A Corollarymentioning

confidence: 99%

Completeness: From Husserl to Carnap

Utrero

2021

Log. Univers.

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In his Doppelvortrag (1901), Edmund Husserl introduced two concepts of “definiteness” which have been interpreted as a vindication of his role in the history of completeness. Some commentators defended that the meaning of these notions should be understood as categoricity, while other scholars believed that it is closer to syntactic completeness. A detailed study of the early twentieth-century axiomatics and Husserl’s Doppelvortrag shows, however, that many concepts of completeness were conflated as equivalent. Although “absolute definiteness” was principally an attempt to characterize non-extendible manifolds and axiom systems (different from Hilbert’s axiom of completeness), an absolutely definite theory has a unique model and, thus, it is non-forkable and semantically complete (decidable). Non-forkability and decidability were formally delimited by Fraenkel and Carnap almost three decades later and, in fact, they mentioned Husserl as precursor of the latter. Therefore, this paper contributes to a reassessment of Husserl’s place in the history of logic.

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“…10 See Awodey and Carus (2001) and Schiemer, Zach, and Reck (2017) for surveys of Carnap's early metatheoretic work. Compare, in particular, Awodey and Reck (2002) for a detailed study of the development of metatheoretic notions in 19th-and early 20th-century mathematics.…”

Section: Pre-syntax Philosophy Of Mathematicsmentioning

confidence: 99%

“…See also Carnap (1929) for a simplified definition. 32 Compare, in particular, Awodey and Carus (2001) and Schiemer, Zach, and Reck (2017) for assessments of Carnap's early metatheory and of the limitations of his approach. relations so determined we will call structures and will say that model M 1 has structure S 1 if "S 1 (M 1 )" is analytic.…”

Section: Model Structuresmentioning

confidence: 99%

Carnap’s Structuralist Thesis

Schiemer¹

2020

The Prehistory of Mathematical Structuralism

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This chapter investigates Carnap’s structuralism in his philosophy of mathematics of the 1920s and early 1930s. His approach to mathematics is based on a genuinely structuralist thesis, namely that axiomatic theories describe abstract structures or the structural properties of their objects. The aim in the present article is twofold: first, to show that Carnap, in his contributions to mathematics from the time, proposed three different (but interrelated) ways to characterize the notion of mathematical structure, namely in terms of (i) implicit definitions, (ii) logical constructions, and (iii) definitions by abstraction. The second aim is to re-evaluate Carnap’s early contributions to the philosophy of mathematics in light of contemporary mathematical structuralism. Specifically, the chapter discusses two connections between his structuralist thesis and current philosophical debates on structural abstraction and the on the definition of structural properties.

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Carnap’s early metatheory: scope and limits

Cited by 5 publications

References 36 publications

Projective Duality and the Rise of Modern Logic

Projective Duality and the Rise of Modern Logic

Completeness: From Husserl to Carnap

Carnap’s Structuralist Thesis

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