The Epsilon-Reconstruction of Theories and Scientific Structuralism

Abstract

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.

Taming conceptual wanderings: Wilson-Structuralism

“…Other promising ways of extending this framework would be to add linguistic and pragmatic contexts in order to model also Wilson's context-adjusting models of predicate extension (Wilson 1982(Wilson , 2012b or to merge Wilson-Structuralism with Carnapian Structuralism (Andreas 2014(Andreas , 2020 and with accounts of deductive reasoning in structuralist frameworks (Andreas 2013) in order to model Wilson's contextual notion of inference validity and logical inconsistencies (Wilson 1994(Wilson , 2000. It would also be interesting to merge Wilson-Structuralism with other reconstructions of scientific theories devised to capture the indeterminacy of theoretical terms such as Carnap's -term methodology (Carnap 1956;Schiemer and Gratzl 2016;Leitgeb and Carus 2020;Leitgeb MS).…”

A choice-semantical approach to theoretical truth

“…relative to a given theoretical context in both approaches. Moreover, both accounts make precise in different ways a central intuition driving systematic work on theoretical terms since Carnap, viz., that the determination of such terms by a scientific theory remains incomplete (see [2,30]).…”

“…See[3] for a survey of different model-theoretic explications of this view in the modern literature 2. Compare[30] for a systematic study of the epsilon reconstruction of theories and its compatibility with scientific structuralism.…”

“…See also[30] for a more detailed presentation of the epsilon reconstruction of scientific theories 6. See[5] and[35] for discussions of the historical development of the epsilon logic.…”

Ramsification and Semantic Indeterminacy

“…Therefore, for non-eliminative structuralists, there does not seem to be any fact of the matter whether the numeral 'i' actually denotes i or whether it rather denotes -i (see [5,63]). The debate about these matters is ongoing and some authors (see [40,41,52,59,61,63,64]) have suggested ways of developing a structuralist semantics that resemble aspects of Ramsey semantics. (I will return to part of that literature later in Section 2.)…”