Shared structure need not be shared set-structure

Abstract

Recent semantic approaches to scientific structuralism, aiming to make precise the concept of shared structure between models, formally frame a model as a type of set-structure. This framework is then used to provide a semantic account of (a) the structure of a scientific theory, (b) the applicability of a mathematical theory to a physical theory, and (c) the structural realist's appeal to the structural continuity between successive physical theories. In this paper, I challenge the idea that, to be so used, the concept of a model and so the concept of shared structure between models must be formally framed within a single unified framework, set-theoretic or other. I first investigate the Bourbaki-inspired assumption that structures are types of set-structured systems and next consider the extent to which this problematic assumption underpins both Suppes' and recent semantic views of the structure of a scientific theory. I then use this investigation to show that, when it comes to using the concept of shared structure, there is no need to agree with French that "without a formal framework for explicating this concept of 'structure-similarity' it remains vague, just as Giere's concept of similarity between models does …" (French, 2000, Synthese, 125, pp. 103-120, p. 114). Neither concept is vague; either can be made precise by appealing to the concept of a morphism, but it is the context (and not any set-theoretic type) that determines the appropriate kind of morphism. I make use of French's (1999, From physics to philosophy (pp. 187-207). Cambridge: Cambridge University Press) own example from the development of quantum theory to show that, for both Weyl and Wigner's programmes, it was the context of considering the 'relevant symmetries' that determined that the appropriate kind of morphism was the one that preserved the shared Lie-group structure of both the theoretical and phenomenological models.

Quantization as a Guide to Ontic Structure

“…We first identified the shared structures between the various formalisms of a theory by isolating the relevant homomorphisms (this would seem to be in keeping with the analysis given in (Landry 2007)). We then proceeded to consider a generalized version of these structures, understood as standing behind the various realisations and underpinning the structural relationships.…”

Mathematics, PISA, and culture: An unpredictable relationship

“…Commentary: In these two short cameos can be seen evidence of an expectation that students are not only exposed to the structures, notation and vocabulary of formal mathematics but that they use them as part of their everyday classroom discourse, a process very close to the original Bourbakian principles (Guedj 1985;Landry 2007). During the former, Pauline formalized some fundamental arithmetical principles that students had previously derived intuitively, in accordance with earlier Belgian perspectives on Bourbakian mathematics (Vanpaemel et al 2012).…”

“…Such behavior reflected a key element of the realistic tradition (Keijzer and Terwel 2003;Wubbels et al 1997). However, her remaining tasks, with their single and essentially nonnegotiable goals and exploitation of number base blocks, were focused explicitly on an understanding of percentage as independent of any real world and were unequivocally Bourbakian (Guedj 1985;Landry 2007). Moreover, while her sequence of tasks could be construed as resonating with the RME tradition of choosing problems "to support a logical sequence of cognitive development" (Gravemeijer 2004, p.107), her aim was the creation of a collective understanding of the same Bourbakian structural relationship (Aubin 1997;Klein 2003), unrelated to any obvious RME privileging of mathematical application (Elbers and De Haan 2005;Gravemeijer 2004;Streefland 1993).…”

“…In so doing she exploited continuously number base blocks "to help children... make the transition from intuitive experiences towards abstract understanding" (Vanpaemel et al 2012, p.425), a key element of school Bourbakian mathematics, which would not have found resonance with RME expectations (Gravemeijer 2004). However, the logical sequencing of Emke's activities, leading to the episode described above in relation to Bourbaki, seemed very much in accord with the RME tradition of choosing problems "to support a logical sequence of cognitive development" (Gravemeijer 2004: 107), which in this case were focused explicitly on Bourbakian multiplicative structures and proportional reasoning (Guedj 1985;Landry 2007). However, her frequent public discussion did not, in this or any other case, privilege, or even encourage, students' non-standard solution processes, as would be expected in an RMEfocused classroom (Van Putten et al 2005).…”

“…The group's aim was the promotion of an axiomatic mathematics, based on set theory, from which all topics would be derived (Guedj 1985;Landry 2007). Through this work Bourbaki expressed his desire "to purify mathematics of any reliance on the external world" (Aubin 1997, p.298) and "uphold the primacy of the pure over the applied, the rigorous over the intuitive" (Weintraub and Mirowski 1994, p. 247/8).…”