How to frame innovation in mathematics.
We discuss conceptual change and progress within mathematics, in particular how tools, structural concepts and representations are transferred between fields that appear to be unconnected or remote from each other. The theoretical background is provided by the frame concept, which is used in linguis...
| Publicado en: | Synthese Vol. 202; no. 4; pp. 1 - 32 |
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| Autores principales: | , , |
| Formato: | Artículo |
| Publicado: |
Springer Nature
Oct2023
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| Acceso en línea: | Ver este registro en EBSCOhost |
| fields | @attributes: recordID: 1 pdfLink: plink: https://search.ebscohost.com/login.aspx?direct=true&db=hlh&AN=172335525&site=ehost-live header: @attributes: shortDbName: hlh uiTerm: 172335525 longDbName: Humanities International Complete uiTag: AN controlInfo: bkinfo: jinfo: jid: 00397857 4LI jtl: Synthese issn: 00397857 maglogo: N pubinfo: dt: Oct2023 vid: 202 iid: 4 pid: 237 pub: Springer Nature artinfo: ui: 172335525 10.1007/s11229-023-04310-3 ppf: 1 ppct: 31 formats: fmt: – @attributes: type: T – @attributes: type: P size: 914KB tig: atl: How to frame innovation in mathematics. aug: au: Fisseni, Bernhard Sarikaya, Deniz Schröder, Bernhard affil: https://ror.org/04mz5ra38 Fakultät für Geisteswissenschaften, Institut für Germanistik, Universität Duisburg-Essen, Campus Essen, Universitätsstraße 12, 45117, Essen, Germany https://ror.org/006e5kg04 Department for History, Archeology, Arts, Philosophy and Ethics (HARP), Centre for Logic and Philosophy of Science, Vrije Universiteit Brussel, Pleinlaan 2 Room 5B425, 1050, Brussels, Belgium sug: keyword: Frames Innovation Mathematical language ab: We discuss conceptual change and progress within mathematics, in particular how tools, structural concepts and representations are transferred between fields that appear to be unconnected or remote from each other. The theoretical background is provided by the frame concept, which is used in linguistics, cognitive science and artificial intelligence to model how explicitly given information is combined with expectations deriving from background knowledge. In mathematical proofs, we distinguish two kinds of frames, namely structural frames and ontological frames. The interaction between both kinds of frames can drive mathematical interpretation. We first discuss two examples where structural frames (formulaic notation) drive ontological development (the discovery or exploration of mathematical objects). The development of Boole’s Boolean algebra may at first appear as a metaphorical treatment of the (then) new area of logic. In the analysis, we discuss how different (aspects of) certain algebraic frames change in the transfer, how arising difficulties are solved and overall argue that Boole uses the numerical algebra frame as a research template for the discovery of a system for calculations in logic. Following Ifrah, we analyse the discovery of zero as an extension to the number ontology as driven by the development of notation. Both structural and ontological frames are extended and simplified as notation progresses. Finally, we discuss two examples from infinite combinatorics, viz. topological graph theory, and one foundational issue. In both examples, the two simultaneous frames about one object are maintained independently. They motivate different research questions, but may also fruitfully interact: shifting between multiple synchronously maintained perspectives acts as a motor of innovation. The analysis shows how a frame-based approach allows to model how different perspectives drive mathematical innovation because they highlight different aspects, questions and heuristics. pubtype: Academic Journal doctype: Article src: R language: English refInfo: copyright: @attributes: flag: Y custom: Synthese is a copyright of Springer, 2023. All Rights Reserved. item: Synthese holder: Springer Nature dt: @attributes: year: 2023 holdings: @attributes: islocal: N |
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