TY - JOUR

T1 - How to frame innovation in mathematics

AU - Sarikaya, Deniz

N1 - Funding Information:
Deniz Sarikaya is thankful for the financial and ideal support of the Studienstiftung des deutschen Volkes and the Claussen-Simon-Stiftung as well as funding by Research Foundation Flanders (Project: FWOAL950). He thanks Karl Heuer for interesting comments and discussions concerning the case of topological infinite graphs. All authors are thankful to the Projekt DEAL for enabling open access.
Publisher Copyright:
© 2023, The Author(s).

PY - 2023/10

Y1 - 2023/10

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=85172372448&partnerID=8YFLogxK

U2 - 10.1007/s11229-023-04310-3

DO - 10.1007/s11229-023-04310-3

M3 - Article

VL - 202

SP - 1

EP - 31

JO - Synthese

JF - Synthese

SN - 0039-7857

IS - 4

M1 - 108

ER -