Abstract
Practical calculation of the limit of a sequence often violates the definition of convergence to a limit as taught in calculus. Together with examples from Euler, Pólya and Poincaré, this fact shows that in mathematics, as in science and in everyday life, we are often obligated to use knowledge that is derived, not rigorously or deductively, but simply by making the best use of available information–plausible reasoning. The “philosophy of mathematical practice” fits into the general framework of “warranted assertibility”, the pragmatist view of the logic of inquiry developed by John Dewey.
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Acknowledgements
In this work, I benefited from suggestions and criticisms by Carlo Cellucci, Richard Epstein, Russell Goodman, Cleve Moler, Peter Lax, Ulf Persson, Vera John-Steiner, and members of the study group on mathematical thinking in Santa Fe, New Mexico.
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Appendix
Appendix
Ap Dijksterhuis and Teun Meurs, Where creativity resides: the generative power of unconscious thought, Social Psychology Program, University of Amsterdam, Roetersstraat 15, 1018 WB Amsterdam, The Netherlands, 2004, 2005.
Abstract
In three experiments, the relation between different modes of thought and the generation of “creative” and original ideas was investigated. Participants were asked to generate items according to a specific instruction (e.g., generate place names starting with an “A”). They either did so immediately after receiving the instruction, or after a few minutes of conscious thought, or after a few minutes of distraction during which “unconscious thought” was hypothesized to take place. Throughout the experiments, the items participants listed under “unconscious thought” conditions were more original. It was concluded that whereas conscious thought may be focused and convergent, unconscious thought may be more associative and divergent.
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Hersh, R. (2013). Mathematical Intuition: Poincaré, Pólya, Dewey. In: de Moura, C., Kubrusly, C. (eds) The Courant–Friedrichs–Lewy (CFL) Condition. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8394-8_2
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