### Abstract

In his reflective writings about mathematics, Reuben Hersh has consistently championed a philosophy of mathematical practice. He argues that if we pay close attention to what mathematicians really do in their research, as they extend mathematical knowledge at the frontier between the known and the conjectured, we see that their work does not only involve deductive reasoning. It also includes plausible reasoning, “analytic” reasoning upward that seeks the conditions of the solvability of problems and the conditions of the intelligibility of mathematical things. We use, he argues, “our mental models of mathematical entities, which are culturally controlled to be mutually congruent within the research community. These socially controlled mental models provide the much-desired “semantics” of mathematical reasoning” (Hersh 2014b, p. 127). Every active mathematician is familiar with a large swathe of established mathematics, “an intricately interconnected web of mutually supporting concepts, which are connected both by plausible and by deductive reasoning,” that include “concepts, algorithms, theories, axiom systems, examples, conjectures and open problems,” and models and applications. Thus, “the body of established mathematics is not a fixed or static set of statements. The new and recent part is in transition” (Ibid, pp. 131-2).

Original language | English (US) |
---|---|

Title of host publication | Humanizing Mathematics and its Philosophy |

Subtitle of host publication | Essays Celebrating the 90th Birthday of Reuben Hersh |

Publisher | Springer International Publishing |

Pages | 97-114 |

Number of pages | 18 |

ISBN (Electronic) | 9783319612317 |

ISBN (Print) | 9783319612300 |

DOIs | |

State | Published - Jan 1 2017 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Humanizing Mathematics and its Philosophy: Essays Celebrating the 90th Birthday of Reuben Hersh*(pp. 97-114). Springer International Publishing. https://doi.org/10.1007/978-3-319-61231-7_10

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*Humanizing Mathematics and its Philosophy: Essays Celebrating the 90th Birthday of Reuben Hersh.*Springer International Publishing, pp. 97-114. https://doi.org/10.1007/978-3-319-61231-7_10

**Reuben Hersh on the growth of mathematical knowledge : Kant, geometry, and number theory.** / Grosholz, Emily Rolfe.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

TY - CHAP

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PY - 2017/1/1

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N2 - In his reflective writings about mathematics, Reuben Hersh has consistently championed a philosophy of mathematical practice. He argues that if we pay close attention to what mathematicians really do in their research, as they extend mathematical knowledge at the frontier between the known and the conjectured, we see that their work does not only involve deductive reasoning. It also includes plausible reasoning, “analytic” reasoning upward that seeks the conditions of the solvability of problems and the conditions of the intelligibility of mathematical things. We use, he argues, “our mental models of mathematical entities, which are culturally controlled to be mutually congruent within the research community. These socially controlled mental models provide the much-desired “semantics” of mathematical reasoning” (Hersh 2014b, p. 127). Every active mathematician is familiar with a large swathe of established mathematics, “an intricately interconnected web of mutually supporting concepts, which are connected both by plausible and by deductive reasoning,” that include “concepts, algorithms, theories, axiom systems, examples, conjectures and open problems,” and models and applications. Thus, “the body of established mathematics is not a fixed or static set of statements. The new and recent part is in transition” (Ibid, pp. 131-2).

AB - In his reflective writings about mathematics, Reuben Hersh has consistently championed a philosophy of mathematical practice. He argues that if we pay close attention to what mathematicians really do in their research, as they extend mathematical knowledge at the frontier between the known and the conjectured, we see that their work does not only involve deductive reasoning. It also includes plausible reasoning, “analytic” reasoning upward that seeks the conditions of the solvability of problems and the conditions of the intelligibility of mathematical things. We use, he argues, “our mental models of mathematical entities, which are culturally controlled to be mutually congruent within the research community. These socially controlled mental models provide the much-desired “semantics” of mathematical reasoning” (Hersh 2014b, p. 127). Every active mathematician is familiar with a large swathe of established mathematics, “an intricately interconnected web of mutually supporting concepts, which are connected both by plausible and by deductive reasoning,” that include “concepts, algorithms, theories, axiom systems, examples, conjectures and open problems,” and models and applications. Thus, “the body of established mathematics is not a fixed or static set of statements. The new and recent part is in transition” (Ibid, pp. 131-2).

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