### Abstract

We apply Benacerraf’s distinction between mathematical ontology and mathematical practice (or the structures mathematicians use in practice) to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves.

Original language | English |
---|---|

Pages (from-to) | 195–238 |

Number of pages | 44 |

Journal | Journal for General Philosophy of Science |

Volume | 48 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 2017 |

### Fingerprint

### Keywords

- Archimedean axiom
- Euler
- Infinite product
- Infinitesimal
- Law of continuity
- Law of homogeneity
- Leibniz
- Mathematical practice
- Ontology
- Principle of cancellation
- Procedure
- Standard part principle

### Cite this

*Journal for General Philosophy of Science*,

*48*(2), 195–238. https://doi.org/10.1007/s10838-016-9334-z

}

*Journal for General Philosophy of Science*, vol. 48, no. 2, pp. 195–238. https://doi.org/10.1007/s10838-016-9334-z

**Interpreting the Infinitesimal Mathematics of Leibniz and Euler.** / Bair, Jacques; Błaszczyk, Piotr; Ely, Robert; Henry, Valérie; Kanovei, Vladimir; Katz, Karin U.; Katz, Mikhail G.; Kutateladze, Semen S.; McGaffey, Thomas; Reeder, Patrick; Schaps, David M.; Sherry, David; Shnider, Steven.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Interpreting the Infinitesimal Mathematics of Leibniz and Euler

AU - Bair, Jacques

AU - Błaszczyk, Piotr

AU - Ely, Robert

AU - Henry, Valérie

AU - Kanovei, Vladimir

AU - Katz, Karin U.

AU - Katz, Mikhail G.

AU - Kutateladze, Semen S.

AU - McGaffey, Thomas

AU - Reeder, Patrick

AU - Schaps, David M.

AU - Sherry, David

AU - Shnider, Steven

PY - 2017/6

Y1 - 2017/6

N2 - We apply Benacerraf’s distinction between mathematical ontology and mathematical practice (or the structures mathematicians use in practice) to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves.

AB - We apply Benacerraf’s distinction between mathematical ontology and mathematical practice (or the structures mathematicians use in practice) to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves.

KW - Archimedean axiom

KW - Euler

KW - Infinite product

KW - Infinitesimal

KW - Law of continuity

KW - Law of homogeneity

KW - Leibniz

KW - Mathematical practice

KW - Ontology

KW - Principle of cancellation

KW - Procedure

KW - Standard part principle

UR - http://www.scopus.com/inward/record.url?scp=84978654577&partnerID=8YFLogxK

U2 - 10.1007/s10838-016-9334-z

DO - 10.1007/s10838-016-9334-z

M3 - Article

VL - 48

SP - 195

EP - 238

JO - Journal for General Philosophy of Science

JF - Journal for General Philosophy of Science

SN - 0925-4560

IS - 2

ER -