The First Modern Definition of the Sum of a Divergent Series: An Aspect of the Rise of 20th Century Mathematics

Ferraro, Giovanni

doi:10.1007/s004070050036

Cited by 14 publications

(18 citation statements)

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“…23 Euler hypothesized that, given an algebraic equation F(x, y) = 0, y was always expressible as a function of x. I think that this hypothetical function served to justify the use of algebraic equations F(x, y) = 0 as (implicit) functions in analysis even when one was not able to transform equations into explicit functions.…”

Section: Conditions For the Representability Of Functional Relations mentioning

confidence: 95%

“…In Euler's mathematics, a definition did not necessarily exhaust the defined notion; it could have an implicit meaning, which, in a sense, was considered as obvious in a given context. Euler's concept of definition is, however, beyond the scope of this paper (on this subject, see [23]). …”

Section: Functions and Functional Relationsmentioning

confidence: 97%

“…22 Some doubts concerned the functions of the kind z c , c being an irrational number: somebody, Euler said, preferred to term it "interscendentes" [14, 1:20]. 23 "[F]unctiones algebraicae saepenumero ne quidem explicite exhiberi possunt, cuiusmodi functio ipsius z est Z , si definiatur per huiusmodi aequationem…”

Section: Conditions For the Representability Of Functional Relations mentioning

confidence: 99%

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Functions, Functional Relations, and the Laws of Continuity in Euler

Ferraro¹

2000

Historia Mathematica

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ALLA MEMORIA DI MIO PADRE IN THE MEMORY OF MY FATHERLa nozione euleriana di funzione presenta un duplice aspetto: essa era, a un tempo, relazione funzionale tra quantità e formula composta da simboli operazionali, da costanti, e da variabili. Queste ultime erano concepite come universali e pertanto godenti di particolari proprietà. Anche se il calcolo di Euler era basato sulla manipolazione delle formule, egli non esitò a ricorrere alle relazioni funzionali se necessario. Inoltre le relazioni funzionali erano essenziali per la costruzione o definizione delle formule analitiche e per le applicazioni dei risultati. Ovviamente un tale modo di intendere le funzioni provocò ambiguità tra l'aspetto intuitivo, geometrico, o empirico dei concetti e la loro rappresentazione simbolica in analisi. C 2000 Academic Press Eulerian functions had two aspects: they were both functional relations between quantities and formulas composed of constants, variables, and operational symbols. The latter were regarded as universal and possessed extremely special properties. Even though Eulerian calculus was based upon the manipulation of formulas, mathematicians did not hesitate to use functional relations when it was necessary. Besides, functional relations were essential to the construction or definition of analytic formulas and application of the results of calculus. This concept of function led to ambiguity between the intuitive, geometrical, or empirical nature of concepts and their symbolic representation in analysis. C

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Section: Conditions For the Representability Of Functional Relations mentioning

confidence: 95%

Section: Functions and Functional Relationsmentioning

confidence: 97%

Section: Conditions For the Representability Of Functional Relations mentioning

confidence: 99%

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Functions, Functional Relations, and the Laws of Continuity in Euler

Ferraro¹

2000

Historia Mathematica

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“…Since this was not a concern for Euler, there was no need to be so precise. For instance, certain characteristics of mathematical objects would be taken as obvious or implicit [Ferraro, 1999[Ferraro, , 2000. Niels Hans Jahnke [2003] declares that it would be considered "ridiculously formalistic" to append x -0 to the function 1…”

Section: Eighteenth Century Definitionsmentioning

confidence: 99%

“…. [Euler, 1770, 2] For eighteenth century mathematicians such as Euler, definitions described the nature of mathematical objects which existed a priori whereas current definitions constitute the exact nature mathematical objects to be studied [Jahnke, 2003;Ferraro, 1998Ferraro, , 1999Fraser, 1989Fraser, , 2003. It is necessary to draw attention to the ontological implications of Euler's definition of a determined quantity.…”

Section: Eighteenth Century Definitionsmentioning

confidence: 99%

Leonhard Euler’s use and understanding of mathematical transcendence

Petrie

2012

Historia Mathematica

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Leonhard Euler primarily applied the term "transcendental" to quantities which could be variable or determined. Analyzing Euler's use and understanding of mathematical transcendence as applied to operations, functions, progressions, and determined quantities as well as the eighteenth century practice of definition allows the author to evaluate claims that Euler provided the first modern definition of a transcendental number. The author argues that Euler's informal and pragmatic use of mathematical transcendence highlights the general nature of eighteenth century mathematics and proposes an alternate perspective on the issue at hand: transcendental numbers inherited their transcendental classification from functions. Ó 2012 Elsevier Inc. All rights reserved. RésuméLeonhard Euler appliqua le terme «transcendantal» surtout aux quantités qui pouvaient être soit variables, soit fixes. En analysant comment Euler appliqua la transcendance mathématique aux opérations, aux fonctions, aux progressions, et aux quantités fixes tout en considérant la manière dont les définitions mathématiques furent élaborées au XVIIIe siècle, l'auteur évalue l'affirmation qu'Euler a fourni la première définition moderne d'un nombre transcendant. L'auteur soutient que le concept de transcendance mathématique employé par Euler, étant informel et pragmatique, met en évidence le caractère générale des mathématiques du XVIIIe siècle, et l'auteur propose ensuite une perspective différente sur la question : nombres transcendants ont hérité de leur classification des fonctions transcendantes.

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Geometry and analysis in Anastácio da Cunha’s calculus

Domingues¹

2023

Arch. Hist. Exact Sci.

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It is well known that over the eighteenth century the calculus moved away from its geometric origins; Euler, and later Lagrange, aspired to transform it into a “purely analytical” discipline. In the 1780 s, the Portuguese mathematician José Anastácio da Cunha developed an original version of the calculus whose interpretation in view of that process presents challenges. Cunha was a strong admirer of Newton (who famously favoured geometry over algebra) and criticized Euler’s faith in analysis. However, the fundamental propositions of his calculus follow the analytical trend. This appears to have been possible due to a nominalistic conception of variable that allowed him to deal with expressions as names, rather than abstract quantities. Still, Cunha tried to keep the definition of fluxion directly applicable to geometrical magnitudes. According to a friend of Cunha’s, his calculus had an algebraic (analytical) branch and a geometrical branch, and it was because of this that his definition of fluxion appeared too complex to some contemporaries.

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The First Modern Definition of the Sum of a Divergent Series: An Aspect of the Rise of 20th Century Mathematics

Cited by 14 publications

References 12 publications

Functions, Functional Relations, and the Laws of Continuity in Euler

Functions, Functional Relations, and the Laws of Continuity in Euler

Leonhard Euler’s use and understanding of mathematical transcendence

Geometry and analysis in Anastácio da Cunha’s calculus

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