Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus

Grabiner, Judith V.

doi:10.1080/00029890.1983.11971185

Cited by 37 publications

(10 citation statements)

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“…These material artifacts may play a role in generating consensus by structuring inference making during discovery and proof. In this vein, we note only in passing that Cauchy was one of the originators of the ε–δ notation (Grabiner, )—and yet failed to deploy those notational resources in any significant way in his contested proof. Imaginative conceptual processes, moreover, interact in largely unexplored ways with these external resources (Kirsh, ).…”

Section: Discussionmentioning

confidence: 99%

“…In arguing for Cauchy's dynamic conceptualization, of course, we are not ruling out the possibility that he also held a static conceptualization—indeed, he was responsible for introducing the notational innovations that are at the heart of the static ε–δ definitions of limits and continuity (Grabiner, ), definitions which rely on notions like preservation of closeness (Núñez & Lakoff, ). Multiple construals of a single conceptual domain are common; we can alternatively think of affection in terms of spatial proximity (“ close friends”) or warmth (“a warm welcome”).…”

Section: Discussionmentioning

confidence: 99%

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The Motion Behind the Symbols: A Vital Role for Dynamism in the Conceptualization of Limits and Continuity in Expert Mathematics

Marghetis

Núñez

2013

Topics in Cognitive Science

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The canonical history of mathematics suggests that the late 19th-century "arithmetization" of calculus marked a shift away from spatial-dynamic intuitions, grounding concepts in static, rigorous definitions. Instead, we argue that mathematicians, both historically and currently, rely on dynamic conceptualizations of mathematical concepts like continuity, limits, and functions. In this article, we present two studies of the role of dynamic conceptual systems in expert proof. The first is an analysis of co-speech gesture produced by mathematics graduate students while proving a theorem, which reveals a reliance on dynamic conceptual resources. The second is a cognitivehistorical case study of an incident in 19th-century mathematics that suggests a functional role for such dynamism in the reasoning of the renowned mathematician Augustin Cauchy. Taken together, these two studies indicate that essential concepts in calculus that have been defined entirely in abstract, static terms are nevertheless conceptualized dynamically, in both contemporary and historical practice.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

The Motion Behind the Symbols: A Vital Role for Dynamism in the Conceptualization of Limits and Continuity in Expert Mathematics

Marghetis

Núñez

2013

Topics in Cognitive Science

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“…Then, it follows from the intermediate value theorem [33] that equation h 1 (t) = 0 has solutions in (0, r). Denote by r 1 the smallest such solution.…”

Section: Local Convergence Analysismentioning

confidence: 99%

Convergence Analysis and Complex Geometry of an Efficient Derivative-Free Iterative Method

2019

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To locate a locally-unique solution of a nonlinear equation, the local convergence analysis of a derivative-free fifth order method is studied in Banach space. This approach provides radius of convergence and error bounds under the hypotheses based on the first Fréchet-derivative only. Such estimates are not introduced in the earlier procedures employing Taylor’s expansion of higher derivatives that may not exist or may be expensive to compute. The convergence domain of the method is also shown by a visual approach, namely basins of attraction. Theoretical results are endorsed via numerical experiments that show the cases where earlier results cannot be applicable.

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“…Note that these conditions cannot be satisfied if we represent time by the ordering of real numbers, the usual understanding of the continuum since the work of Richard Dedekind (1872) and Karl Weierstrass (cf. Grabiner, 1983).…”

Section: The Dividing Instant Problemmentioning

confidence: 99%

Axiomatic theories of the ontology of time in GFO

Baumann

Loebe

Herre³

2014

Applied Ontology

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Time is a pervasive notion of high impact in information systems and computer science altogether. Respective understandings of the domain of time are fundamental for numerous areas, frequently in combination with closely related entities such as events, changes and processes. The conception and representation of time entities and reasoning about temporal data and knowledge are thus significant research areas. Each representation of temporal knowledge bears ontological commitments concerning time. Thus it is important to base temporal representations on a foundational ontology that covers general categories of time entities.In this article we introduce and discuss two consecutive ontologies of time that have been developed for the top-level ontology General Formal Ontology (GFO). The first covers intervals, named chronoids, and time boundaries of chronoids, as a kind of time points. One important specialty of time boundaries is their ability to coincide with other time boundaries. The second theory extends the first one by additionally addressing time regions, i.e., mereological sums of chronoids.Both ontologies are partially inspired by ideas of Franz Brentano, especially from his writings about the continuum. In particular, we view continuous time intervals as a genuine phenomenon which should not be identified with intervals (sets) of real numbers. On these grounds the resulting ontologies allow for proposing novel contributions to several problematic issues in temporal representation and reasoning, among others, the Dividing Instant Problem and the problem of persistence and change.Following our general approach to ontology development, both ontologies are axiomatized as formal theories in first-order logic and are analyzed metalogically. We prove the consistency of both ontologies, and completeness and decidability for one. Moreover, standard time theories with points and intervals are covered by both theories.

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Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus

Cited by 37 publications

References 1 publication

The Motion Behind the Symbols: A Vital Role for Dynamism in the Conceptualization of Limits and Continuity in Expert Mathematics

The Motion Behind the Symbols: A Vital Role for Dynamism in the Conceptualization of Limits and Continuity in Expert Mathematics

Convergence Analysis and Complex Geometry of an Efficient Derivative-Free Iterative Method

Axiomatic theories of the ontology of time in GFO

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