Gregory’s Sixth Operation

Bascelli, Tiziana; Błaszczyk, Piotr; Kanovei, Vladimir; Katz, Karin U.; Katz, Mikhail G.; Kutateladze, S. S.; Nowik, Tahl; Schaps, David; Sherry, David

doi:10.1007/s10699-016-9512-9

Cited by 11 publications

(16 citation statements)

References 22 publications

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“…18 It is still the most popular vehicle for the practice of nonstandard analysis; see for example 17 The subset R ⊆ * R is a counterexample: it is bounded above by every positive infinitely large number L, but it does not have a least upper bound: if L is an upper bound for R, then L − 1 is similarly an upper bound.…”

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confidence: 99%

Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others

Fletcher¹,

Hrbacek²,

Kanovei³

et al. 2017

Real Analysis Exchange

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Abstract. This is a survey of several approaches to the framework for working with infinitesimals and infinite numbers, originally developed by Abraham Robinson in the 1960s, and their constructive engagement with the Cantor-Dedekind postulate and the Intended Interpretation hypothesis. We highlight some applications including (1) Loeb's approach to the Lebesgue measure, (2) a radically elementary approach to the vibrating string, (3) true infinitesimal differential geometry. We explore the relation of Robinson's and related frameworks to the multiverse view as developed by Hamkins.Keywords: axiomatisations, infinitesimal, nonstandard analysis, ultraproducts, superstructure, set-theoretic foundations, multiverse, naive integers, intuitionism, soritical properties, ideal elements, protozoa. The last third of the 19th century saw (at least) two dynamic innovations: set theory and the automobile. Consider the following parable.A silvery family sedan is speeding down the highway. It enters heavy traffic. Every epsilon of the road requires a new delta of patience on APPROACHES TO ANALYSIS WITH INFINITESIMALS 3 the part of the passengers. The driver's inquisitive daughter Sarah, 1 sitting in the front passenger seat, discovers a mysterious switch in the glove compartment. With a click, the sedan spreads wings and lifts off above the highway congestion in an infinitesimal instant. Soon it is a mere silvery speck in an infinite expanse of the sky. A short while later it lands safely on the front lawn of the family's home.Sarah's cousin Georg 2 refuses to believe the story: true Sarah's father is an NSA man, but everybody knows that Karl Benz's 1886 PatentMotorwagen had no wing option! At a less parabolic level, some mathematicians feel that, on the one hand, "It is quite easy to make mistakes with infinitesimals, etc." (Quinn [137, p. 31]) while, as if by contrast, "Modern definitions are completely selfcontained, and the only properties that can be ascribed to an object are those that can be rigorously deduced from the definition. . . Definitions that are modern in this sense were developed in the late 1800s." (ibid., p. 32).We will have opportunity to return to Sarah, cousin Georg, switches, and the heroic "late 1800s" in the sequel; see in particular Section 7.4. IntroductionThe 2.1. Audience. The text presupposes some curiosity about infinitesimals in general and Robinson's framework in particular. While the text is addressed to a somewhat informed audience not limited to specialists in the field, an elementary introduction is provided in Section 3.Professional mathematicians and logicians curious about Robinson's framework, as well as mathematically informed philosophers are one possible (proper) subset of the intended audience, as are physicists, engineers, and economists who seem to have few inhibitions about using terms like infinitesimal and infinite number. , and (2) the Intended Interpretation hypothesis (II), entailing an identification of a standard N in its set-theoretic context, on the one hand, with ord...

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confidence: 99%

Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others

Fletcher¹,

Hrbacek²,

Kanovei³

et al. 2017

Real Analysis Exchange

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“…Thus the difference is not merely a distinct approach to infinitesimals 17 See Section 4.3 at note 30. 18 The transfer principle is a type of theorem that, depending on the context, asserts that rules, laws or procedures valid for a certain number system, still apply (i.e., are "transfered") to an extended number system. Thus, the familiar extension Q ֒→ R preserves the property of being an ordered field.…”

Section: Leibniz's Response To Varignonmentioning

confidence: 99%

Leibniz�s well-founded fictions and their interpetations

Bair

Błaszczyk

Ely³

et al. 2018

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“…Meanwhile, by postulating a so-called Cauchy-Weierstrass foundation (see Section 4.4), Fraser precisely yanks Cauchy right out of his historical milieu, and inserts him in the heroic 1870s alongside C. Boyer's great triumvirate. 9 Robinson was one of the first to express the sentiment that the Aframework is inadequate to account for Cauchy's infinitesimal mathematics. Grattan-Guinness points out that Cauchy's proof of the sum theorem is difficult to interpret in an Archimedean framework, due to Cauchy's use of infinitesimals.…”

Section: Fraser Continuesmentioning

confidence: 99%

“…This procedure bears analogy to Cauchy's representation of the B-continuum. Indeed, Cauchy gives an example of a variable quantity as a sequence at the start of his Cours d'Analyse[23] 9. Historian Carl Boyer described Cantor, Dedekind, and Weierstrass as the great triumvirate in[22, p. 298].…”

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Cauchy, infinitesimals and ghosts of departed quantifiers

Bair

Błaszczyk

Ely

et al. 2017

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. Cauchy, infinitesimals and ghosts of departed quantifiers, Mat. Stud. 47 (2017), 115-144.Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson's frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibniz's distinction between assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robinson's framework, while Leibniz's law of homogeneity with the implied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide parallel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibniz's infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Euler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infinite exponent. Such procedures have immediate hyperfinite analogues in Robinson's framework, while in a Weierstrassian framework they can only be reinterpreted by means of paraphrases departing significantly from Euler's own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinson's framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of departed quantifiers in his work. Cauchy's procedures in the context of his 1853 sum theorem (for series of continuous functions) are more readily understood from the viewpoint of Robinson's framework, where one can exploit tools such as the pointwise definition of the concept of uniform convergence.As case studies, we analyze the approaches of Craig Fraser and Jesper Lützen to Cauchy's contributions to infinitesimal analysis, as well as Fraser's approach toward Leibniz's theoretical strategy in dealing with infinitesimals. The insights by philosophers Ian Hacking and others into the important roles of contextuality and contingency tend to undermine Fraser's interpretive framework.

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Gregory’s Sixth Operation

Cited by 11 publications

References 22 publications

Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others

Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others

Leibniz�s well-founded fictions and their interpetations

Cauchy, infinitesimals and ghosts of departed quantifiers

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