To be or not to be constructive, that is not the question

Sanders, Sam

doi:10.1016/j.indag.2017.05.005

Cited by 8 publications

(6 citation statements)

References 44 publications

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“…We stress that G 2 in SFF(Θ) may be discontinuous and that Kohlenbach has argued for the study of discontinuous functionals in higher-order RM (see Section 2.2). As it turns out, Θ-functionals are intimately connected to Tao's notion of metastability, as explored in [57].…”

Section: The Special and Intuitionistic Fan Functionalsmentioning

confidence: 99%

Computability Theory, Nonstandard Analysis, and Their Connections

Normann¹,

Sanders²

2019

J. symb. log.

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We investigate the connections between computability theory and Nonstandard Analysis. In particular, we investigate the two following topics and show that they are intimately related.(T.1) A basic property of Cantor space$2^ $ is Heine–Borel compactness: for any open covering of $2^ $, there is a finite subcovering. A natural question is: How hard is it to compute such a finite subcovering? We make this precise by analysing the complexity of so-called fan functionals that given any $G:2^ \to $, output a finite sequence $\langle f_0 , \ldots ,f_n \rangle $ in $2^ $ such that the neighbourhoods defined from $\overline {f_i } G\left( {f_i } \right)$ for $i \le n$ form a covering of $2^ $.(T.2) A basic property of Cantor space in Nonstandard Analysis is Abraham Robinson’s nonstandard compactness, i.e., that every binary sequence is “infinitely close” to a standard binary sequence. We analyse the strength of this nonstandard compactness property of Cantor space, compared to the other axioms of Nonstandard Analysis and usual mathematics.Our study of (T.1) yields exotic objects in computability theory, while (T.2) leads to surprising results in Reverse Mathematics. We stress that (T.1) and (T.2) are highly intertwined, i.e., our study is holistic in nature in that results in computability theory yield results in Nonstandard Analysis and vice versa.

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Section: The Special and Intuitionistic Fan Functionalsmentioning

confidence: 99%

Computability Theory, Nonstandard Analysis, and Their Connections

Normann¹,

Sanders²

2019

J. symb. log.

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“…7 Recently the Bishop-Connes critique has been challenged. Namely, the presence of ideal objects (in particular infinitesimals) in Robinson's framework arguably yields the ubiquitous presence of computational content; see e.g., [Sanders 2017], [Sanders 2018].…”

Section: Reception and Meaningmentioning

confidence: 99%

A footnote to The crisis in contemporary mathematics

Katz¹,

Katz

Sanders³

2018

Historia Mathematica

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We examine the preparation and context of the paper "The Crisis in Contemporary Mathematics" by Errett Bishop, published 1975 in Historia Mathematica. Bishop tried to moderate the differences between Hilbert and Brouwer with respect to the interpretation of logical connectives and quantifiers. He also commented on Robinson's Non-standard Analysis, fearing that it might lead to what he referred to as 'a debasement of meaning.' The 'debasement' comment can already be found in a draft version of Bishop's lecture, but not in the audio file of the actual lecture of 1974. We elucidate the context of the 'debasement' comment and its relation to Bishop's position vis-a-vis the Law of Excluded Middle.

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“…For an analysis of further constructive and algorithmic aspects of the hyperreal framework see Sanders [83], [84]. [85]. 6.6.…”

Section: Does Calculus Become Algorithmic Over the Hyperreals?mentioning

confidence: 99%

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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To be or not to be constructive, that is not the question

Cited by 8 publications

References 44 publications

Computability Theory, Nonstandard Analysis, and Their Connections

Computability Theory, Nonstandard Analysis, and Their Connections

A footnote to The crisis in contemporary mathematics

Cauchy, infinitesimals and ghosts of departed quantifiers

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