From Pythagoreans and Weierstrassians to True Infinitesimal Calculus

Katz, Mikhail G.; Polev, Luie

doi:10.5642/jhummath.201701.07

Cited by 12 publications

(15 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Accordingly, over the past few years we have trained over 400 freshmen using Keisler's infinitesimal calculus textbook [107], and summarized the results in the study Katz-Polev [99]. At the high school level infinitesimal calculus has been taught in Geneva for the past twelve years, based on the approach developed in [81].…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

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

Fletcher¹,

Hrbacek²,

Kanovei³

et al. 2017

Real Analysis Exchange

View full text Add to dashboard Cite

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...

show abstract

Section: Resultsmentioning

confidence: 99%

“…It is often thought worthwhile (for example, for pedagogical reasons; see [99]) to develop the subject purely from general principles that make the nonstandard arguments succeed. This approach is similar to, say, deriving results from the axioms for algebraically closed fields rather than arguing about these mathematical structures directly.…”

Section: 4mentioning

confidence: 99%

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

Fletcher¹,

Hrbacek²,

Kanovei³

et al. 2017

Real Analysis Exchange

View full text Add to dashboard Cite

show abstract

“…[W]hile the hyperreals are meaningful objects worthy of their own study, there are other contexts (particularly dealing with real analysis, and the physical theories using it) where they are mere tools for avoiding some complexity in dealing with quantifier structure. [33, p. 12] In mathematical pedagogy particularly at freshman level, one of the main advantages of the hyperreals is providing a simplification of such "quantifier structure" (see e.g., [54]). However, ET's reference to physical theories makes it clear that they are not limiting their sweeping "mere tool" claim echoing Connes (see [50]) to pedagogy.…”

Section: Quantifier Structure Et Writementioning

confidence: 99%

On mathematical realism and applicability of hyperreals

Bottazzi¹,

Kanovei²,

Katz³

et al. 2019

View full text Add to dashboard Cite

We argue that Robinson's hyperreals have just as much claim to applicability as the garden variety reals.In a recent text, Easwaran and Towsner (ET) analyze the applicability of mathematical techniques in the sciences, and introduce a distinction between techniques that are applicable and those that are merely instrumental. Unfortunately the authors have not shown that their distinction is a clear and fruitful one, as the examples they provide are superficial and unconvincing. Moreover, their analysis is vitiated by a reliance on a naive version of object realism which has long been abandoned by most philosophical realists in favor of truth-value realism. ET's argument against the applicability of hyperreals based on automorphisms of hyperreal models involves massaging the evidence and is similarly unconvincing. The purpose of the ET text is to argue that Robinson's infinitesimal analysis is merely instrumental rather than applicable. Yet in spite of Robinson's techniques being applied in physics, probability, and economics (see e.g., [70, Chapter IX], [1], [76], [60]), ET don't bother to provide a meaningful analysis of even a single case in which these techniques are used. Instead, ET produce page after page of speculations mainly imitating Connesian chimera-type arguments 'from first principles' against Robinson. In an earlier paper Easwaran endorsed real applicability of the σ-additivity of measures, whereas the ET text rejects real applicability of the axiom of choice, voicing a preference for ZF. Since it is consistent with ZF that the Lebesgue measure is not σ-additive, Easwaran is thereby walking back his earlier endorsement. We note a related inaccuracy in the textbook Measure Theory by Paul Halmos. ET's arguments are unacceptable to mathematicians because they ignore a large body of applications of infinitesimals in science, and massage the evidence of some crucial mathematical details to conform with their philosophical conclusions.

show abstract

“…10 Hyperreals for Limit Analysis ǫ-δ proofs have long been the bane of calculus students. While they do present an interesting mathematical technique, the retention rate for understanding ǫ-δ proofs is so low as to hardly be worth doing (Katz and Polev, 2017). Instead, the method which helps students understand the process of limits the most is to use the hyperreal number line.…”

Section: Apply the Rulementioning

confidence: 99%