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

Abstract

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...

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

“…B) be the formula in square brackets in (4.34) (resp. (4.37)) and note that the existence of standard µ 2 as in (4.34) implies Π 38) and bringing outside the standard quantifiers, we obtain 39) which is a normal form as the formula in square brackets is internal. Now applying Theorem 4.2 to 'P ⊢ (4.39)' yields a term t such that E-PA ω proves (∀x…”

“…33 As noted above and discussed at length in [39], one does not have to share Nelson's view to study IST: the latter is a logical system while the former is a possible philosophical point of view regarding IST. However, like Nelson, we view the mathematics and this point of view as one.…”

On mathematical realism and applicability of hyperreals

“…In Nelson's framework, one works within the ordinary real line and finds numbers that behave like infinitesimals there via a foundational adjustment. Such an adjustment involves an enrichment of the language of set theory through the introduction of a oneplace predicate st called "standard," together with additional axioms governing its interaction with the axioms of Zermelo-Fraenkel set theory (ZFC); see Fletcher et al [38], Katz-Kutateladze [52], Lawler [58] for details.…”

Leibniz�s well-founded fictions and their interpetations

“…. [82] 38 Leibniz's reference to Archimedes in various texts typically is a reference to the method of exhaustion, and is sometimes accompanied by a claim that infinitesimals violate Euclid Definition V.4 (see e.g., Section 3.3). The latter is called today the Archimedean property (but was not in Leibniz's time).…”