On mathematical realism and applicability of hyperreals

Abstract

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.

Infinitesimals via Cauchy sequences: Refining the classical equivalence

“…7. Arguably, Robinson's framework is the most successful theory of infinitesimals in applications to natural science, probability, and related fields; see [28] as well as [29], [30].…”

Procedures of Leibnizian infinitesimal calculus: An account in three modern frameworks

“…RA appear to be aware of the distinction and even mention the term terminata three times in their article, but don't fully appreciate its significance. 26 We propose a possible formalisation of the bounded/unbounded distiction in modern mathematics in Section 4.4.…”

“…Sanders catalogs many such comments in [108, p. 459]. A rebuttal of the claims by Easwaran-Towsner and Väth appears in [26]. A rebuttal of related criticisms by Pruss [102] appears in [27], [28].…”

“…It is worth noting that bounded/unbounded is not the same distinction as potential/actual infinity. Bounded infinity is a term Leibniz reserves mainly to characterize the quantities used in his infinitesimal calculus, namely the infinitely small and infinitely large 26. Furthermore, RA misinterpret Leibnizian bounded infinities when they compare them to compactifications in modern mathematics; see Section 3.8.…”

The Mathematical Structure Of Business Success

“…I.e. in any situation there are subsets that are, as parts of a situation, not hers, and therefore, do not exist for the situation (Bottazzi et al, 2019). A regime is needed to prevent the formation of such Hausa zones.…”