On mathematical realism and applicability of hyperreals
Journal Title: Математичні Студії - Year 2019, Vol 51, Issue 2
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.
Authors and Affiliations
E. Bottazzi, V. Kanovei, M. G. Katz, T. Mormann, D. Sherry
Keywords
Related Articles
- EP ID EP609521
- DOI 10.15330/ms.51.2.200-224
- Views 76
- Downloads 0
Review of the book by Mariusz Urbanek, “Genialni – Lwowska Szko la Matematyczna"( Polish) [Geniuses – the Lvov school of mathematics], Wydawnictwo Iskry, Warsaw 2014, 283 pp. ISBN: 978-83-244-0381-3
This review is an extended version of my short review of Urbanek's book that was published in MathSciNet. Here it is written about his book in greater detail, which was not possible in the short review. I will present fa...
Problem of determining of minor coefficient and right-hand side function in semilinear ultraparabolic equation
The problem of determining of the right-hand side function and the time depended minor coefficient in semilinear ultraparabolic equation from the initial, boundary and overdetermination conditions, is considered in this...
Hahn’s pairs and zero inverse problem
We prove that for a function α0:[0,1]→R there exists a separately continuous function f:[0,1]2→R such that E0(fx)=α0(x) on [0,1] if and only if α0 is the nonnegative lower semicontinuous function, where fx(y)=f(x,y) for...
The Pfeiffer-Lax-Sato type vector field equations and the related integrable versal deformations
We study versal deformatiions of the Pfeiffer-Lax-Sato type vector field equations, related with a centrally extended metrized Lie algebra as the direct sum of vector fields and differential forms on torus.
An alternative look at the structure of graph inverse semigroups
For any graph inverse semigroup G(E) we describe subsemigroups D0=D∪{0} and J0=J∪{0} of G(E) where D and J are arbitrary D-class and J-class of G(E), respectively. In particular, we prove that for each D-class D of a gra...