On mathematical realism and applicability of hyperreals
email@example.com, firstname.lastname@example.org, email@example.com, firstname.lastname@example.org, David.Sherry@nau.edu5
1) Department of Civil Engineering and Architecture, the University of Pavia, Italy; 2) IPPI RAS, Moscow, Russia; 3) Department of Mathematics, Bar Ilan University, Ramat Gan, Israel; 4) Univ. of the Basque Country UPV/EHU, Donostia San Sebastian, Spain; 5) Department of Philosophy, Northern Arizona University, Flagstaff, USA
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], , , ), 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 $\sigma$-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 $\sigma$-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.
object realism; truth-value realism; applicability; hyperreals; infinitesimals; instrumentalism; rigidity; automorphism; Lotka–Volterra model; Lebesgue measure; $\sigma$-additivity
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