On mathematical realism and applicability of hyperreals 

Author 
emanuele.bottazzi@unipv.it^{1}, kanovei@googlemail.com^{2}, katzmik@macs.biu.ac.il^{3}, thomasarnold.mormann@ehu.eus^{4}, David.Sherry@nau.edu^{5}
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

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 truthvalue 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 chimeratype 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.

Keywords 
object realism; truthvalue realism; applicability; hyperreals; infinitesimals; instrumentalism; rigidity;
automorphism; Lotka–Volterra model; Lebesgue measure; $\sigma$additivity

DOI 
doi:10.15330/ms.51.2.200224

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Pages 
200224

Volume 
51

Issue 
2

Year 
2019

Journal 
Matematychni Studii

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