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 59
- Downloads 0
Lattices of coarse structures
We consider the lattice of coarse structures on a set X and study metrizable, locally finite and cellular coarse structures on X from the lattice point of view.
Fast growing meromorphic solutions of the systems of linear differential equations
Systems of linear differential equations that allow for dimension decrease are considered. Growth estimates for meromorphic vector-solutions are obtained. An essentially new feature is that there are no additional constr...
Pfluger-type theorem for functions of refined regular growth
Without a priori assumptions on zero distribution we prove that if an entire function f of noninteger order ρ has an asymptotic of the form log|f(reiθ)|=rρhf(θ)+O(rρδ(r)), E∌reiθ→∞, where h is the indicator of f, δ is a...
Nonlocal multipoint problem for an ordinary differential equations of even order involution
We study a nonlocal multipoint problem for an ordinary differential equation of even order with coefficients containing an involution operator. The spectral properties of a self-adjoint operator with boundary conditions...
Unique range sets for powers of meromorphic functions
The prime concern of the paper is to deal with the notion of the unique range set for powers of meromorphic functions. As a consequence, we show that the lower bound of URSM (URSE) and URSM-IM (URSE-IM) can be significan...