  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, 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.
