ON MATHEMATICAL REALISM AND THE APPLICABILITY OF HYPERREALS

EMANUELE BOTTAZZI, VLADIMIR KANOVEI, MIKHAIL G. KATZ,

THOMAS MORMANN, AND DAVID SHERRY

【摘要】我们认为Robinson的超实数和普通实数一样具有适用性。

在最近的一篇文章中，Easwaran和Towsner(ET)分析了数学技术在科学中的适用性，并介绍了可应用的技术和仅仅是基本的技术之间的区别。不幸的是，作者没有表明它们的区别是明确和富有成果的，因为他们提供的例子是肤浅和没有说服力的。此外，他们的分析由于依赖于一种朴素的对象实在论而变得苍白无力，而这种实在论早已被大多数哲学现实主义者所抛弃，取而代之的是真实价值实在论。ET反对基于超实模型自同构的超实数的适用性的论点涉及到篡改证据，同样也没有说服力。ET文本的目的是论证罗宾逊的无穷小分析仅仅是工具性的，而不是适用性的。

然而,在尽管Robinson的技术被应用于物理学、概率论和经济学中(参见[70,Chapter IX]，[1]，[76]，[60])，但ET没有对这些技术使用领域中的任何一个案例给出有意义的分析。相反，ET写了一页又一页的推测，主要模仿的是Connesian反对Robinson的嵌合体类型 “来自第一原则” 的论点。Easwaran在一篇早期论文中支持了测度的σ-可加性的实用性，然而ET的文章反对选择公理的实用性，而是更支持ZF。由于勒贝格测度不是σ-可加的观点是和ZF一致的，Easwaran因此重新回到了他早期的支持观点。我们在Paul Halmos的教科书《测度论》中发现了一个相关的错误。ET的论点对数学家来说是不可接受的，因为他们忽略了无穷小在科学中的大量应用，并篡改一些关键数学细节的证据，以符合他们的哲学结论。

关键词：对象实在论，真实价值实在论，适用性，超现实，无穷小量，工具主义，严格性，自同构，Lotka-Volterra模型，勒贝格测度，σ-可加性

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 intro- duce 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 real- ism 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 infinitesmal 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 σ-additivity of measures, whereas the ET text rejects real applicability of the ax- iom 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 con- form with their philosophical conclusions.

Keywords: object realism; truth-value realism; applicability; hyperreals; infinitesimals; instrumentalism; rigidity; automorphisms; Lotka–Volterra model; Lebesgue measure; σ-additivity

http://u.math.biu.ac.il/~katzmik/realismreprint19b.pdf​u.cs.biu.ac.ilhttps://arxiv.org/abs/1907.07040​arxiv.orgMatches for: MR=3988243​mathscinet.ams.org