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The question concerning the applicability of mathematics is one of the focal points of interest in the philosophy of mathematics. One of the areas where the mathematical models are considered as descriptively inadequate is the science of human behavior. The so-called rational choice theory for example, is usually considered as a purely normative theory which poses insurmountable difficulties when applied to real world situations. A collection of counterexamples that aim to show this were introduced by Amartya Sen. He uncovers the difficulties that arise from the classical idea that rational behavior reduces to maximization of an acyclic ordering. The choices that people make and which we treat as rational in the intuitive sense of this word are contextually dependent and are affected by the presence of alternatives, which bear information about the nature of the objects of choice and the choice situation itself. The paper offers a formal reconstruction of the functioning of these alternatives – as “flags” that signal the transition from one ordering to another, that mark the conceptualization of alternatives as chances or as risks. This gives rise to an interesting mathematical structure, related to two of the key concepts of order theory (ideal and filter). This structure can be presented axiomatically by means of simple axioms. This shows, in accord with the position of David Hilbert, that there are no areas where mathematics is inapplicable, just fields of research where it is still not appropriately applied.
Keywords: applicability of mathematics, behavioral sciences, choice operators, rational choice, revealed preferences.
Summary:The paper sketches a peculiar view of Wittgenstein's philosophical legacy after the publication of the Tractatus – as an (inescapably doomed) attempt to construct some form of philosophical grammar which complements the purely negative doctrine of the early Wittgenstein.
Keywords: Wittgenstein, Tractatus, silence, grammer