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**1 - 2**of**2**### MEASUREMENT-THEORETIC OBSERVATIONS ON FIELD’S INSTRUMENTALISM AND THE APPLICABILITY OF MATHEMATICS

"... In this paper I examine Field’s account of the applicability of mathematics from a measurementtheoretic perspective. Within this context, I object to Field’s instrumentalism, arguing that it depends on an incomplete analysis of applicability. I show in particular that, once the missing piece of anal ..."

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In this paper I examine Field’s account of the applicability of mathematics from a measurementtheoretic perspective. Within this context, I object to Field’s instrumentalism, arguing that it depends on an incomplete analysis of applicability. I show in particular that, once the missing piece of analysis is provided, the role played by numerical entities in basic empirical theories must be revised: such revision implies that instrumentalism should be rejected and mathematical entities be regarded not merely as useful tools but also as conceptual schemata by means of which we can articulate our understanding of experience.

### On a bounded version of Hölder’s Theorem and its application to the permutability equation

, 2014

"... This chapter is dedicated to Patrick Suppes, whose works and counsel have shaped much of my scientific life. The permutability equation G(G(x, y), z) = G(G(x, z), y) is satisfied by many scientific and geometric laws. A few examples among many are: The Lorentz-FitzGerald Contraction, Beer’s Law, th ..."

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This chapter is dedicated to Patrick Suppes, whose works and counsel have shaped much of my scientific life. The permutability equation G(G(x, y), z) = G(G(x, z), y) is satisfied by many scientific and geometric laws. A few examples among many are: The Lorentz-FitzGerald Contraction, Beer’s Law, the Pythagorean Theorem, and the formula for computing the volume of a cylinder. We prove here a representation theorem for the permutability equation, which generalizes a well-known result. The proof is based on a bounded version of Hölder’s Theorem. Holder’s Theorem on ordered groups is a foundation stone of measurement theory (c.f. Krantz et al., 1971; Suppes et al., 1989; Luce et al., 1990), and so, of much of quantitative science. There are several renditions of it. Whatever the version, the theorem concerns an algebraic structure (X, ◦,-), in which X is a set, ◦ is an operation on X, and- is a weak order on X (transitive, connected), which may be a simple order (antisymmetric). The axioms imply the existence of a function f: X → R such that x- y ⇐ ⇒ f(x) ≤ f(y) f(x ◦ y) = f(x) + f(y) (whenever x ◦ y is defined). Most formulations of this theorem have one or both of two drawbacks. Hypothesis 1. The elements of X can be arbitrarily large. Hypothesis 2. The elements of X can be arbitrarily small. From the standpoint of social sciences applications, both of these hypotheses are un-warranted because the sensory mechanisms of humans and animals restrict the range of usable stimuli. In psychophysics, for example, small stimuli are undetectable by the sensory mechanisms, and large ones would damage them. Even in physics (relativity) the hypothesis that infinitely large quantities exist is inconsistent with current theories. In the axiomatization of Luce and Marley (1969) (see also Krantz et al., 1971, page 84), arbitrarily large elements need not exist. However, they use the following solvability axiom, which essentially asserts the existence of arbitrarily small elements. If x ≺ y, then there is some z such that x ◦ z- y. ar