Michael Miller

It has been argued that the quantity valuation scheme invoked by spontaneous collapse interpretations of quantum mechanics has the unacceptable consequence of rendering violations of arithmetic, and of conjunction introduction more generally, physically possible. We argue that what is ultimately at issue in this argument and the debate that followed has nothing to do with spontaneous collapse theories, their quantity valuation schemes, nor arithmetic. Instead, we show that the argument in question hinges on an implausible assumption about the relation between the semantics of conjunction and the representation of quantum-mechanical quantities. We argue that this episode holds important lessons for how to treat the semantics of Hilbert space theories in particular, and the semantics of physical theories in general.

The representational theory of measurement provides a collection of results that specify the conditions under which an attribute admits of numerical representation. The original architects of the theory interpreted the formalism operationally and explicitly acknowledged that some aspects of their representations are conventional. There have been a number of recent efforts to reinterpret the formalism to arrive at a more metaphysically robust account of physical quantities. In this paper we argue that the conventional elements of the representations afforded by the representational theory of measurement require careful scrutiny as one moves toward such an interpretation. To illustrate why, we show that there is a sense in which the very number system in which one represents a physical quantity such as mass or length is conventional. We argue that this result does not undermine the project of reinterpreting the representational theory of measurement for metaphysical purposes in general, but it does undermine a certain class of inferences about the nature of physical quantities that some have been tempted to draw.