Theory of Dimensions

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Sterrett, S
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This chapter concerns dimensions as the term is used in the physical sciences today. Some key points made are: (i) Quantities of the same kind have the same dimension; but that two quantities have the same dimension does not necessarily mean they are of the same kind. (ii) The dimension of a quantity is not determined for a single quantity in isolation, but relative to a system of quantities and the relations that hold between them. (iii) Dimensions, units, and quantities are distinct notions. In this article, I explain how dimensions, units, and quantities are involved in the design of coherent systems of units; the account involves the equations of physics. When the use of a coherent system of units can be presumed, dimensional analysis is a powerful logico-mathematical method for deriving equations and relations in physics, and for parameterizing equations in terms of dimensionless parameters, which allows identifying physically similar systems. The source of the information yielded by dimensional analysis is not yet well understood in philosophy of physics. This chapter aims to reveal the role of dimensions not only in applications of dimensional analysis to obtain information by involving the principle of dimensional homogeneity, but to the role of dimensions in encoding information about physical relationships in the language of dimensions, specifically via the feature of coherence of a system of units. Philosophers of mathematics and philosophers of science have been concerned to address the question of the effectiveness of mathematics in science. It is argued here that no philosophical analysis of the question of the applicability of mathematics to science is complete without including dimensions and dimensional analysis in the picture.