Abstract This chapter focuses on empirical structures, with an ordering attribute, leading to numerical or geometric representations. A historical sketch of physical, behavioral, and social science measurement until 1950 is followed by modern approaches to representational measurement in the behavioral sciences. Among these structures are: difference, additive conjoint, averaging, and non‐additive concatenation and conjoint structures. Derived measurement is described by distributive linkages between two structures with a common attribute and is exemplified by physical, utility, and psychophysical examples. Two mathematical results are needed in each case: theexistenceof structure‐preserving numerical representations and theiruniqueness. The latter has been analyzed generically. Using these ideas, models of magnitude estimation and production are summarized. The nature of axiomatization is discussed in terms of types of logical languages, the role of certain second‐order axioms, and issues of consistency and independence of axioms. Important results about the impossibility of simple axiomatizations of classes of finite structures are described along with mention of some problems.
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