Foundations of Measurement: Geometrical, threshold, and probabilistic representations
All of the sciences―physical, biological, and social―have a need for quantitative measurement. This influential series, Foundations of Measurement, established the formal foundations for measurement, justifying the assignment of numbers to objects in terms of their structural correspondence.
Volume I introduces the distinct mathematical results that serve to formulate numerical representations of qualitative structures. Volume II extends the subject in the direction of geometrical, threshold, and probabilistic representations, and Volume III examines representation as expressed in axiomatization and invariance.
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absolute geometry additive segments affine aﬂine alternatives assume asymmetric automorphism axiomatic axioms binary relation Chapter choice probabilities codes collinear color concept conﬁgurations congruence construction convex convex cone coordinates curve deﬁned Deﬁnition denote dimension distance EBA model elements elliptic elliptic geometry empirical Equation equilibrium structure equivalence Euclidean geometry example exists Falmagne ﬁeld ﬁnite ﬁrst ﬁxed graph Grassmann structure hence homogeneous hyperbolic identiﬁcation iﬁ iﬂ implies independent inequality interval order Lemma linear mapping match metameric metric space monotonic nonempty one-dimensional ordered geometry pair perception points projective geometry projective plane proof properties proximity structure qualitative random variables random-utility model real numbers real-valued function representation theorem satisﬁes satisfy scalar product scale Section semiorder sequence solvability speciﬁc stimuli strictly increasing subset Suppose ternary theory three-dimensional tion transformation triangle triangle inequality Tversky unique vector space viewing conditions visual space weak order