Additive and Polynomial 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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a o b A X A additive conjoint structure additive representation algebra algebra of sets Archimedean axiom Archimedean property assume assumption axiomatization axioms of Deﬁnition binary operation binary relation Chapter components concatenation conditional decision constant construct countable deﬁned denote dimensional analysis dimensionally invariant dimensions elements empirical Equation equivalence classes essential maximum example exists extensive measurement extensive structure ﬁnd ﬁnite ﬁrst ﬁxed formulate gambles hence holds hypothesis iﬁ implies independent indifference curves inequalities inﬁnite integer interval scale laws Lemma linear monotonicity multiplicative nonempty obtain pair physical quantities polynomial positive problem proof of Theorem prove qualitative probability ratio scale real numbers real-valued function representation theorem restricted solvability result satisﬁes Section semigroup sign dependence similar solution Speciﬁcally standard sequence strictly increasing subset sufﬁcient Suppose Theorem theory transformations variables vector velocity weak order yields