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- Joel Michell (1994). Numbers as Quantitative Relations and the Traditional Theory of Measurement. British Journal for the Philosophy of Science 45 (2):389-406.The thesis that numbers are ratios of quantities has recently been advanced by a number of philosophers. While adequate as a definition of the natural numbers, it is not clear that this view suffices for our understanding of the reals. These require continuous quantity and relative to any such quantity an infinite number of additive relations exist. Hence, for any two magnitudes of a continuous quantity there exists no unique ratio. This problem is overcome by defining ratios, and hence real numbers, as binary relations between infinite standard sequences. This definition leads smoothly into a new definition of measurement consonant with the traditional view of measurement as the discovery or estimation of numerical relations. The traditional view is further strengthened by allowing that the additive relations internal to a quantity are distinct from relations observed in the behaviour of objects manifesting quantities. In this way the traditional theory can accommodate the theory of conjoint measurement. This is worth doing because the traditional theory has one great strength lacked by its rivals: measurement statements and quantitative laws are able to be understood literally. 1 This paper was completed in 1990-1. while the author was a visiting scholar at the Irvine Research Unit in Mathematical Behavioral Sciences. University of California. Irvine. The author wishes to thank the Director. Professor R. D. Luce, for the generous provision of space and facilities within the Unit and for his critical comments upon some of the ideas expressed herein: Professor L. Narens. for his trenchant criticisms: and the University of Sydney, for granting Special Study Leave and financial assistance to make the visit possible.Reading list | Discuss | Edit | Categorize |My bibliography
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Modern philosophy of mathematics has been dominated by Platonism and nominalism, to the neglect of the Aristotelian realist option. Aristotelianism holds that mathematics studies certain real properties of the world – mathematics is neither about a disembodied world of “abstract objects”, as Platonism holds, nor it is merely a language of science, as nominalism holds. Aristotle’s theory that mathematics is the “science of quantity” is a good account of at least elementary mathematics: the ratio of two heights, for example, is a perceivable and measurable real relation
between properties of physical things, a relation that can be shared by the ratio of two weights or two time intervals. Ratios are an example of continuous quantity; discrete quantities, such as whole numbers, are also realised as relations between a heap and a unit-making universal.
For example, the relation between foliage and being-a-leaf is the number of leaves on a tree, a relation that may equal the relation between a heap of shoes and being-a-shoe. Modern higher mathematics, however, deals with some real properties that are not naturally seen as quantity, so that the “science of quantity” theory of mathematics needs supplementation. Symmetry, topology and similar structural properties are studied by mathematics, but are about pattern, structure or arrangement rather than quantity.
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| | Scholar | At my library | More options ... Extensive measurement theory is developed in terms of theratio of two elements of an arbitrary (not necessarily Archimedean) extensive structure; thisextensive ratio space is a special case of a more general structure called aratio space. Ratio spaces possess a natural family of numerical scales (r-scales) which are definable in non-representational terms; ther-scales for an extensive ratio space thus constitute a family of numerical scales (extensive r-scales) for extensive structures which are defined in a non-representational manner. This is interpreted as involving arelational theory of quantity which contrasts in certain respects with thequalitative theory of quantity implicit in standard representational extensive measurement theory. The representational properties of extensiver-scales are investigated, and found to coincide withweak extensive measurement in the sense of Holman. This provides support for the thesis (developed in a separate paper) that weak extensive measurement is a more natural model of actual physical extensive scales than is the standard model using strong extensive measurement. Finally, the present apparatus is applied to slightly simplify the existing necessary and sufficient conditions for strong extensive measurement.
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| | Other links: jstor.org | Scholar | At my library | More options ... Quantities are naturally viewed as functions, whose arguments may be construed as situations, events, objects, etc. We explore the question of the range of these functions: should it be construed as the real numbers (or some subset thereof)? This is Carnap's view. It has attractive features, specifically, what Carnap views as ontological economy. Or should the range of a quantity be a set of magnitudes? This may have been Helmholtz's view, and it, too, has attractive features. It reveals the close connection between measurement and natural law, it makes dimensional analysis intelligible, and explains the concern of scientists and engineers with units in equations. It leaves the philosophical problem of the relation between the structure of magnitudes and the structure of the reals. What explains it? And is it always the same? We will argue that on the whole, construing the values of quantities as magnitudes has some advantages, and that (as Helmholtz seems to suggest in "Numbering and Measuring from an Epistemological Viewpoint") the relation between magnitudes and real numbers can be based on foundational similarities of structure.
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| | Other links: jstor.org journals.uchicago.edu dx.doi.org | Scholar | At my library | More options ... A description is given of the quantitative-qualitative distinction for terms in theories of measurable attributes, and, adjoined to that account, a suggestion is made concerning the sense in which empirical relational systems have an empirical attribute as their topic or focus. Since this characterization of quantitative terms, relative to a partition, makes no explicit reference to numbers, concatenation operations, or ordering relations, we show how our results are related to some standard theorems in the literature. Analogs of representation and uniqueness theorems are proved, and the notions of exact quantitative term and the underlying attribute of a quantitative term, are described and studied.
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| | Scholar | At my library | More options ... The paper considers a new type of objects – blinking fractals – that are not covered by traditional theories studying dynamics of self-similarity processes. It is shown that the new approach allows one to give various quantitative characteristics of the newly introduced and traditional fractals using infinite and infinitesimal numbers proposed recently. In this connection, the problem of the mathematical modelling of continuity is discussed in detail. A strong advantage of the introduced computational paradigm consists of its well-marked numerical character and its own instrument – Infinity Computer – able to execute operations with infinite and infinitesimal numbers.
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| | Scholar | At my library | More options ... Hale proposes a neo-logicist definition of real numbers by abstraction as ratios defined on a complete ordered domain of quantities (magnitudes). I argue that Hale's definition faces insuperable epistemological and ontological difficulties. On the epistemological side, Hale is committed to an explanation of measurement applications of reals which conflicts with several theorems in measurement theory. On the ontological side, Hale commits himself to the necessary and a priori existence of at least one complete ordered domain of quantities, which is extremely implausible because science treats the logical structure of quantities as subject to experimentally and theoretically motivated refinements and revisions.
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| | Scholar | At my library | More options ... This paper presents a novel semantic analysis of unit names (like pound and meter ) and gradable adjectives (like tall, short and happy ), inspired by measurement theory (Krantz et al. In Foundations of measurement: Additive and Polynomial Representations, 1971). Based on measurement theory’s four-way typology of measures, I claim that different adjectives are associated with different types of measures whose special characteristics, together with features of the relations denoted by unit names, explain the puzzling limited distribution of measure phrases, as well as unit-based comparisons between predicates (as in the table is longer than it is wide ). All considered, my analyses support the view that the grammar of natural languages is sensitive to features of measurement theory.
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| | Scholar | At my library | More options ... Unlike the standard representational theory of measurement, which takes the real numbers as a pregiven numerical domain, the approach presented in this paper is based on an abstract concept of a procedure of measurement, and ‘values of measurement’ are understood in terms of such procedures. The resulting ‘type approach’ makes use of elementary model-theoretic notions and emphasizes the constructibility of scales. It provides a natural starting point for a systematic discussion of issues that tend to be neglected in the standard framework (such as the relation between measurement and computation). At the same time it is perfectly compatible with the modern representational theory of measurement and helps elucidate a number of issues central to that theory (e.g. the role of Archimedean axioms).
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| | Scholar | At my library | More options ... The concept system around ‘quantity’ and ‘quantity value’ is fundamental for measurement science, but some very basic issues are still open on such concepts and their relations. This paper proposes a duality between quantities and quantity values, a proposal that simplifies their characterization and makes it consistent.
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| | Scholar | At my library | More options ... The transition from the traditional to the representational theory of measurement around the turn of the century was accompanied by little sustained criticism of the former. The most forceful critique was Bertrand Russell''s 1897 Mind paper, On the relations of number and quantity. The traditional theory has it that real numbers unfold from the concept of continuous quantity. Russell''s critique identified two serious problems for this theory: (1) can magnitudes of a continuous quantity be defined without infinite regress; and (2) can additive relations between such magnitudes exist if magnitudes are not divisible? The present paper shows how the traditional theory answers these questions and compares the traditional and representational theories as contributions to our understanding of the logic of application.
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| | Other links: springerlink.com dx.doi.org jstor.org | Scholar | At my library | More options ... Discussion of Joel Michell, Numbers as quantitative relations and the traditional theory of measurement
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