Absolute Measurable Spaces (Encyclopedia of Mathematics and its Applications)

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A comprehensive monograph presenting a unified systematic exposition of the large deviations theory Mathematical tools for the study of generalisations of graphs appearing in the modelling of molecular structures. Mathematical tools for the study of generalisations of graphs appearing in the modelling of Emphasizes topological, geometrical and analytical properties of absolute measurable spaces; of interest for real analysis, set theory and measure theory.

Emphasizes topological, geometrical and analytical properties of absolute measurable spaces; of A unified, coherent account of the algebraic aspects and uses of the Ziegler spectrum. Full account of Euler's work on continued fractions and orthogonal polynomials; illustrates the significance of his work on mathematics today. Full account of Euler's work on continued fractions and orthogonal polynomials; illustrates the Toggle navigation.

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Absolute Measurable Spaces (Encyclopedia of Mathematics and its Applications)

From these assumptions, or axioms, the authors of RTM derive the representational adequacy of each scale type, as well as the family of permissible transformations making that type of scale unique. In this way RTM provides a conceptual link between the empirical basis of measurement and the typology of scales. On the issue of measurability, the Representational Theory takes a middle path between the liberal approach adopted by Stevens and the strict emphasis on concatenation operations espoused by Campbell.

Like Campbell, RTM accepts that rules of quantification must be grounded in known empirical structures and should not be chosen arbitrarily to fit the data.

However, RTM rejects the idea that additive scales are adequate only when concatenation operations are available Luce and Suppes Instead, RTM argues for the existence of fundamental measurement operations that do not involve concatenation. Here, measurements of two or more different types of attribute, such as the temperature and pressure of a gas, are obtained by observing their joint effect, such as the volume of the gas. Luce and Tukey showed that by establishing certain qualitative relations among volumes under variations of temperature and pressure, one can construct additive representations of temperature and pressure, without invoking any antecedent method of measuring volume.

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This sort of procedure is generalizable to any suitably related triplet of attributes, such as the loudness, intensity and frequency of pure tones, or the preference for a reward, it size and the delay in receiving it Luce and Suppes Under this new conception of fundamentality, all the traditional physical attributes can be measured fundamentally, as well as many psychological attributes Krantz et al. Above we saw that mathematical theories of measurement are primarily concerned with the mathematical properties of measurement scales and the conditions of their application.

A related but distinct strand of scholarship concerns the meaning and use of quantity terms. A realist about one of these terms would argue that it refers to a set of properties or relations that exist independently of being measured. An operationalist or conventionalist would argue that the way such quantity-terms apply to concrete particulars depends on nontrivial choices made by humans, and specifically on choices that have to do with the way the relevant quantity is measured. Note that under this broad construal, realism is compatible with operationalism and conventionalism. That is, it is conceivable that choices of measurement method regulate the use of a quantity-term and that, given the correct choice, this term succeeds in referring to a mind-independent property or relation.

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Nonetheless, many operationalists and conventionalists adopted stronger views, according to which there are no facts of the matter as to which of several and nontrivially different operations is correct for applying a given quantity-term. These stronger variants are inconsistent with realism about measurement. This section will be dedicated to operationalism and conventionalism, and the next to realism about measurement.

The strongest expression of operationalism appears in the early work of Percy Bridgman , who argued that. Length, for example, would be defined as the result of the operation of concatenating rigid rods. According to this extreme version of operationalism, different operations measure different quantities. Nevertheless, Bridgman conceded that as long as the results of different operations agree within experimental error it is pragmatically justified to label the corresponding quantities with the same name Operationalism became influential in psychology, where it was well-received by behaviorists like Edwin Boring and B.

Skinner As long as the assignment of numbers to objects is performed in accordance with concrete and consistent rules, Stevens maintained that such assignment has empirical meaning and does not need to satisfy any additional constraints. Nonetheless, Stevens probably did not embrace an anti-realist view about psychological attributes.

Instead, there are good reasons to think that he understood operationalism as a methodological attitude that was valuable to the extent that it allowed psychologists to justify the conclusions they drew from experiments Feest For example, Stevens did not treat operational definitions as a priori but as amenable to improvement in light of empirical discoveries, implying that he took psychological attributes to exist independently of such definitions Stevens Operationalism met with initial enthusiasm by logical positivists, who viewed it as akin to verificationism.

Nonetheless, it was soon revealed that any attempt to base a theory of meaning on operationalist principles was riddled with problems. Among such problems were the automatic reliability operationalism conferred on measurement operations, the ambiguities surrounding the notion of operation, the overly restrictive operational criterion of meaningfulness, and the fact that many useful theoretical concepts lack clear operational definitions Chang Accordingly, most writers on the semantics of quantity-terms have avoided espousing an operational analysis.

A more widely advocated approach admitted a conventional element to the use of quantity-terms, while resisting attempts to reduce the meaning of quantity terms to measurement operations. Mach noted that different types of thermometric fluid expand at different and nonlinearly related rates when heated, raising the question: which fluid expands most uniformly with temperature?

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According to Mach, there is no fact of the matter as to which fluid expands more uniformly, since the very notion of equality among temperature intervals has no determinate application prior to a conventional choice of standard thermometric fluid. Conventionalism with respect to measurement reached its most sophisticated expression in logical positivism.

These a priori , definition-like statements were intended to regulate the use of theoretical terms by connecting them with empirical procedures Reichenbach 14—19; Carnap Ch. In accordance with verificationism, statements that are unverifiable are neither true nor false. Instead, Reichenbach took this statement to expresses an arbitrary rule for regulating the use of the concept of equality of length, namely, for determining whether particular instances of length are equal Reichenbach At the same time, coordinative definitions were not seen as replacements, but rather as necessary additions, to the familiar sort of theoretical definitions of concepts in terms of other concepts Under the conventionalist viewpoint, then, the specification of measurement operations did not exhaust the meaning of concepts such as length or length-equality, thereby avoiding many of the problems associated with operationalism.

Realists about measurement maintain that measurement is best understood as the empirical estimation of an objective property or relation.

A few clarificatory remarks are in order with respect to this characterization of measurement. Rather, measurable properties or relations are taken to be objective inasmuch as they are independent of the beliefs and conventions of the humans performing the measurement and of the methods used for measuring. For example, a realist would argue that the ratio of the length of a given solid rod to the standard meter has an objective value regardless of whether and how it is measured.

Third, according to realists, measurement is aimed at obtaining knowledge about properties and relations, rather than at assigning values directly to individual objects. This is significant because observable objects e.

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Knowledge claims about such properties and relations must presuppose some background theory. By shifting the emphasis from objects to properties and relations, realists highlight the theory-laden character of measurements. Realism about measurement should not be confused with realism about entities e. Nor does realism about measurement necessarily entail realism about properties e.

Nonetheless, most philosophers who have defended realism about measurement have done so by arguing for some form of realism about properties Byerly and Lazara ; Swoyer ; Mundy ; Trout , These realists argue that at least some measurable properties exist independently of the beliefs and conventions of the humans who measure them, and that the existence and structure of these properties provides the best explanation for key features of measurement, including the usefulness of numbers in expressing measurement results and the reliability of measuring instruments. The existence of an extensive property structure means that lengths share much of their structure with the positive real numbers, and this explains the usefulness of the positive reals in representing lengths.

Moreover, if measurable properties are analyzed in dispositional terms, it becomes easy to explain why some measuring instruments are reliable. A different argument for realism about measurement is due to Joel Michell , , who proposes a realist theory of number based on the Euclidean concept of ratio. According to Michell, numbers are ratios between quantities, and therefore exist in space and time. Specifically, real numbers are ratios between pairs of infinite standard sequences, e. Measurement is the discovery and estimation of such ratios.

An interesting consequence of this empirical realism about numbers is that measurement is not a representational activity, but rather the activity of approximating mind-independent numbers Michell Realist accounts of measurement are largely formulated in opposition to strong versions of operationalism and conventionalism, which dominated philosophical discussions of measurement from the s until the s.

In addition to the drawbacks of operationalism already discussed in the previous section, realists point out that anti-realism about measurable quantities fails to make sense of scientific practice. By contrast, realists can easily make sense of the notions of accuracy and error in terms of the distance between real and measured values Byerly and Lazara 17—8; Swoyer ; Trout A closely related point is the fact that newer measurement procedures tend to improve on the accuracy of older ones. If choices of measurement procedure were merely conventional it would be difficult to make sense of such progress.

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In addition, realism provides an intuitive explanation for why different measurement procedures often yield similar results, namely, because they are sensitive to the same facts Swoyer ; Trout Finally, realists note that the construction of measurement apparatus and the analysis of measurement results are guided by theoretical assumptions concerning causal relationships among quantities. The ability of such causal assumptions to guide measurement suggests that quantities are ontologically prior to the procedures that measure them. While their stance towards operationalism and conventionalism is largely critical, realists are more charitable in their assessment of mathematical theories of measurement.

Brent Mundy and Chris Swoyer both accept the axiomatic treatment of measurement scales, but object to the empiricist interpretation given to the axioms by prominent measurement theorists like Campbell and Ernest Nagel ; Cohen and Nagel Ch.