British Journal for the Philosophy of Science 45 (2):389-406 (1994)
AbstractThe 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.
Similar books and articles
Quantity and quantity value.Alessandro Giordani & Luca Mari - 2011 - Proc. TC1-TC7-TC13 14th IMEKO Joint Symposium.
What do numbers measure? A new approach to fundamental measurement.Reinhard Niederée - 1992 - Mathematical Social Sciences 24:237-276.
Some measurement-theoretic concerns about Hale's ‘reals by abstraction';.Vadim Batitsky - 2002 - Philosophia Mathematica 10 (3):286-303.
Blinking fractals and their quantitative analysis using infinite and infinitesimal numbers.Yaroslav Sergeyev - 2007 - Chaos, Solitons and Fractals 33 (1):50-75.
Quantity and Quality: Some Aspects of Measurement.Arnold Koslow - 1982 - PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1982:183 - 198.
Bertrand Russell's 1897 critique of the traditional theory of measurement.Joel Michell - 1997 - Synthese 110 (2):257-276.
Added to PP
Historical graph of downloads
Citations of this work
Old and New Problems in Philosophy of Measurement.Eran Tal - 2013 - Philosophy Compass 8 (12):1159-1173.
An Aristotelian Realist Philosophy of Mathematics: Mathematics as the science of quantity and structure.James Franklin - 2014 - London and New York: Palgrave MacMillan.
Mathematics as a science of non-abstract reality: Aristotelian realist philosophies of mathematics.James Franklin - 2022 - Foundations of Science 27 (2):327-344.
References found in this work
Grundlagen der Arithmetik: Studienausgabe mit dem Text der Centenarausgabe.Gottlob Frege - 1884 - Breslau: Wilhelm Koebner Verlag.
The Nature of Mathematical Knowledge.Philip Kitcher - 1983 - Oxford, England: Oxford University Press.
Die Grundlagen der Arithmetik: Eine logische mathematische Untersuchung über den Begriff der Zahl.Gottlob Frege - 2018 - Hansebooks.
A Combinatorial Theory of Possibility.David Malet Armstrong - 1989 - Cambridge and New York: Cambridge University Press.