Abstract
A formal theory of quantity T Q is presented which is realist, Platonist, and syntactically second-order (while logically elementary), in contrast with the existing formal theories of quantity developed within the theory of measurement, which are empiricist, nominalist, and syntactically first-order (while logically non-elementary). T Q is shown to be formally and empirically adequate as a theory of quantity, and is argued to be scientifically superior to the existing first-order theories of quantity in that it does not depend upon empirically unsupported assumptions concerning existence of physical objects (e.g. that any two actual objects have an actual sum). The theory T Q supports and illustrates a form of naturalistic Platonism, for which claims concerning the existence and properties of universals form part of natural science, and the distinction between accidental generalizations and laws of nature has a basis in the second-order structure of the world.
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Bibliography
Armstrong, D.: 1978, Universals and Scientific Realism (Cambridge), two volumes.
Armstrong, D.: 1983, What is a Law of Nature? (Cambridge).
Boyer, C.: 1949, The Concepts of the Calculus (Hafner); reprinted as The History of the Calculus and Its Conceptual Development (Dover, New York, 1959).
Byerly, H.: 1974, ‘Realist foundations of measurement’, in Schaffner, K., and Cohen (eds.), PSA 1972 (D. Reidel, Dordrecht, Holland), pp. 375–384.
Byerly, H., and Lazara, V.: 1973, ‘Realist foundations of measurement’, Philosophy of Science 40, pp. 10–28 (this paper is an expanded version of the preceding one).
Campbell, N.: 1920, Physics: The Elements (Cambridge); reprinted by Dover as Foundations of Science (1957).
Campbell, N.: 1928, Principles of Measurement and Calculation (Longmans Green, London).
Carnap, R.: 1966, Philosophical Foundations of Physics (Basic Books, New York); reprinted as An Introduction to the Philosophy of Science (Basic Books, New York, 1974).
Colonius, H.: 1978, ‘On weak extensive measurement’, Philosophy of Science 45, pp. 303–308.
Dretske, F.: 1977, ‘Laws of nature’, Philosophy of Science 44, pp. 248–68.
Ellis, B.: 1966, Basic Concepts of Measurement (Cambridge).
Field, H.: 1980, Science Without Numbers (Princeton University Press, Princeton).
Glymour, C.: 1980, Theory and Evidence (Princeton University Press, Princeton, N.J.).
Helmholtz, H.: 1887, ‘Numbering and measuring from an epistemological viewpoint’, English translation in Helmholtz, H., Epistemological Writings (D. Reidel, Dordrecht, 1977).
Henkin, L.: 1950, ‘Completeness in the theory of types’, Journal of Symbolic Logic 15, pp. 81–91; reprinted in Hintikka, J. (ed.), The Philosophy of Mathematics (Oxford, 1969), pp. 51–63.
Hölder, O.: 1901, ‘Die Axiome der Quantität und die Lehre vom Mass’, Berichte über die Verhandlungen der Königliche Sachsischen Gesellschaft der Wissenschaften zu Leipzig, Mat.-Phys. Klasse 53, pp. 1–64.
Holman, E. W.: 1969, ‘Strong and weak extensive measurement’, Journal of Mathematical Psychology 6, pp. 286–293.
Krantz, D., Luce, R., Suppes, P., and Tversky, A.: 1971, Foundations of Measurement, Vol. 1 (Academic Press, New York).
Luce, R. D., and Marley, A.: 1969, ‘Extensive measurement when concatenation is restricted and maximal elements may exist’, in Morgenbesser, S. et al. (eds.), Philosophy, Science, and Method: Essays in Honor of Ernest Nagel (St. Martin's Press, New York), pp. 235–249.
Mundy, B.: (a), ‘Faithful representation, physical extensive measurement theory and archimedean axioms’, to appear in Synthese.
Mundy, B.: (b), ‘Extensive measurement and ratio functions’, to appear in Synthese.
Nagel, E.: 1932, ‘Measurement’, Erkenntnis II, pp. 313–33, reprinted in Danto, A. and Morgenbesser, S. (eds.), Philosophy of Science (New American Library, New York, 1960), pp. 121–40.
Quine, W.: 1960, Word and Object (MIT Press, Cambridge, Massachusetts).
Quine, W.: 1976, ‘Whither physical objects?’, in Cohen, R. S. (ed.), Essays in Memory of Imre Lakatos (D. Reidel, Dordrecht), pp. 497–504.
Roberts, F.: 1979, Measurement Theory, Encyclopedia of Mathematics and Its Applications, Vol. 7 (Addison Wesley, Reading, Massachusetts).
Russell, B.: 1903, The Principles of Mathematics, second edition, (Norton, New York, 1938).
Salmon, W.: 1976, ‘Foreword’, in Reichenbach, H., Laws, Modalities, and Counterfactuals (U. of California Press, Berkeley).
Sanford, D.: 1980, ‘Armstrong's Theory of Universals’, British Journal for the Philosophy of Science 31, pp. 69–79.
Scott, D. and Suppes, P.: 1958, ‘Foundational aspects of theories of measurement’, Journal of Symbolic Logic 23, pp. 113–128; reprinted in Suppes, P., Studies in the Methodology and Foundations of Science (D. Reidel, Dordrecht, Holland, 1969), pp. 46–64.
Suppes, P.: 1951, ‘A set of independent axioms for extensive quantities’, Portugalia Mathematica 10, pp. 163–172; reprinted in Suppes, P., Studies in the Methodology and Foundations of Science (D. Reidel, Dordrecht, Holland, 1969), pp. 36–45.
Swoyer, C.: 1982, ‘The nature of natural laws’, Australasian Journal of Philosophy 60, pp. 203–223.
Swoyer, C.: 1983, ‘Realism and explanation’, Philosophical Inquiry 5, pp. 14–28.
Tooley, M.: 1977, ‘The nature of laws’, Canadian Journal of Philosophy 7, pp. 667–698.
Van Benthem, J., and Doets, K.: 1983, ‘Higher-order logic’, in D. Gabbay and F. Guenther (eds.), Hardbook of Philosophical Logic, Vol. 1 (D. Reidel, Dordrecht), pp. 275–329.
Whitney, H.: 1968, ‘The mathematics of physical quantities’, American Mathematical Monthly 75, pp. 115–138, pp. 227–256.
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Mundy, B. The metaphysics of quantity. Philosophical Studies 51, 29–54 (1987). https://doi.org/10.1007/BF00353961
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DOI: https://doi.org/10.1007/BF00353961