Abstract
One of the most efficient methodological tools, Occam’s celebrated razor, is the maxim that it is vain to do with more what can be done with fewer. This principle is primarily ontological: Entities must not be multiplied beyond necessity. But the razor may also be construed as a semantic maxim opposing the use of synonyms. Occam’s principle is often called the Law of Parsimony.
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References
Cf. the author’s papers ‘The idea of variable and function,’ Proc. Natl. Acad. Sci. U.S.A., 39 (1953), pp. 956–961; `On variables in mathematics and in natural science’ Br. Journ. Phil. Sci.,5 (1954), pp. 134–142; `Mensuration and other mathematical connections of observable material,’ in Measurement: Definitions and Theories,ed. Churchman and Ratoosh, John Wiley, New York, 1959, pp. 97–128; ‘An axiomatic Theory of functions and fluents,’ in The Axiomatic Method,ed. Henkin, Suppes, and Tarski. North-Holland Publ. Co.. Amsterdam, 1959, pp. 454–473, and the author’s book Calculus. A modern Approach,Ginn, Boston 1955; mimeo-editions Chicago 1952 and 1953.
A forthcoming paper is devoted to semantic uses of the principle.
Ever since the 18th century, mathematicians have tried to reconcile the interpretation of s and t as variables with the fact that, for all significant scopes of the variables, (3’) is false. Their attempts have centered on the idea of so-called dependent variables. In (3’), only t is regarded as replaceable with any value of the time (and called an independent variable) while s is said to be dependent since, after a value of t has been chosen, the value of s is determined. But a variable (i.e., a symbol that may be replaced with the designation of any element of its scope) which is dependent (i.e., cannot be replaced with the designation of any element of its scope) is a plain contradiction in terms.
I have stressed the difference between these concepts and number variables in the Preface to the 1952 edition of my book Calculus. A Modern Approach and have elaborated the theory in Chapter VII of the 1953 and 1955 editions as well as in the papers quoted in 1) on p. 80. Carnap defined what he called physical quantities as associations of numbers with quadruples of numbers (thus as 4-place functions) in The logical syntax of language,London and New York 1937, pp. 149 sq. and as number variables in Foundations of Logic and Mathematics,Encycl. of Unified Science, vol. I, Chicago, 1939. I thus am glad to see that he expresses views more similar to those stressed loc. cit. 1) on p. 80 in his Einführung in die symbolische Logik, Wien,1954 and Introduction to symbolic Logic,New York 1958, especially p. 168 sq.
Methodus fluxionum et serierum infinitarum, 1737. Apparently influenced by Descartes, Newton continues: hasque representabo per ultimas alphabeti litteras u, x, y, et z ut discerni possint ab aliis quantitatibus quae in equationibus considerantur tamquam cognitae et determinatae.
One may compare Newton’s quoted description of fluents with de la Vallée Poussin’s description of a variable as une quantité qui passe par une infinité des valeurs, distinctes ou non in his Cours d’Analyse vol. 1. It should be observed that those descriptions were not elaborated either by rigorous explicit definitions or by postulates defining the concept implicitly.
Cf., e.g., Courant, Differential and Integral Calculus, vol. 1, p. 16.
Cf. Mc Kinsey, Sugar, and Suppes, `Axiomatic foundations of classical particle mechanics’, Journ. Rat. Mech. Anal., 2. 1953, pp. 253–272, and Artin, Calculus and Analytic Geometry, 1957, p. 70.
What laws, if any, govern the association of the value of a quantity with its object; whether the association must be relative to some frame of reference; and if so, what changes of the value correspond to changes of the frame — these questions will be discussed elsewhere. But whatever the answers to these questions may be, strong emphasis must be laid on the (usually neglected) objects of quantities, which remain permanent in the said changes. Weyl’s elaborate definition of a quantity in The Classical Groups,Princeton 1939, pp. 16 sq. does not seem to include a clear reference to an object.
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Menger, K. (1962). A Counterpart of Occam’s Razor in Pure and Applied Mathematics; Ontological Uses. In: Logic and Language. Synthese Library, vol 5. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2111-0_8
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