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What is Length?

Published online by Cambridge University Press:  14 March 2022

Frank Henmueller  and
Karl Menger
Frank Henmueller
Affiliation:
Illinois Institute of Technology
Karl Menger
Affiliation:
Illinois Institute of Technology
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Abstract

One does not only talk about the length in inches of this sheet of paper but also about the length of this sheet, about length in inches and about length. A clarification of these and related concepts results from a combination of the theory of the length in a definite unit as a fluent, developed by one of the authors, with the other's concept of 2-place fluents. The length ratio L is defined by pairing a number L (α, β) to any two objects of a certain kind, α, β (in a definite order). L thus may be described as the class of all pairs (α, β), L (α, β) for any objects α, β of the said kind. Length of this sheet and length in inches are specializations of this 2-place fluent.

Type
Research Article
Information
Philosophy of Science , Volume 28 , Issue 2 , April 1961 , pp. 172 - 177
Copyright
Copyright © Philosophy of Science Association 1961

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References

1 Procedures assumed to be familiar to readers interested in the diameter of the sun or of an atomic nucleus also include, in Bridgman's terminology, certain paper-and-penciloperations.

2 Cf. K. Menger, “On Variables in Mathematics and in Natural Science,” Brit. Journ. Phil. Sci., 5, 1954, pp. 134-142; Calculus. A Modern Approach, Ginn and Co., Boston, 1955, especially, Chapter VII; “An Axiomatic Theory of Functions and Fluents” in The Axiomatic Method, North-Holland Publishing Co., Amsterdam 1959, pp. 454-473.

3 This typographical convention has been introduced in the book on calculus quoted in2.

4 Functions of any number of places, functionals, and fluents may be combined into functors—mappings into the class of real numbers. But only functions lend themselves to the substitution of other functors (functions or fluents). One may substitue the lenght into the function log but one cannot substitute that function into the length. There is a logarithm of the length but there is no length of the logarithm.

5 F. Henmueller, Functions of Several Places, Master's Thesis Illinois Institute of Technology, 1960.

6 Cf. K. Menger “Mensuration and Other Mathematical Connections of Observable Material” in Measurement: Definitions and Theories, John Wiley, New York 1959, pp. 97-128.