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Mathematical Formalisms and their Realizations

Published online by Cambridge University Press:  25 February 2009

G. T. Kneebone
Affiliation:
Bedford College for Women, University of London.

Extract

In a short article, published in an earlier volume of Philosophy1 under the title “Philosophy and Mathematics,” I tried to explain the current conception of pure mathematics as the study of abstract structure by construction and elaboration of appropriate axiomatic formalisms. In the present paper I propose to consider certain philosophical problems, of interest to philosophers and mathematicians alike, which have their origin in the relation between such formalisms and any applications to experience that they may possess. Consideration of problems of this kind is no new undertaking, and in Reichenbach's Wahrscheinlichkeitslehre, for instance, a considerable amount of space is devoted to the Anwendungsproblem or problem of application of the formal calculus under consideration. Most such discussions, however, are at bottom an appendage to an account of the formalism itself, and the author's interest is primarily mathematical. The result is that the philosophical issues involved are not given due weight; and it is these philosophical issues that I wish to discuss in the present paper.

Type
Research Article
Copyright
Copyright © The Royal Institute of Philosophy 1952

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References

page 138 note 1 Philosophy, XXII, 1947, pp. 231–39.Google Scholar

page 140 note 1 For a convenient version of the Peano formalism see Hilbert and Bernays: Grundlagen der Mathematik.

page 141 note 1 See Russell: Introduction to Mathematical Philosophy.

page 142note 1 See, for instance, Russell: Human Knowledge.