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Some notes on nominalism

Published online by Cambridge University Press:  12 March 2014

Leon Henkin
Leon Henkin*
Affiliation:
University of Southern California
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Extract

In two papers appearing in successive issues of this Journal, Quine and Goodman, impelled by a philosophical standpoint which forswears all traffic in abstract entities, have explored the possibility of reinterpreting the language of classical mathematics so as to give it meaning (consistent with the observed overt pattern of standard usage) in terms of the narrow domain of entities which they recognize. The present paper offers several comments on this program.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 18 , Issue 1 , March 1953 , pp. 19 - 29
Copyright
Copyright © Association for Symbolic Logic 1953

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References

1 Quine, W. V., On universals, this Journal, vol. 12 (1947), pp. 74–84Google Scholar. This will be referred to in the text as Goodman, Q. Nelson and Quine, W. V., Steps toward a constructive nominalism, this Journal, vol. 12 (1947), pp. 105–122Google Scholar. This will be referred to in the text as GQ.

2 Henkin, Leon, Completeness in the theory of types, this Journal, vol. 15 (1950), pp. 81–91Google Scholar. This will be referred to in the text as H1. We shall use H2 as a reference symbol for The completeness of the first-order functional calculus, this Journal, vol. 14 (1949), pp. 159–166Google Scholar.

3 See Gödel, Kurt, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198CrossRefGoogle Scholar, and Tarski, Alfred, Der Wahrheitsbegriff in den Formalisierten Sprachen, Studio Philosophica, vol. 1 (1936), pp. 261–405Google Scholar.

4 Certain reviewers and critics of the work of Quine and Goodman have pointed to the use of the term “abstract entities”, by authors who do not believe in abstract entities, as constituting an inherent inconsistency in the nominalist position. (Yet the use of the word “unicorn” is recognized as compatible with non-belief in the existence of unicorns!) The fact is that there is not much difficulty in giving a nominalistic sense to the predicate “is used as the name of an abstract entity”. However, the predicate “is finite” presents greater difficulties.

5 The a priori possibility here noted can be proved in fact to occur by constructing specific examples.