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A note on nominalism and recursive functions

Published online by Cambridge University Press:  12 March 2014

R. M. Martin
R. M. Martin*
Affiliation:
Bryn Mawr College
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Extract

The purpose of this note is (i) to point out an important similarity between the nominalistic system discussed by Quine in his recent paper On universals and the system of logic (the system н) developed by the author in A homogeneous system for formal logic, (ii) to offer certain corrections to the latter, and (iii) to show that that system (н) is adequate for the general theory of ancestrale and for the definition of any general recursive function of natural numbers.

Nominalism as a thesis in the philosophy of science, according to Quine, is the view that it is possible to construct a language adequate for the purposes of science, which in no wise admits classes, properties, relations, or other abstract objects as values for variables.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 14 , Issue 1 , 16 May 1949 , pp. 27 - 31
Copyright
Copyright © Association for Symbolic Logic 1949

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References

1 This Journal, vol. 12 (1947), pp. 74–84.

2 This Journal, vol. 8 (1943), pp. 1–23. This paper will subsequently be referred to as H.L.

3 Quine, W. V., Mathematical logic, New York, 1940, p. 121.Google Scholar

4 See Church, A., Introduction to mathematical logic. Part I, Princeton, 1944, Chapter II, esp. pp. 37–38.Google Scholar

5 See Quine, W. V., Designation and existence, The journal of philosophy, vol. 36 (1939), pp. 701–709.CrossRefGoogle Scholar

6 The author is indebted to Dr. J. C. C. McKinsey for pointing out the need for these corrections. See this Journal, vol. 8 (1943), p. 54. The error in R8(4) was genuine whereas the omission in R8(5) was an omission in copying due to making a change of notation suggested by the referee.

7 See Frege, G., Grundgesetze der Arithmetik, Jena, vol. 1, 1893, p. 54 ff.Google Scholar, and Whitehead, A. N. and Russell, B., Principia mathematica, 2nd ed., Cambridge, vol. 1, 1925, p. 543 ff.Google Scholar

8 See Kleene, S. C., General recursive functions of natural numbers, Mathematische Annalen, vol. 112 (1936), pp. 727–742.CrossRefGoogle Scholar Cf. also Hilbert, D. und Bernays, P., Grundlagen der Mathematik, Berlin, vol. 1, 1934, esp. pp. 286–382Google Scholar, and vol. 2, 1939, pp. 392–421.

9 See Kleene, S. C., A note on recursive functions, Bulletin of the American Mathematical Society, vol. 42 (1936), pp. 544–646.CrossRefGoogle Scholar