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The formalization of mathematics1

Published online by Cambridge University Press:  12 March 2014

Hao Wang
Hao Wang*
Affiliation:
Oxford University
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Extract

Zest for both system and objectivity is the formal logician's original sin. He pays for it by constant frustrations and by living ofttimes the life of an intellectual outcaste. The task of squeezing a large body of stubborn facts into a more or less rigid system can be a painful one, especially since the facts of mathematics are among the most stubborn of all facts. Moreover, the more general and abstract we get, the farther removed we are from the raw mathematical experience. As intuition ceases to operate effectively, we fall into many unexpected traps. The formal logician gets little sympathy for his frustrations. He is regarded as too rigid by his philosophical colleagues and too speculative by his mathematical friends. The life of an intellectual outcaste may be a result partly of temperament and partly of the youthfulness of the logic profession. The unfortunate lack of wide appeal of logic may, however, be prolonged partly on account of the fact that very little of the well-established techniques of mathematics seems applicable to the treatment of serious problems of logic.

The axiomatic method is well suited to provide results which are both exact and systematic. How attractive would it be if we could get an axiom system in which all the axioms and deductions were intuitively clear and all theorems of mathematics were provable? Such a system would undoubtedly satisfy Descartes who admits solely intuition and deduction, which are, for him, the only “mental operations by which we are able, wholly without fear of illusion, to arrive at the knowledge of things.” Indeed, according to Descartes, intuition and deduction “are the most certain routes to knowledge, and the mind should admit no others. All the rest should be rejected as suspect of error and dangerous.”

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 19 , Issue 4 , December 1954 , pp. 241 - 266
Copyright
Copyright © Association for Symbolic Logic 1954

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Footnotes

1

I wish to thank the referee and Professor Max Black for useful comments on an earlier draft of this paper.

References

2 The rules for the direction of the mind, Rule III.

3 Du rôle de l'intuition et de la logique en mathématiques, Compte-rendu du IIième Congrès International des Mathéaticiens, 1900, Paris (1902), pp. 200–202Google Scholar.

4 See, e.g., du Bois-Reymond, P., Über asymptotische Werte, infinitäre Approximationen und infinitäre Auflösung von Gleichungen, Mathematische Annalen, vol. 8 (1875), pp. 363–414Google Scholar.

5 Fitch, Frederic B., The hypothesis that infinite classes are similar, this Journal, vol. 4 (1939), pp. 159–162Google Scholar.

Chwistek, Leon, Über die Hypotheses der Mengenlehre, Mathematische Zeitschrift, vol. 25 (1926), pp. 439–473Google Scholar.

6 E.g., as in Bernays' series of articles on axiomatic set theory in this Journal, vol. 2 (1937); vol. 6 (1941); vol. 7 (1942); vol. 8 (1943); vol. 13 (1948); vol. 19 (1954).

7 The consistency of the ramified Principia, this Journal, vol. 3 (1938), pp. 140–149Google Scholar.

8 Lorenzen, Paul, Algebraische und logistische Untersuchungen über freie Verbände, this Journal, vol. 16 (1951), pp. 81–106Google Scholar; Schütte, Kurt, Beweistheoretische Untersuchung der verzweigten Analysis, Mathematische Annalen, vol. 124 (1951–1952), pp. 123–147Google Scholar.

9 Hölder, Otto, Der angebliche circulus vitiosus und die sogenannte Grundlagenkrise in der Analysis, Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-physische Klasse, vol. 78 (1926), pp. 243–250Google Scholar.

10 A. N. Whitehead and Bertrand Russell, Vol. I, p. 41.

11 See Kleene, S. C., On notation for ordinal numbers, this Journal, vol. 3 (1938), pp. 150–155Google Scholar, and Church, Alonzo, The constructive second number class, Bulletin of the American Mathematical Society, vol. 44 (1938), pp. 224–232Google Scholar.

12 Compare Prolegomena to any future metaphysics, § 52, c.

13 Ibid, §57.

14 See, e.g., Hilbert, D. and Bernays, P., Grundlagen der Mathematik, vol. II, p. 373Google Scholar.

15 Gödel, Kurt, Russell's mathematical logic, The philosophy of Bertrand Russell, The Library of Living Philosophers, Vol. V (1944), edited by Schilpp, P. A., p. 137Google Scholar.