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Postulational Methods. I

Published online by Cambridge University Press:  14 March 2022

Louis Osgood Kattsoff
Louis Osgood Kattsoff*
Affiliation:
University of Virginia.
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Abstract

The keynote of our investigation is suggested by a remark made by Enriques, “the young student of mathematics would indeed look in vain to the classical logic into which he was initiated for an adequate conception of the structure of a deductive science like Geometry let alone for an explanation of the value and meaning of the principles of such a discipline. What are axioms, postulates, definitions? What place do they occupy in the edifice of the theory? What criteria determine their choice and how can we establish whether they are acceptable or not?” These are in general the questions we shall seek to answer in part and then we shall make a special analysis of two of the criteria determining the choice of the axioms.

Type
Research Article
Information
Philosophy of Science , Volume 2 , Issue 2 , April 1935 , pp. 139 - 163
Copyright
Copyright © Philosophy of Science Association 1935

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References

1 The Historic Development of Logic, Frederigo Enriques, trans. by J. Rosenthal, New York, 1929, p. 3.

2 Cf. Chwistek, Ann. de la Soc. Polonaise de Math., 1923/25 who feels that Hilbert's metamathematics cannot be essentially different from the logical calculus although it is true that Hilbert believed his metamathematics was beyond Logic.

3 But cf. Weiss' on the definitions of Implication which appeared in the Monist. Hilbert accepts the definition of Whitehead and Russell that p → q = p′ + q which is a special case, as H. B. Smith has shown.

4 Cf. Hilbert and Ackermann, Grundzüge der Theoretische Logik.

5 Consistency in Mathematics, Rice Institute Pamphlets, 16, 1929, p. 250.

6 Cf. the definition of consistency given by Wavre, in Revue de Metaphysique et de Morale, 31, 1924, p. 460.

7 Cf. Becker, Mathematische Existenz, Jahrbuch für Phil. VIII, Halle, 1927, p. 457.

8 Cf. Hilbert, Mathematische Annalen., 88, p. 154–155.

9 The law of excluded middle is proven consistent with all these axioms in a similar fashion by Ackermann, in Mathematische Annalen, 93, pp. 1ff.

10 Brouwer himself accepted the names Intuitionism and Formalism. Cf. Bull. American Math. Soc., 1913.

11 But cf. Wavre, “Y a-t-il une crise des Mathematiques?” Rev. de Metaph. et de Morale, v. 31, p. 435.

12 Brouwer essentially makes use of the law in proving that for Finite sets, a proper subset cannot be of the same cardinal number as the finite set itself. He assumes that a proper subset of a finite set is either of the same cardinal number as the set or it is not. To assume that it is leads to a contradiction; therefore it is not. Cf. Begründung der Mengenlehre, etc. Amsterdam 1918, p. 6.

13 It is true that in “Intuitionism and Formalism” and in “Intuitionistische Mengenlehre” Brouwer explicitly rejects the law of excluded middle. But that he does not mean to reject it “in toto” is evident from the statement that the law does hold for finite spheres. Certainly from the article in Sitzungsberichte d. Preuss. Akad. Wiss., 1928, one might hastily conclude that Brouwer intends to reject the Tertium.

14 Cf. Wavre—Logique Formelle et Logique Empriste, in Rev. de Metaph. et de Morale, 1926, p. 66 and p. 69ff.

15 In this connection it is interesting to quote from the De Interpretatione of Aristotle—“From the proposition it may be, it follows that it is contingent, and the relation is reciprocal.” Cf. 22a, 14–15.

16 Such an assumption is not inconsistent with the other logical assumptions and hence no inconsistent results are reached; however a great many important distinctions are wiped out, and it is these which Brouwer is insisting upon. Fraenkel's ad hominem argument in Erkenntnis 1, 1930, p. 292, is therefore no argument at all. (Die Heutigen Gegensätze in der Grundlegung der Mathematik.)

17 Brouwer emphasizes again and again that now we can construct mathematics on a rigorous foundation. Heyting further sets up a new Projective Geometry on Intuitionistic foundations. Cf. Sitz. Preuss. Akad. Wiss., 1928, p. 51 for Brouwer's remarks and Math. Ann., 98, for those of Heyting.

18 Sitz, der Preuss. Akad. der Wiss. Phys.-Math. Klasse, 1928, p. 48ff.

The discussion of postulate sets will be continued in later issues of this journal. We shall in the next article begin to dissect the Set.