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The New Logic

Published online by Cambridge University Press:  14 March 2022

Karl Menger
Translated by
H. B. Gottlieb  and
J. K. Senior
Karl Menger*
Affiliation:
University of Chicago
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Extract

The rapid development of physics, the result of observations made and ideas introduced within the last few decades, has brought about a change in the whole system of physical concepts. This fact is common knowledge, and has already attracted the attention of philosophers. It is less well known that geometry too has had its crises, and undergone a reconstruction. For centuries, so-called “geometrical intuition” was used as a method of proof. In geometrical demonstrations, certain steps were allowed because they were “self-evident,” because the correctness of the conclusion was “shown by the adjoined diagram,” etc. A crisis occurred in geometry because such intuition proved to be untrustworthy. Many of the propositions regarded as self-evident or based upon the consideration of diagrams turned out to be false. And so Euclidean geometry was reconstructed by methods free from all intuitive elements and strictly logical in nature. Moreover, for more than a century, various other geometries have been devised purely as logical constructions. Since they start with assumptions different from those of Euclid, and lead to conclusions partly in contradiction with his theorems, they are called “non-Euclidean.” Nevertheless, each one of these geometries is a closed system of propositions exempt from contradiction. Recently, some of them have even found application in physics.

Type
Research Article
Information
Philosophy of Science , Volume 4 , Issue 3 , July 1937 , pp. 299 - 336
Copyright
Copyright © Philosophy of Science Association 1937

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Footnotes

∗

Translator's Note: This paper is a revised version of a lecture delivered in 1932 and published in “Krise und Neuaufbau in den exakten Wissenschaften.” (Deuticke, Vienna, 1933.)

References

1 See Boole, “The Mathematical Analysis of Logic,” Cambridge (1847), and “An Investigation of the Laws of Thought,” London (1854) and Chicago (1916); Schröder, “Vorlesungen über die Algebra der Logik,” 3 vols. Teubner, Leipzig (1890-1910) and “Abriss der Logik,” ibid. (1909-1910). See also the first chapters of Lewis and Langford “Symbolic Logic,” The Century Company, New York.

2 According to Ladd Franklin, these can moreover be combined into a single theorem, namely the theorem that for the three classes A, B and C, it cannot be simultaneously true that the intersection of A and B is the null class, the intersection of not-B and C is the null class, and the intersection of A and C is not the null class. In the case of the mode Barbara, the premises are: All M is P and all S is M. That is, the intersection of the class M with the class not-P is the null class, and the intersection of the class S with the class not-M is the null class. Then, according to the Ladd Franklin formula, it is impossible that the intersection of the classes S and not-P should not be the null class. In other words, the intersection of the class S and the class not-P is the null class. That is, all S is P, and the assertion of the mode Barbara is thereby proved.

3 The calculus of classes is not only methodologically the same as an axiomatic geometry. It can, in fact, as I have on occasion remarked, be brought formally under one heading with elementary geometrical theories. That is, there exists a theory including elementary geometry and the calculus of classes. From this theory, each of the two more special theories may be obtained by specific additional axioms. Cf. Jahresbericht der deutschen Mathematiker-Vereinigung, 37, 309 (1928), and Annals of Mathematics, 37, 456 (1936).

4 More detailed expositions of the calculus of propositions as well as of the calculus of functions are to be found in Frege, “Begriffsschrift,” Halle (1879); Whitehead-Russell, “Principia Mathematica,” Cambridge University Press (1915); Hilbert-Ackermann, “Grundzüge der Theoretischen Logik, Springer, Berlin (1928); Carnap, “Abriss der Logistik,” Springer, Vienna (1929).

5 An easily comprehensible exposition of the logical foundations of mathematics is Russell's “Introduction to Mathematical Philosophy,” Allen and Unwin, London (1920).

6 Tarski, “Der Wahrheitsbegriff in den formalistischen Sprachen,” Studia Philosophica, Lespoli (1935). This important paper contains the first successful treatment of a classical problem in philosophy by means of modern logic.

7 Cf. particularly Russell's “Introduction” cited in (4).

8 I gave an account of the historical development of this school of thought with many literature references in an article “Der Intuitionismus” Blätter f. deutsche Philosophie, 4, 311 (1930). More recent results are summed up by Heyting in “Ergebnisse d. Mathematik u. ihrer Grenzgebiete,” vol. 3, Springer, Berlin (1934). An account of the discussions of foundation questions together with many literature references is to be found in Fränkel's “Mengenlehre,” 3rd ed., Springer, Berlin (1928).

9 See Gödel in “Ergebnisse eines mathematischen Kolloquiums,” 4, 34, Deuticke, Vienna (1933).

10 Jahresbericht d. deutschen Mathematiker-Vereinigung, 37, 213 (1928).

11 Heyting, Sitzungsberichte d. preussischen Akademie der Wissenschaften, pp. 42, 57, 158, (1930).

12 Easily comprehended expositions: Bernays, Blätter f. deutsche Philosophie, 4, 326, (1936); Herbrand, Revue de metaphysique et de Morale 37, 343 (1930); Hilbert “Die Grundlagen der Mathematik,” Hamburger Mathematische Einzclschriften, vol. 5, Teubner, Leipzig (1928); von Neumann, Erkenntnis, 2, 116 (1931).

13 Metamathematics itself has recently been developed as a deductive theory in some important papers by Tarski, Monatashefte f. Mathematik u. Physik, 37, 361 (1930); Fundamenta Mathematics 25, 503 (1935) and 26, 283 (1936).

14 Post, American Journal of Mathematics 43, 163 (1921); Lukasiewicz, Compt. Rend. Soc. d. Sciences et d. Lettres Warsaw, 23, 51 (1930); Lukasiewicz and Tarski, ibid, p. 11. See also the older literature there quoted.

15 Cf. Lewis and Langford's book quoted in (1).

16 Cf. Gödel, Monatshefte f. Mathematik u. Physik, 38, 173 (1930).

17 Gentzen, Mathematische Annalen, 112, 493 (1936). Transfinite methods in the theory or proofs are also used by Church.

18 Cf. Gödel, Ergebnisse eines mathematischen Kolloquiums, 7, 13 (1936). Cf. also Church, American Journal of Mathematics, 58, 345 (1936).

19 Cf. Gödels paper quoted in (16). Skolem (Norsk. Matem. Forenings Skrifter, p. 73 (1933)) proved that it is impossible to characterize the sequence of natural numbers by a finite number of axioms if the law of the excluded middle is one of them.

20 For this reason, the frequent statement that mathematics is a great tautology or system of tautologies does not seem to me an adequate description of the situation. The concept of tautology has so far been defined only within the calculus of propositions. This concept might, it is true, be defined by methods lying outside this calculus, but it might be defined in different ways, whereas the words “Mathematics is a system of tautologies,” suggest a reference to an absolute logic.

21 “Die neue Logik.” Krise und Neuaufbau in den exakten Wissenschaften. Deuticke, Vienna, 1933.

22 An attempt in this direction is my paper, “Einige neuere Fortschritte in der exakten Behandlung sozialwissenschaftlicher Probleme.” Neuere Fortschritte in den exakten Wissenschaften, Deuticke, Vienna, 1936.

23 After introducing the term “principle of logical tolerance,” Carnap (logische Syntax der Sprache,” Springer, Vienna (1934)) suggests that, in first explicitly formulating this principle, I was expressing the attitude of most mathematicians. I should be glad if Carnap were right, but since the prominent mathematicians (Poincaré, the Paris school at the beginning of the century, Hilbert, Weyl, Brouwer) who have dealt with the foundations of mathematics have explicitly expressed opinions which, divergent as they are from one another, are all diametrically opposed to the above mentioned principle, I am afraid that I must bear the responsibility alone.

I take this opportunity to emphasize that I do not agree with the pronouncements on metaphysics which come from the group of which Carnap is a member. It is true that an expert logician can easily find logical errors, and very elementary errors at that, in many metaphysical theories, just as he can in many economic and sociological ones. Such errors are, however, objections only to special existant systems. If all metaphysics is rejected because its statements cannot be tested, then, I am afraid, very extensive parts of mathematics must likewise be discarded. Thus one arrives at arbitrary statements which resemble some of the critiques of classic mathematics discussed in this paper.