Max Black

From The Nature of Mathematics, 1933 by Max Black

The results of Dedekind, Frege, and Peano had covered in conjunction the whole filed of elementary pure mathematics,¹ and by reducing the real numbers to integers, integers to entities occurring in logic, had supplied all the materials for the logistic thesis. There was still needed a synthesis to co-ordinate these results and remedy the imperfections of these early proofs. This was begun by Bertrand Russell in Principles of Mathematics, 1903, and continued in Principia Mathematica (first edition, 1910) written in collaboration with Alfred North Whitehead. These two books are at the apex of the second period in the logistic movement ; they profess to prove, rigorously and with the utmost detail, the identity of mathematics and logic.

The first is a philosophical and polemical discussion of the logistic theories ; the second, written except for a minimum of incidental explanation entirely in mathematical symbols, a proof of the theories.

Since Principia Mathematica little advance has been made by the logistic school and time has shown serious defects in that work, so that the third period has been one of successive attempts to consolidate a position which at one time Whitehead and Russell appeared to have reached triumphantly.

Among the most notable of these attempts are H. Weyl’s Das Kontinuum, 1918, L. Chwistek’s Theory of Constructive Types, 1923-5 ; and F. P. Ramsey’s Foundations of Mathematics, 1927. All these defend a logistic position. In addition there remains the remarkable Tractatus Logico-Philosophicus, 1922, of L. Wittgenstein, a former pupil of Russell, whose conclusions [are] in many respects unfavourable to Principia Mathematica, (etc).

 

From Bibliography, (selections)

BROUWER, L. E. J.   "Intuitionism and Formalism" (Inaugural address at the University of Amsterdam. 1912).   Reprinted in Bulletin of American Mathematical Society, vol. xx. 1913
———   " Zur Begründung der intuitionistichen Mathematik " Mathematische Annalen, vol. xciii, p. 244.   1923.
———   " Mathematik, Wissenschaft und Sprache " : Monatshefte für Mathematik und Physik, vol. xxxi, p. 153.   1929.

[Carnap, “Des Logische Aufbau der Welt" , Berlin 1928.]

CASSIRER, E.   Philosophie der Symbolischen Formen, vol. iii.   Berlin, 1923-31.

CHWISTEK,L,   Theory of Constructive Types   Cracow, 1923.
———   " Une méthode métamathématique d'analyse " : Comptes-Rendus du Premier Congrès des Mathématiciens des Pays Slaves.   Warsaw, 1929.
———   " Neue Grundlagen der Logik und Mathematik " : Mathematische Zeitschrift.   1929 and 1932.
[end p. 211.]

DEDEKIND, R.   Stetigkeit und irrationale Zahlen.   Fifth Edition, Brunswick, 1918. Originally appeared in 1872.
———   Was sind und was sollen die Zahlen ?   Fourth Edition, Brunswick, 1918. Originally appeared in 1887.

FREGE, G.   Die Grundlagen der Arithmetik.   Breslau. 1884.
———   Grundgesetze der Arithmetick,   Jena : vol. I, 1893 : vol. ii, 1903.

LESNIEWSKI, ST.   “Grundzüge eines neuen Systems der Grundlagen der Mathematik ” : Fundamenta Mathematica, pp. 1-81. [Warsaw] 1929.
{end p. 212]

DE MORGAN, A.   Formal Logic : or the Calculus of Inference, Necessary and Probable.   1847 : Edited by A. E. Taylor, 1926.

PEANO, G.   Formulaire de Mathématiques (in collaboration). Five editions from 1893-1908.   Turin.

PEIRCE, C. S.   Collected Papers.   Edited by Charles Hartshorne and Paul Weiss.   Cambridge, Mass.   1931-5.

POINCARÉ, H.   Science and Hypothesis.   Translated by W. J. G. London and Newcastle-upon-Tyne. 1903.
———   Science and Method. Translated by Francis Maitland. London, 1914.
———   La Valeur de la Science.   Paris, 1904.

RAMSEY,. F. P.   The Foundations of Mathematics.   Kegan Paul, 1931.
[end p. 213.]

WEISS, P.   “ The Theory of Types  “ : Mind, vol. xxxvii.   1928.

WEYL, H.   Das Kontinuum.   Leipzig. 1918.
———   “ Der circulus vitiosus in der heutigen Begründung der Analysis ” : Jahresb. d. Deutschen Mathematikes-Vereinigung. vol. xxviii. 1919.
[p. 214]
———   Philosophie des Mathematik und Naturewissenschaften. (Section IIA of the Handbuch der Philosophie, [..]) Munich and Berlin,. 1926.

WITTGENSTEIN, L.   Tractatus Logico-Philosophicus.   London, Kegan Paul, 1922.
———   “ Some Remarks on Logical Form ” : Aristotelian Society Supplementary Volume IX. 1929. (p. 211)

Note, Tractatus Logico-Philosophicus, 1922 L. Wittgenstein (7th reprint 1958)

The Editor* is indebted to Professor Max Black for providing this Index, which was originally prepared for the use of his students at Cornell University.

London : Routledge & Kegan Paul,
7th impression 1958, p. 191.

  * Apparently Charles Kay Ogden (1889-1957).     (WPT)

From The Labyrinth of Language, 1968 by Max Black

. . . General Semantics . . . an educational movement established in America . . . by the Polish Count, Alfred Korzybski, whose aims are still actively fostered by the International Society for General Semantics.   (Etc.) *

New York etc. : Praeger 1968, note on p. 118.

    * Professor Black clearly had no knowledge of certain developments on the described scene.   Please confer the letter by Korzybski of May 1946 to the Editor of ETC (Hayakawa) ; also the letters reproduced in Collected Writings (1990). — (WPT)

Selected bibliographic

Author Black, Max, 1909- Title A companion to Wittgenstein's Tractatus. Publisher London : Cambridge University press, 1964. Description xv, 450 p. ; 24 cm. Note Bibliography: p. 416-421.

Author Black, Max, 1909- Title Language and philosophy; studies in method Publisher Ithaca, Cornell University Press, 1949 Description xiii, 264 p. 24 cm

Author Black, Max, 1909- Title The nature of mathematics; a critical survey, by Max Black. Publisher London : K. Paul, Trench, Trubner & Co., ltd.; New York, Harcourt, Brace, and Company, 1934. Description xiv, 219 p. 22 cm. Note Bibliography: p. 211-214.