Statistics

STATISTICS and PROBABILITY) should not be lost sight of ; the leading fields for these applications are insurance, sociology, varia tion in zoology and economics.

BIBLIOGRAPHY.-References

to expositions of branches of mathe matics are given in the appropriate articles. We refer here to sources in which the subject is considered as one whole. Most philosophers refer to mathematics more or less cursorily, either in the treatment of the ideas of number and magnitude, or in their consideration of the alleged a priori and necessary truths. A bibliography of such references would be in effect a bibliography of metaphysics, or rather of epistemology. The founder of the modern point of view, explained in this article, was Leibnitz, who, however, was so far in advance of contemporary thought that his ideas remained neglected and undeveloped until recently ; cf. Opuscules et fragments inidits de Leibnitz. Extraits des manuscrits de la bibliotheque royale de Hanovre, by Louis Couturat (Paris, 1903), especially pp. 356-399, "Generates inquisitiones de analysi notionum et veritatum" (written in i686) ; also cf. La Logique de Leibnitz, already referred to. For the modern authors who have rediscovered and improved upon the position of Leibnitz, cf. Grundgesetze der Arith metik, begriffsschriftlich ebgeleitet von Dr. G. Frege, a.o. Professor an der Univ. Jena (Bd. 1893 ; Bd. ii., 1903, Jena) ; also cf. Frege's earlier works, Begriffsschrift eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (Halle, 1879), and Die Grundlagen der Arithmetik (Breslau, 1884) ; also cf. Bertrand Russell, The Principles of Mathematics (Cambridge, 1903) , and his article on "Mathematical Logic" in Amer. Quart. Jaunt. of Math. (vol. xxx.,

1908). Also the following works are of importance, though not all expressly expounding the Leibnitzian point of view: cf. G. Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre," Math. Annal., vol. xxi. (1883) and subsequent articles in vols. xlvi. and xlix.; also R. Dedekind, Stetigkeit and irrationale Zahlen (1st ed., 1872), and Was sind and was sollen die Zahlen? (1st ed., 1887), both tracts translated into English under the title Essays on the Theory of Numbers (Chicago, 1901). These works of G. Cantor and Dedekind were of the greatest importance in the progress of the subject. Also cf. G. Peano (with various collaborators of the Italian school), Formulaire de mathematiques (Turin, various editions, 1894-1908; the earlier editions are the more interesting philosophically) ; Felix Klein, Lectures on Mathematics (New York, 1894) ; W. K. Clifford, The Common Sense of the exact Sciences (London, 1885) ; H. Poincare, La Science et l'hypothese (Paris, ist ed., 1902), English translation under the title, Science and Hypothesis (London, 1905) ; L. Couturat, Les Principes des mathematiques (Paris, 1905) ; E. Mach, Die Mechanik in ihrer Entwickelung (Prague, 1883), English trans lation under the title, The Science of Mechanics (London, 1893) K. Pearson, The Grammar of Science (London, 1st ed., 1892 ; and ed., 1900, enlarged) ; A. Cayley, Presidential Address (Brit. Assoc., 1883) ; B. Russell and A. N. Whitehead, Principia Mathematica (Cambridge, 1911). (A. N. W.)