Every branch of physics gives rise to an application of mathematics. A prophecy may be hazarded that in the future these applications will unify themselves into a mathematical theory of a hypothetical substructure of the universe, uniform under all the diverse phenomena. This reflection is suggested by the following articles: [Aether]; [Molecule]; [Capillary Action]; [Diffusion]; [Radiation, Theory of]; and others.
The applications of mathematics to statistics (see [Statistics] and [Probability]) should not be lost sight of; the leading fields for these applications are insurance, sociology, variation in zoology and economics.
The History of Mathematics.—The history of mathematics is in the main the history of its various branches. A short account of the history of each branch will be found in connexion with the article which deals with it. Viewing the subject as a whole, and apart from remote developments which have not in fact seriously influenced the great structure of the mathematics of the European races, it may be said to have had its origin with the Greeks, working on pre-existing fragmentary lines of thought derived from the Egyptians and Phœnicians. The Greeks created the sciences of geometry and of number as applied to the measurement of continuous quantities. The great abstract ideas (considered directly and not merely in tacit use) which have dominated the science were due to them—namely, ratio, irrationality, continuity, the point, the straight line, the plane. This period lasted[11] from the time of Thales, c. 600 B.C., to the capture of Alexandria by the Mahommedans, A.D. 641. The medieval Arabians invented our system of numeration and developed algebra. The next period of advance stretches from the Renaissance to Newton and Leibnitz at the end of the 17th century. During this period logarithms were invented, trigonometry and algebra developed, analytical geometry invented, dynamics put upon a sound basis, and the period closed with the magnificent invention of (or at least the perfecting of) the differential calculus by Newton and Leibnitz and the discovery of gravitation. The 18th century witnessed a rapid development of analysis, and the period culminated with the genius of Lagrange and Laplace. This period may be conceived as continuing throughout the first quarter of the 19th century. It was remarkable both for the brilliance of its achievements and for the large number of French mathematicians of the first rank who flourished during it. The next period was inaugurated in analysis by K. F. Gauss, N. H. Abel and A. L. Cauchy. Between them the general theory of the complex variable, and of the various “infinite” processes of mathematical analysis, was established, while other mathematicians, such as Poncelet, Steiner, Lobatschewsky and von Staudt, were founding modern geometry, and Gauss inaugurated the differential geometry of surfaces. The applied mathematical sciences of light, electricity and electromagnetism, and of heat, were now largely developed. This school of mathematical thought lasted beyond the middle of the century, after which a change and further development can be traced. In the next and last period the progress of pure mathematics has been dominated by the critical spirit introduced by the German mathematicians under the guidance of Weierstrass, though foreshadowed by earlier analysts, such as Abel. Also such ideas as those of invariants, groups and of form, have modified the entire science. But the progress in all directions has been too rapid to admit of any one adequate characterization. During the same period a brilliant group of mathematical physicists, notably Lord Kelvin (W. Thomson), H. V. Helmholtz, J. C. Maxwell, H. Hertz, have transformed applied mathematics by systematically basing their deductions upon the Law of the conservation of energy, and the hypothesis of an ether pervading space.
Bibliography.—References to the works containing expositions of the various branches of mathematics are given in the appropriate articles. It must suffice here to refer to sources in which the subject is considered as one whole. Most philosophers refer in their works 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 inédits de Leibnitz. Extraits des manuscrits de la bibliothèque royale de Hanovre, by Louis Couturat (Paris, 1903), especially pp. 356-399, “Generales inquisitiones de analysi notionum et veritatum” (written in 1686); also cf. La Logique de Leibnitz, already referred to. For the modern authors who nave rediscovered and improved upon the position of Leibnitz, cf. Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet von Dr G. Frege, a.o. Professor an der Univ. Jena (Bd. i., 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. Journ. 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 und irrationales Zahlen (1st ed., 1872), and Was sind und 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 mathématiques (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. Poincaré, La Science el l’hypothèse (Paris, 1st ed., 1902), English translation under the title, Science and Hypothesis (London, 1905); L. Couturat, Les Principes des mathématiques (Paris, 1905); E. Mach, Die Mechanik in ihrer Entwickelung (Prague, 1883), English translation under the title, The Science of Mechanics (London, 1893); K. Pearson, The Grammar of Science (London, 1st ed., 1892; 2nd ed., 1900, enlarged); A. Cayley, Presidential Address (Brit. Assoc., 1883); B. Russell and A. N. Whitehead, Principia Mathematica (Cambridge, 1911). For the history of mathematics the one modern and complete source of information is M. Cantor’s Vorlesungen über Geschichte der Mathematik (Leipzig, 1st Bd., 1880; 2nd Bd., 1892; 3rd Bd., 1898; 4th Bd., 1908; 1st Bd., von den ältesten Zeiten bis zum Jahre 1200, n. Chr.; 2nd Bd., von 1200-1668; 3rd Bd., von 1668-1758; 4th Bd., von 1795 bis 1790); W. W. R. Ball, A Short History of Mathematics (London 1st ed., 1888, three subsequent editions, enlarged and revised, and translations into French and Italian).
(A. N. W.)
[1] Cf. La Logique de Leibnitz, ch. vii., by L. Couturat (Paris, 1901).
[2] Cf. The Principles of Mathematics, by Bertrand Russell (Cambridge, 1903).
[3] Cf. Formulaire mathématique (Turin, ed. of 1903); earlier formulations of the bases of arithmetic are given by him in the editions of 1898 and of 1901. The variations are only trivial.
[4] Cf. Russell, loc. cit., pp. 199-256.