Classes and Relations.—The foregoing account of the nature of mathematics necessitates a strict deduction of the general propertiesof classes and relations from the ultimate logical premisses. In the course of this process, undertaken for the first time with the rigour of mathematicians, some contradictions have become apparent. That first discovered is known as Burali-Forti’s contradiction,6and consists in the proof that there both is and is not a greatest infinite ordinal number. But these contradictions do not depend upon any theory of number, for Russell’s contradiction7does not involve number in any form. This contradiction arises from considering the class possessing as members all classes which are not members of themselves. Call this class w; then to say that x is a w is equivalent to saying that x is not an x. Accordingly, to say that w is a w is equivalent to saying that w is not a w. An analogous contradiction can be found for relations. It follows that a careful scrutiny of the very idea of classes and relations is required. Note that classes are here required in extension, so that the class of human beings and the class of rational featherless bipeds are identical; similarly for relations, which are to be determined by the entities related. Now a class in respect to its components is many. In what sense then can it be one? This problem of “the one and the many” has been discussed continuously by the philosophers.8All the contradictions can be avoided, and yet the use of classes and relations can be preserved as required by mathematics, and indeed by common sense, by a theory which denies to a class—or relation—existence or being in any sense in which the entities composing it—or related by it—exist. Thus, to say that a pen is an entity and the class of pens is an entity is merely a play upon the word “entity”; the second sense of “entity” (if any) is indeed derived from the first, but has a more complex signification. Consider an incomplete proposition, incomplete in the sense that some entity which ought to be involved in it is represented by an undetermined x, which may stand for any entity. Call it a propositional function; and, if φx be a propositional function, the undetermined variable x is the argument. Two propositional functions φx and ψx are “extensionally identical” if any determination of x in φx which converts φx into a true proposition also converts ψx into a true proposition, and conversely for ψ and φ. Now consider a propositional function Fχin which the variable argument χ is itself a propositional function. If Fχis true when, and only when, χ is determined to be either φ or some other propositional function extensionally equivalent to φ, then the proposition Fφis of the form which is ordinarily recognized as being about the class determined by φx taken in extension—that is, the class of entities for which φx is a true proposition when x is determined to be any one of them. A similar theory holds for relations which arise from the consideration of propositional functions with two or more variable arguments. It is then possible to define by a parallel elaboration what is meant by classes of classes, classes of relations, relations between classes, and so on. Accordingly, the number of a class of relations can be defined, or of a class of classes, and so on. This theory9is in effect a theory of theuseof classes and relations, and does not decide the philosophic question as to the sense (if any) in which a class in extension is one entity. It does indeed deny that it is an entity in the sense in which one of its members is an entity. Accordingly, it is a fallacy for any determination of x to consider “x is an x” or “x is not an x” as having the meaning of propositions. Note that for any determination of x, “x is an x” and “x is not an x,” are neither of them fallacies but are both meaningless, according to this theory. Thus Russell’s contradiction vanishes, and an examination of the other contradictions shows that they vanish also.
Classes and Relations.—The foregoing account of the nature of mathematics necessitates a strict deduction of the general propertiesof classes and relations from the ultimate logical premisses. In the course of this process, undertaken for the first time with the rigour of mathematicians, some contradictions have become apparent. That first discovered is known as Burali-Forti’s contradiction,6and consists in the proof that there both is and is not a greatest infinite ordinal number. But these contradictions do not depend upon any theory of number, for Russell’s contradiction7does not involve number in any form. This contradiction arises from considering the class possessing as members all classes which are not members of themselves. Call this class w; then to say that x is a w is equivalent to saying that x is not an x. Accordingly, to say that w is a w is equivalent to saying that w is not a w. An analogous contradiction can be found for relations. It follows that a careful scrutiny of the very idea of classes and relations is required. Note that classes are here required in extension, so that the class of human beings and the class of rational featherless bipeds are identical; similarly for relations, which are to be determined by the entities related. Now a class in respect to its components is many. In what sense then can it be one? This problem of “the one and the many” has been discussed continuously by the philosophers.8All the contradictions can be avoided, and yet the use of classes and relations can be preserved as required by mathematics, and indeed by common sense, by a theory which denies to a class—or relation—existence or being in any sense in which the entities composing it—or related by it—exist. Thus, to say that a pen is an entity and the class of pens is an entity is merely a play upon the word “entity”; the second sense of “entity” (if any) is indeed derived from the first, but has a more complex signification. Consider an incomplete proposition, incomplete in the sense that some entity which ought to be involved in it is represented by an undetermined x, which may stand for any entity. Call it a propositional function; and, if φx be a propositional function, the undetermined variable x is the argument. Two propositional functions φx and ψx are “extensionally identical” if any determination of x in φx which converts φx into a true proposition also converts ψx into a true proposition, and conversely for ψ and φ. Now consider a propositional function Fχin which the variable argument χ is itself a propositional function. If Fχis true when, and only when, χ is determined to be either φ or some other propositional function extensionally equivalent to φ, then the proposition Fφis of the form which is ordinarily recognized as being about the class determined by φx taken in extension—that is, the class of entities for which φx is a true proposition when x is determined to be any one of them. A similar theory holds for relations which arise from the consideration of propositional functions with two or more variable arguments. It is then possible to define by a parallel elaboration what is meant by classes of classes, classes of relations, relations between classes, and so on. Accordingly, the number of a class of relations can be defined, or of a class of classes, and so on. This theory9is in effect a theory of theuseof classes and relations, and does not decide the philosophic question as to the sense (if any) in which a class in extension is one entity. It does indeed deny that it is an entity in the sense in which one of its members is an entity. Accordingly, it is a fallacy for any determination of x to consider “x is an x” or “x is not an x” as having the meaning of propositions. Note that for any determination of x, “x is an x” and “x is not an x,” are neither of them fallacies but are both meaningless, according to this theory. Thus Russell’s contradiction vanishes, and an examination of the other contradictions shows that they vanish also.
Applied Mathematics.—The selection of the topics of mathematical inquiry among the infinite variety open to it has been guided by the useful applications, and indeed the abstract theory has only recently been disentangled from the empirical elements connected with these applications. For example, the application of the theory of cardinal numbers to classes of physical entities involves in practice some process of counting. It is only recently that thesuccessionof processes which is involved in any act of counting has been seen to be irrelevant to the idea of number. Indeed, it is only by experience that we can know that any definite process of counting will give the true cardinal number of some class of entities. It is perfectly possible to imagine a universe in which any act of counting by a being in it annihilated some members of the class counted during the time and only during the time of its continuance. A legend of the Council of Nicea10illustrates this point: “When the Bishops took their places on their thrones, they were 318; when they rose up to be called over, it appeared that they were 319; so that they never could make the number come right, and whenever they approached the last of the series, he immediately turned into the likeness of his next neighbour.” Whatever be the historical worth of this story, it may safely be said that it cannot be disproved by deductive reasoning from the premisses of abstract logic. The most we can do is to assert that a universe in which such things are liable to happen on a large scale is unfitted for the practical application of the theory of cardinal numbers. The application of the theory of real numbers to physical quantities involves analogous considerations. In the first place, some physical process of addition is presupposed, involving some inductively inferred law of permanence during that process. Thus in the theory of masses we must know that two pounds of lead when put together will counterbalance in the scales two pounds of sugar, or a pound of lead and a pound of sugar. Furthermore, the sort of continuity of the series (in order of magnitude) of rational numbers is known to be different from that of the series of real numbers. Indeed, mathematicians now reserve “continuity” as the term for the latter kind of continuity; the mere property of having an infinite number of terms between any two terms is called “compactness.” The compactness of the series of rational numbers is consistent with quasi-gaps in it—that is, with the possible absence of limits to classes in it. Thus the class of rational numbers whose squares are less than 2 has no upper limit among the rational numbers. But among the real numbers all classes have limits. Now, owing to the necessary inexactness of measurement, it is impossible to discriminate directly whether any kind of continuous physical quantity possesses the compactness of the series of rationals or the continuity of the series of real numbers. In calculations the latter hypothesis is made because of its mathematical simplicity. But, the assumption has certainly no a priori grounds in its favour, and it is not very easy to see how to base it upon experience. For example, if it should turn out that the mass of a body is to be estimated by counting the number of corpuscles (whatever they may be) which go to form it, then a body with an irrational measure of mass is intrinsically impossible. Similarly, the continuity of space apparently rests upon sheer assumption unsupported by any a priori or experimental grounds. Thus the current applications of mathematics to the analysis of phenomena can be justified by no a priori necessity.
In one sense there is no science of applied mathematics. When once the fixed conditions which any hypothetical group of entities are to satisfy have been precisely formulated, the deduction of the further propositions, which also will hold respecting them, can proceed in complete independence of the question as to whether or no any such group of entities can be found in the world of phenomena. Thus rational mechanics, based on the Newtonian Laws, viewed as mathematics is independent of its supposed application, and hydrodynamics remains a coherent and respected science though it is extremely improbable that any perfect fluid exists in the physical world. But this unbendingly logical point of view cannot be the last word upon the matter. For no one can doubt the essential difference between characteristic treatises upon “pure” and “applied” mathematics. The difference is a difference in method. In pure mathematics the hypotheses which a set of entities are to satisfy are given, and a group of interesting deductions are sought. In “applied mathematics” the “deductions” are given in the shape of the experimental evidence of natural science, and the hypotheses from which the “deductions” can be deduced are sought. Accordingly, every treatise on applied mathematics, properly so-called, is directed to the criticism of the “laws” from which the reasoning starts, or to a suggestion of results which experiment may hope to find. Thus if it calculates the result of some experiment, it is not the experimentalist’s well-attested results which are on their trial, but the basis of the calculation. Newton’sHypotheses non fingowas a proud boast, but it rests upon an entire misconception of the capacities of the mind of man in dealing with external nature.
Synopsis of Existing Developments of Pure Mathematics.—A complete classification of mathematical sciences, as they at present exist, is to be found in theInternational Catalogue of Scientific Literaturepromoted by the Royal Society. The classification in question was drawn up by an international committee of eminent mathematicians, and thus has the highest authority. It would be unfair to criticize it from an exacting philosophical point of view. The practical object of the enterprise required that the proportionate quantity of yearly output in the various branches, and that the liability of various topics as a matter of fact to occur in connexion with each other, should modify the classification.Section A deals with pure mathematics. Under the general heading “Fundamental Notions” occur the subheadings “Foundations of Arithmetic,” with the topics rational, irrational and transcendental numbers, and aggregates; “Universal Algebra,” with the topics complex numbers, quaternions, ausdehnungslehre, vector analysis, matrices, and algebra of logic; and “Theory of Groups,” with the topics finite and continuous groups. For the subjects of this general heading see the articlesAlgebra, Universal;Groups, Theory of;Infinitesimal Calculus;Number;Quaternions;Vector Analysis. Under the general heading “Algebra and Theory of Numbers” occur the subheadings “Elements of Algebra,” with the topics rational polynomials, permutations, &c., partitions, probabilities; “Linear Substitutions,” with the topics determinants, &c., linear substitutions, general theory of quantics; “Theory of Algebraic Equations,” with the topics existence of roots, separation of and approximation to, theory of Galois, &c.; “Theory of Numbers,” with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers. For the subjects of this general heading see the articlesAlgebra;Algebraic Forms;Arithmetic;Combinatorial Analysis;Determinants;Equation;Fraction, Continued;Interpolation;Logarithms;Magic Square;Probability. Under the general heading “Analysis” occur the subheadings “Foundations of Analysis,” with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; “Theory of Functions of Complex Variables,” with the topics functions of one variable and of several variables; “Algebraic Functions and their Integrals,” with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; “Other Special Functions,” with the topics Euler’s, Legendre’s, Bessel’s and automorphic functions; “Differential Equations,” with the topics existence theorems, methods of solution, general theory; “Differential Forms and Differential Invariants,” with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; “Analytical Methods connected with Physical Subjects,” with the topics harmonic analysis, Fourier’s series, the differential equations of applied mathematics, Dirichlet’s problem; “Difference Equations and Functional Equations,” with the topics recurring series, solution of equations of finite differences and functional equations. For the subjects of this heading see the articlesDifferential Equations;Fourier’s Series;Continued Fractions;Function;Function of Real Variables;Function Complex;Groups, Theory of;Infinitesimal Calculus;Maxima and Minima;Series;Spherical Harmonics;Trigonometry;Variations, Calculus of. Under the general heading “Geometry” occur the subheadings “Foundations,” with the topics principles of geometry, non-Euclidean geometries, hyperspace, methods of analytical geometry; “Elementary Geometry,” with the topics planimetry, stereometry, trigonometry, descriptive geometry; “Geometry of Conics and Quadrics,” with the implied topics; “Algebraic Curves and Surfaces of Degree higher than the Second,” with the implied topics; “Transformations and General Methods for Algebraic Configurations,” with the topics collineation, duality, transformations, correspondence, groups of points on algebraic curves and surfaces, genus of curves and surfaces, enumerative geometry, connexes, complexes, congruences, higher elements in space, algebraic configurations in hyperspace; “Infinitesimal Geometry: applications of Differential and Integral Calculus to Geometry,” with the topics kinematic geometry, curvature, rectification and quadrature, special transcendental curves and surfaces; “Differential Geometry: applications of Differential Equations to Geometry,” with the topics curves on surfaces, minimal surfaces, surfaces determined by differential properties, conformal and other representation of surfaces on others, deformation of surfaces, orthogonal and isothermic surfaces. For the subjects under this heading see the articlesConic Sections;Circle;Curve;Geometrical Continuity;Geometry,Axioms of;Geometry,Euclidean;Geometry,Projective;Geometry,Analytical;Geometry,Line;Knots, Mathematical Theory of;Mensuration;Models;Projection;Surface;Trigonometry.This survey of the existing developments of pure mathematics confirms the conclusions arrived at from the previous survey of the theoretical principles of the subject. Functions, operations, transformations, substitutions, correspondences, are but names for various types of relations. A group is a class of relations possessing a special property. Thus the modern ideas, which have so powerfully extended and unified the subject, have loosened its connexion with “number” and “quantity,” while bringing ideas of form and structure into increasing prominence. Number must indeed ever remain the great topic of mathematical interest, because it is in reality the great topic of applied mathematics. All the world, including savages who cannot count beyond five, daily “apply” theorems of number. But the complexity of the idea of number is practically illustrated by the fact that it is best studied as a department of a science wider than itself.Synopsis of Existing Developments of Applied Mathematics.—Section B of theInternational Cataloguedeals with mechanics. The heading “Measurement of Dynamical Quantities” includes the topics units, measurements, and the constant of gravitation. The topics of the other headings do not require express mention. These headings are: “Geometry and Kinematics of Particles and Solid Bodies”; “Principles of Rational Mechanics”; “Statics of Particles, Rigid Bodies, &c.”; “Kinetics of Particles, Rigid Bodies, &c.”; “General Analytical Mechanics”; “Statics and Dynamics of Fluids”; “Hydraulics and Fluid Resistances”; “Elasticity.” For the subjects of this general heading see the articlesMechanics;Dynamics, Analytical;Gyroscope;Harmonic Analysis;Wave;Hydromechanics;Elasticity;Motion, Laws of;Energy;Energetics;Astronomy(Celestial Mechanics);Tide. Mechanics (including dynamical astronomy) is that subject among those traditionally classed as “applied” which has been most completely transfused by mathematics—that is to say, which is studied with the deductive spirit of the pure mathematician, and not with the covert inductive intention overlaid with the superficial forms of deduction, characteristic of the applied mathematician.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 (seeStatisticsandProbability) should not be lost sight of; the leading fields for these applications are insurance, sociology, variation in zoology and economics.
Synopsis of Existing Developments of Pure Mathematics.—A complete classification of mathematical sciences, as they at present exist, is to be found in theInternational Catalogue of Scientific Literaturepromoted by the Royal Society. The classification in question was drawn up by an international committee of eminent mathematicians, and thus has the highest authority. It would be unfair to criticize it from an exacting philosophical point of view. The practical object of the enterprise required that the proportionate quantity of yearly output in the various branches, and that the liability of various topics as a matter of fact to occur in connexion with each other, should modify the classification.
Section A deals with pure mathematics. Under the general heading “Fundamental Notions” occur the subheadings “Foundations of Arithmetic,” with the topics rational, irrational and transcendental numbers, and aggregates; “Universal Algebra,” with the topics complex numbers, quaternions, ausdehnungslehre, vector analysis, matrices, and algebra of logic; and “Theory of Groups,” with the topics finite and continuous groups. For the subjects of this general heading see the articlesAlgebra, Universal;Groups, Theory of;Infinitesimal Calculus;Number;Quaternions;Vector Analysis. Under the general heading “Algebra and Theory of Numbers” occur the subheadings “Elements of Algebra,” with the topics rational polynomials, permutations, &c., partitions, probabilities; “Linear Substitutions,” with the topics determinants, &c., linear substitutions, general theory of quantics; “Theory of Algebraic Equations,” with the topics existence of roots, separation of and approximation to, theory of Galois, &c.; “Theory of Numbers,” with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers. For the subjects of this general heading see the articlesAlgebra;Algebraic Forms;Arithmetic;Combinatorial Analysis;Determinants;Equation;Fraction, Continued;Interpolation;Logarithms;Magic Square;Probability. Under the general heading “Analysis” occur the subheadings “Foundations of Analysis,” with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; “Theory of Functions of Complex Variables,” with the topics functions of one variable and of several variables; “Algebraic Functions and their Integrals,” with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; “Other Special Functions,” with the topics Euler’s, Legendre’s, Bessel’s and automorphic functions; “Differential Equations,” with the topics existence theorems, methods of solution, general theory; “Differential Forms and Differential Invariants,” with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; “Analytical Methods connected with Physical Subjects,” with the topics harmonic analysis, Fourier’s series, the differential equations of applied mathematics, Dirichlet’s problem; “Difference Equations and Functional Equations,” with the topics recurring series, solution of equations of finite differences and functional equations. For the subjects of this heading see the articlesDifferential Equations;Fourier’s Series;Continued Fractions;Function;Function of Real Variables;Function Complex;Groups, Theory of;Infinitesimal Calculus;Maxima and Minima;Series;Spherical Harmonics;Trigonometry;Variations, Calculus of. Under the general heading “Geometry” occur the subheadings “Foundations,” with the topics principles of geometry, non-Euclidean geometries, hyperspace, methods of analytical geometry; “Elementary Geometry,” with the topics planimetry, stereometry, trigonometry, descriptive geometry; “Geometry of Conics and Quadrics,” with the implied topics; “Algebraic Curves and Surfaces of Degree higher than the Second,” with the implied topics; “Transformations and General Methods for Algebraic Configurations,” with the topics collineation, duality, transformations, correspondence, groups of points on algebraic curves and surfaces, genus of curves and surfaces, enumerative geometry, connexes, complexes, congruences, higher elements in space, algebraic configurations in hyperspace; “Infinitesimal Geometry: applications of Differential and Integral Calculus to Geometry,” with the topics kinematic geometry, curvature, rectification and quadrature, special transcendental curves and surfaces; “Differential Geometry: applications of Differential Equations to Geometry,” with the topics curves on surfaces, minimal surfaces, surfaces determined by differential properties, conformal and other representation of surfaces on others, deformation of surfaces, orthogonal and isothermic surfaces. For the subjects under this heading see the articlesConic Sections;Circle;Curve;Geometrical Continuity;Geometry,Axioms of;Geometry,Euclidean;Geometry,Projective;Geometry,Analytical;Geometry,Line;Knots, Mathematical Theory of;Mensuration;Models;Projection;Surface;Trigonometry.
This survey of the existing developments of pure mathematics confirms the conclusions arrived at from the previous survey of the theoretical principles of the subject. Functions, operations, transformations, substitutions, correspondences, are but names for various types of relations. A group is a class of relations possessing a special property. Thus the modern ideas, which have so powerfully extended and unified the subject, have loosened its connexion with “number” and “quantity,” while bringing ideas of form and structure into increasing prominence. Number must indeed ever remain the great topic of mathematical interest, because it is in reality the great topic of applied mathematics. All the world, including savages who cannot count beyond five, daily “apply” theorems of number. But the complexity of the idea of number is practically illustrated by the fact that it is best studied as a department of a science wider than itself.
Synopsis of Existing Developments of Applied Mathematics.—Section B of theInternational Cataloguedeals with mechanics. The heading “Measurement of Dynamical Quantities” includes the topics units, measurements, and the constant of gravitation. The topics of the other headings do not require express mention. These headings are: “Geometry and Kinematics of Particles and Solid Bodies”; “Principles of Rational Mechanics”; “Statics of Particles, Rigid Bodies, &c.”; “Kinetics of Particles, Rigid Bodies, &c.”; “General Analytical Mechanics”; “Statics and Dynamics of Fluids”; “Hydraulics and Fluid Resistances”; “Elasticity.” For the subjects of this general heading see the articlesMechanics;Dynamics, Analytical;Gyroscope;Harmonic Analysis;Wave;Hydromechanics;Elasticity;Motion, Laws of;Energy;Energetics;Astronomy(Celestial Mechanics);Tide. Mechanics (including dynamical astronomy) is that subject among those traditionally classed as “applied” which has been most completely transfused by mathematics—that is to say, which is studied with the deductive spirit of the pure mathematician, and not with the covert inductive intention overlaid with the superficial forms of deduction, characteristic of the applied mathematician.
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 (seeStatisticsandProbability) 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 lasted11from the time of Thales,c.600B.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), andDie Grundlagen der Arithmetik(Breslau, 1884); also cf. Bertrand Russell,The Principles of Mathematics(Cambridge, 1903), and his article on “Mathematical Logic” inAmer. 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), andWas sind und was sollen die Zahlen?(1st ed., 1887), both tracts translated into English under the titleEssays 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’sVorlesungen 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).
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), andDie Grundlagen der Arithmetik(Breslau, 1884); also cf. Bertrand Russell,The Principles of Mathematics(Cambridge, 1903), and his article on “Mathematical Logic” inAmer. 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), andWas sind und was sollen die Zahlen?(1st ed., 1887), both tracts translated into English under the titleEssays 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’sVorlesungen 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.)
1Cf.La Logique de Leibnitz, ch. vii., by L. Couturat (Paris, 1901).2Cf.The Principles of Mathematics, by Bertrand Russell (Cambridge, 1903).3Cf.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.4Cf. Russell,loc. cit., pp. 199-256.5The first unqualified explicit statement ofpartof this definition seems to be by B. Peirce, “Mathematics is the science which draws necessary conclusions” (Linear Associative Algebra, § i. (1870), republished in theAmer. Journ. of Math., vol. iv. (1881)). But it will be noticed that the second half of the definition in the text—“from the general premisses of all reasoning”—is left unexpressed. The full expression of the idea and its development into a philosophy of mathematics is due to Russell,loc. cit.6“Una questione sui numeri transfiniti,”Rend. del circolo mat. di Palermo, vol. xi. (1897); and Russell,loc. cit., ch. xxxviii.7Cf. Russell,loc. cit., ch. x.8Cf.Pragmatism: a New Name for some Old Ways of Thinking(1907).9Due to Bertrand Russell, cf. “Mathematical Logic as based on the Theory of Types,”Amer. Journ. of Math.vol. xxx. (1908). It is more fully explained by him, with later simplifications, inPrincipia mathematica(Cambridge).10Cf. Stanley’sEastern Church, Lecture v.11Cf.A Short History of Mathematics, by W. W. R. Ball.
1Cf.La Logique de Leibnitz, ch. vii., by L. Couturat (Paris, 1901).
2Cf.The Principles of Mathematics, by Bertrand Russell (Cambridge, 1903).
3Cf.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.
4Cf. Russell,loc. cit., pp. 199-256.
5The first unqualified explicit statement ofpartof this definition seems to be by B. Peirce, “Mathematics is the science which draws necessary conclusions” (Linear Associative Algebra, § i. (1870), republished in theAmer. Journ. of Math., vol. iv. (1881)). But it will be noticed that the second half of the definition in the text—“from the general premisses of all reasoning”—is left unexpressed. The full expression of the idea and its development into a philosophy of mathematics is due to Russell,loc. cit.
6“Una questione sui numeri transfiniti,”Rend. del circolo mat. di Palermo, vol. xi. (1897); and Russell,loc. cit., ch. xxxviii.
7Cf. Russell,loc. cit., ch. x.
8Cf.Pragmatism: a New Name for some Old Ways of Thinking(1907).
9Due to Bertrand Russell, cf. “Mathematical Logic as based on the Theory of Types,”Amer. Journ. of Math.vol. xxx. (1908). It is more fully explained by him, with later simplifications, inPrincipia mathematica(Cambridge).
10Cf. Stanley’sEastern Church, Lecture v.
11Cf.A Short History of Mathematics, by W. W. R. Ball.
MATHER, COTTON(1663-1728), American Congregational clergyman and author, was born in Boston, Massachusetts, on the 12th of February 1663. He was the grandson of Richard Mather, and the eldest child of Increase Mather (q.v.), and Maria, daughter of John Cotton. After studying under the famous Ezekiel Cheever (1614-1708), he entered Harvard College at twelve, and graduated in 1678. While teaching (1678-1685), he began the study of theology, but soon, on account of an impediment in his speech, discontinued it and took up medicine. Later, however, he conquered the difficulty and finished his preparation for the ministry. He was elected assistant pastor in his father’s church, the North, or Second, Church of Boston, in 1681 and was ordained as his father’s colleague in 1685. In 1688, when his father went to England as agent for the colony, he was left at twenty-five in charge of the largest congregation in New England, and he ministered to it for the rest of his life. He soon became one of the most influential men in the colonies. He had much to do with the witchcraft persecution of his day; in 1692 when the magistrates appealed to the Boston clergy for advice in regard to the witchcraft cases in Salem he drafted their reply, upon which the prosecutions were based; in 1689 he had writtenMemorable Providences Relating to Witchcraft and Possessions, and even his earlier diaries have many entries showing his belief in diabolical possession and his fear and hatred of it. Thinking as he did that the New World had been the undisturbed realm of Satan before the settlements were made in Massachusetts, he considered it natural that the Devil should make a peculiar effort to bring moral destruction on these godly invaders. He used prayer and fasting to deliver himself from evil enchantment; and when he saw ecstatic and mystical visions promising him the Lord’s help and great usefulness in the Lord’s work, he feared that these revelations might be of diabolic origin. He used his great influence to bring the suspected persons to trial and punishment. He attended the trials, investigated many of the cases himself, and wrote sermons on witchcraft, theMemorable ProvidencesandThe Wonders of the Invisible World(1693), which increased the excitement of the people. Accordingly, when the persecutions ceased and the reaction set in, much of the blame was laid upon him; the influence of Judge Samuel Sewall, after he had come to think his part in the Salem delusion a great mistake, was turned against the Mathers; and the liberal leaders of Congregationalism in Boston, notably the Brattles, found this a vulnerable point in Cotton Mather’s armour and used their knowledge to much effect, notably by assisting Robert Calef (d.c.1723) in the preparation ofMore Wonders of the Invisible World(1700) a powerful criticism of Cotton Mather’s part in the delusion at Salem.
Mather took some part as adviser in the Revolution of 1689 in Massachusetts. In 1690 he became a member o£ the Corporation (probably the youngest ever chosen as Fellow) of Harvard College, and in 1707 he was greatly disappointed at his failure to be chosen president of that institution. He received the degree of D.D. from the University of Glasgow in 1710, and in 1713 was made a Fellow of the Royal Society. Like his father he was deeply grieved by the liberal theology and Church polity of the new Brattle Street Congregation, and conscientiously opposed its pastor Benjamin Colman, who had been irregularly ordained in England and by a Presbyterian body; but with his father he took part in 1700 in services in Colman’s church. Harvard College was now controlled by the Liberals of the Brattle Street Church, and as it grew farther and farther away from Calvinism, Mather looked with increasing favour upon the college in Connecticut; before September 1701 he had drawn up a “scheme for a college,” the oldest document now in the Yale archives; and finally (Jan. 1718) he wrote to a London merchant, Elihu Yale, and persuaded him to make a liberal gift to the college, which was named in his honour. During the small-pox epidemic of 1721 he attempted in vain to have treatment by inoculation employed, for the first time in America; and for this he was bitterly attacked on all sides, and his life was at one time in danger; but, nevertheless, he used the treatment on his son, who recovered, and he wroteAn Account of the Method and further Success of Inoculating for the Small Pox in London(1721). In addition he advocated temperance, missions, Bible societies, and the education of the negro; favoured the establishing of libraries for working men and of religious organizations for young people, and organized societies for other branches of philanthropic work. His later years were clouded with many sorrows and disappointments; his relations with Governor Joseph Dudley were unfriendly; he lost much of his former prestige in the Church—his own congregation dwindled—and in the college; his uncle John Cotton was expelled from hischarge in the Plymouth Church; his son Increase turned out a ne’er-do-well; four of his children and his second wife died in November 1713; his wife’s brothers and the husbands of his sisters were ungodly and violent men; his favourite daughter Katherine, who “understood Latin and read Hebrew fluently,” died in 1716; his third wife went mad in 1719; his personal enemies circulated incredible scandals about him; and in 1724-1725 he saw a Liberal once more preferred to him as a new president of Harvard. He died in Boston on the 13th of February 1728 and is buried in the Copps Hill burial-ground, Boston. He was thrice married—to Abigail Phillips (d. 1702) in 1686, to Mrs Elizabeth Hubbard (d. 1713) in 1703, and in 1715 to Mrs Lydia George (d. 1734). Of his fifteen children only two survived him.
Though self-conscious and vain, Cotton Mather had on the whole a noble character. He believed strongly in the power of prayer and repeatedly had assurances that his prayers were heard; and when he was disappointed by non-fulfilment his grief and depression were terrible. His spiritual nature was high-strung and delicate; and this condition was aggravated by his constant study, his long fasts and his frequent vigils—in one year, according to his diary, he kept sixty fasts and twenty vigils. In his later years his diaries have less and less of personal detail, and repeated entries prefaced by the letters “G.D.” meaning Good Device, embodying precepts of kindliness and practical Christianity. He was remarkable for his godliness, his enthusiasm for knowledge, and his prodigious memory. He became a skilled linguist, a widely read scholar—though much of his learning was more curious than useful—a powerful preacher, a valued citizen, and a voluminous writer, and did a vast deal for the intellectual and spiritual quickening of New England. He worked with might and main for the continuation of the old theocracy, but before he died it had given way before an increasing Liberalism—even Yale was infected with the Episcopalianism that he hated.
Among his four hundred or more published works, many of which are sermons, tracts and letters, the most notable is hisMagnalia Christi Americana: or the Ecclesiastical History of New England, from Its First Planting in the Year 1620 unto the Year of Our Lord, 1698. Begun in 1693 and finished in 1697, this work was published in London, in 1702, in one volume, and was republished in Hartford in 1820 and in 1853-1855, in two volumes. It is in seven books and concerns itself mainly with the settlement and religious history of New England. It is often inaccurate, and it abounds in far-fetched conceits and odd and pedantic features. Its style, though in the main rather unnatural and declamatory, is at its best spontaneous, dignified and rhythmical; the book is valuable for occasional facts and for its picture of the times, and it did much to make Mather the most eminent American writer of his day. His other writings includeA Poem Dedicated to the Memory of the Reverend and Excellent Mr Urian Oakes(1682);The Present State of New England(1690);The Life of the Renowned John Eliot(1691), later included in Book III. of theMagnalia; The Short History of New England(1694);Bonifacius, usually known asEssays To Do Good(Boston, 1710; Glasgow, 1825; Boston, 1845), one of his principal books and one which had a shaping influence on the life of Benjamin Franklin;Psalterium Americanum(1718), a blank verse translation of the Psalms from the original Hebrew;The Christian Philosopher: A Collection of the Best Discoveries in Nature, with Religious Improvements(1721);Parentator(1724), a memoir of his father;Ratio Disciplinae(1726), an account of the discipline in New England churches;Manuductio ad Ministerium: Directions for a Candidate of the Ministry(1726), one of the most readable of his books. He also left a number of works in manuscript, including diaries, a medical treatise and a huge commentary on the Bible, entitled “Biblia Americana.”SeeThe Life of Cotton Mather(Boston, 1729), by his son, Samuel Mather; William B. O. Peabody,The Life of Cotton Mather(1836) (in Jared Sparks’s “Library of American Biography,” vol. vi.); Enoch Pond,The Mather Family(Boston, 1844); John L. Sibley,Biographical Sketches of Graduates of Harvard University, vol. iii. (Cambridge, 1885); Barrett Wendell,Cotton Mather, the Puritan Priest(New York, 1891), a remarkably sympathetic study and particularly valuable for its insight into (and its defence of) Mather’s attitude toward witchcraft; Abijah P. Marvin,The Life and Times of Cotton Mather(Boston, 1892); M. C. Tyler,A History of American Literature during the Colonial Period, vol. ii. (New York, 1878); and Barrett Wendell,A Literary History of America(New York, 1900).
Among his four hundred or more published works, many of which are sermons, tracts and letters, the most notable is hisMagnalia Christi Americana: or the Ecclesiastical History of New England, from Its First Planting in the Year 1620 unto the Year of Our Lord, 1698. Begun in 1693 and finished in 1697, this work was published in London, in 1702, in one volume, and was republished in Hartford in 1820 and in 1853-1855, in two volumes. It is in seven books and concerns itself mainly with the settlement and religious history of New England. It is often inaccurate, and it abounds in far-fetched conceits and odd and pedantic features. Its style, though in the main rather unnatural and declamatory, is at its best spontaneous, dignified and rhythmical; the book is valuable for occasional facts and for its picture of the times, and it did much to make Mather the most eminent American writer of his day. His other writings includeA Poem Dedicated to the Memory of the Reverend and Excellent Mr Urian Oakes(1682);The Present State of New England(1690);The Life of the Renowned John Eliot(1691), later included in Book III. of theMagnalia; The Short History of New England(1694);Bonifacius, usually known asEssays To Do Good(Boston, 1710; Glasgow, 1825; Boston, 1845), one of his principal books and one which had a shaping influence on the life of Benjamin Franklin;Psalterium Americanum(1718), a blank verse translation of the Psalms from the original Hebrew;The Christian Philosopher: A Collection of the Best Discoveries in Nature, with Religious Improvements(1721);Parentator(1724), a memoir of his father;Ratio Disciplinae(1726), an account of the discipline in New England churches;Manuductio ad Ministerium: Directions for a Candidate of the Ministry(1726), one of the most readable of his books. He also left a number of works in manuscript, including diaries, a medical treatise and a huge commentary on the Bible, entitled “Biblia Americana.”
SeeThe Life of Cotton Mather(Boston, 1729), by his son, Samuel Mather; William B. O. Peabody,The Life of Cotton Mather(1836) (in Jared Sparks’s “Library of American Biography,” vol. vi.); Enoch Pond,The Mather Family(Boston, 1844); John L. Sibley,Biographical Sketches of Graduates of Harvard University, vol. iii. (Cambridge, 1885); Barrett Wendell,Cotton Mather, the Puritan Priest(New York, 1891), a remarkably sympathetic study and particularly valuable for its insight into (and its defence of) Mather’s attitude toward witchcraft; Abijah P. Marvin,The Life and Times of Cotton Mather(Boston, 1892); M. C. Tyler,A History of American Literature during the Colonial Period, vol. ii. (New York, 1878); and Barrett Wendell,A Literary History of America(New York, 1900).
Cotton Mather’s son,Samuel Mather(1706-1785), also a clergyman, graduated at Harvard in 1723, was pastor of the North Church, Boston, from 1732 to 1742, when, owing to a dispute among his congregation over revivals, he resigned to take charge of a church established for him in North Bennett Street.
Among his works areThe Life of Cotton Mather(1729);An Apology for the Liberties of the Churches in New England(1738), andAmerica Known to the Ancients(1773).
Among his works areThe Life of Cotton Mather(1729);An Apology for the Liberties of the Churches in New England(1738), andAmerica Known to the Ancients(1773).
(W. L. C.*)
MATHER, INCREASE(1639-1723), American Congregational minister, was born in Dorchester, Massachusetts, on the 21st of June 1639, the youngest son of Richard Mather.1He entered Harvard in 1651, and graduated in 1656. In 1657, on his eighteenth birthday, he preached his first sermon; in the same year he went to visit his eldest brother in Dublin, and studied there at Trinity College, where he graduated M.A. in 1658. He was chaplain to the English garrison at Guernsey in April-December 1659 and again in 1661; and in the latter year, refusing valuable livings in England offered on condition of conformity, he returned to America. In the winter of 1661-1662 he began to preach to the Second (or North) Church of Boston, and was ordained there on the 27th of May 1664. As a delegate from Dorchester, his father’s church, to the Synod of 1662, he opposed the Half-Way Covenant adopted by the Synod and defended by Richard Mather and by Jonathan Mitchell (1624-1668) of Cambridge; but soon afterwards he “surrendered a glad captive” to “the truth so victoriously cleared by Mr Mitchell,” and like his father and his son became one of the chief exponents of the Half-Way Covenant. He was bitterly opposed, however, to the liberal practices that followed the Half-Way Covenant and (after 1677) in particular to “Stoddardeanism,” the doctrine of Solomon Stoddard (1643-1729) that all “such Persons as have a good Conversation and a Competent Knowledge may come to the Lord’s Supper,” only those of openly immoral life being excluded. In May 1679 Mather was a petitioner to the General Court for the call of a Synod to consider the reformation in New England of “the Evils that have Provoked the Lord to bring his Judgments,”2and when the “Reforming Synod” met in September it appointed him one of a committee to draft a creed; this committee reported in May 1680, at the Synod’s second session, of which Mather was moderator, the Savoy Declaration (slightly modified, notably in ch. xxiv., “Of the Civil Magistrate”), which was approved but was not made mandatory on the churches by the General Court, and in 1708 was reaffirmed at Saybrook, Connecticut. With the Cambridge Platform of 1646, drafted by his father, the Confession of 1680, for which Increase Mather was largely responsible, was printed as a book of doctrine and government for the churches of Massachusetts.
After the threat of aQuo Warrantowrit in 1683 for the surrender of the Massachusetts charter, Mather used all his tremendous influence to persuade the colonists not to give up the charter; and the Boston freemen unanimously voted against submission. The royal agents immediately afterwards sent to London a treasonable letter, falsely attributed to Mather; but its spuriousness seems to have been suspected in England and Mather was not “fetch’d over and made a Sacrifice.” He became a leader in the opposition to Sir Edmund Andros, to his secretary Edward Randolph, and to Governor Joseph Dudley. He was chosen by the General Court to represent the colony’s interests in England, eluded officers sent to arrest him,3and in disguise boarded a ship on which he reached Weymouth on the 6th of May 1688. In London he acted with Sir Henry Ashurst, the resident agent, and had two orthree fruitless audiences with James II. His first audience with William III. was on the 9th of January 1689; he was active in influencing the Commons to vote (1689) that the New England charters should be restored; and he publishedA Narrative of the Miseries of New-England, By Reason of an Arbitrary Government Erected there Under Sir Edmund Andros(1688),A Brief Relation for the Confirmation of Charter Privileges(1691), and other pamphlets. In 1690 he was joined by Elisha Cooke (1638-1715) and Thomas Oakes (1644-1719), additional agents, who were uncompromisingly for the renewal of the old charter. Mather, however, was instrumental in securing a new charter (signed on Oct. 7, 1691), and prevented the annexation of the Plymouth Colony to New York. The nomination of officers left to the Crown was reserved to the agents. Mather had expressed strong dissatisfaction with the clause giving the governor the right of veto, and regretted the less theocratic tone of the charter which made all freemen (and not merely church members) electors. With Sir William Phips, the new governor, a member of Mather’s church, he arrived in Boston on the 14th of May 1692. The value of his services to the colony at this time is not easily over-estimated. In England he won the friendship of divines like Baxter, Tillotson and Burnet, and effectively promoted the union in 1691 of English Presbyterians and Congregationalists. He was at heavy expense throughout his stay, and even greater than his financial loss was his loss of authority and control in the church and in Harvard College because of his absence.
Mather had been acting president of Harvard College in 1681-1682, and in June 1685 he again became acting president (or rector), but still preached every Sunday in Boston and would not comply with an order of the General Court that he should reside in Cambridge. In 1701 after a short residence there he returned to Boston and wrote to the General Court to “think of another President for the Colledge.” The opposition to him had been increasing in strength, his resignation was accepted, and Samuel Willard took charge of the college as vice-president, although he also refused to reside in Cambridge. That Mather’s administration of the college was excellent is admitted even by his harsh critic, Josiah Quincy, in hisHistory of Harvard University.4The Liberal party, which now came into control in the college repeatedly disappointed the hopes of Cotton Mather (q.v.) that he might be chosen president, and by its ecclesiastical laxness and its broader views of Church polity forced the Mathers to turn from Harvard to Yale as a truer school of the prophets.
The Liberal leaders, John Leverett (1662-1724), William Brattle (1662-1713)—who graduated with Leverett in 1680, and with him as tutor controlled the college during Increase Mather’s absence in England—William Brattle’s eldest brother, Thomas Brattle (1658-1713), and Ebenezer Pemberton (1671-1717), pastor of the Old South Church, desired an “enrichment of the service,” and greater liberality in the matter of baptism. In 1697 the Second Boston Church, in which Cotton Mather had been his father’s colleague since 1685, upbraided the Charlestown Church “for betraying the liberties of the churches in their late putting into the hands of the whole inhabitants the choice of a minister.” In 1699 Increase Mather publishedThe Order of the Gospel, which severely (although indirectly), criticized the methods of the “Liberals” in establishing the Brattle Street Church and especially the ordination of their minister Benjamin Colman by a Presbyterian body in London; the Liberals replied withThe Gospel Order Revived, which was printed in New York to lend colour to the (partly true) charge of its authors that the printers of Massachusetts would print nothing hostile to Increase Mather.5The autocracy of the Mathers in church, college, colony and press, had slipped from them. The later years of Mather’s life were spent almost entirely in the work of the ministry, now beginning to be a less varied career than when he entered on it. He died on the 23rd of August 1723. He married in 1662 Maria, daughter of Sarah and John Cotton. His first wife died in 1714; and in 1715 he married Ann Lake, widow of John Cotton, of Hampton, N.H., a grandson of John Cotton of Boston.
Increase Mather was a great preacher with a simple style and a splendid voice, which had a “Tonitruous Cogency,” to quote his son’s phrase. His style was much simpler and more vernacular than his son’s. He was an assiduous student, commonly spending sixteen hours a day among his books; but his learning (to quote Justin Winsor’s contrast between Increase and Cotton Mather) “usually left his natural ability and his education free from entanglements.” He was not so much self-seeking and personally ambitious as eager to advance the cause of the church in which he so implicitly believed. That it is a mistake to consider him a narrow churchman is shown by his assisting in 1718 at the ordination of Elisha Callender in the First Baptist Church of Boston. Like the most learned men of his time he was superstitious and a firm believer in “praesagious impressions”; hisEssay for the Recording of Illustrious Providences: Wherein an Account is Given of many Remarkable and very Memorable Events which have Hapned in this Last Age, Especially in New England(1684) shows that he believed only less thoroughly than his son in witchcraft, though in hisCases of Conscience Concerning Evil Spirits(1693) he considered some current proofs of witchcraft inadequate. The revulsion of feeling after the witchcraft delusion undermined his authority greatly, and Robert’s Calef’sMore Wonders of the Spiritual World(1700) was a personal blow to him as well as to his son. With Jonathan Edwards, than whom he was much more of a man of affairs, and with Benjamin Franklin, whose mission in England somewhat resembled Mather’s, he may be ranked among the greatest Americans of the period before the War of Independence.