Chapter XXI: Part 21

_Cardinal Numbers._--A one-one relation between the members of two
classes [alpha] and [beta] is any method of correlating all the
members of [alpha] to all the members of [beta], so that any member of
[alpha] has one and only one correlate in [beta], and any member of
[beta] has one and only one correlate in [alpha]. Two classes between
which a one-one relation exists have the same cardinal number and are
called cardinally similar; and the cardinal number of the class
[alpha] is a certain class whose members are themselves
classes--namely, it is the class composed of all those classes for
which a one-one correlation with [alpha] exists. Thus the cardinal
number of [alpha] is itself a class, and furthermore [alpha] is a
member of it. For a one-one relation can be established between the
members of [alpha] and [alpha] by the simple process of correlating
each member of [alpha] with itself. Thus the cardinal number one is
the class of unit classes, the cardinal number two is the class of
doublets, and so on. Also a unit class is any class with the property
that it possesses a member _x_ such that, if _y_ is any member of the
class, then _x_ and _y_ are identical. A doublet is any class which
possesses a member _x_ such that the modified class formed by all the
other members except _x_ is a unit class. And so on for all the finite
cardinals, which are thus defined successively. The cardinal number
zero is the class of classes with no members; but there is only one
such class, namely--the null class. Thus this cardinal number has only
one member. The operations of addition and multiplication of two given
cardinal numbers can be defined by taking two classes [alpha] and
[beta], satisfying the conditions (1) that their cardinal numbers are
respectively the given numbers, and (2) that they contain no member in
common, and then by defining by reference to [alpha] and [beta] two
other suitable classes whose cardinal numbers are defined to be
respectively the required sum and product of the cardinal numbers in
question. We need not here consider the details of this process.

With these definitions it is now possible to _prove_ the following six
premisses applying to finite cardinal numbers, from which Peano[3] has
shown that all arithmetic can be deduced:--

i. Cardinal numbers form a class.

ii. Zero is a cardinal number.

iii. If a is a cardinal number, a + 1 is a cardinal number.

iv. If s is any class and zero is a member of it, also if when x is a
cardinal number and a member of s, also x+1 is a member of s, then the
whole class of cardinal numbers is contained in s.

v. If a and b are cardinal numbers, and a + 1 = b + 1, then a = b.

vi. If a is a cardinal number, then a + 1 [/=] 0.

It may be noticed that (iv) is the familar principle of mathematical
induction. Peano in an historical note refers its first explicit
employment, although without a general enunciation, to Maurolycus in
his work, _Arithmeticorum libri duo_ (Venice, 1575).

But now the difficulty of confining mathematics to being the science
of number and quantity is immediately apparent. For there is no
self-contained science of cardinal numbers. The proof of the six
premisses requires an elaborate investigation into the general
properties of classes and relations which can be deduced by the
strictest reasoning from our ultimate logical principles. Also it is
purely arbitrary to erect the consequences of these six principles
into a separate science. They are excellent principles of the highest
value, but they are in no sense the necessary premisses which must be
proved before any other propositions of cardinal numbers can be
established. On the contrary, the premisses of arithmetic can be put
in other forms, and, furthermore, an indefinite number of propositions
of arithmetic can be proved directly from logical principles without
mentioning them. Thus, while arithmetic may be defined as that branch
of deductive reasoning concerning classes and relations which is
concerned with the establishment of propositions concerning cardinal
numbers, it must be added that the introduction of cardinal numbers
makes no great break in this general science. It is no more than an
interesting subdivision in a general theory.

_Ordinal Numbers._--We must first understand what is meant by "order,"
that is, by "serial arrangement." An order of a set of things is to be
sought in that relation holding between members of the set which
constitutes that order. The set viewed as a class has many orders.
Thus the telegraph posts along a certain road have a space-order very
obvious to our senses; but they have also a time-order according to
dates of erection, perhaps more important to the postal authorities
who replace them after fixed intervals. A set of cardinal numbers have
an order of magnitude, often called _the_ order of the set because of
its insistent obviousness to us; but, if they are the numbers drawn in
a lottery, their time-order of occurrence in that drawing also ranges
them in an order of some importance. Thus the order is defined by the
"serial" relation. A relation (R) is serial[4] when (1) it implies
diversity, so that, if x has the relation R to y, x is diverse from y;
(2) it is transitive, so that if x has the relation R to y, and y to
z, then x has the relation R to z; (3) it has the property of
connexity, so that if x and y are things to which any things bear the
relation R, or which bear the relation R to any things, then _either_
x is identical with y, _or_ x has the relation R to y, _or_ y has the
relation R to x. These conditions are necessary and sufficient to
secure that our ordinary ideas of "preceding" and "succeeding" hold in
respect to the relation R. The "field" of the relation R is the class
of things ranged in order by it. Two relations R and R´ are said to be
ordinally similar, if a one-one relation holds between the members of
the two fields of R and R´, such that if x and y are any two members
of the field of R, such that x has the relation R to y, and if x´ and
y´ are the correlates in the field of R´ of x and y, then in all such
cases x´ has the relation R´ to y´, and conversely, interchanging the
dashes on the letters, i.e. R and R´, x and x´, &c. It is evident that
the ordinal similarity of two relations implies the cardinal
similarity of their fields, but not conversely. Also, two relations
need not be serial in order to be ordinally similar; but if one is
serial, so is the other. The relation-number of a relation is the
class whose members are all those relations which are ordinally
similar to it. This class will include the original relation itself.
The relation-number of a relation should be compared with the cardinal
number of a class. When a relation is serial its relation-number is
often called its serial type. The addition and multiplication of two
relation-numbers is defined by taking two relations R and S, such that
(1) their fields have no terms in common; (2) their relation-numbers
are the two relation-numbers in question, and then by defining by
reference to R and S two other suitable relations whose
relation-numbers are defined to be respectively the sum and product of
the relation-numbers in question. We need not consider the details of
this process. Now if n be any finite cardinal number, it can be proved
that the class of those serial relations, which have a field whose
cardinal number is n, is a relation-number. This relation-number is
the ordinal number corresponding to n; let it be symbolized by n.
Thus, corresponding to the cardinal numbers 2, 3, 4 ... there are the
ordinal numbers 2, 3, 4.... The definition of the ordinal number 1
requires some little ingenuity owing to the fact that no serial
relation can have a field whose cardinal number is 1; but we must omit
here the explanation of the process. The ordinal number 0 is the class
whose sole member is the null relation--that is, the relation which
never holds between any pair of entities. The definitions of the
finite ordinals can be expressed without use of the corresponding
cardinals, so there is no essential priority of cardinals to ordinals.
Here also it can be seen that the science of the finite ordinals is a
particular subdivision of the general theory of classes and relations.
Thus the illusory nature of the traditional definition of mathematics
is again illustrated.

_Cantor's Infinite Numbers._--Owing to the correspondence between the
finite cardinals and the finite ordinals, the propositions of cardinal
arithmetic and ordinal arithmetic correspond point by point. But the
definition of the cardinal number of a class applies when the class is
not finite, and it can be proved that there are different infinite
cardinal numbers, and that there is a least infinite cardinal, now
usually denoted by [aleph]0, where [aleph] is the Hebrew letter aleph.
Similarly, a class of serial relations, called _well-ordered_ serial
relations, can be defined, such that their corresponding
relation-numbers include the ordinary finite ordinals, but also
include relation-numbers which have many properties like those of the
finite ordinals, though the fields of the relations belonging to them
are not finite. These relation-numbers are the infinite ordinal
numbers. The arithmetic of the infinite cardinals does not correspond
to that of the infinite ordinals. The theory of these extensions of
the ideas of number is dealt with in the article NUMBER. It will
suffice to mention here that Peano's fourth premiss of arithmetic does
not hold for infinite cardinals or for infinite ordinals. Contrasting
the above definitions of number, cardinal and ordinals, with the
alternative theory that number is an ultimate idea incapable of
definition, we notice that our procedure exacts a greater attention,
combined with a smaller credulity; for every idea, assumed as
ultimate, demands a separate act of faith.

_The Data of Analysts._--Rational numbers and real numbers in general
can now be defined according to the same general method, If m and n
are finite cardinal numbers, the rational number m/n is the relation
which any finite cardinal number x bears to any finite cardinal number
y when n × x = m × y. Thus the rational number one, which we will
denote by 1_r, is not the cardinal number 1; for 1_r is the relation
1/1 as defined above, and is thus a relation holding between certain
pairs of cardinals. Similarly, the other rational integers must be
distinguished from the corresponding cardinals. The arithmetic of
rational numbers is now established by means of appropriate
definitions, which indicate the entities meant by the operations of
addition and multiplication. But the desire to obtain general
enunciations of theorems without exceptional cases has led
mathematicians to employ entities of ever-ascending types of
elaboration. These entities are not created by mathematicians, they
are employed by them, and their definitions should point out the
construction of the new entities in terms of those already on hand.
The real numbers, which include irrational numbers, have now to be
defined. Consider the serial arrangement of the rationals in their
order of magnitude. A real number is a class ([alpha], say) of
rational numbers which satisfies the condition that it is the same as
the class of those rationals each of which precedes at least one
member of [alpha]. Thus, consider the class of rationals less than
2_r; any member of this class precedes some other members of the
class--thus 1/2 precedes 4/3, 3/2 and so on; also the class of
predecessors of predecessors of 2_r is itself the class of
predecessors of 2_r. Accordingly this class is a real number; it will
be called the real number 2_R. Note that the class of rationals less
than or equal to 2_r is not a real number. For 2_r is not a
predecessor of some member of the class. In the above example 2_R is
an integral real number, which is distinct from a rational integer,
and from a cardinal number. Similarly, any rational real number is
distinct from the corresponding rational number. But now the
irrational real numbers have all made their appearance. For example,
the class of rationals whose squares are less than 2_r satisfies the
definition of a real number; it is the real number [root]2. The
arithmetic of real numbers follows from appropriate definitions of the
operations of addition and multiplication. Except for the immediate
purposes of an explanation, such as the above, it is unnecessary for
mathematicians to have separate symbols, such as 2, 2_r and 2_R, or
2/3 and (2/3)_R. Real numbers with signs (+ or -) are now defined. If
a is a real number, +a is defined to be the relation which any real
number of the form x + a bears to the real number x, and -a is the
relation which any real number x bears to the real number x + a. The
addition and multiplication of these "signed" real numbers is suitably
defined, and it is proved that the usual arithmetic of such numbers
follows. Finally, we reach a complex number of the nth order. Such a
number is a "one-many" relation which relates n signed real numbers
(or n algebraic complex numbers when they are already defined by this
procedure) to the n cardinal numbers 1, 2 ... n respectively. If such
a complex number is written (as usual) in the form x1e1 + x2e2 + ... +
x_n e_n, then this particular complex number relates x1 to 1, x2 to 2,
... x_n to n. Also the "unit" e1 (or e2) considered as a number of the
system is merely a shortened form for the complex number (+1) e1 + 0e2
+ ... + 0e_n. This last number exemplifies the fact that one signed
real number, such as 0, may be correlated to many of the n cardinals,
such as 2 ... n in the example, but that each cardinal is only
correlated with one signed number. Hence the relation has been called
above "one-many." The sum of two complex numbers x1e1 + x2e2 + ... +
x_n e_n and y1e1 + y2e2 + ... + y_n e_n is always defined to be the
complex number (x1 + y1)e1 + (x2 + y2)e2 + ... + (x_n + y_n)e_n. But
an indefinite number of definitions of the product of two complex
numbers yield interesting results. Each definition gives rise to a
corresponding algebra of higher complex numbers. We will confine
ourselves here to algebraic complex numbers--that is, to complex
numbers of the second order taken in connexion with that definition of
multiplication which leads to ordinary algebra. The product of two
complex numbers of the second order--namely, x1e1 + x2e2 and y1e1 +
y2e2, is in this case defined to mean the complex (x1y1 - x2y2)e1 +
(x1y2 + x2y1)e2. Thus e1 × e1 = e, e2 × e2 = -e1, e1 × e2 = e2 × e1 =
e2. With this definition it is usual to omit the first symbol e1, and
to write i or [root]-1 instead of e2. Accordingly, the typical form
for such a complex number is x + yi, and then with this notation the
above-mentioned definition of multiplication is invariably adopted.
The importance of this algebra arises from the fact that in terms of
such complex numbers with this definition of multiplication the utmost
generality of expression, to the exclusion of exceptional cases, can
be obtained for theorems which occur in analogous forms, but
complicated with exceptional cases, in the algebras of real numbers
and of signed real numbers. This is exactly the same reason as that
which has led mathematicians to work with signed real numbers in
preference to real numbers, and with real numbers in preference to
rational numbers. The evolution of mathematical thought in the
invention of the data of analysis has thus been completely traced in
outline.

_Definition of Mathematics._--It has now become apparent that the traditional field of mathematics in the province of discrete and continuous number can only be separated from the general abstract theory of classes and relations by a wavering and indeterminate line. Of course a discussion as to the mere application of a word easily degenerates into the most fruitless logomachy. It is open to any one to use any word in any sense. But on the assumption that "mathematics" is to denote a science well marked out by its subject matter and its methods from other topics of thought, and that at least it is to include all topics habitually assigned to it, there is now no option but to employ "mathematics" in the general sense[5] of the "science concerned with the logical deduction of consequences from the general premisses of all reasoning."

_Geometry._--The typical mathematical proposition is: "If x, y, z ... satisfy such and such conditions, then such and such other conditions hold with respect to them." By taking fixed conditions for the hypothesis of such a proposition a definite department of mathematics is marked out. For example, geometry is such a department. The "axioms" of geometry are the fixed conditions which occur in the hypotheses of the geometrical propositions. The special nature of the "axioms" which constitute geometry is considered in the article GEOMETRY (_Axioms_). It is sufficient to observe here that they are concerned with special types of classes of classes and of classes of relations, and that the connexion of geometry with number and magnitude is in no way an essential part of the foundation of the science. In fact, the whole theory of measurement in geometry arises at a comparatively late stage as the result of a variety of complicated considerations.

_Classes and Relations._--The foregoing account of the nature of
mathematics necessitates a strict deduction of the general properties
of 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,[6] and
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 contradiction[7] does 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.[8] All 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 [phi]x be a propositional function,
the undetermined variable x is the argument. Two propositional
functions [phi]x and [psi]x are "extensionally identical" if any
determination of x in [phi]x which converts [phi]x into a true
proposition also converts [psi]x into a true proposition, and
conversely for [psi] and [phi]. Now consider a propositional function
F_[chi] in which the variable argument [chi] is itself a propositional
function. If F_[chi] is true when, and only when, [chi] is determined
to be either [phi] or some other propositional function extensionally
equivalent to [phi], then the proposition F_[phi] is of the form which
is ordinarily recognized as being about the class determined by [phi]x
taken in extension--that is, the class of entities for which [phi]x 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 theory[9] is in
effect a theory of the _use_ of 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 the _succession_ of 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 Nicea[10] illustrates 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's _Hypotheses non fingo_ was 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 the _International Catalogue of Scientific Literature_
promoted 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 articles ALGEBRA, 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 articles ALGEBRA; 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
articles DIFFERENTIAL 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 articles CONIC 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 the _International Catalogue_ deals 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 articles MECHANICS; 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 (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 Phoenicians. 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.)

FOOTNOTES:

[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.

[5] The first unqualified explicit statement of _part_ of this
definition seems to be by B. Peirce, "Mathematics is the science
which draws necessary conclusions" (_Linear Associative Algebra_, §
i. (1870), republished in the _Amer. 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.

[7] Cf. Russell, _loc. cit._, ch. x.

[8] Cf. _Pragmatism: a New Name for some Old Ways of Thinking_
(1907).

[9] Due 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, in
_Principia mathematica_ (Cambridge).

[10] Cf. Stanley's _Eastern Church_, Lecture v.

[11] Cf. _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 written _Memorable 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, the _Memorable Providences_ and _The 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 of _More 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 wrote _An 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 his charge 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 his _Magnalia 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 include _A 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 the _Magnalia; The Short
History of New England_ (1694); _Bonifacius_, usually known as _Essays
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."

See _The 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 are _The Life of Cotton Mather_ (1729); _An Apology
for the Liberties of the Churches in New England_ (1738), and _America
Known to the Ancients_ (1773). (W. L. C.*)