**MATHEMATICS** (Gr. μαθηματική, *sc*. τέχνη or ἐπιστήμη; from μάθημα, “learning” or “science”), the general term for the various applications of mathematical thought, the traditional field of which is number and quantity. It has been usual to define mathematics as “the science of discrete and continuous magnitude.” Even Leibnitz,^{[1]} who initiated a more modern point of view, follows the tradition in thus confining the scope of mathematics properly so called, while apparently conceiving it as a department of a yet wider science of reasoning. A short
consideration of some leading topics of the science will exemplify both
the plausibility and inadequacy of the above definition.
Arithmetic, algebra, and the infinitesimal calculus, are sciences
directly concerned with integral numbers, rational (or fractional)
numbers, and real numbers generally, which include incommensurable
numbers. It would seem that “the general theory
of discrete and continuous quantity” is the exact description of
the topics of these sciences. Furthermore, can we not complete
the circle of the mathematical sciences by adding geometry?
Now geometry deals with points, lines, planes and cubic contents.
Of these all except points are quantities: lines involve lengths,
planes involve areas, and cubic contents involve volumes. Also,
as the Cartesian geometry shows, all the relations between
points are expressible in terms of geometric quantities. Accordingly,
at first sight it seems reasonable to define geometry in
some such way as “the science of dimensional quantity.”
Thus every subdivision of mathematical science would appear
to deal with quantity, and the definition of mathematics as
“the science of quantity” would appear to be justified. We
have now to consider the reasons for rejecting this definition
as inadequate.

*Types of Critical Questions*.—What are numbers? We can
talk of five apples and ten pears. But what are “five” and
“ten” apart from the apples and pears? Also in addition to
the cardinal numbers there are the ordinal numbers: the fifth
apple and the tenth pear claim thought. What is the relation
of “the fifth” and “the tenth” to “five” and “ten”?
“The first rose of summer” and “the last rose of summer”
are parallel phrases, yet one explicitly introduces an ordinal
number and the other does not. Again, “half a foot” and
“half a pound” are easily defined. But in what sense is there
“a half,” which is the same for “half a foot” as “half a
pound”? Furthermore, incommensurable numbers are defined
as the limits arrived at as the result of certain procedures with
rational numbers. But how do we know that there is anything
to reach? We must know that √2 exists before we can prove
that any procedure will reach it. An expedition to the North
Pole has nothing to reach unless the earth rotates.

Also in geometry, what is a point? The straightness of a
straight line and the planeness of a plane require consideration.
Furthermore, “congruence” is a difficulty. For when a triangle
“moves,” the points do not move with it. So what is it that
keeps unaltered in the moving triangle? Thus the whole
method of measurement in geometry as described in the elementary
textbooks and the older treatises is obscure to the last
degree. Lastly, what are “dimensions”? All these topics
require thorough discussion before we can rest content with the
definition of mathematics as the general science of magnitude;
and by the time they are discussed the definition has evaporated.
An outline of the modern answers to questions such as the above
will now be given. A critical defence of them would require a
volume.^{[2]}

*Cardinal Numbers*.—A one-one relation between the members of two classes α and β is any method of correlating all the members
of α to all the members of β, so that any member of α has one and
only one correlate in β, and any member of β has one and only one
correlate in α. 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 α 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 α exists.
Thus the cardinal number of α is itself a class, and furthermore α
is a member of it. For a one-one relation can be established between
the members of α and α by the simple process of correlating each
member of α 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 α and β, 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 α and β 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 familiar 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 *ṅ*. 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 ȯ
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 ℵ_{0}, where ℵ 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 Analysis*.—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 (α, 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 α. 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 √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 *n*th 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 *x*_{1}*e*_{1}+*x*_{2}*e*_{2}+. . .+*x*_{n}*e*_{n}, then this particular
complex number relates *x*_{1} to 1, *x*_{2} to 2, . . . *x*_{n} to *n*. Also
the “unit” *e*_{1} (or *e*_{s}) considered as a number of the system is merely
a shortened form for the complex number (+1) *e*_{1}+o*e*_{2}+. . .+o*e*_{n}.
This last number exemplifies the fact that one signed real number,
such as o, 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 *x*_{1}*e*_{1}+*x*_{2}*e*_{2}+. . .+*x*_{n}*e*_{n}
and *y*_{1}*e*_{1}+*y*_{2}*e*_{2}+. . .+*y*_{n}*e*_{n} is always defined to be the complex
number (*x*_{1}+*y*_{1})*e*_{1}+(*x*_{2}+*y*_{2})*e*_{2}+. . .+(*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, *x*_{1}*e*_{1}+*x*_{2}*e*_{2} and *y*_{1}*e*_{1}+*y*_{2}*e*_{2}, is in this case
defined to mean the complex (*x*_{1}*y*_{1}−*x*_{2}*y*_{2})*e*_{1}+(*x*_{1}*y*_{2}+*x*_{2}*y*_{1})*e*_{2}. Thus
*e*_{1}×*e*_{1} = *e*_{1}, *e*_{2}×*e*_{2} = −*e*_{1}, *e*_{1}×*e*_{2} = *e*_{2}×*e*_{1} = *e*_{2}. With this definition
it is usual to omit the first symbol *e*_{1}, and to write *i* or √−1
instead of *e*_{2}. 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 φ*x*
be a propositional function, the undetermined variable *x* is the
argument. Two propositional functions φ*x* and ψ*x* are “extensionally
identical” if any determination of *x* in φ*x* which converts
φ*x* into a true proposition also converts ψ*x* into a true proposition,
and conversely for ψ and φ. Now consider a propositional function
Fχ in which the variable argument χ is itself a propositional function.
If Fχ is true when, and only when, χ is determined to be either φ or
some other propositional function extensionally equivalent to φ,
then the proposition Fφ is of the form which is ordinarily recognized
as being about the class determined by φ*x* taken in extension—that
is, the class of entities for which φ*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; Fraction, Continued; 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 Phœnicians. The Greeks
created the sciences of geometry and of number as applied to the
measurement of continuous quantities. The great abstract ideas
(considered directly and not merely in tacit use) which have
dominated the science were due to them—namely, ratio, irrationality,
continuity, the point, the straight line, the plane. This
period lasted^{[11]} from the time of Thales, *c*. 600 B.C., to the capture
of Alexandria by the Mahommedans, A.D. 641. The medieval
Arabians invented our system of numeration and developed
algebra. The next period of advance stretches from the Renaissance
to Newton and Leibnitz at the end of the 17th century.
During this period logarithms were invented, trigonometry and
algebra developed, analytical geometry invented, dynamics
put upon a sound basis, and the period closed with the magnificent
invention of (or at least the perfecting of) the differential
calculus by Newton and Leibnitz and the discovery of gravitation.
The 18th century witnessed a rapid development of analysis,
and the period culminated with the genius of Lagrange and
Laplace. This period may be conceived as continuing throughout
the first quarter of the 19th century. It was remarkable both
for the brilliance of its achievements and for the large number
of French mathematicians of the first rank who flourished during
it. The next period was inaugurated in analysis by K. F. Gauss,
N. H. Abel and A. L. Cauchy. Between them the general
theory of the complex variable, and of the various “infinite”
processes of mathematical analysis, was established, while other
mathematicians, such as Poncelet, Steiner, Lobatschewsky and
von Staudt, were founding modern geometry, and Gauss inaugurated
the differential geometry of surfaces. The applied
mathematical sciences of light, electricity and electromagnetism, and of heat, were now largely developed. This school of mathematical
thought lasted beyond the middle of the century, after
which a change and further development can be traced. In the
next and last period the progress of pure mathematics has been
dominated by the critical spirit introduced by the German
mathematicians under the guidance of Weierstrass, though foreshadowed
by earlier analysts, such as Abel. Also such ideas as
those of invariants, groups and of form, have modified the
entire science. But the progress in all directions has been too
rapid to admit of any one adequate characterization. During
the same period a brilliant group of mathematical physicists,
notably Lord Kelvin (W. Thomson), H. V. Helmholtz, J. C.
Maxwell, H. Hertz, have transformed applied mathematics by
systematically basing their deductions upon the Law of the
conservation of energy, and the hypothesis of an ether pervading
space.

Bibliography.—References to the works containing expositions
of the various branches of mathematics are given in the appropriate
articles. It must suffice here to refer to sources in which the subject
is considered as one whole. Most philosophers refer in their works
to mathematics more or less cursorily, either in the treatment of
the ideas of number and magnitude, or in their consideration of the
alleged a priori and necessary truths. A bibliography of such
references would be in effect a bibliography of metaphysics, or
rather of epistemology. The founder of the modern point of view,
explained in this article, was Leibnitz, who, however, was so far
in advance of contemporary thought that his ideas remained
neglected and undeveloped until recently; cf. *Opuscules et fragments*
*inédits de Leibnitz. Extraits des manuscrits de la bibliothèque*
*royale de Hanovre*, by Louis Couturat (Paris, 1903), especially
pp. 356–399, “Generales inquisitiones de analysi notionum et
veritatum” (written in 1686); also cf. *La Logique de Leibnitz*, already
referred to. For the modern authors who have 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 et 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 1799*); 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.)

- ↑ Cf.
*La Logique de Leibnitz*, ch. vii., by L. Couturat (Paris, 1901). - ↑ Cf.
*The Principles of Mathematics*, by Bertrand Russell (Cambridge, 1903). - ↑ 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. - ↑ Cf. Russell,
*loc. cit.*, pp. 199–256. - ↑
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.* - ↑
“Una questione sui numeri transfiniti,”
*Rend. del circolo mat. di**Palermo*, vol. xi. (1897); and Russell,*loc. cit.*, ch. xxxviii. - ↑
Cf. Russell,
*loc. cit.*, ch. x. - ↑
Cf.
*Pragmatism: a New Name for some Old Ways of Thinking*(1907). - ↑
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). - ↑
Cf. Stanley’s
*Eastern Church*, Lecture v. - ↑
Cf.
*A Short History of Mathematics*, by W. W. R. Ball.