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See also:MATHEMATICS (Gr. /saBnµar1Kil, Sc. vOcvn or E7rlQTt'µn; from p iN.La, "learning" or "See also:science ") , the See also:general See also:term for the various applications of mathematical thought, the traditional See also: The highest See also:form of the doctrine is scientific materialism, by which term is meant the doctrine so commonly adopted by the physicist, zoologist and biologist.
It may perhaps be fail-1y said that materialism is at See also:present a necessary methodological postulate of natural-scientific inquiry. The business of the scientist is to explain everything by the See also:physical causes which are comparatively well understood and to exclude the interference of spiritual causes. It was the See also:great See also:work of See also:Descartes to exclude rigorously from science all explanations which were not scientifically verifiable; and the prevalence of materialism at certain epochs, as in the enlightenment of the 18th century and in the See also:German See also:philosophy of the See also:middle 19th, were occasioned by See also:special need to vindicate the scientific position, in the former See also:case against the See also: 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 See also: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 See also:Critical Questions.—What are numbers? We can talk of five apples and ten See also:pears. But what are " five " and " ten " apart from the apples and pears? Also in addition to the See also:cardinal numbers there are the ordinal numbers: the fifth See also:apple and the tenth See also:pear claim thought. What is the relation of " the fifth " and " the tenth " to " five " and " ten "? " The first See also: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, " See also:half a See also:foot " and " half a See also: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 -l2 exists before we can prove that any See also:procedure will reach it. An expedition to the See also:North See also:Pole has nothing to reach unless the See also:earth rotates. Also in geometry, what is a point? The straightness of a straight See also:line and the planeness of a See also: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 See also:treatises is obscure to the last degree. Lastly, what are " dimensions " ? All these topics require thorough discussion before we can See also:rest content with the definition of mathematics as the general science of magnitude; and by the See also: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 See also:defence of them would require a See also:volume.' Cardinal Numbers.—A one-one relation between the members of two classes a and 19 is any method of correlating all the members of a to all the members of 0, so that any member of a has one and only one correlate in {4, and any member of /3 has one and only one correlate in a. 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 a 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 a exists. Thus the cardinal number of a is itself a class, and furthermore a is a member of it. For a one-one relation can be established between the members of a and a by the See also:simple See also:process of correlating each member of a 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 See also: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 'Cf. The Principles of Mathematics, by See also:Bertrand See also:Russell (See also:Cam-See also:bridge, 1903).number has only one member. The operations of addition and multiplication of two given cardinal numbers can be defined by taking two classes a and $, satisfying the conditions (1) that their cardinal numbers are respectively the given numbers, and (2) that they contain no member in See also:common, and then by defining by reference to a and 3 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 See also:definitions it is now possible to prove the following six premisses applying to finite cardinal numbers, from which Peano 2 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+t, then a=b. vi. If a is a cardinal number, then a+t +o. It may be noticed that (iv) is the familar principle of mathematical See also:induction. Peano in an See also:historical See also:note refers its first explicit employment, although without a general enunciation, to Maurolycus in his work, Arithmeticorum libri duo (See also: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 See also: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 See also:separate science. They are excellent principles of the highest value, but they are in no sense the necessary pre-misses which must be proved before any other propositions of cardinal numbers can be established. On the contrary, the pre-misses 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 See also:branch of deductive reasoning concerning classes and relations which is concerned with the See also: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 See also: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 See also:telegraph posts along a certain road have a space-order very obvious to our senses; but they have also a time-order according to See also: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 See also:drawn in a lottery, their time-order of occurrence in that See also:drawing also ranges them in an order of some importance. Thus the order is defined by the " serial " relation. A relation (R) is serial 8 when (I) 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 See also: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 See also: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 See also: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 See also: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 (I) their fields have no 2 Cf. Formulaire mathematique (See also:Turin, ed. of 1903) ; earlier formulations of the bases of arithmetic are given by him in the See also:editions of 1898 and of 1901. The See also:variations are only trivial. 2 Cf. Russell, loc. cit., pp. 199-256. 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, ¢ . . . The definition of the ordinal number i 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 o is the class whose See also: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 See also:Infinite Numbers.—Owing to the See also: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 H„ where s is the See also:Hebrew See also: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 corre; spend to that of the infinite ordinals. The theory of these extensions of the ideas of number is dealt with in the See also:article NUMBER. It will suffice to mention here that Peano's See also: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 See also:idea incapable of definition, we See also:notice that our procedure exacts a greater See also:attention, combined with a smaller credulity; for every idea, assumed as ultimate, demands a separate See also:act of faith. The Data of See also: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 nXx=mXy. Thus the rational number one, which we will denote by I„ is not the cardinal number I ; for 1, is the relation I/I 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 See also: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 See also: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 (a, say) of rational numbers which satisfies the See also:condition that it is the same as the class of those rationals each of which precedes at least one member of a. Thus, consider the class of rationals less than 2,; 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, is itself the class of predecessors of 2,. Accordingly this class is a real number; it will be called the real number 2R. Note that the class of rationals less than or equal to 2, is not a real number. For 2, is not a predecessor of some member of the class. In the above example 2R 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 See also:appearance. For example, the class of rationals whose squares are less than 2, satisfies the definition of a real number; it is the real number AI 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, and 2R, 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 s+a. The addition and multiplication of these " signed " realnumbers is suitably defined, and it is proved that the usual arith. metic 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 x1e,+x2e2+... +x„e,,, then this particular complex number relates xi to I, x2 to 2, . . x„ to n. Also the " unit " el (or e,) considered as a number of the system is merely a shortened form for the complex number (+I) el+oe2+...+oe,,. 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 . It 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 xiei+x2e2+ • • • +x,,e,, and y1ei-Fyie;+ • . • +yse„ is always defined to be the complex number (xi+yl)e;+(x2-f-y2)e2+• . •+(x„+y„)en. But an in-definite 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, xiel+x2e2 and yie1+y2e2, is in this case defined to mean the complex (xlyl-x2y2)ei+(xiy2+x2y1)e2. Thus el Xel = el, e2Xe2 = — el, el X e2 =e2 X e1=e2. With this definition it is usual to omit the first See also:symbol el, and to write i or Al -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 See also: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 See also: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 See also: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 See also: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 See also:assumption that " mathematics " is to denote a science well marked out by its subject See also: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 See also:option but to employ " mathematics " in the general sense' of the " science concerned with the logical See also: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 See also: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 See also:part of the See also:foundation of the science. In fact, the whole theory of measurement in geometry arises at a comparatively See also:late See also:stage as the result of a variety of complicated considerations. Classes and Relations.—The foregoing See also:account of the nature of mathematics necessitates a strict deduction of the general properties ' The first unqualified explicit statement of part of this definition seems to be by B. See also:Peirce, " Mathematics is the science which draws necessary conclusions " (Linear Associative Algebra, § i. (187o), re-published in the Amer. Journ. of Math., vol. iv. (1881) ). But it will be noticed that the second half of the definition in the See also:text—" from the general premisses of all reasoning "—is See also:left unexpressed. The full expression of the idea and its development into a philosophy of mathematics is due to Russell, loc. cit. 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 See also:contradiction,' 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 contradiction2 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. See also:Call this class w; then to say that x is a w is See also: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 See also:scrutiny of the very idea of classes and relations is required. Note that classes are here required in See also: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.' 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 See also:pen is an entity and the class of pens is an entity is merely a See also: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 he involved in it is represented by an undetermined x, which may stand for any entity. Call it a propositional See also:function; and, if ¢x be a propositional function, the undetermined variable x is the See also:argument. Two propositional functions ¢x and ,'x are " extensionally identical " if any determination of x in ¢x which converts 4)x into a true proposition also converts +1'x into a true proposition, and conversely for 4) and ¢. Now consider a propositional function Fx in which the variable argument x is itself a propositional function. If Fx is true when, and only when, x is determined to be either ¢ or some other propositional function extensionally equivalent to 4), then the proposition F¢ is of the form which is ordinarily recognized as being about the class determined by 49x taken in extension—that is, the class of entities for which ¢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 theory4 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 See also: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 See also: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 See also:legend of the See also:Council of Nicea' illustrates this point: " When the Bishops took their " Lna questione sui numeri transfiniti," Rend. del circolo See also:mat. di See also:Palermo, vol. xi. (1897); and Russell, loc. cit., ch. xxxviii. 2 Cf. Russell, loc. cit., ch. x. 'Cf. See also:Pragmatism: a New Name for some Old Ways of Thinking (1907). ' Due to Bertrand Russell, cf. " Mathematical See also:Logic as based on the Theory of Types," Amer. Journ. of Math. vol. See also:xxx. (1908). It is more fully explained by him, with later simplifications, in Principia mathematica (See also:Cambridge). a Cf. See also:Stanley's Eastern Church, Lecture v.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 See also:series, he immediately turned into the likeness of his next See also:neighbour." Whatever be the historical See also:worth of this See also: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 See also:scale is unfitted for the See also: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 See also:place, some physical process of addition is presupposed, involving some inductively inferred See also:law of permanence during that process. Thus in the theory of masses we must know that two pounds of See also:lead when put together will counterbalance in the scales two pounds of See also: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 See also: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 See also: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 See also:base it upon experience. For example, if it should turn out that the See also: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 See also:necessity. In one sense there is no science of applied mathematics. When once the fixed conditions which any hypothetical See also: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 See also:independence of the question as to whether or no any such group of entities can be found in the See also:world of phenomena. Thus rational See also:mechanics, based on the Newtonian See also:Laws, viewed as mathematics is See also:independent of its supposed application, and See also: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 See also:evidence of natural science, and the hypotheses from which the " deductions " can be deduced are sought. Accordingly, every See also:treatise on applied mathematics, properly so-called,. is directed to the See also:criticism of the " laws " from which the reasoning starts, or to a See also:suggestion of results which experiment may See also: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. See also:Newton's Hypotheses non See also:fingo was a proud boast, but it rests upon an entire misconception of the capacities of the mind of See also:man in dealing with See also:external nature. Synopsis of Existing Developments of Pure Mathematics.—A See also:corn- a special property. Thus the modern ideas, which have so powerplete See also:classification of mathematical sciences, as they at present exist, Is to be found in the See also:International See also:Catalogue of Scientific Literature promoted by the Royal Society. The classification in question was drawn up by an international See also: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 See also: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. See also:Section A deals with pure mathematics. Under the general heading " Fundamental Notions" occur the subheadings " See also:Foundations of Arithmetic," with the topics rational, irrational and transcendental numbers, and aggregates; " Universal Algebra," with the topics complex numbers, See also:quaternions, ausdehnungslehre, vector analysis, matrices, and algebra of logic; and " Theory of See also: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 See also:Galois, &c. " Theory of Numbers," with the topics congruences, quadratic residues, See also:prime numbers, particular irrational and transcendental numbers. For the subjects of this general heading see the articles ALGEBRA; ALGEBRAIC FORMS; ARITHMETIC; COMBINATORIAL ANALYSIS; DETERMINANTS; See also:EQUATION; FRACTION, CONTINUED; See also:INTERPOLATION; LOGARITHMS; MAGIC SQUARE; See also: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 See also: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 See also:Euler's, See also:Legendre's, See also:Bessel's and automorphic functions; " Differential Equations," with the topics existence theorems, methods of See also: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; " See also:Analytical Methods connected with Physical Subjects," with the topics See also:harmonic analysis, See also: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 See also: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; See also:MAXIMA AND MINIMA; SERIES; SPHERICAL HARMONICS; See also: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 See also:Con-figurations," 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 con-figurations in hyperspace; " Infinitesimal Geometry: applications of Differential and Integral Calculus to Geometry," with the topics kinematic geometry, curvature, rectification and See also: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 See also:representation of surfaces on others, deformation of surfaces, orthogonal and isothermic surfaces. For the subjects under this heading see the articles CONIC SECTIONS; CIRCLE; See also:CURVE; GEOMETRICAL CONTINUITY; GEOMETRY, Axioms of; GEOMETRY, Euclidean; GEOMETRY, Projective; GEOMETRY, Analytical; GEOMETRY, Line; KNOTS, MATHEMATICAL THEORY OF; See also:MENSURATION; See also:MODELS; See also:PROJECTION; See also: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 fully 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 See also:interest, because it is in reality the great topic of applied mathematics. All the world, including savages who cannot See also: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 See also:units, measurements, and the See also:constant of See also:gravitation. The topics of the other headings do not require See also:express mention. Additional information and CommentsThere are no comments yet for this article.
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