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Mathematics




MATHEMATICS. Any conception which is definitely and completely determined by means of a finite number of specifications, say by assigning a finite number of elements, is a mathematical conception. Mathematic has for its function to develop the consequences involved in the definition of a group of mathematical conceptions. Interdependence and mutual logical consistency among the members of the group are postulated, otherwise the group would either have to be treated as several distinct groups, or would lie beyond the sphere of mathematics.

As an example of a mathematical conception we may take "a triangle"; regarded without reference to its position in space, this is determined when three elements are specified say its three sides ; or we may take a "colour sensation," which, on Young’s theory, is determined when the amounts of the three fundamental colour sensations that enter into it are stated. As an example of a non-mathematical conception we may take "a man," "a mineral," "iron," no one of which admits of being so determined by a finite number of specifications that all its properties can be truly said to be deducible from the definition.

A mathematical conception is, from its very nature, abstract ; indeed its abstractness is usually of a higher order than the abstractness of the logician. Thus, for instance, we may neglect the other attributes of a body and consider merely its form ; we thus reach the abstract idea of "form." But the form of a irregular fragment of stone does not admit of being finitely specified, and is therefore not susceptible of mathematical treatment. If, however, we have a carefully squared cubical block of granite to deal with, for most practical purposes its form is specified by stating that it is a cube, and assigning one element, viz., an edge of the abstract mathematical cube by which we replace it. This example, illustrates at once the limits of mathematical reasoning and the nature of the bearing of mathematics on practice

A variety of words have been used to denote the dependence of a mathematical conception upon its elements. It is frequently said, for instance ,that the conception is a "function" of its elements. One word has recently come into use which is very convenient, inasmuch as it draws attention at once to the fundamental idea involved in mathematical conception and to the prime object mathematical contemplation, viz., "manifoldness."

Number is involved in the notion of a manifoldness both directly, as any one can see, and also indirectly in a manner which the mind untrained to mathematical thinking does not so readily understand. Take on the one hand the case of a triangle considered without reference to its position but merely as composed of three limited straight lines, it may be completely determined in various ways by assigning three elements. A triangle may therefore be called a triple discrete manifoldness. A plane quadrilateral considered in the same way (being fully determined when four sides and a diagonal are known) is a quintuple discrete manifoldness ; and a plane polygon on n sides a (2n – 3) –ple discret manifoldness. Consider on the other hand the assemblage of points on a given straight line, they are infinite in number yet so related that any one of them is singled out by assigning its distance from an arbitrarily chosen fixed point on the line. Such an assemblage is called a onefold continuous manifoldness, or simply a onefold manifoldness; another example of the same kind is the totality of instants in a period of time. The assemblage of points on a surface is a twofold manifoldness; the assemblage of points in tridimensional space is a threefold manifoldness; the values of a continuous function of n arguments an n-fold manifoldness.

It should be observed that he distinction between discrete and continuous manifoldness is not of necessity inherent in the conception. For one purpose we may treat a conception as a discrete manifoldness, for another as a continuous manifoldness. Thus we have seen that an unlimited straight line may be treated as a one-fold continuous manifoldness; but, if we regard it as a whole, and with reference to the fact that its position in space is determined by four data, it becomes a quadruple discrete manifoldness.

The primary, although not the only, operation in the treatment of a discrete manifoldness is numbering or counting; hence arises the pure mathematical science of number, comprehending (abstract) Arithmetic and its higher branch commonly called the Theory of Numbers. Without entering into a discussion of the definitions and axioms of the science of number, it will be sufficient here to remark that all numerical operations are reducible to three fundamental laws commonly called the commutative, associate, and distributive laws. The four fundamental processes, or four species, as they are sometimes called, two of which, addition and multiplication, are direct, and two , subtraction and division, inverse, are solely defined by the derive their meaning from the three laws of operation just mentioned.






A careful consideration of the methods in vogue for dealing with continuous manifoldness shows that they reduce themselves to two, which may be called the synoptic method and the analytic method. In the synoptic method we declare the properties of a manifoldness by contemplating it as a whole, aiding our understanding, when it is necessary to do so, by a diagram, a model, or any other concrete device more or less refined according to circumstances. In the analytic method we fix our attention upon the individual elements of the manifoldness, usually defining each element by a definite number of specifications the variation of which leads us from element to element of the given manifoldness. We examine the properties of an element in the most general manner, and from them, we predicate the properties of the manifoldness as a whole.

The best and most familiar examples of the synoptic treatment of manifoldness are the different varieties of pure geometry. Among these we may mention the apagogic geometry of the Greeks, which starts with a collection of definitions and axioms, enunciates and proves proposition after proposition with great attention to strict logical form and with continual reference to the grounds of inference, but pays little attention to the ordering of theorems with a view to mutual illustration, and carefully suppresses all traces of the method of the method by which the propositions were or might have been discovered. It is true that the Greeks were in possession of a method, called by them analysis, which had for its object the discovery of geometrical truth. But this consisted merely in taking any proposition suspected to be true and tracing its consequences until one was reached which either contradicted a known proposition or else was true and capable of leading by a direct process of reasoning (synthesis) to the proposition in question. In this we have no trace of the systematic development of geometric truth, and the method was apparently regarded by the ancients themselves as imperfect, for it makes no figure in such of their systematic treatises as have reached us. In somewhat sharp contrast with the Grecian geometry, but still essentially synoptic in method, stand the different varieties of modern geometry,—which aims at greater generality in its definitions, pays less explicit attention to logical form, but arranges geometrical propositions as much as possible in the natural order of development or discovery, and above all makes extensive use of the principle of continuity. As examples of the modern geometry may be cited the descriptive geometry (Géométrie Descriptive, Darstellende Geometrie) of Monge; the projective geometry (Géométrie Projective, Geometrie der Lage) of Poncelet, Steiner, and Von Staudt; and the geometry of transformation in general, of which projective geometry is but a particular case. There is one other highly interesting form of modern geometry, which, although analytic in some of its development, and often exhibited in close alliance with other analytical methods, is nevertheless synoptic as to its fundamental principles, viz., arithmic geometry (Abzählende Geometrie) or theory of characteristics, which originated in the characteristic equations of Plücker, and was developed into a powerful special method by Chasles and others See GEOMETRY and CURVE.

Geometry, however, is not the only field for the synoptic treatment of manifoldness. This is obvious if we reflect that any magnitude whatever may be represented by a line; so that any function of not more than two elements may be represented by a geometrical construction and treated by any method applicable in geometry. Since the famous dissertation of Riemann, On the Hypothese that form the Basis of Geometry, mathematicians have been familiar with the fact that the methods of geometry suitably generalized can be applied to the treatment of an n-fold manifoldness; and in point of fact the synoptic treatment of manifoldness under the name of n-dimensional geometry has been usefully employed by Cayley and others as an adjunct to the analytic method.

The fundamental characteristic of the analytical treatment of an n-fold manifoldness is the specification of an element by means of n continuously varying quantities or variable (see MEASUREMENT). For dealing with continuous as distinguished form discrete quantity we have the special analytical method of the INFINITESIMAL CALCULUS (q.v.), built upon the notion of a limit, with its various branches, viz., the differential calculus, the integral calculus including differential equations, the calculus of functions, and the theory of functions in general (see FUNCTION). But, whether we make use of the algorithm of the infinitesimal calculus or not, we find upon examination that all analytical operations with continuous quantity fall under the three laws of commutation, association, and distribution, so that they are fundamentally identical with the operations with discrete quantity; the difference so far as there is any consists simply in the greater generality of the operand. The same fact may be looked at instructively in another light. Whether we consider analytical processes in concrete applications or look at them abstractly, we are equally led to the notion of a unit, by the multiplication or subdivision of which all the other quantities that enter into our calculus are derived. The exigencies of continuity are met by allowing that the multiplication or subdivision of the unit can be carried on to an unlimited extent; but in any case where analytical formulae have to be reduced to arithmetical calculation (in which of course only a finite number of figures or arithmetical symbols can be used) the subdivision (or multiplication) of the unit actually stops short at a certain point; in other words, all our methods are, short at a certain point; in words, all our methods are, in practise at least, discrete. Here therefore we have the meeting point of discrete and continuous quantity, and on this ground alone we might infer the fundamental identity of their laws of operation.





The abstract science of quantity which we have just seen to be the essential part of the analytic treatment of manifoldness receives the name of ALGEBRA (q.v.). It was found very early in the history of that science that the full development of which it is capable could not be attained without great extension of the idea of quantity. This necessity first arose in connexion with the inverse operations, such as subtraction, the extraction of roots, and the numerical solution of algebraical equations (see EQUATION), of which root extraction is merely a particular case. In this way arose essentially negative quantities, and the so-called impossible or imaginary quantities. The former may be said to depend on a new abstract unit—1, and the latter upon new units ±&Mac195;¯¯-1. The numbers having ± 1 for abstract unit are usually classed as real numbers, and in that care we may regard the ordinary imaginaries of algebra as depending on the new unit + &Mac195;¯¯-1, or _, defined by the equation _2 + 1 + 0. But the extension was soon carried father by the classical researches of Hamilton and Grassmann.1 The theory of sets and the QUATERNIONS (q.v.) of the former and the Ausdehnungslehre of the latter opened up a boundless field for algebra, and led to a total revolution in our ideas of quantity.

In view of the great extension thus effected in the meaning of quantity, it becomes an interesting if somewhat difficult undertaking to define the word. The following may be taken as a provisional definition—Quantity is that which is operated with according to fixed mutually consistent laws. Both operator and operand must derive their meaning from the laws of operation. In the case of ordinary algebra these are the three laws already indicated, in the algebra of quaternions the same save the law of commutation for multiplication and division, and so on. It may be questioned whether this definition is sufficient, and it may be objected that it is vague; but the reader will do well to reflect that any definition must include the linear algebras of Pierce, the algebras of logic, and others that may be easily imagined, although they have not yet been developed. This general definition of quantity enables us to see how operators may be treated as quantities, and thus to understand the rationale of the so-called symbolical methods. In combining operations, it is often observed that the combinations of operators fall under a few simple laws, in some cases, in fact under the three laws of ordinary algebra: these operators are therefore quantities according to the general definition, and can be treated as such.

From the historical development of the analytic method there is little danger of the error arising that its application is peculiar to any special kind of manifoldness. As examples of its use in deducing the properties of tridimensional space we may cite the Cartesian geometry, the Géométrie de Position of carnot, and the line geometry of Plücker (see GEOMETRY). Its use in the various branches of applied mathematics, of which geometry is merely one of the simplest, is far more common than that of the synoptic method, although most branches of applied mathematics are mixtures using the one or the other, as happens to be convenient. In addition to those already mentioned, we may enumerate the following as among the more important departments of applied mathematics:—Kinematics; Abstract Dynamics, including Statics and Kinetics whether of a Particle, of a Rigid Solid, or an Elastic Solid, of a Fluid, or of a Chain; Statistical Mathematics, as exemplified in the Theory of Annuities, and the Kinetic Theory of gases; the Mathematical Theory of Diffusion whether of Heat or of Matter; the Theory of Diffusion whether of Heat or of Matter; the Theory of Potential; and so on. See MECHANICS, HYDROMECHANICS, ANNUITIES, HEAT, ELECTRICITY, MAGNETISM, &c.

The two great methods employed in the investigation of manifoldness must of course be, as bottom, identical; and every conclusion arrived at by the one must be reachable by the other. The exact nature of the connexion between them will be well seen by studying two instances. One of these is the treatment of areas by Euclid and the treatment by the analytical method, which are carefully compared in the article GEOMETRY, vol. x. p. 379. the other is the connexion between the descriptive and the metrical properties of loci. The former include all properties such as intersection, tangency, &c., depending on position merely, and are obviously the natural product of the synoptic method. The latter include all relations involving the lengths of lines and the magnitudes of angles, they depend therefore on expression in terms of unit, and are the natural product of the analytic method. Nevertheless the analytic method furnishes descriptive properties of loci, and by the introduction of "the absolute" descriptive theorems are made to furnish metrical relations, as has been fully shown by Cayley, Clifford, and Klein (see MEASUREMENT). (G. CH.)


Footnote

FOOTNOTE (page. 630)

1 In this connexion should be mentioned the great services of De Morgan, whose bold speculations on the fundamental principles of mathematical science have perhaps met with less than their due share of appreciation.



The above article was written by: George Chrystal, M.A., LL.D.; Hon. Fellow of Corpus Christi College, Cambridge; Professor of Mathematics, Edinburgh University, form 1879; author of Treatise on Algebra and Introduction to Algebra.




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