Function (mathematics)

“ In fact, even if we could integrate every equation of the first order relating to a single function, that would not be sufficient for obtaining the final integral of an equation of any order whatever, inasmuch as every differential equation is not reducible to that of an immediately inferior order. ”

“ Functions, definition of, 356; theory of, 268, 269, 345, 356–362; arbitrary, 262, 283. See Elliptic functions, Abelian functions, Hyperelliptic functions, Theta functions, Beta function, Gamma function, Omega function, Sigma function, Bessel's function, Potential. ”

“ To present this comparison under a new point of view, we may say concrete mathematics has a philosophical character, which is essentially experimental, physical, phenomenal; while that of abstract mathematics is purely logical, rational. ”

“ In seeking to determine with precision in what determinations of values properly consist, we easily recognize that they are nothing else but veritable transformations of the functions to be valued; transformations which, in spite of their special end, are none the less essentially of the same nature as all those taught by analysis. In this point of view, the calculus of values might be simply conceived as an appendix, and a particular application of the calculus of functions, so that arithmetic would disappear, so to say, as a distinct section in the whole body of abstract mathematics. ”

“ Mathematics is based upon nothing, and, consequently, arises out of nothing.
"Out of nothing, therefore, it is possible for something to arise; for mathematics, consisting of propositions, is something, in relation to 0." ”

“ The point which I now want to make is that this dominance of the idea of functionality in the abstract sphere of mathematics found itself reflected in the order of nature under the guise of mathematically expressed laws of nature. ”

“ I will explain presently the general artifice, so profoundly ingenious, by which the human mind has succeeded in diminishing, in a remarkable degree, this fundamental difficulty which is presented by the relation of the concrete to the abstract in mathematics, without, however, its having been necessary to multiply the number of these analytical elements. ”

“ Avoiding, however, the mathematical form of expression, one may here try to make clear the metaphysically important nature of theorems of this sort very much as follows: In pure mathematics, the student deals with certain objects that, upon their face, are the products of purely arbitrary definitions. ”

“ I dispute the availability, and thus the value, of that reason which is cultivated in any special form other than the abstractly logical. I dispute, in particular, the reason educed by mathematical study. The mathematics are the science of form and quantity; mathematical reasoning is merely logic applied to observation upon form and quantity. The great error lies in supposing that even the truths of what is called pure algebra are abstract or general truths. ”

“ The mathematical questions which have given birth to the Calculus of Variations consist generally in the investigation of the maxima and minima of certain indeterminate integral formulas, which express the analytical law of such or such a phenomenon of geometry or mechanics, considered independently of any particular subject. ”

“ Before we can undertake it, we must consider certain other parts of the philosophy of mathematics, which we have hitherto ignored. In a synthetic treatment, the parts which we shall now be concerned with come first: they are more fundamental than anything that we have discussed hitherto. ”

“ If philosophers cannot consent to relegate the criticism of these categories to any of the special sciences, it is essential, at any rate, that they should know the precise meaning that the science of mathematics, in which these concepts play so large a part, assigns to them. ”

“ The Calculus of Values, or Arithmetic, would appear, at first view, to present a field as vast as that of algebra, since it would seem to admit as many distinct questions as we can conceive different algebraic formulas whose values are to be determined. ”

“ If we had been embarking upon mathematical philosophy in earlier days, we should have had to start with something less abstract than the series of natural numbers, which we should reach as a stage on our backward journey. When the logical foundations of mathematics have grown more familiar, we shall be able to start further back, at what is now a late stage in our analysis. But for the moment the natural numbers seem to represent what is easiest and most familiar in mathematics. ”

“ It is not necessary to agree with what he there suggests as to the readjustment of the field of philosophy by the transference from it to mathematics of such problems as those of class, continuity, infinity, in order to perceive the bearing of the definitions and discussions that follow on the work of "traditional philosophy." ”

“ The exercise of logical reason is always concerned with these absolutely general conditions. In its broadest sense, the discovery of mathematics is the discovery that the totality of these general abstract conditions, which are concurrently applicable to the relationships among the entities of any one concrete occasion, are themselves inter-connected in the manner of a pattern with a key to it. ”

“ The reality of mathematics consists in the universality of its quantities; viz. numbers or figures. Every number, and every thing which belongs to mathematics, can be derived from no other source than zero. ”

“ Mathematics in its pure form, as arithmetic, algebra, geometry, and the applications of the analytic method, as well as mathematics applied to matter and force, or statics and dynamics, furnishes the peculiar study that gives to us, whether as children or as men, the command of nature in this its quantitative aspect ”

“ But when at last they have been found, they are seen to be few in number, and to be the sole source of everything in pure mathematics. The discovery that all mathematics follows inevitably from a small collection of fundamental laws is one which immeasurably enhances the intellectual beauty of the whole ”

“ In the most simple questions, and when we have really to resolve only a single equation with a single unknown quantity, this numerical resolution is none the less a very imperfect method, even when it is strictly sufficient. ”

“ The definite object of our researches in concrete mathematics being the discovery of the equations which express the mathematical laws of the phenomenon under consideration, and these equations constituting the true starting point of the calculus, which has for its object to obtain from them the determination of certain quantities by means of others, I think it indispensable, before proceeding any farther, to go more deeply than has been customary into that fundamental idea of equation, the continual subject, either as end or as beginning, of all mathematical labours. ”

“ It is composed of what is called the Calculus, [2] taking this word in its greatest extent, which reaches from the most simple numerical operations to the most sublime combinations of transcendental analysis. The Calculus has the solution of all questions [Pg 34] relating to numbers for its peculiar object. Its starting point is, constantly and necessarily, the knowledge of the precise relations, i.e., of the equations, between the different magnitudes which are simultaneously considered; that which is, on the contrary, the stopping point of concrete mathematics. ”

“ We have thus far viewed mathematical science only as a whole, without paying any regard to its divisions. We must now, in order to complete this general view, and to form a just idea of the philosophical character of the science, consider its fundamental division. ”

“ But mathematics uses these facts to deduce more and more complicated theorems, while philosophy seeks, by analysis, to go behind these facts to others, simpler, more fundamental, and inherently more fitted to form the premisses of the science of arithmetic. The question, “What is a number?” is the pre-eminent philosophic question in this subject, but it is one which the mathematician as such need not ask, provided he knows enough of the properties of numbers to enable him to deduce his theorems. We, since our object is philosophical, must grapple with the philosopher’s question. ”

“ The close study of pure mathematics, by directing the mind to processes of calculation rather than to phenomena, induces that sublime indifference to facts which has characterized the purely mathematical intellect of all ages. ”

“ It enables one to readily understand numbers and their combinations. Its activity takes place, whenever there is a departure from unity. A large endowment of this organ is not essential to the algebraist and geometrician, its functions being limited to arithmetical calculations.
Predominant.—One having Calculation predominant, will reckon in his head almost any arithmetical problem that can be proposed to him. If Causality and Comparison are large, he will excel in the higher branches of mathematics, and possess a great fondness for these studies. ”

“ To complete this rapid general enumeration of the different essential parts of the calculus of direct functions, I must, lastly, mention expressly one of the most fruitful and important theories of algebra proper, that relating to the transformation of functions into series by the aid of what is called the Method of indeterminate Coefficients. ”

“ What we know in Algebra. In the present condition of algebra, the complete resolution of the equations of the first four degrees, of any binomial equations, of certain particular equations of the higher degrees, and of a very small number of exponential, logarithmic, or circular equations, constitute the fundamental methods which are presented by the calculus of direct functions for the solution of mathematical problems. ”

“ Pure mathematics is concerned with the investigation of the logical consequences of certain exactly stateable postulates or hypotheses—such, for instance, as the postulates upon which arithmetic and analysis are founded, or such as the postulates that lie at the basis of any type of geometry. ”

“ We have sufficiently defined the character of the primitive ideas in terms of which all the ideas of mathematics can be defined, but not of the primitive propositions from which all the propositions of mathematics can be deduced. This is a more difficult matter, as to which it is not yet known what the full answer is.
We may take the axiom of infinity as an example of a proposition which, though it can be enunciated in logical terms, [Pg 202] cannot be asserted by logic to be true. ”

“ No doubt this logical purpose is its application in the world of objects and the mathematical concepts must thus fit the objective world so absolutely that mathematics can be conceived as a description of the world after abstracting not only from the will-relations, as physics does, but also from the content. ”

“ 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. ”

“ The complete solution of every question of the calculus, from the most elementary up to the most transcendental, is necessarily composed of two successive parts, whose nature is essentially distinct. In the first, the object is to transform the proposed equations, so as to make apparent the manner in which the unknown quantities are formed by the known ones: it is this which constitutes the algebraic question. ”

“ According to such a definition, the spirit of mathematics consists in always regarding all the quantities which any phenomenon can present, as connected and interwoven with one another, with the view of deducing them from one another. Now there is evidently no phenomenon which cannot give rise to considerations of this kind; whence results the naturally indefinite extent and even the rigorous logical universality of mathematical science. We shall seek farther on to circumscribe as exactly as possible its real extension. ”

“ So the most fantastic forms of mathematics, which themselves seem to bear no relation to actual phenomena, just because the relations involved in them are the relations that have been derived from dealing with an actual world, may contribute to the solutions of problems in other forms of calculus, or even to the creation of new forms of mathematics. ”

“ Every problem will constantly require a certain preliminary labour to be performed, in which the calculus can be of no assistance, and which, by its nature, cannot be subjected to abstract and invariable rules; it is that which has for its special object the establishment of equations, which form the indispensable starting point of all analytical researches. ”

“ Important as it is to comprehend the rigorous universality, in a logical point of view, of mathematical science, it is no less indispensable to consider now the great real limitations, which, through the feebleness of our intellect, narrow in a remarkable degree its actual domain, in proportion as phenomena, in becoming special, become complicated. ”

“ It is even certain that the last special improvements in mathematical analysis have contributed to extend our resources in that respect, by introducing within the domain of the calculus certain definite integrals, which in some respects supply the place of new simple functions, although they are far from fulfilling all the necessary conditions, which has prevented me from inserting them in the table of true analytical elements. ”