Mathematical analysis

“ In the mere act of drawing a curve to represent an analytical function we make unconsciously a host of assumptions which are difficult not merely to prove, but even to formulate precisely. It is now sought to establish the whole fabric of mathematical analysis on a strictly arithmetical basis. ”

“ The terms synthesis and analysis are used in mathematics in a more special sense than in logic. In ancient mathematics they had a different meaning from what they now have. The oldest definition of mathematical analysis as opposed to synthesis is that given in Euclid, XIII. 5, which in all probability was framed by Eudoxus: "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth; synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it." ”

“ Mathematical analysis is, then, the true rational basis of the entire system of our actual knowledge. ”

“ There is a second reason why we cannot bring complicated phenomena under the dominion of mathematical analysis. Even if we could ascertain the mathematical law which governs each agent, taken by itself, the combination of so great a number of conditions would render the corresponding mathematical problem so far above our [Pg 39] feeble means, that the question would remain in most cases incapable of solution. ”

“ It is true that in the greatest number of important cases its use requires to be preceded and prepared by that of the Calculus of indirect Functions, which is intended to facilitate the establishment of equations. But, although algebra has then only a secondary office to perform, it has none the less a necessary part in the complete solution of the question, so that the Calculus of direct Functions must continue to be, by its nature, the fundamental base of all mathematical analysis. ”

“ We know that the application of mathematical analysis, from its nature, can never commence any science whatever, since evidently it cannot be employed until the science has already been sufficiently cultivated to establish, with respect to the phenomena considered, some equations which can serve as starting points for the analytical operations. ”

“ This necessary perfection of geometry, obtained essentially by the application of mathematical analysis, which it so eminently admits, is apt to produce erroneous views of the real nature of this fundamental science, which most minds at present conceive to be a purely logical science quite independent of observation. ”

“ 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. ”

“ Now it must not be thought that the introduction of the mathematical instrument into our study of nature creates any essential departure from the commonplace method of ordinary deductive and inductive reasoning. There is no particular mystery about mathematical analysis ”

“ The high relative perfection of mathematical analysis is as easily perceptible. This perfection is not due, as some have thought, to the nature of the signs which are employed as instruments of reasoning, eminently concise and general as they are. In reality, all great analytical ideas have been formed without the algebraic signs having been of any essential aid, except for working them out after the mind had conceived them. ”

“ 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. ”

“ The same error was afterward committed, in the infancy of the infinitesimal calculus, in relation to the integration of differential equations.
[9] The fundamental principle on which reposes the theory of equations, and which is so frequently applied in all mathematical analysis—the decomposition of algebraic, rational, and entire functions, of any degree whatever, into factors of the first degree—is never employed except for functions of a single variable, without any one having examined if it ought to be extended to functions of several variables. ”

“ Such is, in substance, the eminently fruitful conception, which the human mind has succeeded in establishing, and which constitutes its most admirable instrument for the mathematical explanation of natural phenomena; the analysis, called transcendental.
As a general philosophical principle, the auxiliary quantities, which are introduced in the place of the primitive magnitudes, or concurrently with them, in order to facilitate the establishment of equations, might be derived according to any law whatever from the immediate elements of the question. ”

“ Thus the number of truly distinct arithmetical operations is only that determined by the number of the elementary abstract functions, the very limited list of which has been given above. The determination of the values of these ten functions necessarily gives that of all the functions, infinite in number, which are considered in the whole of mathematical analysis, such at least as it exists at present. There can be no new arithmetical operations without the creation of really new analytical elements, the number of which must always be extremely small. ”

“ Considering the infinite diversity which must necessarily exist in this respect in the external world, we easily understand how far below the true difficulty our conceptions must frequently be found, especially if we add that as these elements of our analysis have been in the first place furnished to us by the mathematical consideration of the simplest phenomena, we have, à priori, no rational guarantee of their necessary suitableness to represent the mathematical law of every other class of phenomena. ”

“ Euclidean mathematics assumes the completeness and invariability of mathematical forms; these forms it describes with appropriate accuracy and enumerates their inherent and related properties with perfect clearness, order, and completeness, that is, Euclidean mathematics operates on forms after the manner that anatomy operates on the dead body and its members.
On the other hand, the mathematics of variable magnitudes—function theory or analysis—considers mathematical forms in their genesis. ”

“ 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. ”

“ We have insisted particularly on this generalising tendency of mathematics, as it is the one which will play a prominent part when we consider the methodology of mathematical physics as applied to the world of reality. ”

“ 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; to those who have been oppressed by the fragmentary and incomplete nature of most existing chains of deduction this discovery comes with all the overwhelming force of a revelation; like a palace emerging from the autumn mist as the traveller ascends an Italian hill-side, the stately storeys of the mathematical edifice appear in their due order and proportion, with a new perfection in every part. ”

“ 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. ”

“ 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. ”

“ We are now able to define mathematical science with precision, by assigning to it as its object the indirect measurement of magnitudes, and by saying it constantly proposes to determine certain magnitudes from others by means of the precise relations existing between them. ”

“ The statement that mathematics is the science of number would be untrue in two different ways. On the one hand, there are recognised branches of mathematics which have nothing to do with number—all geometry that does not use co-ordinates or measurement, for example: projective and descriptive geometry, down to the point at which co-ordinates are introduced, does not have to do with number, or even with quantity in the sense of greater and less. ”

“ The great inventions of the seventeenth century—Analytical Geometry and the Infinitesimal Calculus—were so fruitful in new results that mathematicians had neither time nor inclination to examine their foundations. Philosophers, who should have taken up the task, had too little mathematical ability to invent the new branches of mathematics which have now been found necessary for any adequate discussion. ”

“ This quantitative reduction is, however, but a first step; and even after it has been accomplished, nature might still defy mathematical investigation. The fact is that mathematics consists in compounding the similar with the similar; wide heterogeneities even among quantities would render an appeal to human mathematics useless. ”

“ No doubt this logical purpose is its application in the world of phenomena, and the mathematical concept must thus fit the world so absolutely that it 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. Mathematics would then be the phenomenalistic science of the form and order of the world. ”

“ The nature of infinity and continuity, for example, belonged in former days to philosophy, but belongs now to mathematics. Mathematical philosophy, in the strict sense, cannot, perhaps, be held to include such definite scientific results as have been obtained in this region; the philosophy of mathematics will naturally be expected to deal with questions on the frontier of knowledge, as to which comparative certainty is not yet attained. But speculation on such questions is hardly likely to be fruitful unless the more scientific parts of the principles of mathematics are known. ”

“ On the contrary, he says, I think that wherever, from the side of the theory of knowledge, or in geometry, or from the theories of natural or physical science, mathematical ideas come up, the problem arises for mathematical science to investigate the principles underlying these ideas and so to establish them upon a simple and complete system of axioms, that the exactness of the new ideas and their applicability to deduction shall be in no respect inferior to those of the old arithmetical concepts. ”

“ All traditional pure mathematics, including analytical geometry, may be regarded as consisting wholly of propositions about the natural numbers. That is to say, the terms which occur can be defined by means of the natural numbers, and the propositions can be deduced from the properties of the natural numbers—with the addition, in each case, of the ideas and propositions of pure logic. ”

“ 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. ”

“ 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. ”

“ ALL objects in the world which can be made the subjects of our contemplation are subordinate to the conditions of Space, Time, and Number; and on this account, the doctrines of pure mathematics have most numerous and extensive applications in every department of our investigations of nature. And there is a peculiarity in these Ideas, which has caused the mathematical sciences to be, in all cases, the first successful efforts of the awakening speculative powers of nations at the commencement of their intellectual progress. ”

“ Even in those sciences which are not yet under the dominion of mathematics, and perhaps never will be, a working copy of the mathematical process has been made. This is not known to the followers of those sciences who are not themselves mathematicians and who very often exalt their horns against the mathematics in consequence. ”