Infinitesimal calculus

“ That which in every application of the transcendental analysis really facilitates the formation of equations, is the infinitesimal method, and not the infinitesimal calculus, which is perfectly distinct from it, although it is its indispensable complement. Such a consideration would, then, give a false idea of the special destination which characterizes the differential calculus in the general system of the transcendental analysis. ”

“ The transcendent results of calculus are, like all the abstractions of the understanding, general signs whose true meaning may be ascertained only by repassing by metaphysical analysis to the elementary ideas which have led to them; this often presents great difficulties, for the human mind tries still less to transport itself into the future than to retire within itself. The comparison of infinitely small differences with finite differences is able similarly to shed great light upon the metaphysics of infinitesimal calculus. ”

“ So for the incommensurable number, the vague idea of continuity, which we owe to intuition, has resolved itself into a complicated system of inequalities bearing on whole numbers. Thus have finally vanished all those difficulties which frightened our fathers when they reflected upon the foundations of the infinitesimal calculus. To-day only whole numbers are left in analysis, or systems finite or infinite of whole numbers, bound by a plexus of equalities and inequalities. Mathematics we say is arithmetized. ”

“ That this is possible is evident, for, the limit of each infinitesimal being zero, each one can be made numerically less than ( being the number of infinitesimals) , and therefore their sum can be made numerically less than .
(2) The product of a constant and an infinitesimal is an infinitesimal. For the numerical value of the product can always be made less than any small positive quantity (as ) by making the numerical value of the infinitesimal less than .
(3) The product of any finite number of infinitesimals is an infinitesimal. ”

“ Critics of new theories are never lacking. On the other hand, as E. W. Hobson has well said, “pertinent criticism of fundamentals almost invariably gives rise to new construction.” In the history of the infinitesimal calculus the 17th and 18th centuries were mainly a period of construction, the 19th century mainly a period of criticism. ”

“ Both of these ideas are somewhat technical, and would hardly demand treatment in a mere introduction to mathematical philosophy but for the fact that, especially through the so-called infinitesimal calculus, wrong views upon our present topics have become so firmly embedded in the minds of professional philosophers that a prolonged and considerable effort is required for their uprooting. ”

“ If φ′ (x) and ψ′ (x) have determinate finite limits when x is increased indefinitely, while φ (x) and ψ (x) are determinately (positively or negatively) infinite, we have the result expressed by the equation
For the meaning of the statement that φ (x) and ψ (x) are determinately infinite reference may be made to the article Function. The evaluation of forms of the type ∞/∞ leads to a scale of increasing “infinities,” each being infinite in comparison with the preceding. ”

“ All the more exact and successful parts of physical science depend upon its use, and yet the mathematicians who have created so great a body of exact truths have never decided upon the basis of the calculus. What is the nature of a limit or the nature of an infinitesimal? Start the question among a knot of mathematicians, and it will be found that hardly two agree, unless it is in regarding the question itself as a trifling one. Some hold that there are no such things as infinitesimals, and that it is all a question of limits. ”

“ Euclid, Archimedes, and Apollonius brought geometry to as high a state of perfection as it perhaps could be brought without first introducing some more general and more powerful method than the old method of exhaustion. A briefer symbolism, a Cartesian geometry, an infinitesimal calculus, were needed. The Greek mind was not adapted to the invention of general methods. Instead of a climb to still loftier heights we observe, therefore, on the part of later Greek geometers, a descent, during which they paused here and there to look around for details which had been passed by in the hasty ascent. ”

“ This result itself shows that every ordinary differential equation of the first order is subject to an infinite number of infinitesimal transformations. But every infinitesimal transformation ξdƒ/dx + ηdƒ/dy can by change of variables (after integration) be brought to the form dƒ/dy, and all differential equations of the first order allowing this group can then be reduced to the form F (x, dy/dx) = 0. ”

“ Then when we take in hand the general theorem it becomes indispensable, for otherwise we should ever be approaching the analytical verification without ever actually reaching it. In this domain of Arithmetic we may think ourselves very far from the infinitesimal analysis, but the idea of mathematical infinity is already playing a preponderating part, and without it there would be no science at all, because there would be nothing general.
VI. The views upon which reasoning by recurrence is based may be exhibited in other forms ”

“ If we want to go so far that not even a Whitworth’s measuring machine would detect it, we should merely have to go to about terms. A microscope would not show even the term! So the infinite number of operations is no such dreadful thing after all. The integral is simply the whole lot. But, as we shall see, there are cases in which the integral calculus enables us to get at the exact total that there would be as the result of an infinite number of operations. ”

“ The philosophy of the infinitesimal, as we have just seen, is mainly negative. People used to believe in it, and now they have found out their mistake. The philosophy of the infinite, on the other hand, is wholly positive. It was formerly supposed that infinite numbers, and the mathematical infinite generally, were self-contradictory. ”

“ It is of course true that these advances have depended upon problems such as those which in the researches of Kepler and Galileo led to the early concepts of the infinitesimal procedure, and upon such undertakings as bringing the combined theories of geometry and algebra to bear upon the experiences of continuous change. For a century after the formulation of the infinitesimal method men were occupied in carrying the new tool of analysis into every field where its use promised advance. ”

“ So on the other hand it would be equally wrong to imagine, that in a geometrical Demonstration we may be allowed to admit any Error, though ever so small, or that it is possible, in the nature of Things, an accurate Conclusion should be derived from inaccurate Principles. Therefore o cannot be thrown out as an Infinitesimal, or upon the Principle that Infinitesimals may be safely neglected. But only because it is destroyed by an equal Quantity with a negative Sign, whence o - qo is equal to nothing. ”

“ We know that mathematicians distinguish between infinitesimals of different orders and that those of the second order are infinitesimal, not only in an absolute way, but also in relation to those of the first order. It is not difficult to imagine infinitesimals of fractional or even of irrational order, and thus we find again that scale of the mathematical continuum which has been dealt with in the preceding pages. ”

“ An absolutely infinite number cannot be expressed in the algebraic or geometrical values; the attempt so to express it limits it in a certain sense, and therefore destroys its absolute infinity. If the expression ∞, represented an absolutely infinite number, it would not be susceptible of any combination which would increase it: to suppose that it may be multiplied by other numbers, finite or infinite, is to take its infinity in another than an absolute sense. ”

“ If the Finite and the Infinite can in turn be deduced from one another, neither conception can be truly called an axiom. The real axiom is existence, which includes both, and which is defined by mathematicians as that which is not self-contradictory.
Now the problem of Infinity includes also that of continuity; in other words, the problem of number includes that of cardinal and ordinal number. It is time to get to the mathematical definition of number, which we have found as a conception can be attached both to the Finite and to the Infinite. What is number in mathematics? ”

“ The theorem therefore holds if F (z) has an infinity up to which it is absolutely integrable; this will, for example, be the case if F (z) near the point C is of the form x (z) (z − c) −μ + ψ (z) , where χ (c) , ψ (c) are finite, and 0 < μ < 1. It is thus seen that ƒ (x) may have a finite number of infinities within the given interval, provided the function is integrable through any one of these points; the function is in that case still representable by Fourier’s Series. ”

“ Such a general verification [Pg 100] is undoubtedly strictly sufficient to dissipate any uncertainty as to the legitimate employment of the analysis of Leibnitz. But the infinitesimal method is so important—it offers still, in almost all its applications, such a practical superiority over the other general conceptions which have been successively proposed—that there would be a real imperfection in the philosophical character of the science if it could not justify itself, and needed to be logically founded on considerations of another order, which would then cease to be employed. ”

“ To infinitesimal values of the parameters, thus chosen, will correspond operations which cause an infinitesimal change in each of the variables. These are called infinitesimal operations. The most general infinitesimal operation of the group is that given by the system
where, in ∂ƒs/∂ai, zero values of the parameters are to be taken. Since a1, a2, ..., ar are independent, the ratios of δa1, δa2, ..., δar are arbitrary. Hence the most general infinitesimal operation of the group may be written in the form
where e1, e2, ..., er are arbitrary constants, and δt is an infinitesimal. ”

“ The mathematical notion of an infinitesimal contains, in truth, the whole paradox of the same and yet the nascent other, of an identity that won't keep except so far as it keeps failing, that won't transfer, any more than the serial relations in question transfer, when you apply them to reality instead of applying them to concepts alone. ”

“ The infinity of rationals does not demand, for its definition or use, any infinite classes or infinite integers. It is not, in actual fact, a very important notion, and we could dispense with it altogether if there were any object in doing so. The Cantorian infinite, on the other hand, is of the greatest and most fundamental importance; the understanding of it opens the way to whole new realms of mathematics and philosophy. ”

“ They are those in which the differential equations are found to be immediately ready for integration, because they contain, at their first formation, only the infinitesimals which relate to the functions sought, or to the really independent variables, without its being necessary to introduce, differentially, other functions as intermediaries. If in these new cases we introduce these last functions, since, by hypothesis, they will enter directly and not by their differentials, ordinary algebra will suffice to eliminate them, and to bring the question to depend on only the integral calculus. ”

“ As a matter of fact, so long as we are content with the arithmetic of finite integers, and do not introduce either infinite integers or infinite classes or series of finite integers or ratios, it is possible to obtain all desired results without the axiom of infinity. ”

“ Our English author, on the other hand, exclaims, let no man give himself leave to talk about infinite number, for infinite number is a contradiction in terms; whatever is once numbered, we all see, cannot be infinite. “I think,” said Mr. Johnson, after a pause, “we must settle the matter thus: numeration is certainly infinite, for eternity might be employed in adding unit to unit; but every number is in itself finite, as the possibility of doubling it easily proves; besides, stop at what point you will, you find yourself as far from infinitude as ever.” ”

“ These preliminary ideas being established, the spirit of the infinitesimal analysis consists in constantly neglecting the infinitely small quantities in comparison with finite quantities, and generally the infinitely small quantities of any order whatever in comparison with all those of an inferior order. ”

“ So that according to them an inch does not barely contain an infinite number of parts, but an infinity of an infinity of an infinity ad infinitum of parts. Others there be who hold all orders of Infinitesimals below the first to be nothing at all; thinking it with good reason absurd to imagine there is any positive quantity or part of extension which, though multiplied infinitely, can ever equal the smallest given extension. And yet on the other hand it seems no less absurd to think the square, cube, or other power of a positive real root, should itself be nothing at all ”

“ A function having an algebraic infinity of the mth order (m > 1) , only for one sign of s when these signs are separable, at (1) z = ∞, (2) z = z0, (3) z = a, is given respectively by (s d/dz) m−1 J1, (s d/dz) m−1 J2, (s d/dz) m−1 J3, as we easily see. If then we have any elliptic integral having algebraic infinities we can, by subtraction from it of an appropriate sum of constant multiples of J1, J2, J3 and their differential coefficients just written down, obtain, as the result, an integral without algebraic infinities. ”