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From Bottazzini's The Higher Calculus:
In his Théorie des fonctions analytiques (1772), Lagrange attempted to
present the principles of the calculus in a systematic manner without making any reference to infinitesimals, evanescent quantities, differentials or limits. He instead stressed the need to reduce the calculus to simple algebraic manipulations of finite quantities. (Bottazzini 48ab)
He defines a function thus:
We call a function of one or more quantities every expression of the calculus in which these quantities enter in any way whatsoever, mixed or not with other quantitites that one takes as having given and invariable values, while the quantitities of the function can have all possible values. (Lagrange, 1797, p. 1; 1813, p. 15, qtd in Bottazzini 48bc)
Langrange's next step was to show that any given function can be expanded as a series. Bottazzini quotes Lagrange as writing:
"We therefore consider a function f (x) of any variable x. If in place of x we put x + i, i being any indeterminate quantity whatever, it becomes f (x + i) and, by the theory of series [Bottazzini's emphasis] we can expand it as a series of this form:
in which the quantities p, q, r, ... , the coefficients of the powers of i, will be new functions of x, derived from the primitive function x and independent of the [indeterminate] quantity i ... (1797, p.2; 1813, pp.21-2, qtd in Bottazzini 48d).
Because Lagrange is interested in the formation and calculation of these different functions, the new calculus for him is differential or fluxional (48d).
Lagrange critiqued early renditions of the calculus, arguing instead that "'the true metaphysics' of the calculus lies in the fact that the errors resulting from neglecting infinitesimals of higher degrees were 'corrected or compensated' by the procedures of the calculus themselves, when they were limited to infinitesimals of the same degree" (49a). Euler and D'Alembert failed to realize this, although they addressed the problem by claiming that infinitely small values equal zero, but the limits of the ratios of indefinite differences have quantitative value (49b). Lagrange critiques Newton's notion of the motion of quantities so to avoid the notion of infinitesimal value by saying
on the one hand, to introduce movement into a calculus that only has algebraic quantities as its object is to introduce a foreign idea which obligates one to think of these quantities as the lines traversed by a moving body. On the other hand, it must be granted that we do not even have a really clear idea of what the velocity of a point is at every instant when this velocity is variable (Lagrange 1797, pp. 3-4; 1813, p.17; qtd in Bottazzini 49bc).
Lagrange thinks that Landen and Arbogast make steps in the right direction, because their aim according to Lagrange was to deal with the functions arising when expanding any given function, and then to apply these "derived" functions in math and science. Lagrange claims that his approach is free of "every illicit supposition" and "all metaphysics," and that it bears "the rigor of the ancient demonstrations" [see Hegel's Science of Logic section 587] by basing it on his method of primitive functions and derivatives (49d). "Nevertheless," Bottazzini writes
the crucial point in his entire construction is his assertion that it is possible to expand any function in a series of ascending powers of an indeterminate increment i. (49d)
Because he wants to offer proof, Lagrange considers the
form of the series that must represent the expansion of every function f (x) when one substutites x + 1 in place of x, and which we have supposed must contain only integral and positive power of i. (Lagrange 1797, p.7; 1813, p. 22; qtd in Bottazzini 50a).
Lagrange wants to establish that this holds for all functions, so he shows that "when x and i remain indeterminate, the series cannot contain fractional or negative powers of i" (50b). But there is no way to prove such an expansion of a function, "there is only the fact that the series contains only positive powers of i," and it is only after the function is expanded that we learn what each terms means. So we begin with
where i is any indeterminate quantity whatever, P is a new function of x and i. We can determine this other function P in terms of x and i by moving fx to the other side of the equation, and dividing both sides by i so to obtain:
We then let i vanish so to separate P insofar as it is independent of i (which does not equal zero as it vanishes), leaving a differential value p. Lagrange then shows that
P = p + iQ.
Q can be further determined, and so on, hence:
But each time we substitute we continue to multiple i's with other i's
Lagrange says we may make i so small that any given term will be larger than those following it in the series.
He recognizes that this expansion does not hold for every possible function.
Lagrange then identifies the functions p, q, r, ... with f 'x, (f ''x/2), (f '''x/3!), ... , where f 'x, f ''x, f '''x, and so on are the successive derivatives of the function fx. [See the Taylor Series]
What is most notable about his technique is that it is purely algebraic.
Lagrange's theory was attacked a year later by Hoëne Wronski. In his Introduction à la philosophie de la mathématique et technie de l'algorithmie, he comes to conclude that all functions may take the form:
"where the
are any functions of x and the determination of the coefficients
of the series depend on the determinants that are today called 'Wronskian'" (54-55).
Wronski then vigorously critiques Lagrange in Réfutation de la théorie des fonctions analytiques de Lagrange. In it, Wronski notes that Lagrange's theory is based on two assumptions:
Wronski attacks three aspects of Lagrange's theory:
1) He wonders what the grounds are for the first formulation. There seem not to be any.
2) Lagrange claims that the second formulation is true because it is verified when i equals one, but Wronski wants to know what about when i does not equal 1?
3) Lagrange defines f '(x) as the coefficient of the second term in the expanded series in the bottom equation. But Wronski notes that the position of a function in a series does not by its place alone determine the meaning of that function; it seems Lagrange slips them together invalidly.
Bottazzini, Umberto. The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass. Transl. Warren Van Egmond. Paris: Springer-Verlag, 1981.
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from Boyer's History of the Calculus
Under the influence of Leibniz' differential method and Kant's transcendental philosophy, Hoëné Wronski objected against Lagrange's ban on the infinite in analysis.
Wronki rightly asks where Lagrange obtained the series
Wronski believed instead that mathematics should be based on what he considered the "suprime algorithmic law:"
"where the quantities
are any functions of the variable x. Being the supreme law of mathematics, the irrecusable truth of this law he held to be not mathematically derived, but given by transcendental philosophy" (Boyer 261c.d).
Wronski was correct to criticize Lagrange's theory for being limited to only expandable functions. But he otherwise held highly unconventional and controversial views on the calculus (262a).
Whereas Lagrange had attempted to give a formal logical justification of the subject, Wronski asserted that the differential calculus constituted a primitive algorithm governing the generation of quantities, rather than the laws of quantities already formed. (262ab).
The calculus' propositions, which express absolute truths, cannot be deduced within the sphere of mathematics. Wronski objected to construing calculus in terms of limits, ultimate ratios, vanishing quantities, and functions, because he thought it was fruitless to abandon the notion of the infinite (262b).
Although mathematicians continued to hold to the limit concept,
Wronski represents an extreme example of a view which we shall find recurring throughout the nineteenth century. In regarding the calculus as a means of explaining the growth of magnitudes, followers of this school of thought were to attempt to retain the concept of the infinitely small, not as an extensive quantity but as an intensive magnitude. Mathematics has excluded the fixed infinitely small because it has failed to establish the notion logically; but transcendental philosophy has sought to preserve primitive intuition in this respect by interpreting it as having an a priori metaphysical reality associated with the generation of magnitude. (262-263).
Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York: Dover Publications, 1949.
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