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The integral

For the whole of the 18th century and into the 19th, integration had been treated as the inverse of differentiation. Cauchy's definition of the derivative given above makes the following clear:

• the derivative will not exist at a point for which the function is discontinuous
• yet the integral may afford no difficulty, since even discontinuous curves may determine a well-defined area.

The fact that the inverse could not always be computed exactly led 18th mathematicians to do much work approximating the values of definite integrals:

• Euler treated sums of the form

$\displaystyle \sum_{k = 0}^n f(x_k) (x_{k+1} - x_k)$ :as approximations to the integral $\int_{x_0}^{x_n} f(x) dx$ * Poisson attempted a proof of the following what he called ''the fundamental proposition of the theory of definite integrals'', which he stated as follows: ::If the integral $F$ is defined as the antiderivative of $f$, and if $b - a = nh$ ::then $F(b) - F(a)$ is the limit of the sum :::$S = hf(a) + hf(a + h) + . . . + hf(a + (n - 1)h)$ ::as $h$ gets small. In effect, Poisson was the first to attempt a proof of the equivalence of the antiderivative and limit-of-sums conceptions of the integral. Following this tradition, Cauchy also defined the definite integral in terms of the limit of the integral sums. Then, having defined the integral independently of differentiation, it was necessary for him to prove the usual relation between the integral and the antiderivative, which he accomplished using the mean value theorem:'"`UNIQ--ref-00000001-QINU`"' :If $f(x)$ is continuous over the closed interval $[a, b]$ and differentiable over the open interval $(a, b)$, then there will be some value $x_0$ such that $a < x_0 < b$ and $f(b) - f(a) = (b — a) f'(x_0 )$.'"`UNIQ--ref-00000002-QINU`"' Cauchy's proof proceeds as follows:'"`UNIQ--ref-00000003-QINU`"' * Defining the integral as the limit of Euler-style sums $\sum f(x_k)(x_{k+1} - x_k)$ for sufficiently small $x_{k + 1} - x_k$ * Assuming explicitly that it was continuous on the given interval (and implicitly that it was uniformly continuous) * Showing that all sums of that form approach a fixed value, called by definition the integral of the function on that interval * Borrowing from Lagrange the mean-value theorem for integrals, proving the [[Fundamental theorem of calculus]]. Similar views were developed at about the same time by Bolzano.'"`UNIQ--ref-00000004-QINU`"' As mentioned above, the existence of continuous nowhere differentiable functions, including of course the "pathological" functions of Bolzano and Weierstrass, contributed to the concerns about the foundations of analysis. Riemann exhibited a function $f(x)$ with the following characteristics: :it is discontinuous at infinitely many points in an interval and yet its integral exists and defines a continuous function $F(x)$ that, for the infinity of points in question, fails to have a derivative Cauchy's definition of the integral was guided largely by geometrical feeling for the area under a curve. Riemann's function made clear that the integral required a more careful definition than that of Cauchy. The present-day definition of the definite integral over an interval in terms of upper and lower sums generally is known as the Riemann integral, in honor of the man who gave necessary and sufficient conditions that a bounded function be integrable.'"`UNIQ--ref-00000005-QINU`"' ==='"`UNIQ--h-1--QINU`"'Theory of irrational numbers=== Weierstrass also developed a method of constructing irrationals, but he did not publish. However, method was made known and in fact published by his students, such as Ferdinand Lindemann and Eduard Heine.<Boyer, Carl S. pp. 606-7</ref> In 1871 Cantor had initiated a third program of arithmetization, similar to those of Meray and Weierstrass. Heine suggested simplifications to Cantor's program, which led to the so-called Cantor-Heine development, published by Heine.... In essence, this scheme resembled that of Meray: irrational numbers are defined as convergent sequences of rational numbers that fail to converge to rational numbers.'"`UNIQ--ref-00000006-QINU`"' Alike with Meray et. al., Dedekind developed a unified treatment of rational and irrational numbers. His approach, however, differed remarkably from other treatments in that its central concept was not convergence, but continuity. Because of the light that Dedekind's approach sheds on the notion and nature of continuity and on the continuum, it is worth noting some of its details. Dedekind treated the system of rational numbers as a whole, i.e. as a complete, infinite set, closed under addition and multiplication. In addition, he identified three fundamental properties of the rationals: # order: if $a>b$ and $b>c$ then $a>c$ # density: if $a\neq b$ then there are infinite rationals between $a$ and $b$ # section: if $a$ is a given rational, then all rationals can be divided into two classes $A_1$ and $A_2$ containing each an infinite number of elements, such that in the first are all the numbers smaller than $a$and in the second all the numbers larger than $a$, and $a$ can be in either the first or the second class.

It is worth noting that both Newton and Leibniz believed that something equivalent to the density property of geometric magnitudes captured their "continuousness". Dedekind, however, realized that this was not the case, since the rationals, too, were dense, but they were not a continuum:

• each rational number corresponds (in a unique, order-preserving way) to a point on a line
• not every point on a line corresponds to a rational number

As far back as 1858, Dedekind recognized that the continuousness of the points on a line, i.e. of geometric magnitudes, is not captured by the density property. His understanding of the continuity not only of a line segment, but also of geometric magnitudes generally, and, hence, of the continuum, turned on a reversal of the view of Newton and Leibniz. Continuity results not from "a vague hang-togetherness, but to an exactly opposite property—the nature of the division of the segment into two parts by a point on the segment."

This continuity property is the inverse of the property (3) above, the "section" property, and has come to be known as the Dedekind axiom or, more properly the Cantor-Dedekind axiom:

continuity: in any division of the points of the segment into two classes such that each point belongs to one and only one class, and such that every point of the one class is to the left of every point in the other, there is one and only one point that brings about the division.

Using this property, as Dedekind expressed it, "we can reach a continuous field [of real numbers by] enlarging the discontinuous field of rational numbers."

Dedekind proved the existence of irrational numbers by constructing an example and, showed also that rational numbers do not satisfy the continuity property:

Let D be a positive integer which is not the square of an integer. Let A2 be the set of all rational numbers whose square is greater than D and A1 be all other rational numbers. Then (A1,A2) is a cut.

In other words, the system of rational numbers is not continuous, i.e. not line-complete.

There is an irony in Dedekind's treatment of real numbers, which has been expressed as follows:

• though geometry had pointed the way to a suitable definition of the concept of continuity
• geometry was, in the end and by design, excluded from the formal arithmetic definition

Primary sources

• Cauchy, A.-L. (1821) Cours d’analyse, Paris.
• Cauchy, A.-L. (1823) Résumé des leçons données à l’école royale polytechnique sur le calcul infinitésimal, Paris.
• Dedekind, Richard (1872) "Stetigkeit und Irrationale Zahlen" ("Continuity and Irrational Numbers") Essays on the Theory of Numbers, New York: Dover Publications, Inc, 1963.
• Heine, Eduard (1872). "Die Elemente der Funktionenlehre" Crelle's Journal.
• Lagrange, Traite de la resolution des squations numeriques de tous les degres, 2nd ed., 1808, in [B24, vol. VIII, pp. 46-7, 163].

References

• "The theory of irrational numbers", The Garden of Archimedes, URL: http://web.math.unifi.it/~archimede/archimede_NEW_inglese/mostra_calcolo/guida/node35.html#b
• Reich, Erich (2011) "Dedekind's Contribution to the Foundations of Mathematics", The Stanford Encyclopedia of Philosophy.
• Snow, Joanne E. "Views on the real numbers and the continuum". The Review of Modern Logic 9 (2003), no. 1-2, 95--113. URL: http://projecteuclid.org/euclid.rml/1081173837.