Arithmetization of analysis

The phrase "arithmetization of analysis" refers to 19th century efforts to create a "theory of real numbers ... using set-theoretic constructions, starting from the natural numbers." These efforts took place over a period of about 50 years, with the following results:

1. the establishment of fundamental concepts related to limits
2. the derivation of the main theorems concerning those concepts
3. the creation of the theory of real numbers.

This article presents a brief history of these efforts.

In setting out the history of any such mathematical development or period, the central question to be answered has been identified as simply this:

How did the present come to be?

In searching for the answer this question, a mathematician looks at the mathematics of the past and seeks to understand how it has led to the mathematics of the present. In their searches, mathematicians ask questions such as these about the mathematics of the past:

• When was a concept first defined and what problems led to its definition?
• Who first proved a theorem, how was it done, and is the proof correct by modern standards?

Thus, "the mathematician begins with mathematics that is important now, and looks backwards for its antecedents." In other words, the suggestion is that, for the mathematician, "all mathematics is contemporary."

Non-mathematical issues

The history of the arithmetization of analysis was complicated by non-mathematical issues. Some authors were very slow to publish and some important results were not published at all during their authors' lifetimes. The work of other authors was, for unknown reasons, completely ignored. As a consequence, some results were achieved multiple times, albeit in slightly different forms or using somewhat different methods, by different authors.

As a first example, consider the work of Bolzano. Only two of his papers dealing with the foundations of analysis were published during his lifetime. Both of these papers remained virtually unknown until after his death. A third work of his, based on a manuscript that dates from 1831-34, but that remained undiscovered until after WWI, was finally published in 1930. This work contains some results fundamental to the foundations of analysis that were re-discovered in the 19th century by others decades after Bolzano completed his manuscript. "We may ask how much Bolzano's work could have changed the way analysis followed, had it been published at the time."

As a second example, consider the work of W.R. Hamilton, in particular his 1837 essay on the foundations of mathematics, in which he attempted to show that analysis (which for Hamilton included algebra) alike with geometry, can be "a Science properly so called; strict, pure, and independent; deduced by valid reasonings from its own intuitive principles." His essay contained the following:

• the notion that analysis "can be constructively inferred from a few intuitively based axioms"
• ideas used much later by others (Peano, Dedekind, others) "including a notion related to the concept of a cut in the rationals"

His essay was ignored by other English mathematicians and had no apparent influence on the work of German mathematicians who completed the process of arithmetization later in the century. Even so, and years before his work in 1837, Hamilton wrote the following:

An algebraist who should thus clear away the metaphysical stumbling blocks that beset the entrance to analysis without sacrificing those concise and powerful methods which constitute its essence and its value would perform a useful work and deserve well of Science.

Thus, though his work was overlooked by other mathematicians of the day, Hamilton grasped the importance of his ideas to the future of analysis.

The origin and need for arithmetization

Mathematicians of the 19th century who laboured on the arithmetization of analysis gave various reasons for their pursuits. One source identifies the following as the chief causes of concern about 19th century mathematics:

• the lack of confidence in operations performed on infinite series
• the lack of any definition of the phrase "real number"

Much if not all of the uneasiness arose from the very productive, yet very suspect methods of the calculus that had emerged during the previous two centuries. An example of a thoroughly modern definition of calculus is as follows:

Calculus is the branch of mathematics that defines and deals with limits, derivatives and integrals of functions.

The same source locates the origin of and need for the arithmetization program in the work of the inventors of the calculus themselves. Newton and Leibniz, driven by their intuitions, based their work on geometric considerations -- for reasons that, it was retrospectively realized, were very legitimate:

Newton's limit operation had already been successfully used in special cases by the Greek mathematician Archimedes (third century B.C.), whose "method of exhaustion" had led him to calculate correctly certain geometrical limits.

Even so, though the final results of their methods remained undisputed, the methods themselves came to be suspect. Central to such concerns was the notion of an infinitely small or indefinitely small quantity, the infinitesimal, which had this very strange property: it was sometimes zero and sometimes non-zero!

Hermann Hankel has been credited with the foresight that "the condition for erecting a universal arithmetic is therefore a purely intellectual mathematics, one detached from all perceptions." In other words, mathematicians needed to view real numbers as "intellectual structures" rather than as "intuitively given magnitudes inherited from Euclid's geometry."

Two pillars of mathematics

The state of mathematics after the invention of the calculus, but prior to 19th century efforts at arithmetization, has been described by modern authors in various ways:

• analysis rested more or less comfortably on two pilars: the discrete side on arithmetic, the continuous side on geometry.
• the source domain of analysis was geometry; that of number theory was arithmetic.

"The analytic work of L. Euler, K. Gauss, A. Cauchy, B. Riemann, and others led to a shift towards the predominance of algebraic and arithmetic ideas. In the late nineteenth century, this tendency culminated in the so-called arithmetization of analysis, due principally to K. Weierstrass, G. Cantor, and R. Dedekind."

The fundamental theorem of algebra

Proofs of The fundamental theorem of algebra have a long history, with dates (currently) ranging from 1608 (Peter Rothe) to 1998 (Fred Richman).

Gauss offered two proofs of the theorem. All proofs offered before his assumed the existence of roots. Gauss' proofs were the first that did not make this assumption:

• In 1799, he offered a proof of the theorem that was largely geometric. This first proof assumed as obvious a geometric result that was actually harder to prove than the theorem itself!
• In 1816, he offered a second proof that was not geometric. This proof assumed as obvious a result known today as the intermediate value theorem.

The significance of Gauss' proofs for the arithmetization program has been explained in various ways:

• the theorem itself involved a discrete result, while his proofs used continuous methods, calling into question the comfortable two-pillar foundation of mathematics.
• using analysis to prove the fundamental theorem of number theory raised a problem about the boundary between number theory and analysis.

Continuous nowhere differentiable functions

From the time of the invention of the calculus through the first half of the 19th century, mathematicians generally assumed that a continuous real function must have a derivative at most points. Allowing for occasional abrupt changes in their direction and discontinuities at isolated points, Newton himself generally assumed that curves were generated by smooth and continuous motions. Certainly solutions of differential equations, power series, Fourier series, and, generally speaking, functions that actually occurred in the real world, were believed to be (almost) everywhere differentiable. Such beliefs were said to have been "blown away" by the publication of examples to the contrary. The following example, by Weierstrass, of a function continuous everywhere, but differentiable nowhere, was published in 1875:

$\displaystyle f(x) = \sum_{n=1}^\infty a^n cos(b^n \pi x)$ where $0 < a < 1$, $b$ is positive odd integer, and $\displaystyle ab > 1+\frac{3}{2}\pi$ Functions such as this that refused to behave as expected were termed "pathological" and their ongoing discovery during the 2nd half of the 19th century was "shocking" to mathematicians. An oft-cited comment is the following:'"`UNIQ--ref-00000014-QINU`"' :I turn away with fear and horror from the lamentable plague of continuous functions which do not have derivatives ... Hermite in a letter to Stieltjes dated 20 May, 1893 Other mathematicians of the second half of the 19th century shared Hermite's opinion, fearing that similar investigations into the foundations of mathematics would lead to harmful results.'"`UNIQ--ref-00000015-QINU`"' As late as 1920, Jasek is said to have created a "sensation" when he revealed Bolzano's example of a continuous function that is neither monotone in any interval nor has a finite derivative at the points of a certain everywhere dense set. It has been pointed out that Bolzano's function is actually nowhere differentiable, though he neither claimed nor proved this. Bolzano discovered/invented this function about 1830, more than 30 years before Weierstrass's example.'"`UNIQ--ref-00000016-QINU`"' Discovery of such functions continued throughout the 20th century, though with less shocking effects! With respect to the arithmetization program, the discovery of these functions did the following: * it served to accentuate the need for analytic rigour in mathematics'"`UNIQ--ref-00000017-QINU`"' * it dealt "a decisive blow to the intuitive picture of the behavior of continuous functions."'"`UNIQ--ref-00000018-QINU`"' A turn of the century address to the American Mathematical Society summarized the situations of the "intuitionist" and the "arithmetician" as follows:'"`UNIQ--ref-00000019-QINU`"' :It is easy to construct continuous functions which have absolutely no derivative at all rational points in a given interval, so that in any little interval there are an infinite number of points with tangents, and an infinite number without. Our intuition is utterly helpless to give us any information in regard to such curves. Indeed our intuition would rather say such curves do not exist. :Any definition [of the fundamental concepts of mathematics] we can give and which will serve as the base for rigorous deduction, can at best be but an approximate interpretation of the hazy and illusive nature of [the notions of our intuition].... The familiar $\varepsilon, \delta$ criterion of Cauchy-Weierstrass ... [allows us to] reason with absolute precision and fineness.... We have now fairly established the justness of the position of the arithmetician. =='"`UNIQ--h-5--QINU`"'The arithmetization program== ==='"`UNIQ--h-6--QINU`"'An early step towards arithmetization=== The "half century of investigation into the nature of function and number" that culminated in the arithmetization of analysis is said to have begun in the year 1822, which saw the following two signal efforts:'"`UNIQ--ref-0000001A-QINU`"' #Fourier's attempt to establish a theoretical foundation for periodic functions #Ohm's attempt to reduce all of analysis to arithmetic Ohm described the motivation for his work as a desire to answer this question: "How may the paradoxes of calculation be most securely avoided?" His answer was "to submit to a very exact examination of the subject of mathematical analysis, its first and simplest ideas, as also the methods of reasoning which are applied to it."'"`UNIQ--ref-0000001B-QINU`"' In 1822, he published the first two volumes of a work that has been described as "the first attempt since Euclid to write down a logical exposition of everything that was more or less basic in contemporary mathematics, starting from scratch ... a completely formalist conception."'"`UNIQ--ref-0000001C-QINU`"' Years later, while still in the midst of this project, Ohm noted as follows, quite retrospectively, several types of "complaints of the want of clearness and rigour in that part of Mathematics" that led him to pursue his decades-long efforts: * contradictions of the theory of "opposed magnitudes" * disquiet by "imaginary quantities" * difficulties in either divergence or convergence of "infinite series" After his two volumes of 1822, Ohm continued for 30 more years and produced ultimately nine volumes. He himself believed that his work had put mathematics on a firm basis.'"`UNIQ--ref-0000001D-QINU`"' ==='"`UNIQ--h-7--QINU`"'Limits=== Beginning perhaps with D'Alembert, it was an oft-repeated statement by 18th century mathematicians that the calculus should be "based on limits." It is not surprising then that the arithmetization program culminated in the establishment of the concept of the limit and of those other fundamental concepts that were connected with it, including convergence and continuity. D'Alembert's own definition of limit was as follows:'"`UNIQ--ref-0000001E-QINU`"' :... the quantity to which the ratio $z/u$ approaches more and more closely if we suppose $z$ and $u$ to be real and decreasing. Nothing is clearer than that. Bolzano and Cauchy are said to have been contemporaries "both chronologically and mathematically."'"`UNIQ--ref-0000001F-QINU`"' They gave similar definitions of limits, convergence, and continuity, and they both developed (independently) a concept of ''limit'' that was an advance over D'Alembert's and over all previous attempts: * it was free from the ideas of motion and velocity and did not depend on geometry * it did not retain the (unnecessary) restriction, that a variable could never surpass its limit For example, Cauchy's definition was constructed using only these three elements * the variable * the limit * the quantity by which the variable differed from the limit and stated simply that the variable and its limit differed by less than any desired quantity, as follows:'"`UNIQ--ref-00000020-QINU`"' :When the successive values attributed to a variable approach indefinitely a fixed value so as to end by differing from it by as little as one wishes, this last [latter fixed value] is called the limit of all the others [successive values]. The effect of this definition was to transform the infinitesimal from a very small number into a dependent variable. Cauchy put this as follows:'"`UNIQ--ref-00000021-QINU`"' :One says that a variable quantity becomes infinitely small when its numerical value decreases indefinitely in such a way as to converge toward the limit zero. Cauchy's definition is wholly verbal, although it has been noted elsewhere that he translated such statements into the precise language of inequalities when he needed them for proofs.'"`UNIQ--ref-00000022-QINU`"' Even so, it was Weierstrass who finally provided a formal $\delta,\varepsilon$ definition of limit. His student Heine published this definition of the limit of a function using notes from Weierstrass's lectures:'"`UNIQ--ref-00000023-QINU`"' :$\displaystyle \lim_{x \to \alpha}f(x) = L$ if and only if, for every $ε > 0$, there exists a $δ > 0$ so that, if $0 < |x - a| < δ$, then $|f(x) - L| < ε$. ==='"`UNIQ--h-8--QINU`"'Convergence=== Working with the notion of a sequence that "converges within itself," Bolzano and Cauchy sought to relate the concepts ''limit'' and ''real number'', somewhat as follows: :If, for a given integer $p$ and for $n$ sufficiently large, $S_{n+p}$ differs from $S_{n}$ by less than any assigned magnitude $\varepsilon$, then $S_{n}$ also converges to the (external) real number $S$, the limit of the sequence. Meray understood the error involved in the circular way that Bolzano and Cauchy had defined the concepts ''limit'' and ''real number'':'"`UNIQ--ref-00000024-QINU`"' * the limit (of a sequence) was defined to be a real number $S$ * a real number was defined as a limit (of a sequence of rational numbers) To avoid this circularity, Meray avoided references to convergence to an (external) real number $S$. Instead, he described convergence using only the rational numbers $n$, $p$, and $\varepsilon$, which is the Bolzano-Cauchy condition. Weierstrass also understood the error involved in earlier ways of defining the concepts ''limit'' and ''irrational number'':'"`UNIQ--ref-00000025-QINU`"' * the definition of the former presupposed the notion of the latter * therefore, the the definition the latter must be independent of the former ==='"`UNIQ--h-9--QINU`"'Continuity=== Bolzano saw that the intermediate value theorem needed to be proved "as a consequence of the definition of continuity." In his 1817 proof, he introduced ''essentially'' the modern condition for continuity of a function $f$ at a point $x$:'"`UNIQ--ref-00000026-QINU`"' :$f(x + h) − f(x)$ can be made smaller than any given quantity, provided $h$ can be made arbitrarily close to zero The caveat ''essentially'' is needed because of his complicated statement of the theorem, as noted above. In effect, the condition for continuity as stated by Bolzano actually applies not at a point $x$, but within an interval. In his 1831-34 manuscript, Bolzano provided a definition of continuity at a point (including one-sided continuity). However, as noted above, this manuscript remained unpublished until eighty years after Bolzano's death and, consequently, it had no influence on the efforts of Weierstrass and others, who completed the arithmetization program.'"`UNIQ--ref-00000027-QINU`"' In 1821, Cauchy added to Bolzano's definition of continuity at a point "the final touch of precision":'"`UNIQ--ref-00000028-QINU`"' :for each $\varepsilon > 0$ there is a $\delta > 0$ such that $|f(x + h) − f(x)| < \varepsilon$ for all $|h| < \delta$ Here it's important to note that, as he stated it, Cauchy's condition for continuity, alike with Bolzano's, actually applies not at a point $x$, but within an interval.'"`UNIQ--ref-00000029-QINU`"' Once again, it was Weierstrass who, working very long after both Bozano and Cauchy, formulated "the precise $(\varepsilon,\delta)$ definition of continuity at a point."'"`UNIQ--ref-0000002A-QINU`"' ==='"`UNIQ--h-10--QINU`"'The intermediate value theorem=== As noted above, Gauss' 1816 proof of the fundamental theorem of algebra assumed as obvious, and hence did not prove, the intermediate value theorem. Bolzano was the first to offer a correct proof of the theorem, which he stated as follows:'"`UNIQ--ref-0000002B-QINU`"' :If a function, continuous in a closed interval, assumes values of opposite signs at the endpoints of this interval, then this function equals zero at one inner point of the interval at least. As has been noted elsewhere:'"`UNIQ--ref-0000002C-QINU`"' * the theorem seems ''intuitively'' plausible, for a continuous curve which passes partly under, partly above the x-axis, ''necessarily'' intersects the x-axis; * it was Bolzano's insight that the theorem needed to be proved ''as a consequence of the definition of continuity.'' Quite independently of Bolzano, Cauchy proved the intermediate value theorem, which he stated in the following more general form: :If $f(x)$ is a continuous function of a real variable $x$ and $c$ is a number between $f(a)$ and $f(b)$, then there is a point $x$ in this interval such that $f(x) = c$. Both Bolzano and Cauchy were influenced by Lagrange's general view that the concepts of the calculus could be made rigorous only if they were defined in terms of algebraic concepts.'"`UNIQ--ref-0000002D-QINU`"' More specifically, with respect to the intermediate value theorem, they were influenced by the specifics of Lagrange's work as follows: * in his proof, Bolzano first proved the same stronger theorem about pairs of continuous functions that Lagrange had stated in his own 1798 proof, then Bolzano derived the intermediate-value theorem as a corollary of that stronger result -- details below;'"`UNIQ--ref-0000002E-QINU`"' * in his proof, Cauchy employed Lagrage's approximation procedure that was used for ''finding'' the roots of a poynomial, and "stood it on its head", converting it into a proof of the ''existence'' of those very roots -- details below.'"`UNIQ--ref-0000002F-QINU`"' In their proofs of the intermediate value theorem, neither Bolzano nor Cauchy identified all the assumptions that underlay their treatment of real numbers. ===='"`UNIQ--h-11--QINU`"'Bolzano's proof==== Bolzano undertook to prove the theorem in his paper of 1817, one year after Gauss's incomplete proof of the fundamental theorem of algebra. Indeed, Bolzano's motivation for proving the theorem was precisely to fill the gap in Gauss's proof.'"`UNIQ--ref-00000030-QINU`"' In the prefatory remarks to his proof, Bolzano discussed in detail previous proofs of the intermediate value theorem. Many of those proofs (alike with Gauss' 1799 proof of the fundamental theorem of algebra) depended "on a truth borrowed from geometry." Bolzano rejected all such proofs in totality and unequivocally:'"`UNIQ--ref-00000031-QINU`"' :It is an intolerable offense against correct method to derive truths of pure (or general) mathematics (i.e., arithmetic, algebra, analysis) from considerations which belong to a merely applied (or special) part, namely, geometry.... A strictly scientific proof, or the objective reason, of a truth which holds equally for all quantities, whether in space or not, cannot possibly lie in a truth which holds merely for quantities which are in space. Other proofs that Bolzano examined and rejected were based "on an incorrect concept of continuity": :No less objectionable is the proof which some have constructed from the concept of the continuity of a function with the inclusion of the concepts of time and motion.... No one will deny that the concepts of time and motion are just as foreign to general mathematics as the concept of space. Bolzano caped his prefatory remarks with the first mathematical achievement of his paper, namely, a formal definition of the continuity of a function of one real variable, which he stated as follows:'"`UNIQ--ref-00000032-QINU`"' :If a function $f(x)$ varies according to the law of continuity for all values of $x$ inside or outside certain limits, then if $x$ is some such value, the difference $f(x + \omega) - f(x)$ can be made smaller than any given quantity provided $\omega$ can be taken as small as we please. Bolzano's proof of the main theorem proceeded as follows: * First, Bolzano introduced the (necessary and sufficient) condition for the (pointwise) convergence of a sequence, known today as the [[Cauchy criteria|Cauchy condition]] (on occasion the Bolzano-Cauchy condition), as follows:'"`UNIQ--ref-00000033-QINU`"' :: If a series [sequence] of quantities :::$F_1x$, $F_2x$, $F_3x$, . . . , $F_nx$, . . . , $F_{n+r}x$, . . . ::has the property that the difference between its.$n$th term $F_nx$ and every later term $F_{n+r}x$, however far from the former, remains smaller than any given quantity if $n$ has been taken large enough, then there is always a certain constant quantity, and indeed only one, which the terms of this series [sequence] approach, and to which they can come as close as desired if the series [sequence] is continued far enough. :As noted elsewhere, Bolzano here demonstrated the ''plausibility'' of the assertion that a sequence satisfying the condition has a limit, but did not provide a proof of its sufficiency. :Bolzano provided here also a proof of the fact that a sequence has at most one limit. The significance of this proof lies not in its achievement (since the proof is very easy) but in the fact that Bolzano may have been the first to realize the need for such a proof.'"`UNIQ--ref-00000034-QINU`"' * Next, Bolzano used the [[Cauchy test|Cauchy condition]] in a proof of the following theorem, namely, that a bounded set of numbers has a least upper bound:'"`UNIQ--ref-00000035-QINU`"' ::If a property $M$ does not belong to all values of a variable $x$, but does belong to all values which are less than a certain $u$, then there is always a quantity $U$ which is the greatest of those of which it can be asserted that all smaller $x$ have property $M$. :In effect, Bolzano here proved the least upper bound theorem. The number $U$ is in fact the greatest lower bound of those numbers which do NOT possess the property $M$.'"`UNIQ--ref-00000036-QINU`"' The theorem proved is the original form of the [[Bolzano-Weierstrass theorem]] and is in fact the original statement of that theorem:'"`UNIQ--ref-00000037-QINU`"' ::Every bounded infinite set has an accumulation point. :A complete proof of the least upper bound theorem, alike with the condition of convergence on which it depends, needed to await the building of the theory of real numbers. However, Bolzano here demonstrated the ''plausibility'' of the theorem. * Next, Bolzano proved the following theorem, which is sometimes called Bolzano's theorem, which Bolzano himself believed to be "a more general truth," and which certainly is stronger than the main theorem he set out to prove:'"`UNIQ--ref-00000038-QINU`"' ::If two functions of $x$, $f(x)$ and $g(x)$, vary according to the law of continuity either for all values $x$ or only for those which lie between $\alpha$ and $\beta$, and if $f(\alpha) < g(\alpha)$ and $f(\beta) > g(\beta)$, then there is always a certain value of $x$ between $\alpha$ and $\beta$ for which $f(x) = g(x)$ :Interestingly, Lagrange used this same theorem as an intermediate result in his own 1798 proof of the intermediate value theorem. Dismissing Lagrange's proof as inadequate, Bolzano nevertheless took very seriously "Lagrange’s call to reduce the calculus to algebra," as his definition of continuous function and his proof of the intermediate value theorem clearly show.'"`UNIQ--ref-00000039-QINU`"' * Finally, Bolzano proved the intermediate value theorem itself, which he stated in terms of the roots of a polynomial equation in one real variable, as follows:'"`UNIQ--ref-0000003A-QINU`"' :If a function of the form ::$x^n + ax^{n-1} + bx^{n-2} + ... + px + q$ :in which $n$ denotes a whole positive number, is positive for $x = \alpha$ and negative for $x = \beta$, then the equation ::$x^n + ax^{n-1} + bx^{n-2} + ... + px + q = 0$ :has at least one real root lying between $\alpha$ and $\beta$. ===='"`UNIQ--h-12--QINU`"'Cauchy's proof==== In his 1821 paper, Cauchy provided a proof of the intermediate value theorem, which some authors identify as [[Cauchy theorem|Cauchy's (intermediate-value) theorem]]. He stated the theorem as follows:'"`UNIQ--ref-0000003B-QINU`"' :Let $f(x)$ be a real function of the variable $x$, continuous with respect to that variable between $x = x{o}$, $x = X$. If the two quantities $f(x{o})$, $f(X)$ have opposite sign, the equation ::(1) $f(x) = 0$ :can be satisfied by one or more real values of $x$ between $x{o}$ and $X$. Cauchy's proof of the theorem included the following elements: * a definition of a continuous function, stated as follows:'"`UNIQ--ref-0000003C-QINU`"' :The function $f (x)$ will be a continuous function of the variable $x$ between two assigned limits ["limit" here means "bound"] if, for each value of $x$ between those limits, the numerical [absolute] value of the difference $f(x + \alpha) - f(x)$ decreases indefinitely with $\alpha$. :This definition of continuity is like and has the same meaning as Bolzano's definition, though it uses slightly different and less precise language.'"`UNIQ--ref-0000003D-QINU`"' * an new manner of defining real numbers: **Working before Cauchy, Lagrange and others assumed the existence of real numbers and used approximations to arrive at the values of those numbers; **Cauchy reversed this, defining real numbers as the limits of approximations and using the convergence of those approximations to prove the existence of the real numbers. Cauchy used both his definition of continuity and his method of defining real numbers in his proof of the intermediate value theorem. As was Bolzano's, Cauchy's proof is not without its problems. His understanding of convergence and continuity assumed, without either proof or statement, the completeness of real numbers:'"`UNIQ--ref-0000003E-QINU`"' # he treated as obvious that a series of positive terms, bounded above by a convergent geometric progression, converges # his proof of the intermediate-value theorem assumes that a bounded monotone sequence has a limit. ==='"`UNIQ--h-13--QINU`"'The derivative and the integral=== It has been suggested that the two most fundamental concepts of mathematics are ''set'' and ''function''. The second of these derives from the calculus, which turns on these two notions:'"`UNIQ--ref-0000003F-QINU`"' # the ''derivative'', representing the instantaneous rate of change of a function at a given point # the ''integral'', allowing for an exact calculation of the portion of a space determined by a given function As discussed above, by the middle of the 19th century, the mathematicians at work on the arithmetization program had not only established rigourous definitions of ''limit'', ''convergence'', and ''continuity'', but also had put those concepts to work in proofs of important theorems of analysis. It remained for them to establish equally rigourous definitions of the ''derivative'' and the ''integral''. ==='"`UNIQ--h-14--QINU`"'Theory of irrational numbers=== "The first modern construction of the irrational numbers" was offered by Hamilton in two separate papers, which were later published as one in 1837. Somewhat later, he began work on a theory of separations of the numbers, similar to Dedekind’s theory of cuts, but he never completed his work on this topic.'"`UNIQ--ref-00000040-QINU`"' In addition to Hamilton, several, including Ohm and Bolzano, attempted to define irrational numbers, all on the basis of using the limit of a sequence of rational numbers. All of their efforts, however, were either incomplete or lacking in rigor or both. Cantor himself pointed out an error with all these attempts:'"`UNIQ--ref-00000041-QINU`"' :the limits of such sequences, if irrational, do not logically exist until the irrational numbers themselves have been defined It was not until 1869 that Charles Méray published "the earliest coherent and rigorous theory of irrational numbers."'"`UNIQ--ref-00000042-QINU`"' Méray's contemporaries in France, however, failed to appreciate the significance of his work, while others in Germany and elsewhere were unaware of it -- this was the period of the Franco-Prussian War. As a result, his great achievement, though the equivalent of Cantor's which followed shortly after, went unacknowledged and had no influence of the direction of mathematics.'"`UNIQ--ref-00000043-QINU`"' In 1872, Cantor published his own theory of irrational numbers, defining them in terms of Cauchy sequences of rational numbers. It was his accomplishment that became known and influenced the work of others, especially Dedekind, and that consequently became celebrated as a significant step in the arithmetization of analysis.'"`UNIQ--ref-00000044-QINU`"' Subsequently, theories of irrationals were published by Dedekind (who mentioned Cantor's achievement) and by students of Weierstrass, who worked from notes taken at his lectures. Common to all three of these very different definitions of irrational numbers was "a well-defined collection of rational numbers."'"`UNIQ--ref-00000045-QINU`"' The differences among the three theories sprang from the quite different motivations of their authors:'"`UNIQ--ref-00000046-QINU`"' *Weierstrass saw the formulation of the real number system as essential to the foundation of the theory of real functions that he himself had developed *Cantor was led to define irrational numbers in terms of convergent sequences of rational numbers after having himself shown that a function of a complex variable can be represented in only one way by a trigonometric series *Dedekind saw the axiomatic characterization of the major number systems as essential for a rigorous foundation for differential calculus Taken together, the result of their efforts has been described as follows:'"`UNIQ--ref-00000047-QINU`"' :geometrical ideas were and are always present and available via Descartes' correspondence between geometry and algebra, but ... though convenient and intuitively useful, these ideas were in no wise logically necessary to the development of analysis. Mathematical analysis was logically independent of geometry. =='"`UNIQ--h-15--QINU`"'Looking back at these efforts== It is interesting, and has been noted elsewhere, that although the theory of real numbers is today the logical starting point (foundation) of analysis in the real domain, the creation of the theory was not achieved historically until the end of the period (program or movement) of arithmetization.'"`UNIQ--ref-00000048-QINU`"' What today are commonplace notions in undergraduate mathematics were anything but commonplace among practicing mathematicians even a quarter century after the 1872 achievements of Cantor, Dedekind, and Weierstrass. In 1899, addressing the American Mathematical Society, James Pierpont spoke to show these two things:'"`UNIQ--ref-00000049-QINU`"' # why arithmetical methods form the only sure foundation in analysis at present known # why arguments based on intuition cannot be considered final in analysis In a later, printed version of his address, Pierpont prefaced his words with the following:'"`UNIQ--ref-0000004A-QINU`"' :We are all of us aware of a movement among us which Klein has so felicitously styled the arithmetization of mathematics. Few of us have much real sympathy with it, if indeed we understand it. It seems a useless waste of time to prove by laborious $\varepsilon$ and $\delta$ methods what the old methods prove so satisfactorily in a few words. Indeed many of the things which exercise the mind of one whose eyes have been opened in the school of Weierstrass seem mere fads to the outsider. As well try to prove that two and two make four!

The term "arithmetization of mathematics," which Pierpont here ascribed to Klein, has also been credited to Kronecker -- perhaps to others as well? In any case, Pierpont ended his 1899 address with this paean to the labours of Weierstrass and others:

The mathematician of to-day, trained in the school of Weierstrass, is fond of speaking of his science as die absolut klare Wissenschaft. Any attempts to drag in metaphysical speculations are resented with indignant energy. With almost painful emotions he looks back at the sorry mixture of metaphysics and mathematics which was so common in the last century and at the beginning of this. The analysis of to-day is indeed a transparent science. Built up on the simple notion of number, its truths are the most solidly established in the whole range of human knowledge.

Writing almost 70 years after Pierpont's address, Carl Boyer gave these as motivations for the arithmetization program:

1. a lack of confidence in operations performed on infinite series
2. a lack of any definition of the phrase "real number"

Boyer credits Hermann Hankel with the foresight that "the condition for erecting a universal arithmetic is therefore a purely intellectual mathematics, one detached from all perceptions." In other words, what was needed was a mathematics that views real numbers as "intellectual structures" rather than as "intuitively given magnitudes inherited from Euclid's geometry."

Primary sources

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• Bolzano, Bernard (1930), Functionenlehre, Royal Bohemian Learned Society, based on a manuscript dating from 1831-34.
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