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 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.

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"

Hamilton's 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

Two pillars of mathematics

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 a very strange property: it was sometimes zero and sometimes non-zero! In 1797, no less than Lagrange himself stated that his intention, in publishing the first theory of functions of a real variable, was to provide the following:

the principles of the differential calculus, freed from all consideration of the infinitely small or vanishing quantities....

Summarizing the mathematical situation in the seventeenth and eighteenth centuries, a historian of the period has contrasted "the powerful techniques of the calculus" with "the relatively unimpressive views put forth to justify them"

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."

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 pillars: 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-00000016-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-00000017-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-00000018-QINU`"' It was with good reason that he is said to have been a "voice crying in the wilderness."'"`UNIQ--ref-00000019-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-0000001A-QINU`"' * it dealt "a decisive blow to the intuitive picture of the behavior of continuous functions."'"`UNIQ--ref-0000001B-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-0000001C-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`"'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-0000001D-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-0000001E-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-0000001F-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-00000020-QINU`"' =='"`UNIQ--h-6--QINU`"'The arithmetization program== ==='"`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-00000021-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-00000022-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-00000023-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-00000024-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-00000025-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-00000026-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 $|f(x) - L| < ε$. There is nothing in this definition of limit but real numbers, the operations $+$ and $-$, and the relationships $<$ and $>$. With their "unequivocal language and symbolism," Weierstrass and Heine "banished from the calculus the notion of variability and rendered unnecessary the persistent resort to fixed infinitesimals."'"`UNIQ--ref-00000027-QINU`"' ==='"`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-00000028-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-00000029-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-0000002A-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-0000002B-QINU`"' In 1821, Cauchy added to Bolzano's definition of continuity at a point "the final touch of precision":'"`UNIQ--ref-0000002C-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-0000002D-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-0000002E-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-0000002F-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-00000030-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-00000031-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-00000032-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-00000033-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-00000034-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-00000035-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-00000036-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-00000037-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-00000038-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-00000039-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-0000003A-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-0000003B-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-0000003C-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-0000003D-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-0000003E-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-0000003F-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-00000040-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-00000041-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-00000042-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=== The concept "function," which is fundamental to mathematics and derives from the calculus, turns on these two notions:'"`UNIQ--ref-00000043-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''. As did other authors of 18th century calculus books, Cauchy provided an explicit verbal definition for the derivative, as follows:'"`UNIQ--ref-00000044-QINU`"' :the derivative of $f(x)$ is the limit, when it exists, of the quotient of differences when $h$ goes to zero More importantly, he also provided and, in fact, pioneered the following:'"`UNIQ--ref-00000045-QINU`"' * a rigourous $\delta,\varepsilon$ definition of the derivative based on an inequality property, which came to him from Lagrange’s work on the Lagrange remainder * associated inequality proof techniques, which were developed largely in the study of algebraic approximations in the eighteenth century. Cauchy's $\delta,\varepsilon$ definition of derivative has been given as follows:'"`UNIQ--ref-00000046-QINU`"' :Let $\delta,\varepsilon$ be two very small numbers; the first is chosen so that for all numerical [i.e., absolute] values of $h$ less than $\delta$ and for any value of $x$ included [in the interval of definition], the ratio $(f(x + h) - f(x))/h$ will always be greater than $f'(x) - \varepsilon$ and less than $f'(x) + \varepsilon$. The same author has noted Cauchy's shortcoming in translating his verbal definition to the rigourous $\delta,\varepsilon$ form, namely, that he assumed his $\delta$ would work for all $x$ on the given interval, an assumption equivalent to that of the uniform convergence of the differential quotient.'"`UNIQ--ref-00000047-QINU`"' ==='"`UNIQ--h-14--QINU`"'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:'"`UNIQ--ref-00000048-QINU`"' * 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-00000049-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-0000004A-QINU`"' Cauchy's proof proceeds as follows:'"`UNIQ--ref-0000004B-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-0000004C-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 honour of the man who gave necessary and sufficient conditions that a bounded function be integrable.'"`UNIQ--ref-0000004D-QINU`"' ==='"`UNIQ--h-15--QINU`"'The 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-0000004E-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-0000004F-QINU`"' :the limits of such sequences, if irrational, do not logically exist until the irrational numbers themselves have been defined In 1869 Charles Méray, following earlier work of Lagrange, published "the earliest coherent and rigorous theory of irrational numbers ...,but gave rigorous proofs of what Lagrange had only conjectured."'"`UNIQ--ref-00000050-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-00000051-QINU`"' Even so, during the period 1872-1894, Meray continued to publish works intended to "remove geometric considerations from analytic proofs."'"`UNIQ--ref-00000052-QINU`"' 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 (Cauchy) sequences of rational numbers that fail to converge to rational numbers.'"`UNIQ--ref-00000053-QINU`"' It was Cantor's 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-00000054-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 principles or properties of the rationals:'"`UNIQ--ref-00000055-QINU`"' # ''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."'"`UNIQ--ref-00000056-QINU`"' Based on what he felt was the continuity of the real line, Dedekind captured what he termed the "essence of continuity" as follows:'"`UNIQ--ref-00000057-QINU`"' :If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions. This ''continuity'' property is a reversal of the property (3) above, the ''section'' property, and has come to be known as the [[Dedekind theorem|Dedekind axiom or principle]]. He proved the existence of irrational numbers by constructing an example using the ''section'' property above:'"`UNIQ--ref-00000058-QINU`"' :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 the process of doing this, he showed that rational numbers do not satisfy the continuity property. In other words, the system of rational numbers is dense, but not continuous, i.e. not ''line-complete''.'"`UNIQ--ref-00000059-QINU`"' Finally, using the continuity property, he proved that the addition of the irrationals to the rationals do form a continuous domain. As Dedekind himself expressed it, "we can reach a continuous field [of real numbers by] enlarging the discontinuous field of rational numbers."'"`UNIQ--ref-0000005A-QINU`"' There is an irony in Dedekind's treatment of real numbers, which has been expressed as follows:'"`UNIQ--ref-0000005B-QINU`"' * 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 Common to all three of these definitions of irrational numbers was "a well-defined collection of rational numbers."'"`UNIQ--ref-0000005C-QINU`"' The differences among the three theories sprang from the quite different motivations of their authors:'"`UNIQ--ref-0000005D-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-0000005E-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-16--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-0000005F-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-00000060-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-00000061-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:'"`UNIQ--ref-00000062-QINU`"' :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. One modern commentator, forthright in his support of the arithmetization program, generally, and of the contribution of his countryman Bolzano, in particular, nevertheless advances somewhat of a caveat in the form of the following contrast:'"`UNIQ--ref-00000063-QINU`"' * it was essential to move analysis off of its intuitive/geometric base onto a rigourous/arithmetic base * mathematics continued to develop during the arithmetization period, somewhat as a "correction" to it In his own words: :[By] the first half of the nineteenth century the building of mathematical analysis was raised to such a height that continuing its construction without fortifying its foundations was unthinkable. This brought a period of great revision of the foundations of analysis...; the development of the other branches of mathematics continued, of course, simultaneously and in mutual interaction. :It seems evident that ... the revision could not follow other direction than that of consequential arithmetization of analysis.... [Yet,] this arithmetization ... was later corrected by the modern development of mathematics; after all, even in the period mentioned the dialectics of this process can be observed: so, for example, B. Riemann who on the one hand contributed considerably to the arithmetization of analysis by his theory of integral, was on the other hand the ingenious builder of the geometric theory of analytic functions. Certainly a very significant "correction" of arithmetization, the development of [[Non-standard analysis|non-standard analysis]], was described generally by another modern commentor as follows:'"`UNIQ--ref-00000064-QINU`"' :Weierstrass' definition of limit appeared to finally nail the coffin of the departed quantities and led to a complete abandonment of the original idea of infinitesimals. However, in the 1960s the ghosts have been resurrected by Abraham Robinson and placed on the sound foundation of the non-standard analysis thus vindicating the intuition of [Newton and Leibniz,] the founding fathers. Here, from the same source, is Robinson's definition of limit: :$\displaystyle \lim_{x \to a} f(x) = L$, if and only if $f(x)$ is infinitely close to $L$ whenever $x ≠ a$ is infinitely close to $a$.

The following contrast highlights a modern issue in our understanding of the arithmetization program:

• to the mathematicians who developed it, the arithmetization program signified efforts to develop a foundation for analysis, i.e. the calculus, in terms of the natural numbers
• after the development of naive set theory by Cantor, the arithmetization program came to signify the set-theoretic definition of function and the set-theoretic construction of the real line

There were several important consequences of this shift in meaning, some of which were and remain contentious.

First, as noted above, it was widely believed that, until the development of non-standard analysis by Abraham Robinson in the 1960s, arithmetization had banished infinitesimals from mathematics. Certainly, the banishment of infinitesimals was considered an important reason for pursuing the arithmetization program In reality, however, the work on non-Archimedean systems continued unabated during and after the period of arithmetization, as documented by P. Ehrlich.

Next, as has been noted by many, arithmetization brought about a shift in emphasis from geometric to algebraic reasoning. Again, this shift was certainly considered an important reason for pursuing arithmetization. What is not widely appreciated is that an important consequence of this shift was a change in the way mathematics is taught today.

Finally, the shift in the meaning of arithmetization led to logicism, a currently prevalent philosophical position that all of mathematics should be derivable from logic and set theory. Logicism, in turn, led to Hilbert's program, to the theorems of Gödel, Turing, and Chaitin on undecidability and incompleteness, and to non-standard analysis.

Primary sources

• Baillaud, B. and H. Bourget. (1905) Correspondance d'Hermite et de Stieltjes, Gauthier-Villars, Paris, Tome II.
• Bolzano, Bernard (1817). Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewahren, wenigstens eine reelle Wurzel der Gleichung liege. ["Purely analytic proof of the theorem that between any two values which give results of opposite sign there lies at least one real root of the equation"]. Prague.
• Bolzano, Bernard (1930), Functionenlehre, Royal Bohemian Learned Society, based on a manuscript dating from 1831-34.
• 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 (1963 [German original 1872]) "Continuity and Irrational Numbers" ("Stetigkeit und Irrationale Zahlen") Essays on the Theory of Numbers, New York: Dover Publications, Inc, 1963.
• du Bois-Reymond, P. (1875) Journal fur die reine und angewandte Mathematik, 79 pp. 29-31.
• Hamilton, W. R. (1837), Algebra as the Science of Pure Time.
• Hankel, Hermann (1867). Theorie der komplexen Zahlen-systeme.
• Heine, Eduard, "Die Elemente der Funktionenlehre," Journal fur die Reine und Angewandte Mathematik (Crelle's Journal), 74 (1872), 172-188.
• Kronecker, Leopold (1901), Vorlesungen Uber Zahlentheorie ("Lecture Notes on Number Theory"), Leipzig, Druck und Verlag von B.G.Teubner.
• Lagrange, J-L, (1797). Théorie des fonctions analytiques, Paris.
• Méray, Charles (1869) Remarques sur la nature des quantités définies par la condition de servir de limites à des variables données.
• Ohm, Martin (1822), Versuch eines vollkommen consequenten Systems der Mathematik ("Attempt at a completely consequential system of mathematics").
• Ohm, Martin (1843 [German original 1842]), The Spirit of Mathematical Analysis and its Relation to a Logical System.
• Pierpont, James (1899). "On the Arithmetization of Mathematics," Bulletin of the American Mathematical Society, (5) No 8, URL: https://projecteuclid.org/download/pdf_1/euclid.bams/1183415834.

References

• Arithmetization, Encyclopedia of Mathematics.
• Fundamental Theorem of Algebra, Encyclopedia of Mathematics.
• Bogomolny, A. "What Is Calculus?" from Interactive Mathematics Miscellany and Puzzles http://www.cut-the-knot.org/WhatIs/WhatIsCalculus.shtml#Alembert, Accessed 27 May 2014
• Bottazzini, Umberto (1986). The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass, Translated from the Italian by Warren Van Emend, New York: Springer-Verlag.
• Boyer, Carl B. (1968). A History of Mathematics, New York: John Wiley & Sons, URL: https://archive.org/stream/AHistoryOfMathematics/Boyer-AHistoryOfMathematics_djvu.txt
• Cauchy theorem, Encyclopedia of Mathematics.
• Dunham, W. (2008). The Calculus Gallery: Masterpieces from Newton to Lebesgue, Princeton University Press.
• "Fundamental Theorem of Algebra," Wikipedia, URL: http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra.
• The Garden of Archimedes, "The theory of irrational numbers", URL: http://web.math.unifi.it/~archimede/archimede_NEW_inglese/mostra_calcolo/guida/node35.html#b
• Grabiner, J. V. (1975). "The Mathemetician, the Historian, and the History of Mathematics," Historia Mathematica (2), pp. 439-447.
• Grabiner, J. V. (1981). The Origins of Cauchy's Rigorous Calculus, Mineola, New York: Dover Publications, ISBN-13: 9780486438153, URL: http://users.uoa.gr/~spapast/TomeasDidaktikhs/Caychy/GrabinerOriginsofCauchysRigorousCalculus.pdf
• Grabiner, J. V. (1983). "Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus." Amer. Math. Monthly 90, pp. 185-194.
• Graves, R. P. (2013, [1882]). Life and Miscellaneous Writings, London: Forgotten Books, URL: http://www.forgottenbooks.org/readbook_text/Life_and_Miscellaneous_Writings_v1_1000543220/325.
• Hatcher, William S. (2000). Foundations of Mathematics: An Overview at the Close of the Second Millenium 3.2 Aritmetization of Analysis, Switzerland: Landegg Academy, URL: http://bahai-library.com/hatcher_foundations_mathematics.
• Jarník, Vojtěch; Novák, Josef; Folta, Jaroslav; Jarník, Jiří (1981). Bolzano and the Foundations of Mathematical Analysis. (English). Praha: Society of Czechoslovak Mathematicians and Physicists, pp. 33--42, URL: http://dml.cz/dmlcz/400082.
• Mathews, Jerold (1978). "William Rowan Hamilton's 1837 Paper on the Arithmetization of Analysis," Archive for History of Exact Sciences, 14. VIII. 19, (2), pp 177-200, URL: http://link.springer.com/article/10.1007%2FBF00328612#.
• O'Connor, J. J. and Robertson, E. F. (2000). Joseph-Louis Lagrange MacTutor History of Mathematics archive, University of St Andrews, URL: http://www-history.mcs.st-andrews.ac.uk/Biographies/Lagrange.html
• O'Connor, J. J. and Robertson, E. F. (2000). "Hugues Charles Robert Méray," MacTutor History of Mathematics archive, University of St Andrews, URL: http://www-history.mcs.st-andrews.ac.uk/Biographies/Meray.html.
• O'Connor, J. J. and Robertson, E. F. (2000). "Martin Ohm," MacTutor History of Mathematics archive, University of St Andrews, URL: http://www-history.mcs.st-andrews.ac.uk/Biographies/Ohm_Martin.html.
• Pinkus, Allan (2000). "Weierstrass and Approximation Theory", Journal of Approximation Theory, 107 (1), pp. 1-66, URL: http://www.math.technion.ac.il/hat/fpapers/wap.pdf.
• Reich, Erich (2011) "Dedekind's Contribution to the Foundations of Mathematics", The Stanford Encyclopedia of Philosophy.
• Robinson, A. Biography in Dictionary of Scientific Biography (New York 1970-1990).
• Russ, S. B. "A Translation of Bolzano's Paper on the Intermediate Value Theorem," Historia Mathematica 7 (1980), 156-185, URL: http://www.sciencedirect.com/science/article/pii/0315086080900361
• 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.
• Stillwell, John Colin (2013). "Arithmetization of Analysis," Encyclopaedia Britannica, URL: http://www.britannica.com/EBchecked/topic/22486/analysis/247690/Complex-exponentials.
• Tensegrity wikispace. URL: http://tensegrity.wikispaces.com/Arithmetization
• Tweddle, J Christopher (2012). "Weierstrass's Construction of the Irrational Numbers".
• Ueno, Yoshiaki (2003). "Kronecker’s idea of arithmetization of mathematics," ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 26 No.1.
• Weisstein, Eric W. "Bolzano's Theorem." From MathWorld--A Wolfram Web Resource. URL: http://mathworld.wolfram.com/BolzanosTheorem.html
• Zerner, M (1989) Review: Martin Ohm (1792-1872): Universitäts und Schulmathematik in der neuhumanistischen Bildungsreform by Bernd Bekemeier, Educational Studies in Mathematics 20 (4), 469-474.