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." ^{[1]}These efforts took place over a period of about 50 years, with the following results:
- the establishment of fundamental concepts related to limits
- the derivation of the main theorems concerning those concepts
- the creation of the theory of real numbers.
This article accepts as a terminus for the arithemetization program the development of the theory of irrationals in what has been termed "the red-letter year"of 1872.^{[2]} An alternative view is premised on the following facts:^{[3]}
- 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 arithmatization program came to signify the set-theoretic definition of function and the set-theoretic construction of the real line
This article presents a brief history of the arithmetization efforts during that 50-year period.^{[4]}
In setting out the history of any such mathematical development or period, the central question to be answered is 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."
Contents
- 1 Non-mathematical issues
- 2 The origin and need for arithmetization
- 3 The arithmetization program
- 4 Looking back at arithmetization
- 5 Notes
- 6 Primary sources
- 7 References
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.^{[5]}
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."^{[6]} 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:^{[7]}
- 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.
As a final example, consider that even Weierstrass had issues with publishing. Notwithstanding the breadth and depth of his contributions to mathematics, he did not publish much of his own work himself and, when he did, was slow to do so.^{[8]} Most remarkably, he failed even to publish two of his signal contributions to the arithmetization program:
- his example of a continuous, nowhere deifferentiable function
- his construction of the irrationals.
Instead, it was his students who, working from their lecture notes, published his results in various and at times conflicting papers. In addition, of course, Heine, in his own paper offering a construction of the irrationals, credits Weierstrass with having developed the fundamental theory.^{[9]}
The origin and need for arithmetization
Two pillars of mathematics
An example of a thoroughly modern definition of calculus is as follows:^{[10]}
- Calculus is the branch of mathematics that defines and deals with limits, derivatives and integrals of functions.
Modern authors generally describe the state of mathematics after the invention of the calculus, but prior to 19th century efforts at arithmetization, somewhat as follows:
- analysis rested uneasily on two pillars: the discrete side on arithmetic, the continuous side on geometry.^{[11]}
- the source domain of analysis was geometry; that of number theory was arithmetic.^{[12]}
Summarizing the mathematical situation in the seventeenth and eighteenth centuries, a modern historian has contrasted "the powerful techniques of the calculus" with "the relatively unimpressive views put forth to justify them"^{[13]}
Mathematicians of the 19th century who laboured on the arithmetization of analysis gave various reasons for their pursuits. One modern source identifies the following as the chief causes of concern among mathematicians about 19th century mathematics:^{[14]}
- 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. Another 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:^{[15]}
- 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:^{[16]}
- the principles of the differential calculus, freed from all consideration of the infinitely small or vanishing quantities....
Among 19th century mathematicians, 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."^{[17]} In other words, mathematicians needed to view real numbers as "intellectual structures" rather than as "intuitively given magnitudes inherited from Euclid's geometry."^{[18]}
The work of Euler, Gauss, Cauchy, and others shifted analysis towards algebraic and arithmetic ideas, culminating, in the "red-letter year" of 1872, when the arithmetization program was (more or less) completed by Meray, Weierstrass, Heine, Cantor, and Dedekind.^{[19]}^{[20]}
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:^{[21]}
- 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.^{[22]}
- using analysis to prove the fundamental theorem of number theory raised a problem about the boundary between number theory and analysis.^{[23]}
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.^{[24]} 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:^{[25]}
- $\displaystyle f(x) = \sum_{n=1}^\infty a^n cos(b^n \pi x)$ 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:^{[26]}
- 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. 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.^{[27]} It was with good reason that he is said to have been a "voice crying in the wilderness."^{[28]} Discovery of such functions continued throughout the 20th century, though with less shocking effects!^{[29]}
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^{[30]}
- it dealt "a decisive blow to the intuitive picture of the behavior of continuous functions."^{[31]}
A turn of the century address to the American Mathematical Society summarized the situations of the "intuitionist" and the "arithmetician" as follows:^{[32]}
- 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.
The arithmetization program
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:^{[33]}
- 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."^{[34]}
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."^{[35]}
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.^{[36]}
Basic concepts of calculus
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.
Limits
D'Alembert's own definition of limit was as follows:^{[37]}
- ... 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."^{[38]} They gave similar definitions of limits, convergence, and continuity. The concept of limit that they each developed (independently) 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:^{[39]}
- 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:^{[40]}
- 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.^{[41]} 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:^{[42]}
- $\displaystyle \lim_{x \to \alpha}f(x) = L$ 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."^{[43]}
Convergence
As was noted above, the uneasiness among 19th century mathematicians had its origin in two issues:
- a lack of clarity to do with operations on infinite series
- the lack of a definition of the phrase "real number"
In the work of Bolzano and Cauchy, the concept of convergence was central to resolving both of these issues.^{[44]}
With respect to the first concern (about operations on infinite series), several specific tests for convergence were known and used to good effect long before the 19th century, for example:^{[45]}
- the integral test for convergence of infinite series (given earlier by Euler in 1732)
- the ratio test for the convergence of infinite series . . . had been given by Waring as early as 1776
Gauss, responding in 1812 to concerns about the need to test infinite series for convergence, "used the ratio test to show that his hypergeometric series converges for |x| < 1 and diverges for |x| > 1."
A general resolution came with the work of Bolzano and Cauchy. Working independently, they clarified the notion of convergence as it applied to an infinite series. Cauchy offered the following definition of a series that was "convergent within itself":
- a series is convergent if, for increasing values of $n$, the sum $S_n$ of the first $n$ terms approaches a limit $S$, called the sum of the series.
This definition of convergence refers to the sum $S$ to which the sequence of partial sums of the series $S_n$ converges. But since the sum of a series is often not explicitly known, what was needed is a criterion for convergence that is not based on the external sum $S$, but on what is internal to the series itself.
Cauchy based a criterion for convergence of a series on the following successive differences of partial sums:^{[46]}
- $S_{n+1} - S_n = u_n$
- $S_{n+2} - S_n = u_n + u_{n+1}$
- . . .
- $S_{n+p} - S_n = u_n + u_{n+1} + . . . + u_{n+p-1}$
- . . .
It was known by Cauchy and others that, for a series to converge to something, a necessary condition is that the first of these differences approach 0. Cauchy's insight was that a sufficient condition is that the above differences of partial sums, as he put it, "taken, from the first, in whatever number one wishes, finish by constantly having an absolute value less than any assignable limit."^{[47]} When both of these (necessary and sufficient) conditions are fulfilled, Cauchy continued, "the convergence of the series is assured."^{[48]}
Using Cauchy's statement above of the successive differences of partial sums, we might restate his insightful conclusion in more modern terminology as follows:^{[49]}
- the series $S_n$ converges IFF for every $\varepsilon > 0$, there exists a $K$ such that
- $|S_{n+p} − S_n| < \varepsilon$ whenever $n+p > n > K$.
This criterion for a convergent series came to be know as the Cauchy criterion. It was known earlier to Bolzano, but as noted previously, his work was unknown to others working on the arithmetization program.
Bolzano's (equivalent) statement of the Cauchy criterion is as follows:^{[50]}
- If a sequence of magnitudes $F_1(x), F_2 (x), F_3(x) ... F_n(x) ... F_{n+r}(x) ...$ is subject to the condition that the difference between its $nth$ member $Fn(x)$ and every later member $F{n+r}(x)$ , no matter how far beyond the $nth$ term the latter may be, is less than any given magnitude if $n$ is taken large enough; then, there is one and only one determined magnitude to which the members of the sequence approach closer, and to which they can get as close as desired, if the sequence is continued far enough.
Neither Cauchy nor Bolzano actually proved the sufficiency of the Cauchy condition, though Bolzano attempted a demonstration of its plausibility.
Bolzano was interested in properties of the real numbers and used the Cauchy criterion to prove the least upper-bound property as a step in his proof of the intermediate-value theorem for continuous functions (see below). Cauchy was interested in infinite series and used the criterion for deriving various convergence tests, including the root test and the ratio test.^{[51]}
With respect to the second concern (about the definition of "real number"), Bolzano and Cauchy, again independently and working with the notion of a sequence that "converges within itself", 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.
Both Meray and Weierstrass understood the error involved in the circular way that Bolzano and Cauchy had defined the concepts limit and real number:
- 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)
In other words, since the definition of the former presupposed the notion of the latter, therefore, the the definition the latter must be independent of the former! 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.
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 $x$:^{[52]}
- $f(x + h) − f(x)$ 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.^{[53]}
In 1821, Cauchy added to Bolzano's definition of continuity at a point "the final touch of precision":^{[54]}
- 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.^{[55]}
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."^{[56]}
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:^{[57]}
- 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:^{[58]}
- 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.^{[59]} 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;^{[60]}
- in his proof, Cauchy employed Lagrange'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.^{[61]}
In their proofs of the intermediate value theorem, neither Bolzano nor Cauchy identified all the assumptions that underlay their treatment of real numbers.
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.^{[62]} 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:^{[63]}
- 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:^{[64]}
- 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 condition (on occasion the Bolzano-Cauchy condition), as follows:^{[65]}
- 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.
- If a series [sequence] of quantities
- 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.^{[66]}
- Next, Bolzano used the Cauchy condition in a proof of the following theorem, namely, that a bounded set of numbers has a least upper bound:^{[67]}
- 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 $M$.^{[68]} The theorem proved is the original form of the Bolzano-Weierstrass theorem and is in fact the original statement of that theorem:^{[69]}
- 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:^{[70]}
- 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.^{[71]}
- 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:^{[72]}
- 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 $\beta$.
Cauchy's proof
In his 1821 paper, Cauchy provided a proof of the intermediate value theorem, which some authors identify as Cauchy's (intermediate-value) theorem. He stated the theorem as follows:^{[73]}
- 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:^{[74]}
- 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.^{[75]}
- 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:^{[76]}
- 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.
Derivatives and integrals
The concept "function," which is fundamental to mathematics and derives from the calculus, turns on these two notions:^{[77]}
- 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.
The derivative
As did other authors of 18th century calculus books, Cauchy provided an explicit verbal definition for the derivative, as follows:^{[78]}
- 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:^{[79]}
- 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:^{[80]}
- 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.^{[81]}
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:^{[82]}
- 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 (Cauchy) 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:^{[83]}
- 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 $f(b) - f(a) = (b — a) f'(x_0 )$.
Cauchy's proof proceeds as follows:^{[84]}
- 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.^{[85]}
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.^{[86]}
The theory of irrational numbers
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.^{[87]}
"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.^{[88]}
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:^{[89]}
- 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."^{[90]} 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.^{[91]} Even so, during the period 1872-1894, Meray continued to publish works intended to "remove geometric considerations from analytic proofs."^{[92]}
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.^{[93]}
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.^{[94]} 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.^{[95]}
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:^{[96]}
- 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."^{[97]}
Based on what he felt was the continuity of the real line, Dedekind captured what he termed the "essence of continuity" as follows:^{[98]}
- 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 axiom or principle.
He proved the existence of irrational numbers by constructing an example using the section property above:^{[99]}
- 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.^{[100]} 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."^{[101]}
There is an irony in Dedekind's treatment of real numbers, which has been expressed as follows:^{[102]}
- 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
In the above, we mean suitable in the sense that Dedekind's definition of continuity was entirely sufficient to define the real numbers and, thus, to complete the arithmetization program.^{[103]}
- For Dedekind, the issue of continuity is one and the same as the real numbers.... Having proved that the real numbers as constructed by himself satisfied the law of continuity, Dedekind believed he had characterized continuity completely.
But Dedekind's definition was not sufficient to capture the essence either of the real line or of the continuum.^{[104]}
Common to all three of these definitions of irrational numbers (by Dedekind, Cantor, and Weierstrass) was "a well-defined collection of rational numbers."^{[105]} The differences among the three theories sprang from the quite different motivations of their authors:^{[106]}
- 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:^{[107]}
- 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.
Looking back at arithmetization
As stated at the outset, this article is intended to be a brief history of the arithmetization program. It seeks to set forth the achievements of mathematicians working in that program that have led to what is important in mathematics today. The history presented has been dubbed the standard account by some, for the simple reason that it is not the only account.^{[108]} In such history articles, it is usual to include alternate views of events within the main narrative. That practice is not adhered to in this article. Instead, summaries of those alternate views are gathered here, at the end of the article, with the intention of providing a platform for inserting additional entries and, if appropriate, for linking to additional articles.
A modest caveat
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:^{[109]}
- 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.
Challenges from non-standard analysis
In 1960, Abraham Robinson developed non-standard analysis. He did this by (rigourously) extending the field of reals to include infinitesimal numbers and infinite numbers. The new, extended field is called the field of hyperreal numbers. His goal was to create a system of analysis that was more intuitively appealing than standard analysis but without losing any of the rigor of standard analysis.^{[110]}
In summary and informally, the definition of the hyperreal field is as follows:^{[111]}
- Axiom A – $\mathbb{R}$ is a complete ordered field.
- Axiom B – $\mathbb{R}^*$ is a proper ordered field extension of $\mathbb{R}$.
- An element, $x$, of $\mathbb{R}^*$ is an infinitesimal if $|x| < r$ for all positive real $r$.
- Two elements $x, y$ of $\mathbb{R}^*$ are said to be infinitely close if $x \approx y$ and if $x - y$ is an infinitesimal. Thus, $x$ is an infinitesimal if and only if $x \approx 0$.
- An element, $x$, of $\mathbb{R}^*$ is an infinite if $|x| > r$ for all real $r$.
A fundamental property of non-standard analysis the transfer principle, which states "every proposition true in classical analysis is also true in non-standard analysis," and vice versa. Here, for example, are two, equivalent definitions of a limit:^{[112]}
- a standard $\varepsilon, \delta$ definition of a limit
- the limit of $f (x)$ as $x \to c$ is L if,
- for every real $\varepsilon > 0$, there is a real $\delta > 0$, such that
- whenever $x$ is real and $0 < |x - c| < | \delta$, then $| f (x) - L | < \varepsilon$
- and an equivalent, non-standard definition:
- whenever $x \approx c$ but $x \neq c$ then $f (x) \approx L$
Given a sentence $\phi$, we may take its *-transform $\phi^*$ as follows:^{[113]}
- replacing a function $f$ with $f^*$
- replacing a relation $R$ with $R^*$, and
- replacing any bound $P$ occurring in $x \in P$ by $x \in$ $P^*$.
The Transfer Principle states that if we are working over the language of $\mathbb{R}$, then, a sentence $\phi$ is true if and only if $\phi^*$ is true.
As an interesting example, we can look at the Archimidean property of the reals:
- $\forall x \in \mathbb{R^+} (\exists n \in \mathbb{N}) (nx >$ $1)$
The *-transfer of this statement is
- $\forall x \in \mathbb{R^*}^+ (\exists n \in \mathbb{N}^*) (nx >$ $1^*$)
This second statement is true, yet it is not equivalent to the Archimidean property. In fact, $\mathbb{R}^*$ is not Archimidean in the standard sense. No repeated addition of $[1/n]$ will ever bring it above $1$. As Robinson himself noted, “In our own theory the answer to the question whether the Archimedes’ axiom is true not only in $\mathbb{R}$ but also in $\mathbb{R}$* is unambiguously, yes – and no!”^{[114]}
The Transfer Principle says that It doesn't really matter whether we work in the standard or the non-standard setting. Yet, there is a lesson here, namely, that some properties transfer very naturally between $\mathbb{R}$ and $\mathbb{R}^*$, while others may transfer in ways that may not be intuitive and/or useful.^{[115]}
An alternate expression of the Archimedean axiom or property with respect to the theory of ordered fields is this:^{[116]}:
- $(\forall x > 0) (\forall \varepsilon > 0) (\exists n \in \mathbb{N}) (n \varepsilon > x)$
or equivalently
- (A) $(\forall \varepsilon > 0) (\exists n \in \mathbb{N}) (n \varepsilon > 1 )$.
A number system satisfying (A) above is an Archimedean continuum.
In the contrary case, there is an element $\varepsilon > 0$ called an infinitesimal such that no finite sum $\varepsilon + \varepsilon + . . . + \varepsilon$ will ever reach 1; in other words,
- (B) $(\exists \varepsilon > 0) (\forall n \in \mathbb{N}) (\varepsilon \leq 1/n)$.
A number system satisfying (B) above is a Bernoullian continuum (i.e., a non-Archimedean continuum).
The development of non-standard analysis has given rise to the following questions, the answers to which challenge the standard account of arithmetization:
- What is the role of intuition in mathematical theory and thinking?
- What is the place of infinitesimals in analysis?
- What is the nature of rigour in mathematical definitions and proofs?
- Are there useful alternate methods of teaching (and doing) analysis?
The remainder of this section relates some answers that have arisen from non-standard analysis.
A role for intuition in mathematics
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 addressed these two questions:^{[117]}
- 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:^{[118]}
- 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:^{[119]}
- 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.
Even so, and in somewhat of an about face, Pierpont concludes with this comment about the "extreme arithmetizations" ushered in by that school of Weierstrass:^{[120]}
- It is, however, not to be overlooked that the price paid for this clearness is appalling, it is total separation from the world of our senses.
Today, more than 100 years after Pierpont's address, intuition is present in mathematics in at least two, quite different, respects:
- as an accompaniment to the reintroduction of infinitesimals by non-standard analysis
- as an essential part of what is involved in mature, high-level mathematical thinking
Intuition in mathematical theory
According to the standard account, the arithmetization program excised intuition from the foundations of mathematics. With the development (or, as it has alternatively been put, the "discovery") of non-standard analysis, intuition was restored to legitimacy in the foundations of mathematics. Robinson maintained that results achieved using non-standard analysis could not have been achieved "just as well" by standard methods and that translation of non-standard results into standard terms "usually complicated matters considerably." He explained the reason for this difficulty as follows:^{[121]}
- Our approach has a certain natural appeal, as shown by the fact that it was preceded in history by a long line of attempts to introduce infinitely small and infinitely large numbers into Analysis.
The "natural appeal" to which Robinson refered is grounded in two factors:^{[122]}
- non-standard analysis is often simpler and more intuitive in a very direct, immediate way than standard approaches
- the concept of infinitesimals had always seemed natural and intuitively preferable to more convoluted and less intuitive sorts of rigor.
Robinson's extension of the number concept has been shown to be of use in various domains, including areas of analysis, the theory of complex variables, mathematical physics, and economics. As a consequence, while the methods of non-standard analysis can be avoided in these domains, the cost of doing so may be more complicated proofs and less intuitive arguments.^{[123]}
The "natural advantages" of using infinitesimals could be exploited, now that non-standard analysis had shown that their use was "safe for consumption" in mathematics!
Intuition in mathematical thinking
In a very different sense, intuition is present in mathematical thinking at the highest levels, quite apart from and irrespective of the nature of foundations theory. In the remarks quoted above, Pierpont noted what may seem to be a very uncontroversial result among several that are said to have proceeded from the arithmetization program, namely, the replacement of "arguments based on intuition" with "arithmetic methods" as the foundation of analysis. Even so, an objection has been raised in the form of a suggestion that an untoward fixation on "rigourous formalism" can lead to a loss of the mathematical intuition that is essential to working at a mature level:^{[124]}
- The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture. . . . So once you are fully comfortable with rigorous mathematical thinking, you should revisit your intuitions on the subject and use your new thinking skills to test and refine these intuitions rather than discard them. . . .
- The ideal state to reach is when every heuristic argument naturally suggests its rigorous counterpart, and vice versa.
A place for infinitesimals in mathematics
The standard account asserts that arithmetization "banished" infinitesimals from mathematics. Certainly, the elimination of infinitesimals was considered an important reason for pursuing the arithmetization program. Here, very briefly, is some history:^{[125]}
- For many centuries, early mathematicians and physicists would solve problems by considering infinitesimally small pieces of a shape, or movement along a path by an infinitesimal amount.... In particular, the construction of calculus was first motivated by this intuitive notion of infinitesimal change.... Although intuitively clear, infinitesimals were ultimately rejected as mathematically unsound, and were replaced with the common $\varepsilon, \delta$ method of computing limits and derivatives.
And here, equally brief, is a bit more:^{[126]}
- When Newton and Leibnitz practiced calculus, they used infinitesimals, which were supposed to be like real numbers, yet of smaller magnitude than any other type of postive real number. In the 19th century, mathematicians realized they could not justify the use of infinitesimals according to their sense of rigor, so they began using definitions involving $\varepsilon$'s and $\delta$'s. Then, in the early 1960's, the logician Abraham Robinson figured out a way to rigorously define infinitesimals, creating a subject now known as non-standard analysis.
Several comments are relevant. First, the work on non-Archimedean systems actually continued unabated both during and after the period of arithmetization.^{[127]} Indeed, there was "a rich and uninterrupted chain of work" on non-Archimedean or infinitesimally-enriched systems or what some have called Bernoullian continua:^{[128]}
Second, as noted above, in 1960 came a very significant "correction" to arithmetization, namely, the development of non-standard analysis:^{[129]}
- 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 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 claim that Weierstrass eliminated infinitesimals has been termed an oversimplification of the history of analysis:^{[130]}
- If the epsilon-delta methods had not been discovered, then infinitesimals would have been postulated entities (just as ‘imaginary’ numbers were for a long time). Indeed, this approach to the calculus–enlarging the real number system–is just as consistent as the standard approach, as we know today from the work of Abraham Robinson [. . . ] If the calculus had not been ‘justified’ Weierstrass style, it would have been ‘justified’ anyway.
As noted above, Cantor refuted Dedekind's view that "the system of real numbers IS the continuous domain" because it "precluded the possibility of other continuous domains."^{[131]} Modern authors working with NSA have seen this more specifically. Their criticism is that Dedekind's principle of continuity "entails a suppression of infinitesimals":^{[132]}
- a pair of distinct, non-rational numbers can define the same Dedekind cut on $Q$, such as $π$ and $π + h$ WITH $h$ infinitesimal
- BUT one cannot have such a pair if one postulates ... continuity, as Dedekind does.
In effect, Dedekind's principle of continuity "steamrollered infinitesimals out of existence"!
There are a variety of possible conceptions of the continuum. Mathematicians have considered at least two different types of continua:
- Archimedean continua, or A-continua for short
- Bernoulli (infinitesimal-enriched) continua, or B-continua for short
Neither an A-continuum nor a B-continuum corresponds to a unique mathematical structure. The notion that there is a single coherent conception of the continuum, and it is a complete, Archimedean ordered field has been called "an academic dogma."^{[133]}
- the collection of all number systems is not a finished totality whose discovery was complete around 1600, or 1700, or 1800, but . . . has been and still is a growing and changing area, sometimes absorbing new systems and sometimes discarding old ones, or relegating them to the attic.
Accordingly, there are not one but two separate tracks for the development of analysis:^{[134]}
- (A) the Weierstrassian approach (in the context of an Archimedean continuum) and
- (B) the approach with indivisibles and/or infinitesimals (in the context of a Bernoullian continuum).
In summary, the effect (at least in some quarters) of Robinson's work in non-standard analysis was to usher in "a rehabilitation of the use of infinitesimals in mathematics" and to show, as Robinson himself proposed, that (at least some of) the infinitesimal methods used by the founders of calculus to be "correct and consistent."^{[135]}
The nature of rigour in definitions and proofs
The standard account of the arithmetization program claims for it an increase in rigour, both in mathematical thinking and in mathematical definitions and proofs:^{[136]}
- A foundational rock of the received history of mathematical analysis is the belief that mathematical rigor emerged starting in the 1870s through the efforts of Cantor, Dedekind, Weierstrass, and others, thereby replacing formerly unrigorous work of infinitesimalists from Leibniz onward.
In his own historical writing, Robinson refuted the claim that increased rigour resulted from of the success of Cauchy-Weierstrassian epsilontics over infinitesimals. Robinson's achievements in non-standard analysis are said to have demonstrated conclusively that the claim of increased rigour was mere historicism.^{[137]} Subsequent achievements of others in non-standard methods have underscored this conclusion.^{[138]}Looking back at arithmetization through the lens of NSA, we see clearly that the elimination of infinitesimals "was not the only way to achieve rigor in analysis. . . , but rather a decision to develop analysis in just one specific way."^{[139]}
The following question and the answer that it provoked challenge the view that rigour can be today (if, indeed, it ever was) paramount in directing our choice of mathematical methods:^{[140]}
- Q - Is there an example of a result that was first proved using non-standard analysis?
- A - All the early fundamental theorems of calculus were first proved via methods using infinitesimals. . . . The early arguments---whatever their level of rigor---are closer to their modern analogues in nonstandard analysis than to their modern analogues in epsilon-delta methods.
More fully set out, the answer argues as follows:
- early fundamental theorems of calculus were proved using infinitesimal methods
- 19th century $\varepsilon, \delta$ methods were developed in response to uncertainty about infinitesimal methods
- today, NSA provides "the missing legitimacy" for infinitesimal methods
The suggestion is that the "original motivation" for $\varepsilon, \delta$ methods has "fallen away" and we are now faced with this new question:
- What today ought to motivate us in our choice to use either standard or non-standard methods?
One thing we know (for sure?): Rigour will not figure in the choices we make!
An alternate method of teaching calculus
There is little dispute that arithmetization brought about a shift in emphasis from geometric to algebraic reasoning. Certainly this shift was 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 was taught.^{[141]}
The development of non-standard analysis and the consequent return of infinitesimals to mathematics have given rise to the possibility of a further shift in the way mathematics is taught. The year 1971 saw the publication of Jerome Keisler's textbook, Elementary Calculus: An Approach Using Infinitesimals. This text used non-standard analysis to explain, in an introductory course, the basic ideas of calculus. In 1973-74, a study was done to determine the pragmatic value of using this textbook in teaching calculus. The study examined this pedagogical claim:^{[142]}
- from this non-standard approach, the definitions of the basic concepts [of the calculus] become simpler and the arguments more intuitive.
The results of the study, both as measured by a calculus test given to the students and by the comments of the instructors, are remarkable in their support of the heuristic value of using non-standard analysis in the classroom.^{[143]}
It is worth looking at the single question in the test that brought out the greatest difference between the experimental group and the control group:
- Define $f(x)$ by this rule: \( f(x) = \begin{cases} x^2 & \text {for } x ≠ 2 \\ 0 & \text {for } x = 2 \end{cases} \)
- Prove using the definition of limit: \( \begin{align} \displaystyle \lim_{x \to 2} f(x) = 4 \end{align} \)
The study results summarized the comments of the instructors as follows:
- The group as a whole responded in a way favorable to the experimental method on every item: the students learned the basic concepts of the calculus more easily, proofs were easier to explain and closer to intuition, and most felt that the students end up with a better understanding of the basic concepts of the calculus.^{[144]}
Notes
- ↑ Arithmetization, Encyclopedia of Mathematics. See also Lakoff & Nunes p. 262 cited in Ueno p. 71, noting a terminological alternative: "In late 19th century Europe,... mathematics had gained an important stature as being the discipline that defined the highest form of reason, with precise, rigorous, and indisputable methods of proof. Anything not formalizable was seen as "vague", "intuitive" (as opposed to "rigorous"), and imprecise.... This movement is later called 'the discretization program'".
- ↑ Boyer p. 604
- ↑ Arithmetization, Tensegrity wikispace
- ↑ Grabiner (1975) p. 439. In her article, Grabiner develops an intriguing contrast between the mathematician's view of a period of mathematical history and the historian's view of the same period. The introductory paragraphs of this article summarize what Grabiner presents as the mathematician's view.
- ↑ Jarnik et. al. pp. 34-35 In the midst of his extensive exposition of Bolzano's contribution to the foundations of analysis, Jarník asks the question, "how much Bolzano's work could have changed the way analysis followed, had it been published at the time." For an extensive comparison of the contributions of Bolzano and Cauchy, see the various works of Grabiner.
- ↑ Hamilton cited in Mathews, Introduction
- ↑ 1828 letter from W. R. Hamilton to John T. Graves cited in Graves, p. 304
- ↑ Pinkus p. 2. "Nevertheless," as Pinkus further notes, "his collected works (Mathematische Werke) contain 7 volumes of well over 2500 pages,.. much of [which] is taken up with a great deal of previously unpublished lecture notes and similarly unpublished talks."
- ↑ Jarnik et. al. p. 37; Snow pp. 99-102; Tweedle p. 1; and many other sources.
- ↑ Bogomolny. See also Differential calculus, Integral calculus, Encyclopedia of Mathematics
- ↑ Stillwell
- ↑ Ueno p. 73. Even at the beginning of the 20th century, Kronecker, though not wholly satisfied with the results of the arithmetization program (see "Some caveats" below in this article) agreed with its intent, as Ueno notes, to "free analysis from its source domain — geometry."
- ↑ Grabiner (1983)
- ↑ Boyer, p. 604 The many references in this article to Boyer's textbook attest to its place as a work, not of mathematics, but of mathematical history. In his Preface, Boyer notes that it is not his fundamental purpose to teach mathematics, but "to present the history of mathematics with fidelity, not only to mathematical structure and exactitude, but also to historical perspective and detail."
- ↑ Hatcher, Sect. 3.2 The Arithmetization of Analysis
- ↑ Lagrange (1797) cited in O'Connor and Robertson
- ↑ Hankel cited in Boyer, p. 605
- ↑ Boyer p. 605
- ↑ Hatcher Sect. 3.2 "The Aritimetization of Analysis"
- ↑ Boyer p. 604
- ↑ "Fundamental Theorem of Algebra," Wikipedia
- ↑ Stillwell
- ↑ Ueno p. 72
- ↑ Boyer p. 565
- ↑ du Bois-Reymond cited in Jarnik p. 37n.; Pinkus p. 3; Schultz "The function concept"; and many other sources.
- ↑ Baillaud and Bourget cited in Pinkus p. 3.
- ↑ Jarnik et. al. p. 37-41
- ↑ Boyer p. 565
- ↑ Thim This paper catalogues and examines the many such "pathological" functions, developed from 1830 (Bolzano) through 2002 (Wen).
- ↑ Pinkus p. 4. Pinkus notes, humourously, "yesterday's pathologies are at times central in today's 'cutting edge' theories and technologies," and, seriously, "without nowhere differentiable functions we would not have Brownian motion, fractals, chaos, or wavelets."
- ↑ Grabiner (1981)
- ↑ Pierpont pp. 398-400
- ↑ Boyer p. 604
- ↑ Ohm 1843 cited in O'Connor and Robertson
- ↑ Zerner cited in O'Connor and Robertson
- ↑ O'Connor and Robertson, Ohm
- ↑ Dunham p. 72 cited in Bogomolny
- ↑ Pinkus, p. 3. As his reason for saying so, Pinkus cites Grabiner (1981) that Bolzano and Cauchy gave similar definitions for the basic concepts of the calculus.
- ↑ Grabiner (1981) p. 80. In her book, Grabiner documents in detail the rigourous manner in which Cauchy, using the algebra of inequalities, formulated the basic definitions on which calculus rests. More importantly, however, Grabiner reveals the source of the ideas on which Cauchy based his rigourous definitions, namely, the approximation techniques and associated methods of estimating errors developed by Lagrange (especially) and others.
- ↑ Boyer p. 563
- ↑ Grabiner (1983) p. 1 and also numerous instances in Grabiner (1981) noting that, when appropriate, Cauchy translated imprecise, verbal definitions into precise, $\delta,\varepsilon$ definitions constructed using the algebra of inequalities.
- ↑ Heine cited in Boyer p. 608
- ↑ Boyer p. 609
- ↑ Boyer, p. 566 ff. and 606. It is worth keeping in mind that Bolzano and Cauchy, in their efforts to clarify the notion of convergence, worked with the notion of a series that "converges within itself". Boyer notes that Cauchy's awareness of the importance of "uniform convergence" of a series did not arise until 1853, that Cauchy was anticipated in this awareness by Stokes (in England) and Weierstrass and Seidel (in Germany), and that it was Dirichlet who provided the test for the uniform convergence of a series. pp. 552 & 610
- ↑ Boyer pp. 470 & 500
- ↑ Grabiner (1981) p. 102
- ↑ Katz, p. 712 cited in Chaitesipaseut
- ↑ Cauchy (1821) cited in Grabiner (1981) p. 102
- ↑ Boyer p. 556
- ↑ Grabiner (1981) p. 102
- ↑ Grabiner (1981) p. 105
- ↑ Stillwell
- ↑ Jarník et. al., p. 38
- ↑ Stillwell
- ↑ Jarník et. al., p. 38
- ↑ Pinkus, p. 2
- ↑ Jarník et. al. p. 35
- ↑ Jarník et. al. p. 36
- ↑ Grabiner (1981) p. 38
- ↑ Grabiner (1981) p. 74
- ↑ Grabiner (1981) p. 70
- ↑ Grabiner (1981) p. 74
- ↑ Russ p. 160 This translation is the first (1980) into English of Bolzano's remarkable 1817 paper, which is identified as "Analytic Proof" in Jarnik's commentary. The paper includes a formal definition of continuity, the criterion for (pointwise) convergence of an infinite series, a proof of the Bolzano-Weierstrass theorem (in its original form), and a proof of Bolzano's theorem (the intermediate value theorem).
- ↑ Russ p. 162
- ↑ Russ p. 171
- ↑ Jarník et. al. p. 36
- ↑ Russ p. 174
- ↑ Jarník et. al. p. 36
- ↑ Russ p.157
- ↑ Russ p. 177
- ↑ Grabiner (1981) p. 11
- ↑ Russ p. 181
- ↑ Grabiner (1983) p. 167
- ↑ Grabiner (1981) p. 87
- ↑ Grabiner (1983) p. 87
- ↑ Grabiner (1983) p. 8
- ↑ Hatcher Sect. 3.1 "The Emergence of Modern Mathematical Analysis"
- ↑ Cauchy (1823) cited in Grabiner (1983) p. 1
- ↑ Grabiner (1975) p. 441
- ↑ Cauchy (1823) cited in Grabiner (1983) p. 1
- ↑ Grabiner (1981) p. 115
- ↑ Grabiner (1983) p. 9
- ↑ Boyer p. 564
- ↑ Cauchy (1823) cited in Grabiner (1983) p. 11
- ↑ Boyer p. 564
- ↑ Boyer p. 604
- ↑ Jarník et. al. p. 34n
- ↑ Tweddle, p. 4 In his article, Tweedle traces the subject of irrationals from its ancient beginnings and then, based on the published notes of his students, develops Weierstrass's contruction of the irrational numbers. In addition, Tweedle offers interesting comments that relate the alternative constructions of irrationals by Dedekind and Cantor to W's own construction. Finally, Tweedle notes the connection (based on the fundamental concept of convergence) between the Meray/Cantor/Heine constructions of irrationals (as Cauchy sequences of rational numbers) to Weierstrass's construction (as convergent sequences of partial sums).
- ↑ Tweddle, p. 4
- ↑ O'Connor and Robertson, Méray
- ↑ Robinson
- ↑ O'Connor and Robertson, Méray
- ↑ Boyer, Carl S. pp. 606-7
- ↑ Boyer p. 607
- ↑ Tweddle, p. 5
- ↑ Dedekind cited in Garden of Archimedes
- ↑ Boyer p. 607
- ↑ Dedekind p. 11 cited in Snow p. 96
- ↑ Dedekind pp. 13-15 cited in Snow p. 89
- ↑ Rech Sect. 2.1 The Foundations of Analysis. Rech's article in the SEP includes an extensive bibliography, both primary works by Dedekind and secondary materials by modern authors.
- ↑ Dedekind cited in Garden of Archimedes
- ↑ Boyer p. 608
- ↑ Snow pp. 107-108
- ↑ Snow pp. 107-108 In her article, Snow explains the precise manner in which Dedekind's concept of continuity is insufficient to do entirely as he wished, namely, as Snow puts it, "to capture arithmetically the continuity that is modeled by an unbroken line ... and to understand a continuous domain." Particularly interesting is Snow's examination of the correspondence on this subject between Dedekind and Cantor, "for [whom] there was more to the concept of a continuum than a characterization of the real numbers." A century after Cantor's criticisms, the developers and proponents of non-standard analysis effectively refuted the assertion that Dedekind had captured the essence of continuity. See below Looking back at arithmetization below.
- ↑ Tweddle, p. 6
- ↑ Bottazzini cited in Tweddle, p. 6. In his review of Bottazzini's work, Truesdell gave high praise for two matters: (1) knowing that "the sources of pure mathematics often lie in works that today's mathematicians would consider to be 'applied' mathematics or 'physics'"; (2) not assuming that "rigor was sought for rigor's sake, which while true of some works of some mathematicians was not at all characteristic of the search for and achievement of rigorous procedures by, for example, Cauchy and Riemann." p. 186-7.
- ↑ Hatcher Sect. 3.2 "The Arithemetization of Analysis"
- ↑ Among proponents of the standard account are two authors whose contributions to the history of the arithmetization program are celebrated in this article, viz. Carl B. Boyer and Judith V. Grabiner
- ↑ Jarník et. al. p. 33
- ↑ Davis, p. 1
- ↑ Keisler, p. 2 cited in Parker, p. 9
- ↑ Keisler, p. 31 & 103, cited in Parker, p. 9
- ↑ O'neill, p. 4-5. Both the statement of the transfer principle and the *-transforms of the Archimedean property are excerpted from O'Neill.
- ↑ Dauben, (1995), p. 351, cited in Parker, p. 9-10.
- ↑ O'Neill, p. 5
- ↑ See, e.g., Hilbert 1899 [51, p. 27], cited in Bair et. al., p. 888
- ↑ Pierpont, p. 394
- ↑ Pierpont, p. 395
- ↑ Pierpont, p. 406
- ↑ Arithmetization, Tensegrity
- ↑ Robinson (1965) cited in Dauben, p. 184
- ↑ Dauben, p. 184
- ↑ Dauben, p. 195
- ↑ Tao, "There's more to math...."
- ↑ Davis, 1. Introduction
- ↑ O'Neill, p. 1
- ↑ Erlich cited in Arithmetization, Tensegrity
- ↑ Blaszczyk, Katz, & Sherry, Abstract
- ↑ Bogomolny, "What is Calculus?"
- ↑ Putnam cited in Blaszczyk, Katz, & Sherry, p. 21
- ↑ Snow, p. 111
- ↑ Blaszczyk, Katz, & Sherry, p. 25
- ↑ Blaszczyk, Katz, & Sherry, p. 23 The authors note challenges to this "dogma" ranging from S. Feferman’s predicative conception of the continuum, to F. William Lawvere’s and J. Bell’s conception in terms of an intuitionistic topos.
- ↑ Klein, p. 214, cited in Bair et. al., p. 887 The authors note that systems of quantities encompassing infinitesimal ones were used by Leibniz, Bernoulli, Euler, and others. The term "Bernoullian" encompasses modern non-Archimedean systems. They explain their choice of the term in the article subsection "Bernoulli, Johann," p. 889.
- ↑ Kvasz The comments in this article are based on an abstract, provided by conference organizers, that described the address Kvasz would be giving. The text of the address itself was not provided.
- ↑ Bair et. al., p. 897
- ↑ Dauben (1985), p. 188
- ↑ Blaszczyk, Katz, & Sherry, p. 26. The authors note, for example, Edward Nelson's creation, in 1977, of an enriched ZFC set-theoretic framework "where the usual construction of the reals produces a number system containing etities that behave like infinitesimals."
- ↑ Blaszczyk, Katz, & Sherry, p. 26.
- ↑ MathOverflow "How helful is non-standard analysis?"
- ↑ Arithmetization, Tensegrity
- ↑ Sullivan, p. 371 cited in Dauben, p. 190
- ↑ Dauben, pp. 190-1
- ↑ Sullivan, pp. 383-84 cited in Dauben, p. 191
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