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Arithmetization of analysis

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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:

  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 the history of these efforts. Judith Grabiner identified the following as the central question in the history of any such mathematical development or period:[2]

How did the present come to be?

In seeking to answer this question, the mathematician looks at the mathematics of the past and seeks to understand how it has led to the mathematics of the present. Grabiner found that mathematicians typically ask these questions about the mathematics of the past:

  • When was this concept first defined?
  • What problems led to its definition?
  • Who first proved this theorem?
  • How was it done?
  • Is the proof correct by modern standards?

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

Non-mathematical issues

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

As a first example, consider the work of Bolzano. Only two of his papers dealing with the foundations of analysis were published during his lifetime. Both of these papers remained virtually unknown until after his death. A third work of his, based on a manuscript that dates from 1831-34, but that remained undiscovered until after WWI, was finally published in 1930. This work contains some results fundamental to the foundations of analysis that were re-discovered in the 19th century by others decades after Bolzano completed his manuscript. "We may ask how much Bolzano's work could have changed the way analysis followed, had it been published at the time."[3]

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

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

His essay was ignored by other English mathematicians and had no apparent influence on the work of German mathematicians who completed the process of arithmetization later in the century.[4] Even so, and years before his work in 1837, Hamilton wrote the following:[5]

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.

Early steps towards arithmetization

Martin Ohm

In 1822, Martin Ohm 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."[6]

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"

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."[7]

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.[8]

The arithmetization program

Two pillars of mathematics

The state of mathematics prior to 19th century efforts at arithmetization has been described by modern authors in various ways:

  • analysis rested more or less comfortably on two pilars: the discrete side on arithmetic, the continuous side on geometry.[9]
  • the source domain of analysis was geometry; that of number theory was arithmetic.[10]

"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."[11]

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).[12]

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:[13]

  • 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 of analysis 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.[14]
  • using analysis to prove the fundamental theorem of number theory raised a problem about the boundary between number theory and analysis.[15]

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:[16]

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:[17]

  • 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 and he undertook to do this in his paper of 1817.

Bolzano's proof

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:[18]

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:[19]

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) condtition for the (pointwise) convergence of an infinite series, known today as the Cauchy condition (on occasion the Bolzano-Cauchy condition), as follows:[20]
If a series 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 approach, and to which they can come as close as desired if the series is continued far enough. :As noted previously, Bolzano demonstrated only the ''plausibility'' of the criterion, 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-00000014-QINU`"' * Next, Bolzano used the [[Cauchy test|Cauchy condition]] in a proof of the following theorem:'"`UNIQ--ref-00000015-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 greatest lower bound theorem. The number $U$ is in fact the greatest lower bound of those numbers which do not possess the property $M$.'"`UNIQ--ref-00000016-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-00000017-QINU`"' ::Every bounded infinite set has an accumulation point. :A complete proof of the greatest lower 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-00000018-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-00000019-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-0000001A-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-8--QINU`"'Cauchy's proof==== Quite independently of Bolzano, Cauchy formulated the intermediate value theorem in 1821 in the following, simpler form:'"`UNIQ--ref-0000001B-QINU`"' :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 $f(x) = c$. Indeed, some authors identify the theorem as Cauchy's (intermediate-value) theorem. ==='"`UNIQ--h-9--QINU`"'The condition for continuity of a function=== 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-0000001C-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-0000001D-QINU`"' Bolzano and Cauchy gave similar defnitions of limits, derivatives, continuity, and convergence. They were contemporaries, "both chronologically and mathematically."'"`UNIQ--ref-0000001E-QINU`"' In 1821, Cauchy added to Bolzano's definition of continuity at a point "the final touch of precision":'"`UNIQ--ref-0000001F-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-00000020-QINU`"' Weierstrass, working very long after both Bozano and Cauchy, formulated "the precise $(\varepsilon,\delta)$ definition of continuity at a point."'"`UNIQ--ref-00000021-QINU`"' ==='"`UNIQ--h-10--QINU`"'Theory of irrational numbers=== "The first modern construction of the irrational numbers" was offered by Hamilton in two separate papers, which were later published as one in 1837. Somewhat later, he began work on a theory of separations of the numbers, similar to Dedekind’s theory of cuts, but he never completed his work on this topic.'"`UNIQ--ref-00000022-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-00000023-QINU`"' :the limits of such sequences, if irrational, do not logically exist until the irrational numbers themselves have been defined It was not until 1869 that Charles Méray published "the earliest coherent and rigorous theory of irrational numbers."'"`UNIQ--ref-00000024-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-00000025-QINU`"' In 1872, Cantor published his own theory of irrational numbers, defining them in terms of Cauchy sequences of rational numbers. It was his accomplishment that became known and influenced the work of others, especially Dedekind, and that consequently became celebrated as a significant step in the arithmetization of analysis.'"`UNIQ--ref-00000026-QINU`"' Subsequently, theories of irrationals were published by Dedekind (who mentioned Cantor's achievement) and by students of Weierstrass, who worked from notes taken at his lectures. Common to all three of these very different definitions of irrational numbers was "a well-defined collection of rational numbers."'"`UNIQ--ref-00000027-QINU`"' The differences among the three theories sprang from the quite different motivations of their authors:'"`UNIQ--ref-00000028-QINU`"' * Dedekind established a rigorous foundation for differential calculus * Cantor was concerned with developing a uniqueness theorem for the representation of a function by trigonometric series * Weierstrass saw the formulation of the real number system as essential to the development of his theory of analytic functions ==='"`UNIQ--h-11--QINU`"'Continuous nowhere differentiable functions=== "The discovery of continuous nowhere differentiable functions shocked the mathematical community. It also accentuated the need for analytic rigour in mathematics."'"`UNIQ--ref-00000029-QINU`"' =='"`UNIQ--h-12--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-0000002A-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-0000002B-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-0000002C-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:[46]

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

Writing almost 70 years after Pierpont's address, Carl Boyer gave these as motivations for the arithmetization program:[47]

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

Boyer credits Hermann Hankel with the foresight that "the condition for erecting a universal arithmetic is therefore a purely intellectual mathematics, one detached from all perceptions."[48] In other words, what was needed was a mathematics that views real numbers as "intellectual structures" rather than as "intuitively given magnitudes inherited from Euclid's geometry."[49]

Notes

  1. ↑ Arithmetization
  2. ↑ Grabiner, (1975) p. 439
  3. ↑ Jarník et. al.
  4. ↑ Hamilton cited in Mathews, Introduction
  5. ↑ Graves, p. 304, 1828 letter from W. R. Hamilton to John T. Graves
  6. ↑ Zerner cited in O'Connor and Robertson
  7. ↑ Ohm 1843 cited in O'Connor and Robertson
  8. ↑ O'Connor and Robertson, Ohm
  9. ↑ Stillwell
  10. ↑ Ueno p. 73
  11. ↑ Hatcher
  12. ↑ "Fundamental Theorem of Algebra," Wikipedia
  13. ↑ "Fundamental Theorem of Algebra," Wikipedia
  14. ↑ Stillman
  15. ↑ Ueno p. 72
  16. ↑ Jarnik
  17. ↑ Jarnik p. 36
  18. ↑ Russ p. 160
  19. ↑ Russ p. 162
  20. ↑ Russ p. 171
  21. ↑ Jarnik p. 36
  22. ↑ Russ p. 174
  23. ↑ Jarnik p. 36
  24. ↑ Russ p.157
  25. ↑ Russ p. 177
  26. ↑ Grabiner (1981) p. 11
  27. ↑ Russ p. 181
  28. ↑ Cauchy theorem
  29. ↑ Stillman
  30. ↑ Jarník et. al., p. 38
  31. ↑ Grabiner (1981) cited in Pinkus, p. 3
  32. ↑ Stillman
  33. ↑ Jarník et. al., p. 38
  34. ↑ Pinkus, p. 2
  35. ↑ Tweddle, p. 4
  36. ↑ Tweddle, p. 4
  37. ↑ O'Connor and Robertson, Méray
  38. ↑ Robinson
  39. ↑ Tweddle, p. 5
  40. ↑ Tweddle, p. 6
  41. ↑ Bottazzini cited in Tweddle, p. 6
  42. ↑ Pinkus p. 4
  43. ↑ Jarník et. al.
  44. ↑ Pierpont, p. 394
  45. ↑ Pierpont, p. 395
  46. ↑ Pierpont, p. 406
  47. ↑ Boyer, p. 604
  48. ↑ Hankel, cited in Boyer
  49. ↑ Boyer, p. 605

Primary sources

  • 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.
  • Hamilton, W. R. (1837), Algebra as the Science of Pure Time.
  • Hankel, Hermann (1867). Theorie der komplexen Zahlen-systeme.
  • Kronecker, Leopold (1901), Vorlesungen Uber Zahlentheorie ("Lecture Notes on Number Theory"), Leipzig, Druck und Verlag von B.G.Teubner.
  • 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

How to Cite This Entry:
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