irrational numbers in english language textbooks, 1890-i 915: … · 2017-01-14 · berlin...

HISTORIA MATHEMATICA 19 (1992), 158-176

irrational Numbers in English Language Textbooks, 1890-I 915: Constructions and Postulates for the

Completeness of the Real Numbers

R. P. BURN

School of Education, Exeter University, Exeter, EXl 2LU, United Kingdom

A comparison is made of English-language books for students of mathematics, dealing with the real numbers and real functions, in the period 1890-1915. Publications in the United Kingdom and in the United States of America are contrasted. Irrational numbers are constructed from rational numbers in books written in the United States some fifteen years before this is done in books by British authors, published in the United Kingdom. Also, within the period we consider, American authors were offering an axiomatic presentation of the real number system while no such account was offered by a British author of the time. The different developments can be explained by the direct experience which many Ameri- can mathematicians had of German universities. Q 1992 Academic Pres. Inc.

Der Aufsatz vergleicht die englischsprachigen Bticher aus der Zeit von 1890-1915 fur Mathematikstudenten, die die reellen Zahlen und die reellen Funktionen behandeln. Er stellt die Veroffentlichungen in GroDbritannien und in den USA einander gegeniiber. Die irra- tionalen Zahlen werden aus rationalen Zahlen in den Btichem ftlnfzehn Jahre splter kon- struiert, die in den USA verfaat wurden, als in Btichern britischer Autoren, die in GroRbri- tannien veroffentlicht wurden. AuRerdem legen im betrachteten Zeitraum die amerikanischen Autoren eine axiomatische Darstellung des Systems der reellen Zahlen vor, wahrend eine solche Rechnung von den britischen Autoren nicht angeboten wird. Man kann die verschiedenen Entwicklungen durch die unmittelbare Erfahrung erklaren. die viele amerikanische Mathematiker an deutschen Universitaten erwarben. o IW? Academic PWK IIK.

HM 19 IRRATIONAL NUMBERS IN ENGLISH LANGUAGE TEXTBOOKS 159

INTRODUCTION

1. The Word “Completeness” and Its Meaning

There is an anachronism in our title, for the widespread use of the term “completeness” to describe the extension of the rational numbers to the real numbers only became conventional from its occurrence in the study of metric spaces, though it was used by Dedekind in [I8721 and in his correspondence with Lip- schitz [Dedekind 18761, and by Hilbert in his axioms for geometry [1899] (in every translation, and every edition after the first). in the period with which we are concerned, the issue would generally have been worded differently. How, it would have been asked, could the continuity of the real line be guaranteed or how could the continuum be characterized? Dedekind had used the term “completeness” in [1872] both to describe the closure of a number field under the four arithmetical operations, and as a synonym for “continuity.” We will retain the modern term “completeness” for the sake of clarity, since the term “continuity,” in modern usage, applies only to functions.

Propositions which become theorems after the rational numbers are completed by adjoining the irrationals include:

any monotonic bounded sequence is convergent; any Cauchy sequence is convergent; any infinite decimal has a limit; each Dedekind cut is made by a number; any nonempty set of numbers which is bounded above has a least upper

bound; and, for any monotonic bounded function f, f(x) tends to a limit as x tends to 00.

If any one of these is added as an axiom to the axioms for an Archimedean ordered field, the others can be proved, thus showing that each could serve as an axiom of completeness.

2. Irrational Numbers in German, French, and Italian Literature

The completeness of the real numbers was first publicly discussed by K. Weier- strass in the lectures which he gave at Berlin University in 1865. Weierstrass’s construction of irrational numbers used infinite sets of positive rationals with bounded partial sums. This construction was first published by Kossak [1872], and further expounded by Pincherle [I8803 and Biermann [1887]. (See Dugac 119731 on the authenticity of Kossak’s exposition.) Weierstrass insisted on the foundational importance of the property that an infinite bounded set has a cluster point and he was the first to claim that a continuous function on a closed interval was bounded and attains its bounds.

The students of Weierstrass, notably H. A. Schwarz, who was a student in Berlin 1859-1861, and G. Cantor, who was a student in Berlin 1863-1866, recog- nized the importance of Weierstrass’s ide.as and sought to present a more accessi- ble construction of irrational numbers. In 1872, both Cantor and Heine (to whom Schwarz had been, and to whom Cantor currently was, assistant at Halle) published constructions of irrational numbers as rational Cauchy sequences. On read-

160 R. P. BURN HM 19

ing their accounts, R. Dedekind [I8721 published his construction of irrational numbers as cuts of the set of rational numbers, which he had worked out in 1858. Dedekind proved that his construction was complete in the sense that the extension of the rationals which he had developed did not admit of any further extension by the same method. Cantor did not prove that his construction was complete, in the sense that a Cauchy sequence of Cauchy sequences was necessarily convergent: he conceived of an ongoing hierarchy of constructions of numbers.

Weierstrass’s lectures and Cantor’s and Dedekind’s publications were in Ger- man. Independently, a parallel development of irrational numbers as Cauchy sequences of rational numbers with known or fictitious limits was published in French by C. MCray [1869].

The first analysis textbook in any language with Weierstrassian rigor was written in Italian by U. Dini [1878]; it was translated into German in 1892. Dini constructed irrational numbers by Dedekind cuts.

Two further texts, which were never translated into English, but which were frequently cited as influential by English speaking authors, were by C. Jordan in Paris and 0. Stolz in Innsbruck. Jordan had only given a construction of irrational numbers in the appendix to Volume 3 of the first edition of his Cours d’analyse [1887]. It was the second and third editions which were to be so influential. In the first volume of the second edition [ 18931, Jordan gave a generalized account of Dedekind cuts as nonunique upper and lower subsets of rationals with arbitrarily close elements. Of this second edition Hardy [ 1922, 24; Collected papers p. 72211 said: “Coming from the futility of Tripos mathematics, I found myself at last in the presence of the real thing.” Hardy graduated in 1900.

The first volume of Stolz [ 18851 dealt with real numbers. Stolz motivated discussion of irrational numbers with monotonic increasing sequences c c;/e’, where e is a positive integer 22, co an arbitrary integer, and ci an integer such that 0 5 ci 5 e - 1, for each i > 0. He discussed “nonconvergent” Cauchy sequences thor- oughly. Then he defined irrational numbers as Cauchy sequences, following Can- tor. He made careful use of closed intervals for theorems on continuous functions. An enlarged second edition, written with J. A. Gmeiner, was published in 1902.

3. Irrational Numbers in English Textbooks

The definitive account of the development of the foundations of real analysis with Weierstrassian rigor was given in two substantial articles by Pringsheim [1898, 18991 which were written as part of F. Klein’s plan for an encyclopaedic documentation of the mathematical achievements of the nineteenth century. The first of these articles cites twenty-one sources, and the second a further twenty- four. Apart from Euler (who wrote in Latin) every other source cited is in Ger- man, French, or Italian. What had happened to the English-speaking mathematicians? Joan Richards [1988, 71 has written:

It is almost a truism in the history of mathematics that in the nineteenth century most interesting developments in pure mathematics took place outside of England: in France, Germany or Italy. Considering the number of Englishmen who seriously studied the subject, this seems strange.


In fact no books written in English before 1890 dealt with the matter. But by 1900 there were research papers in English on the theory of functions of a real variable written on both sides of the Atlantic, with the Americans contributing a few years before the British. By 1907, with Osgood [1907a] and W. H. and G. C. Young [1906], both American and British authors had written treatises in the area which earned international repute. Some American textbooks had offered a formal construction of irrational numbers from 1891. This did not become common practice in Britain until 1914. By this date there were publications in English on both sides of the Atlantic, J. W. Young [ 191 lb] and Jourdain [ 19121, dealing with the foundations of the number system in a manner which was intended to be read by teachers of mathematics in secondary schools. It is the purpose of this article to document the transition with comments and quotations from books published for a student readership during the intervening twenty-five years. Other sources will be quoted when such sources give insight into the conventions used in the textbooks of the time. The prehistory of our story is too complex to be discussed fully here.

There are really two stories to tell: one about crossing the Channel and one about crossing the Atlantic. One would expect the two stories to interact, because of the shared language, and because several publishers either had offices on both sides of the Atlantic or had established trans-Atlantic pairings with other publishers. But the interactions, at a textbook level, were limited. North American publications including a construction of irrational numbers date from 1891. In Britain, despite Lamb’s important excursion into the elements of real analysis [1897], the first British book to contribute significantly to our understanding of irrational numbers was Russell [1903], which was read by American scholars though not recommended for undergraduates. In the other direction, we know (from his preface [1906]) that W. H. Young was in correspondence with Veblen, but not until 1908 does a British writer on functions of a real variable (Hardy) commend an American text (Osgood [1907b]) in this field. The relationship between English-speaking mathematicians and German mathematics is the critical factor in the development and this operated quite differently in the two countries. I start with the United Kingdom and give an account of North American developments thereafter.

THE CONSTRUCTION OF IRRATIONAL NUMBERS IN MATHEMATICAL BOOKS WRITTEN IN THE UNITED KINGDOM,

UP TO 1915

4. The Acknowledgement of Continental Developments in British Books

The story begins with the publication of Chrystall18891. G. Chrystal lectured in Edinburgh, Scotland. He gave a thorough account of the convergence of infinite series including most of Gauss’ work on the hypergeometric series [I8131 and Cauchy’s Cours d’analyse [1821]. He had read Stolz 118851, but used Stolz’s historical references, not his treatment of Cauchy sequences. Chrystal gave a “proof’ that Cauchy sequences are convergent, based on Bolzano’s idea [1817] that such a sequence cannot contain subsequences converging to two distinct


limits. The second edition of this part of Chrystal’s book was published in 1900. It contained, as a supplement, a construction of irrational numbers as Dedekind cuts, but the proof of the convergence of Cauchy sequences in the first edition was allowed to stand!

In 1891 we have the sole instance in the period under consideration of the translation of a continental text on the real numbers or on functions of a real variable being made and published in the British Isles. This was the English translation of Harnack [1881] by G. L. Cathcart of Trinity College, Dublin, Ire- land. Harnack defined real numbers as fictitious limits of Cauchy sequences. Chrystal and Harnack-Cathcart are the only British books dealing with aspects of real analysis which are cited in the American texts of our period. Harnack- Cathcart is not cited by any non-American English-speaking author except Carslaw [ 19061.

The first edition of A. R. Forsyth’s Theory of Functions of a Complex Variable was published in Cambridge in 1893. Forsyth claimed to present functions of a complex variable according to Cauchy, Riemann, and Weierstrass, but said that the theory of functions of a real variable was satisfactorily treated by Dini [1878, Italian], Stolz [ 1885, German], Tannery [ 1886, French] and Chrystal! He offered no construction of the real numbers.

The first indication of an awareness of the issues surrounding the use of irrational numbers in a widely used textbook in the U.K. comes with the publication of Lamb [ 18971. Lamb had been at Cambridge and then in Adelaide, Australia for ten years before his appointment to Manchester in 1885. He read widely in German, French, and Italian. In his review of Lamb in the Mathematical Gazette, E. T. Whittaker [1898] of Cambridge said that Lamb’s was the first calculus book in English to take account of continental developments. He then chastised Lamb for considering possibly nondifferentiable functions!

In his preface Lamb wrote:

Considerable attention has been paid to the logic of the subject. Writers of text-books, however elementary, cannot remain permanently indifferent to the investigations of the modem Theory of Functions (of a real variable), although opinions may differ widely as to the character and extent of the influence which these should exert. It is not claimed that the proofs of fundamental propositions which are here offered have the formal precision of statement which is de rigeur in the theory referred to; but it is hoped that in substance they will be found to be correct.

Right at the beginning of the book Lamb investigated the convergence of an increasing bounded sequence and showed that such a sequence partitioned the points of a real line. He then claimed convergence (without a definition of Dede- kind cuts) and said that this theorem was called “the Fundamental Theorem of the Calculus” ! Lamb gave a remarkable introduction to real analysis including Bolzano’s [1817] proof of the Intermediate Value Theorem. He presumed that continuous functions on closed intervals were bounded and then proved that they attained their bounds, so Rolle’s Theorem and the Mean Value Theorem followed. He used the exponential series to define real exponents of any positive number. He also affirmed that continuous functions were integrable, but only proved that


monotonic functions were integrable for which Newton’s well-known argument [1687 Bk. 1, Lemmas 2 and 31 sufficed.

The first edition of Whittaker’s Modern Analysis was published in 1902, while he was still at Cambridge. There was no construction or definition of real numbers in the book, but Whittaker gave a sharpened version of Bolzano’s “proof’ that Cauchy sequences of real numbers converged, due to Jordan [1893], by showing that the terms of a Cauchy sequence lay progressively in a set of nested intervals with length tending to 0. Jordan was able to complete the proof because he had defined real numbers using his version of Dedekind cuts. Whittaker had not. Whittaker left Cambridge in 1906 and moved to Edinburgh in 1912. The second edition of this book was published in 1915 with the collaboration of G. N. Watson, who had studied under Whittaker and was at Cambridge until 1916. The second edition contained a construction of the real numbers with Dedekind cuts.

Some of the flavor of the times can be gathered from Hobson’s presidential address to the London Mathematical Society [1902], when he said:

Many British mathematicians, absorbed as they rightly are in the technique of their science, and in the work of applying it to the quantitative description of natural phenomena, have not yet fully appreciated the results of recent movements of mathematical thought . . . . In some of the text-books in common use in this country, the symbol 30 is still used as if it denoted a number, and one in all respects on a par with the finite numbers. The foundations of the integral calculus are treated as if Riemann had never lived and worked. The order in which double limits are taken is treated as immaterial, and in many other respects the critical results of the last century are ignored.

This can be illustrated by considering one of the better books of the period, by G. A. Gibson of Glasgow [ 1901,2nd ed. 19061. Gibson gave a visual “proof’ that monotonic bounded sequences converged and said that Rolle’s Theorem and the Mean Value Theorem were “obvious” in the context of continuous derivatives. The second edition of this book was specially commended by G. H. Hardy [I9081 and Bromwich (in the Appendix to Bromwich, 1st ed. [1908]) because of its treatment of infinite integrals and of 1’Hopital’s Rule.

5. The Discussion of Irrational Numbers and Their Construction in British Books

After leaving Cambridge in 1894, Bertrand Russell spent some of the year 1895 in Berlin where he heard of the work of Weierstrass for the first time. In 1900 he attended the International Congress of Philosophy in Paris where he met and was greatly impressed by Peano. Despite the fact that it was the first British book to exhibit a detailed understanding of the continental developments, the publication of Russell [ 19031 inhibited developments in Britain because of Russell’s criticisms of both Dedekind’s and Cantor’s work. Russell identified unstated assumptions in Dedekind’s work (relating to the denseness of the set on which cuts are constructed) and said that a Dedekind cut did not construct a point which filled a pre- existent gap in the rational number line. (If one only reads Dedekind [1872] it is not obvious that Russell’s is a valid criticism, but (unknown to Russell) Dedekind clarified his position in a letter to Weber [Dedekind 18881; in the light of this


clarification, Russell’s critique stands.) Russell also criticized Cantor because if a real number is defined to be a Cauchy sequence of rational numbers then no rational number is real unless a sequence is confused with its limit. He recommended Peano’s definition [1899 Sect. 62, 1’, 931 of real numbers as subsets of rationals which are bounded above. British mathematicians were shaken by his criticisms (see the preface to Hobson [1907]), but were unwilling to follow his recommendations. Peano’s construction was reiterated by Jourdain [I9121 (who had studied under Russell in Cambridge in 1902) and by Whitehead and Russell [1913 Sect. 310,316-3191 but not elsewhere in British books of the time. Jourdain, in fact, detected that Peano’s construction had been previously used by Pasch [1882] (see Grattan-Guinness [1977, 1291).

The year 1906 marks a new dawn in English textbooks with the publications of H. S. Carslaw, and of W. H. and G. C. Young; none of them was a full time English resident. Carslaw had studied in Glasgow and Cambridge and was appointed lecturer in Glasgow in 1896. He studied in Gottingen in 1897, and went to Sydney, Australia in 1903. After his appointment to Sydney, he periodically returned to Cambridge for sabbatical leave, and his book on Fourier series [Carslaw 19061 was published in London. In it, he gave many historical references. He acknowledged a special debt to J. Tannery 119041. His bibliography included Dedekind [English trans. 19011, Dini [German trans. 18921, Harnack [English trans. 18911, Stolz [1902], Fine [1891], and Pringsheim [1898, 18991. Carslaw constructed real numbers as Dedekind cuts. Even in the 195Os, there were those who claimed that his book contained the best English introduction to the theory of functions of a real variable [Jaeger 19561.

Grace Chisolm had studied in Cambridge from 1889 to 1893 where W. H. Young had been one of her supervisors. From 1893 to 1895 she studied under Klein in Gottingen where she was awarded her doctorate. Chisholm and Young married in 1896, and the Youngs moved from Cambridge to Gottingen in 1897 to work in association with Klein, but without an appointment. From 1900, W. H. Young returned periodically to Cambridge in term time and was a part-time lecturer in Liverpool from 1905. The Youngs’ home was in Gijttingen until 1908 when they moved to Switzerland. Their research monograph [1906] on “sets of points” contained 112 German references, 75 French, 62 Italian, in addition to 28 papers of their own, 21 British papers (of which just one, on the Riemann integral, by H. J. S. Smith of Oxford in 1876, was published before 1900), 18 American papers, 2 from Sweden, 2 from Czechoslovakia, 1 from Poland, and 1 from the Nether- lands. The absence of any account of the real numbers in British undergraduate texts at this time put them under obligation to try to provide their readers with a rigorous starting point. They defined real numbers as fictitious limits of Cauchy sequences of rationals. In the preface, Veblen (Princeton) is thanked for help with proofreading. For further information about the Youngs’ mathematical partner- ship see Grattan-Guinness [ 19721. Their relationship with German mathematicians was unique among British mathematicians of the time.

In 1907 a thorough treatment of the contruction of real numbers both by Dede- kind cuts and by equivalence classes of Cauchy sequences (the first such in


English, though still without modern terminology) was given by E. W. Hobson of Cambridge [1907, 26-351. Hobson acknowledged Russell’s criticism of the Cauchy sequence definition of real numbers (that a sequence, even if convergent, is not a number) and believed he had answered it. He also proved that a Cauchy sequence of rational Cauchy sequences was convergent. He expressed gratitude for the texts of Dini [German tr. 18921, Stolz [ 18931, and Schoenflies [ 19001. Hobson was the first fully resident Englishman to publish a construction of the real numbers. In his obituary of Hobson, Hardy 119341 says that Hobson and Young were the first English mathematicians to see the significance of the new ideas in the functions of a real variable.

In 1906, T. J. I’A. Bromwich moved from Queen’s College, Galway, Ireland to St. John’s College, Cambridge, where he published his text [1908] on infinite series. The substance of the book consisted of lectures he had given each year in Galway from 1902 in which he had set out to prove all the theorems on convergence cited in Pringsheim [ 18981 except for those on continued fractions. The final chapter on divergent series and the appendices were written in Cambridge with Hardy’s help. In Berry’s review [ 19081 of this book in the Muthematical Gazette, he suggested that it was superior to any continental text on the subject, which, true or not, must have been the first time for well over a century that this could have been thought of any book in English in this field. In the appendix, real numbers were defined as Dedekind cuts (which were said to coincide with the definition of ratio in Euclid, Book V). Bromwich seemed to echo Russell in saying, despite Hobson, that the attempt to define real numbers by rational Cauchy sequences had not been carried through (Appendix, paragraph 151, 376). It may be noted that, after Young and Hobson, no British author proposed the Cauchy sequence definition of irrational numbers for at least half a century.

6. Hardy’s Pure Mathematics

One could be forgiven for presuming that after the publications of Carslaw, Hobson, and Bromwich, the best British textbooks would offer some formal basis for irrational numbers. The year 1908 saw the publication by the Cambridge Uni- versity Press of the first edition of G. H. Hardy’s Pure Mathematics [1908]. So many British mathematicians have been brought up on the later editions of this book that the distinctive features of the first edition have been forgotten.

In the preface Hardy wrote: “I have not attempted to include any account of any purely arithmetic theory of irrational numbers, since I believe all such theories to be entirely unsuitable for elementary teaching.” He then proceded to give a lengthy account of his pedagogical concerns in writing the book. In his first chapter, Hardy matched the points of a line with the real numbers and in Chaper 4 “proved” the convergence of an increasing bounded sequence in exactly the way Lamb had done. A. N. Whitehead was named as his consultant for Chapters 1 and 4 [ 1908, Preface xl. Hardy expressed appreciation of Lamb [ 18971, Gibson [ 19061, Osgood [1907b], Bromwich [1908], and J. Tannery [1906].

Berry [1910] reviewed Hardy’s book for the Mathematical Gazette. He was appalled at the arrogance of its title, but he appreciated the fact that Hardy had


brought together, under one cover, material that could normally only be found by consulting at least four different books.

Also in the year 1908, a rather compressed mathematical and philosophical discussion of how to introduce irrational numbers was given by P. E. B. Jourdain [1908] in the Mathematical Gazette. The year 1912 saw the publication of Jour- dain’s small book aiming at a wider audience which contained a brief chapter on irrational numbers. He explained, simply and clearly, the need for some formal construction of irrational numbers to define the continuum [Jourdain 19121. After mentioning Dedekind’s method, he commended Peano’s method, following Rus- sell. In August 1912 the International Congress of Mathematicians met in Cam- bridge and heard Jourdain present a clear description of the logical errors commit- ted in evading an overt treatment of irrational numbers [Jourdain 19131. There is no direct evidence that Hardy was influenced by Jourdain, though they would both have been at the International Congress.

Hardy was elected a Fellow of the Royal Society in 1910, and his collaboration with Littlewood began in 1911. In 1914 Hardy brought out his second edition. In it he introduced an account of Dedekind cuts, Weierstrass’s theorem that bounded infinite sets have a cluster point, lim sup and lim inf, the general principle of convergence, the Heine-Bore1 Theorem, Heine’s theorem that a continuous function on a closed interval is uniformly continuous, in fact, all the basic theory that stems from the completeness of the real numbers. The change from [1908] was profound. Yet this did not represent a change in Hardy’s knowledge since we know that he had long been an admirer of Jordan [Hardy 1922,241 and he had been involved in writing the appendix to Bromwich [ 19081. What it did represent was an acknowledgment that pedagogical criteria had changed and later editions were but minor variants of the 1914 edition. In the preface to the 2nd edition, Hardy wrote: “The result of all these alterations is, no doubt, to make the book a little more difficult. I have been encouraged to make them by the feeling that it is no longer necessary to apologize for treating mathematical analysis as a serious subject worthy of study for its own sake,” None of the later editions gave axioms for the real numbers, but our modem preference for an axiomatic definition can be a way of evading any discussion of the foundations of the number system and in that respect is in sympathy with the instinctive declaration of Hardy’s first preface.

By 1910, it was possible for W. H. Young to presume, in writing a graduate text on differential calculus [1910], that undergraduates had seen the real numbers constructed, at least so he says in his preface. The changing awareness is quietly marked in the bibliographies to the articles which Mathews wrote for the 10th and 1 lth editions of the Encyclopaedia Britannica [1902] and [1911]. The second article is longer than the first because of an extensive account of number theory. But the accounts of the number system itself in the two articles, up to the affirma- tion of the completeness of the system of Dedekind cuts and its equivalence to a construction by Cauchy sequences, are identical, word for word, but for the substitution of “manifold” in [191 l] where “aggregate” appears in [1902]. The bibliography for the first article cites many of Cantor’s papers, Dedekind [ 18721, Heine [1872], Weber [2nd ed. 18981, Stolz [1885], Dini [1878] and [1892], and


Pringsheim [1898]. The relevant part of the bibliography of the second article mentions only Pringsheim [ 18981 and the Royal Society [ 19081. Public discussion of the status of irrational numbers had become orthodox. The second editions of Hardy [I9141 and of Whittaker and Watson [1915], each of which contained a construction of irrationals by Dedekind cuts, show that conventions had changed.

THE CONSTRUCTION OF IRRATIONAL NUMBERS AND THE USE OF POSTULATES OF COMPLETENESS IN MATHEMATICAL BOOKS

WRITTEN IN THE UNITED STATES OF AMERICA UP TO 1911

7. American Links with German Mathematics and the Construction of Irrational Numbers

In 1904, in his presidential address to the American Mathematical Society [1905], T. S. Fiske said that ten percent of the members of the A.M.S. had received their doctorates from German universities, and twenty per cent of the members had pursued some mathematical studies in Germany. It is impossible to exaggerate the significance of this in relation to the subject we are considering.

Almost all the American authors who wrote about irrational numbers befor 1911 had clear connections with German mathematics. H. B. Fine studied with Felix Klein in Leipzig 1884-1885, and with Kronecker in Berlin 1890-1891. W. F. Osgood studied with Klein in Gottingen 1887-1889, and in Erlangen 1889-1890 where he received his doctorate. Osgood was then appointed to Harvard. J. Pierpont received his doctorate from Vienna in 1894 and was then appointed to Yale. E. V. Huntington studied under Osgood at Harvard 1893-1897 and then obtained his doctorate from Strasbourg in 1901 and returned to Harvard. E. R. Hedrick had studied under Osgood at Harvard 1897-1899 and then took his doctorate at Gottingen. 0. Veblen did not study in Europe, but he was at Harvard under Osgood during 1899-1900 and then at Chicago. He received his doctorate there in 1903, studying under E. H. Moore (who had been in Gottingen and Berlin in 1885), 0. Bolza (doctorate, Gottingen, 1886) and H. Maschke (doctorate, Got- tingen, 1880). For these and other biographical data, see Duren et al. [1989].

In contrast to the United Kingdom, where the construction of irrational numbers was not seriously discussed in undergraduate texts before 1906, the first text in the United States to give a construction of the real numbers was by Fine [ 18911 of Princeton. The book was about numbers, not functions. Fine gave Cantor’s [1872] construction of irrational numbers as rational Cauchy sequences, and defined ax as ex,lOg a.

The first analysis book by J. Harkness of Bryn Mawr, Pennsylvania and F. Morley of Haverford, Pennsylvania was 118931. Reviewing this book for the Bulle- tin of the New York Mathematical Society, Maschke [1894] wrote:

It has been impossible to obtain a fair knowledge of this subject without consulting quite a number of different German and French treatises and original papers. There did not exist any English text book on the subject at all (I should here perhaps except some chapters of Chrystal’s Algebra, which bear on subjects belonging to the theory of functions). But there is also no Continental work which deals with the theory of functions in any degree of complete-


ness. Either Cauchy’s or Riemann’s or Weierstrass’s method is given alone, whereas a combination of all three methods is needed.

Maschke continued: “An urgent demand was met therefore . . .” with the publications of Forsyth [1893] and the present text. Both Harkness and Morley had been Cambridge Wranglers, Harkness in 1885 and Morley in 1883. Harkness then studied mathematics and law in London before his appointment to Bryn Mawr in 1888. Morley was a schoolmaster in Bath, England, until his appointment to Haverford in 1887. The Bryn Mawr college library began to subscribe to Ger- man mathematical journals in 1886 (under the influence of Charlotte Angas Scott). In [1893], Harkness and Morley cited over 200 continental European authors. They recommended particular authors at the end of each chapter. For example, they quoted Riemann, Schwarz, Stolz [1885], Pincherle [1880], and Biermann [1887], and particularly commended Dedekind [1872], Harnack [English trans. 18911, J. Tannery [1886] and Du Bois-Reymond [1882]. The authors constructed irrational numbers from Cauchy sequences of rational numbers following Heine [1872].

In 1898 the same authors published a complementary text [1898] in which they briefly indicated how irrational numbers may be constructed by Dedekind cuts. In this text they recommended Burkhardt [ 18971, Thomae [ 18801, Picard [ 1891/96], Jordan [1893-18961, Chrystal [1889], Stolz [1885], and J. Tannery [1886]. Al- though this text was also published by Macmillan, it was printed at the Cambridge University Press and proofread in Cambridge by A. Berry. A considerably im- pro.ved development of the real variable theorems in Harkness and Morley was given by J. Pierpont [1905], who constructed irrational numbers as Cauchy sequences of rational numbers.

The strong German connection explains the confident relationship with the mathematics of continental Europe and the readiness of Americans to publish English translations of outstanding texts. The translation of Dedekind [1872] by W. W. Beman of the University of Michigan was published in 1901, and, on the advice of W. F. Osgood, the English translation of Goursat [1902] by E. R. Hedrick of Missouri was published in 1904. In his preface, Hedrick wrote: “The lack of standard texts on mathematical subjects in the English language is too well known to require insistence.” Goursat claimed completeness by postulating as a “Self-evident assumption” that every monotonic bounded function has a limit. The third notable translation in this field was of Cantor’s work [1895, 18971 by Jourdain, which was published by Open Court of Chicago in 1915 at the close of the period we are considering. Open Court had already published Dedekind [En- glish trans. 19011 and Hilbert [English trans. 19021.

8. Postulates for Completeness

In 1897 Osgood published a pamphlet on infinite series [1897] which could well serve as a text today. In it, he stated as a “Fundamental Principle” that an increasing bounded sequence was convergent, and proved, in an appendix, a generalized form of the Cauchy criterion, namely that iff(x) - f(y) + 0 as x and y + ~4, thenf(x) tends to a limit as x + ~0. This appears to be the first instance of


obtaining the irrational numbers, by postulate, in an English language text. It may be noted that Osgood contributed an article [1901] (in German) on complex variables for the Encyklopiidie der mathematischen Wissenschaften.

We have already noted a readiness by Osgood and Goursat to introduce completeness by an appropriate postulate, a practice that was not to be tried in the U.K. until the 1930s (e.g. by Ferrar [1938]). In 1905, articles by E. V. Huntington [1905a] and 0. Veblen [1905] (written while he was still at Chicago) offered two different axiomatic definitions of the real numbers. Huntington used the postulate that a nonempty set of real numbers which is bounded above has a least upper bound. This postulate has been widely adopted as the standard axiom of completeness of the real numbers since the late 1950s. Veblen postulated the existence of numbers for Dedekind cuts (cf. Dedekind [1872 Sect. III]). When irrational numbers are constructed from rational numbers, Archimedean order is part of the given context. With an axiomatic development, Archimedean order must be con- sciously built into the arithmetic structure. Huntington proved Archimedean order from his least upper bound postulate, whereas Veblen stated a “Pseudo- Archimedean Postulate.” Huntington’s postulate was not incorporated into a textbook of the period. The same year Huntington wrote two expository articles [1905b] in which he used Veblen’s “Dedekind Axiom” in preference to his own least upper bounds. These two articles were reprinted the following year in book form [1906]. Through the papers of Huntington and Veblen, the axiomatic approach was available for consideration, and in varying ways was considered and used in most American publications in this field which followed. On the role of Huntington and Veblen in initiating an axiomatic approach to mathematics in North America, see Scanlan [to appear].

From the late 1890s Fine had given courses at Princeton discussing the foundations of the number system. The year in which Veblen became Fine’s colleague at Princeton, Fine [1905] was published. Fine cited Stolz and Gmeiner [1902] but gave his own treatment of the number system both with axioms and by construction, the axioms bearing an obvious relation to those of Veblen’s paper [ 19051. The rational numbers were given by axioms, with a lengthy footnote describing how they may be constructed from the Peano postulates. The irrational numbers were constructed by Dedekind cuts, with a lengthy footnote offering an Archimedean axiom and an “Axiom of Continuity” postulating the existence of a point which makes a Dedekind cut, as in Veblen’s paper. In the following year H. P. Manning [1906] of Brown (better known for his books on geometry), who had studied at Brown and then at Johns Hopkins for his Ph.D. in 1891, cited the “law of Archi- medes” for rational numbers, and postulated a point corresponding to a Dedekind cut, again as in Veblen’s axioms [I9051 and Huntington’s expository articles [ 1905b]. Manning carefully avoided mention of infinity. His bibliography consisted of Weber [1895], Dedekind [English trans. 19011, Fine [1891, 19051, Pierpont [ 19051, Huntington [ 1905b], and Veblen and Lennes [ 19071. Manning had attended meetings of the American Mathematical Society in 1902, 1903, and 1904 at which Huntington had read papers expounding a postulational approach to the real numbers.


The first American book in this field to earn an international reputation was Osgood [1907a], which he wrote in German. After stating the theorems about continuous functions for which completeness was needed, Osgood proved that an increasing bounded sequence converged by constructing its limit as an infinite decimal. Osgood’s calculus book [1907b] in English was commended by Hedrick and Hardy. After a visual discussion of monotonic bounded sequences, he stated their convergence as a “General Principle,” as in his earlier pamphlet on infinite series [1897].

A remarkably modern treatment was offered in Veblen and Lennes 119071. In their preface they stated:

The general aim has been to obtain rigor of logic with a minimum of elaborate machinery. . When this work was undertaken there was no convenient source in English containing a

rigorous and systematic treatment of the body of theorems usually included in even an elementary course on real functions, and it was necessary to refer to the French and German treatises. Since then one treatise, at least [they mean Pierpont [1905]], has appeared in English on the theory of functions of real variables.

They did not work with the axioms of Veblen’s paper, but in a presentation which combined Peano’s construction of the real numbers [1899] with Huntington’s postulate [1905a], they adopted the axiom that every set of rational numbers bounded above had a least upper bound, and deduced the least upper bound property for real numbers. They commended the discussions of irrational numbers in Manning [1906], Fine [1905], Dedekind [English trans. 19011, and Pierpont [1905]. They credited Veronese [I8911 with the recognition that infinitesimals were excluded by an Archimedean axiom. The authors used notation which distin- guished carefully between open, closed, and half-open intervals, and made per- sistent use of the Heine-Bore1 theorem.

In reviewing Veblen and Lennes, Jourdain wrote that this book had “. . . almost the first acknowledgement in a textbook that a real number is to be defined at all” [Jourdain 1907, 88-901. From an American reviewer such a comment would be indefensible; but from an English author, especially one as expert in the field as Jourdain, there could be no sharper underlining of the thesis of this paper.

In 1909, J. W. Young, who had collaborated with Veblen in writing a geometry book, gave a series of lectures at the University of Illinois which were published as [1911a] and reprinted many times. These lectures incorporated Hilbert’s axioms for geometry, including completeness, and an account of Dedekind’s work. Young says: “Dedekind’s method of treating irrational numbers consisted simply in defining every cut in the system of rational numbers to be a number.” [Young, J. W. 1911a, 103-41.

In the same year, Young [I91 lb] edited a collection of monographs on matters relating to the foundations of mathematics intended for those teaching mathematics in secondary schools. One of the contributors was Huntington, who offered postulates for the real numbers including: every nonempty set of real numbers which is bounded above has a least upper bound. This was a reversion to his [1905a] which, in this context, he called “Dedekind’s postulate”!


CONCLUDING REMARKS

9. Chronological Concordance of Books Published in Europe, the United Kingdom, and the United States of America

In Table I I list the authors mentioned in this paper comparing their dates of publication with those of the authors they cite from continental Europe. Entries which are neither treatises nor textbooks are bracketed. A number following an author’s name indicates that more than one of his publications appears in the list. The numbering follows the sequence in the bibliography. Hyphenated authors indicate translations.

10. Conclusion

The strong connections which American mathematicians had with German mathematics brought the major mathematical developments which had taken place in continental Europe during the nineteenth century to universities in the United States before they found a comparable place in British universities. Some of the non-British writers (represented in Table I by Osgood, Goursat, Huntington, Veblen, Manning, and Fine 2) exhibit a readiness to cut the Gordian knot and obtain completeness by postulate, and it is this convention which has become dominant since the late 1950s. This approach was not used by British writers until the 1930s.

Most English-speaking writers of the period who constructed the real numbers from the rationals either followed Cantor with rational Cauchy sequences, or followed Dedekind. Other constructions of the real numbers were devised, two by Jordan [ 1887, 18931, another by Pasch [ 18821 and Peano [ 18991 from upper bounds of sets of rationals, but neither these constructions nor Weierstrass’s construction were repeated in English texts of the period, despite Russell’s commendation of Peano. The British contribution in these matters consists of Russell’s criticisms and Hobson’s equivalence classes of Cauchy sequences. The American contribution was the development of the axiomatic approach. In the 1930s the construction of the reals from infinite decimals was proposed, but it must be said that from the late 1930s the preference for an axiomatic treatment has grown, and in recent times only the constructivists have held out for a Cantor-like approach.

The books listed here provide the means of exploring developments parallel to the foundational matters on which we have focussed. Other matters relating to the arithmetization of analysis which immediately come to mind are the use of infinity as a possible value for a function (which was standard practice in the nineteenth century), the significance of closed intervals, the extension of the variety of fami- lies of functions whose properties are considered worthy of serious attention, upper and lower limits, and the definitions of exponential and trigonometric functions. The Axiom of Choice, however, is not mentioned in any of the textbooks of real analysis that we have considered.


TABLE I

Continental Europe U.S.A. U.K. -

1878 1880 1881 1882

1885 1886 1887

Dini Thomae Hamack Jordan 1st ed. Vol. 1 Du Bois Reymond Stolz 1 1st ed. Vol. 1 Tannery 1 Jordan 1st ed. Vol. 3 with appendix Biennann

1889 1891 1892 1893

1895 1897 1898

1899 1900 1901 1902

Veronese Dini-Liiroth Jordan 2nd ed. vol. 1 St012 2 Weber Burkhardt Pringsheim Weber 2nd ed. Peano Schoenflies

Goursat Stolz 1 and Gmeiner 2nd ed

1903 1904 1905

Tannery 1 2nd ed.

1906 Tannery 2

1907 Osgood 2

1908

1909 de la ValMe Poussin 2nd ed. Jordan 3rd ed. vol. 1

1910 1911 1912 1913

Stolz 1 and Gmeiner 3rd ed. Young J. I, 2

1914 1915

Fine 1

Harkness and Morley I

Osgood 1 Harkness and Morley 2

Dedekind-Beman

Goursat-Hedrick Pierpont (Huntington) 1, 2 (Veblen) Fine 2 Huntington 3=2 Manning

Osgood 3 Veblen and Lennes

Cantor-Jourdain

Chrystal 1st ed. Hamack-Cathcart

Forsyth

Lamb 1st ed.

Chrystal2nd ed.

Whittaker 1st ed. Lamb 2nd ed. Russell

Carslaw 1st ed. Gibson 2nd ed. Young W. + G. Hobson 1st ed.

Bromwich Hardy 1st ed.

Young W.

Jourdain (Inter. Congress) WhiteheadlRussell v. 3 Hardy 2nd ed. Whittaker 2nd ed.


ACKNOWLEDGMENTS

I am grateful to Caroline Rittenhouse, archivist of Bryn Mawr College, for information about J. Harkness and acquisitions by the Bryn Mawr library before 1893; to Karen Parshall of the University of Virginia for information about meetings of the American Mathematical Society during the first decade of this century; to Frank Smithies for directing me toward Peano’s Formulaire and for reminis- cences of G. H. Hardy; and to Ivor Grattan-Guinness for extensive and detailed advice which has enhanced almost every paragraph of this paper.

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