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ernardus Placidus Johann Nepomuk Bolzano was born in Prague, Bohemia (now part of the Czech Republic), on 5 O c t o b e r 1781. His h o u s e o f birth d o e s n o t exist any more, b u t it w a s on the site o f the p r e s e n t Maria Square (Marihnsk~ nfim~st0 in the Old Town. His mother, Caecilia Maurer, w a s a d a u g h t e r o f a h a r d w a r e t r a d e s m a n in Prague; at the age of twenty-two she m a r r i e d the e l d e r Bernard Bolzano, a n Italian i m m i g r a n t who e a r n e d a m o d est living as an art dealer. Both p a r e n t s w e r e p i o u s Christians. Bernardus was the fourth of twelve children, t e n of w h o m died b e f o r e reaching adulthood. He grew up with a high m o r a l c o d e and a belief in holding to his principles. It w a s this b a c k g r o u n d that a t t r a c t e d him to the c h u r c h and the p r i e s t l y life. F r o m 1791 to 1796 he w a s a pupil in the Piarist Gymnasium, a n d in 1796 he e n t e r e d the philos o p h i c a l faculty at Charles University ( e s t a b l i s h e d b y Charles IV [ 1315-1378], Holy R o m a n E m p e r o r and B o h e m i a n King in 1348), w h e r e he f o l l o w e d c o u r s e s in philosophy, physics, a n d m a t h e m a t i c s . Bolzano's i n t e r e s t in m a t h e m a t i c s was stimulated b y reading A. G. Kfistner's Anfangsgri~nde der Mathematik, mainly b e c a u s e Kfistner t o o k c a r e to prove s t a t e m e n t s that w e r e c o m m o n l y u n d e r s t o o d as e v i d e n t in o r d e r to m a k e clear the a s s u m p t i o n s on w h i c h t h e y d e p e n d e d [12, p. 273]. After having finished his studies in phil o s o p h y in 1800, Bolzano e n t e r e d the theological faculty and was o r d a i n e d a Catholic p r i e s t in 1804 [4]. In 1805 E m p e r o r Franz I o f the Austro-Hungarian Empire, of which B o h e m i a w a s then a part, d e c i d e d that a chair in the philosophy o f religion would be established at each university. The r e a s o n s for this were mainly political. The empire was comprised of m a n y different ethnic groups that w e r e p r o n e to nationalistic movements for independence. The e m p e r o r feared the fruits of Enlightenment e m b o d i e d in the F r e n c h Revolution. The authorities con-
THE MATHEMATICAL INTELUGENCER 9 200t SPRINGER-VERLAG NEW YORK
I
sidered the Catholic Church to be conservative and h o p e d it would control the liberal thinldng of the time in Bohemia. Bolzano was called to the new chair at Charles University in 1805 [12, p. 273]. Bolzano, t h o u g h a priest, spiritually belonged to the Enlightenment. He w a s a "free thinker"; his a p p o i n t m e n t w a s received in Vienna with suspicion a n d was not a p p r o v e d until 1807. F o r the next 14 y e a r s Bolzano taught at the university, lecturing mainly on ethics, social questions, a n d the links b e t w e e n m a t h e m a t i c s a n d p h i l o s o p h y [4]. In 1815 he b e c a m e a m e m b e r of the K6niglichen B 6 h m i s c h e n Gesellschaft der Wissenschaften and, in 1818, Dean of the p h i l o s o p h i c a l faculty. However, the Austro-Hungarian authorities bec a m e d i s p l e a s e d with his liberal views. On 24 D e c e m b e r 1819, he was susp e n d e d from his professorship, forbidden to publish, a n d p u t u n d e r p o l i c e supervision. Bolzano refused to b a c k down, and in 1825 t h e action c a m e to an end through t h e intervention o f the influential nationalist l e a d e r J o s e f Dobrovsk~ (1753-1829). The latter h a d b e e n e d u c a t e d for the R o m a n Catholic p r i e s t h o o d and d e v o t e d himself to scholarship after t h e 1773 dissolution of the Jesuit Order. He was an important Enlightenment figure, and his textual criticism of the Bible led him to study Old Church Slavonic and subsequently the Slavic languages as a group. F r o m 1823 on, Bolzano s p e n t summers on the estate o f his friend J. Hoffmann, n e a r the village of T~chobuz in Southern Bohemia. He lived there permanently from 1831 until the death o f Mrs. Hoffmann in 1842. He then ret u r n e d to Prague w h e r e he c o n t i n u e d his m a t h e m a t i c a l a n d philosophical studies until his d e a t h on 18 December, 1848. A small pension, and the generosity of Count Leo Thun-Hohenstein (1811-1888, the B o h e m i a n a r i s t o c r a t and later Austrian s t a t e s m a n ) relieved him of all m o n e t a r y c a r e [7]. In Prague, Bolzano and his b r o t h e r s t a y e d in the h o u s e that had b e l o n g e d
Figure 1. The house at 25 Celetnd Street, showing the Bolzano plaque centered over the keystone arch.
to their p a r e n t s at 25 Celetmi Street (Celetn~ ulice 25) n e a r Old T o w n Square (StaromSstsk~ n ~ n ~ s t 0 . The p h o t o g r a p h s (Figs. 1, 2, t a k e n by the author) are of the h o u s e and its Bolzano plaque. This h o u s e is o w n e d by the City o f Prague, and all the s p a c e in the h o u s e is filled with apartments. A r o u n d the turn o f the n i n e t e e n t h century, E u r o p e a n m a t h e m a t i c i a n s w e r e mainly c o n c e r n e d with the status of Euclid's parallel p o s t u l a t e a n d with the p r o b l e m o f providing a solid foundation for m a t h e m a t i c a l analysis. Bolzano t r i e d his h a n d at b o t h p r o b lems. In 1804 Bolzano p u b l i s h e d a t h e o r y of parallel lines, which a n t i c i p a t e d Adrien Legendre's well-known theory. It is c o m m o n l y a c c e p t e d that Bolzano in his m a n u s c r i p t Anti-Euklid, w a s the first to state the t h e o r e m ( n o w k n o w n as the "Jordan Curve Theorem") that a simple c l o s e d curve divides the p l a n e into t w o p a r t s [12, p. 274]. The introduction of infmitesimals b y Isaac Newton and Gottfried Leibniz in the seventeenth century had m e t with violent resistance from philosophers and mathematicians. To o v e r c o m e the difficulties p r e s e n t e d by infmitesimals, Joseph-Louis Lagrange p r o p o s e d to b a s e analysis on the existence o f B r o o k Taylor's e x p a n s i o n for functions, while J e a n d ' A l e m b e r t p r o p o s e d to found differential calculus on the notion o f limit.
Among the first to d o u b t the rigor of Lagrange's e x p o s i t i o n of the calculus were Abel Btirja (1752-1816) of Berlin, the Poles J. M. Ho~n6-Wron3(si (17761853) and J. B. Sniadecki (1756-1830), and Bolzano [3, p. 258], who d e v o t e d his m a n u s c r i p t R e i n analytischer Bewe/s (1817) to a p r o o f of the "Bolzano I n t e r m e d i a t e Value Theorem." Bolzano argues that a s o u n d p r o o f of this theorem requires a s o u n d definition of continuity. His definition is the first that does not involve infinitesimals. The definition as it w a s f o r m u l a t e d in
Volume I of his m a n u s c r i p t Functionenlehre reads: If F ( x + hx) - F(x) in a b s o l u t e value b e c o m e s less t h a n an arb i t r a r y given fraction 1/N, if o n e takes Ax small enough, and r e m a i n s so the s m a l l e r one t a k e s Ax, the function F(x) is said to be c o n t i n u o u s in x [12, p. 275]. Bolzano's definition of continuity was r e p l a c e d in 1821 b y Augustin-Louis C a u c h y ' s elegant a n d generally acc e p t e d definition. Also in his p r o o f of the "Intermediate Value Theorem," Bolzano uses a l e m m a that later proved to be a cornerstone of the theory of real n u m b e r s - - h e introduced the concept of the infimum of a n o n e m p t y set. His 1817 manuscript also contains the theorem that is known as "Cauchy's Criterion for Convergence of Sequences." The proofs given by Bolzano were incomplete, but he was a w a r e of the difficulties involved. A fairly c o m p l e t e t h e o r y of real functions is c o n t a i n e d in Bolzano's Functionenlehre, including m a n y of the fundamental results that w e r e rediscovered in the s e c o n d half of the nineteenth century through the work of Karl Weierstrass (1815-1897) and others. Bolzano proved in this manuscript that a function that is u n b o u n d e d on a closed interval [a, b] cannot be continuous on [a, b] [12, p. 275]. In proving this, Bolzano used the so-called BolzanoWeierstrass Theorem, that a b o u n d e d infinite set has a cluster point. This theo-
Figure 2. Close-up of the Bolzano plaque on the facade of the house at 25 Celetna Street, Prague.
VOLUME 23, NUMBER 2, 2001
53
rem rests on the "Bolzano-Weierstrass Selection Principle," a method frequently employed in mathematical analysis, which consists of successive subdivision of a segment into halves, one of which is selected as the new initial segment [6, pp. 419-420]. The most remarkable result of the Functionenlehre is the construction of the "Bolzano function." Bolzano constructs a function which is continuous, but nowhere differentiable, on the interval [0, 1]. This example preceded by some forty years that of Weierstrass. In August 1830, Charles X of France was forced to emigrate by the Revolution of 1830; in the autumn of 1832, he left his refuge in Scotland and took his family to Prague, where Emperor Franz I placed part of the Hradschin Palace (HradSany) at his disposal. In September 1830, Cauchy left France and went into voluntary exile, losing all his public positions in the process [1, p. 147]. Cauchy first went to Fribourg, Switzerland, then was appointed as professor in sublime physics (that is, mathematical physics) in January 1832 at the University of Turin, Italy. In the summer of 1833 Cauchy was invited by Charles X to help with the education of his grandson, the Duke of Bordeaux, in Prague. While in Prague, Cauchy seems to have had only tenuous relations with the Prague scientific community. It is a matter of importance to inquire whether he knew Bolzano during his years (1833-1836) in Bohemia. The question remained unresolved for years. To the Struiks [11] it seemed rather improbable that there existed any interaction between Bolzano and Cauchy, even though both were members of the Royal Bohemian Society. Cauchy was a famous French acaddmicien whose new method was already studied and followed in all parts of Europe, but he was also associated with a banished court, which maintained the severest reserve in a hospitable but foreign country. Bolzano, for his part, had been removed from his professorship since 1819, and he lived secluded from society and public notice in T~chobuz. Cauchy would have risked offending the imperial authorities of Austria if he communicated with the compromised Bolzano. An autobi-
54
THE MATHEMATICAL INTELLIGENCER
ography of Bolzano published by his pupil J. M. Fesl in 1836 (see [11, p. 365]) mentions nothing about any influence of Cauchy on Bolzano. Cauchy had already completed long before, as had Bolzano, his works on the foundations of the theory of real functions. He had published his investigations in 1821 and 1823. By 1834 he was chiefly occupied with theoretical physics, such as his investigations on the dispersion of light. His works do not contain any reference to Bolzano, not even to the latter's earlier definition of continuity. Bolzano, on the other hand, did not publish any pure mathematics after 1817, and was, about 1835, probably occupied by philosophical questions concerning theology, or problems in mechanics. His Versuch einer objektiven Begri~ndung der Lehre von der Zusammensetzung der Krdfte was not published until 1842, and neither that paper nor another paper on the aberration of light suggests the possibility of a connection between him and Cauchy. The Struiks concluded in [11] that only a few letters of either Cauchy, or Bolzano had been published to that date, and that the contents of unpublished letters might clarify the relation between Cauchy and Bolzano. That is exactly what happened. In a letter to F. P~ihonsk~, dated 24 April 1833, at T~chobuz, Bolzano writes about his esteem for Cauchy, stating that he would like to meet Cauchy in September of that year, hopefully accompanied by P~ihonsl~. This letter was published in 1936 by E. Winters; see [8, p. 164] for a passage from it. It was found in 1962 by I. Seidlerov~ that there is a letter, dated 18 December 1843, from Bolzano to his student Fesl, in which he mentions several meetings with Cauchy in Prague, see [8, p. 164]. This was also confirmed by Winter in 1965, see [9, p. 99]. It is also mentioned in [1, p. 172] that around 1834 a meeting between Cauchy and Bolzano took place. That meeting appears to have been sought by Bolzano, who had sent Cauchy a tract on the problem of the rectification of curves (ideas developed by Bolzano in 1817 in Die drey Probleme der Rektifikation, Komplanation und Ku-
b i e r u n g , . . . ) , which he had written in French for Cauchy's benefit [9, p. 99]. I. Grattan-Guinness assumed that Cauchy had plagiarized Bolzano's definition of continuity, but H. Freudenthal and H. Sinaceur clarified the differences in approach of the two mathematicians [1, p. 255], [9, p. 99]. Boyer [2, p. 564] is of the opinion that the similarity in Bolzano's and Cauchy's arithmetization of the calculus, of their definitions of limit, derivative, continuity, and convergence were only coincidental. L. E. J. Brouwer in a 1923 paper giving examples of theorems whose proofs require the law of the excluded middle for infinite sets, mentioned in particular the Bolzano-Weierstrass theorem and the result on the existence of a maximum of a continuous function on a closed interval; see [5, p. 238]. Charles X and his court left the Hradschin Palace and the city of Prague in May 1836 for Toeplitz, to make way for the new Emperor Ferdinand to come to Prague to receive his investiture as King of Bohemia. Charles X died in GSritz on 6 November 1836. The Duke of Bordeaux reached his eighteenth birthday in September 1838, and that ended Cauchy's duties with the exiled court. In October 1838, Cauchy and his family returned to Paris, where a new period in his life began. Like most of Bolzano's mathematical work, Functionenlehre remained in manuscript form and was published for the first time only in 1962. As a result, this bold enterprise failed to exercise any influence on the development of mathematics; Bolzano was "a voice crying in the wilderness" [2, p. 565], and many of his results had to be rediscovered in the second half of the nineteenth century. H. A. Schwarz in 1872 [3, pp. 367, 368] looked upon Bolzano as the inventor of a line of reasoning developed by K. Weierstrass. Bolzano is buried in Olsany Cemetery (Olsansk~ h~bitovy), Cemetery III, Part 9, Grave 107, in Prague. There is also a Bolzano Street in Prague, some 100 meters north of the Main Railway Station. The Bolzano stamp displayed in Figure 3 was issued by Czechoslovakia in
Karel Segeth, Director of the Mathematical Institute of the Academy of Science in Prague, and the assistance received from Professor Marcel Wild, Mathematics Department, University of Stellenbosch.
[7] Leimkuhler, M. Bernard Bolzano, The Catholic Encyclopedia.
Volume II. Robert
Appleton Company, 1907. Transcribed by Thomas J. Bress. Retrieved June 3, 2000 from the World Wide Web: http://www. newadvent,org/cathen/02643c,htm [8] Rychlik, K. Sur les contacts personnels de Cauchy et de Bolzano, Revue d’Hi.sfoire
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[I 0] Stanley Gibbons Simplified Cafalogue, Stamps of the World.
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Figure 3. The 1981 Czechoslovakian stamp
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Private Bag X 1
of Mafhe-
ACKNOWLEDGMENTS
mafics.
The author gratefully acknowledges the information supplied by Professor
Hazewinkel (Ed.). Reidel, Kluwer Aca-
7602 South Africa
demic Publishers, Dordrecht, 1988.
e - m a i l :
[email protected]
Volume 1,
pp.
419-420.
M.
Matieland
