# Bernard Bolzano's manuscripts

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*Grössenlehre*(Theory of quantity) which was intended to be an introduction to mathematics covering many different areas of mathematics such as numbers, elementary geometry, geometry in general, function theory, methodology, and the ideas of quantity and space. Specific parts such as

*Functionenlehre*(Theory of functions), and

*Zahlenlehre*(Theory of Numbers) were written, much was in a less complete form with workings and reworkings of parts of his ambitious project. Towards the end of his life Bolzano realised that he would never complete the work, and he tried to find someone who would be able to understand what he had written and also be able to continue developing parts which were unfinished. He chose Robert Zimmermann who was 24 years old at the time of Bolzano's death.

Zimmermann received the collection of Bolzano's manuscripts, having been left them in Bolzano's will. Had his interests been more inclined towards mathematics, it is indeed possible that he would have found such gems within the manuscripts that he would have devoted much of his life to them. However, Zimmermann's interests were more in the area of philosophy and indeed he was appointed to the chair of philosophy in Prague in 1852, only four years after Bolzano's death. One of Bolzano's collaborators, Franz Prihonsky, had ensured that not all of Bolzano's mathematical work lay unpublished in Zimmermann's possession, for he published Bolzano's *Paradoxien des Unendlichen,* a study of paradoxes of the infinite, in 1851. It may be considered as the first book on set theory, and certainly Cantor praises the work highly in his own, much more developed, contributions to the topic. In 1909 Charles Peirce wrote that Bolzano's *Paradoxien des Unendlichen* conferred:-

Except for this noteworthy publication, after Bolzano's death the remaining manuscripts were in Zimmermann's possession until 1882 when he gifted them to the Austrian Academy of Sciences (Kaiserliche Akademie der Wissenschaften in Wien). Ten years later the Austrian Academy of Sciences gifted the Bolzano manuscripts to the manuscript section of the Austrian National Library. That no complete works of Bolzano had been published during this period was not due to lack of interest. Several attempts had been made but the size of the task, and the difficulty of tracing various improvements that Bolzano had continued to make, meant that every attempt had failed.... a singular benefit upon humanity.

Martin Jasek was a secondary school teacher of mathematics in Pilsen. He worked with the Bolzano manuscripts in the National Library in Vienna, and became fascinated with the discoveries he made. He published four papers published between 1920 and 1923 and gave three lectures to the Union of Czech Mathematicians and Physicists on 3 December 1921, 14 January 1922, and 4 December 1922. Jasek had discovered that the unpublished *Functionenlehre* contained some important results in analysis showing that Bolzano had made certain discoveries in that topic well before similar results had been discovered by others. Jasek's lectures interested other Czech mathematicians, particularly Karel Petr and Karel Rychlik, and on 5 March 1924 the Czech Academy of Sciences set up the Bolzano Committee to:-

The chairman of the committee was Karel Petr, the secretary was Martin Jasek, while Karel Rychlik was one of eight other members. Some manuscripts were in Prague but the main collection was in the Austrian National Library in Vienna. The Czech Academy of Sciences paid for Jasek to spend about eighteen months working with the manuscripts in Vienna where he made photocopies (in an early form which produced white writing on a black background), bringing the copies back to Prague where the main task of preparing the work for publication was to take place. The Committee set itself the task of publishing the first volume of Bolzano's manuscripts, which they intended to be... acquire, unify, and publish Bolzano's manuscripts.

*Functionenlehre,*in 1925 and then the remaining material over the following five years.

The task which the Committee set itself proved far more difficult than they had first anticipated. Financial support for the project was initially quite strong but, as publication was delayed, it became harder to persuade sponsors to put up further finance. *Functionenlehre* appeared in 1930, five years behind schedule and at the time when the Committee had expected the whole project to be completed. Progress for a while was at least steady with *Zahlentheorie* being published in 1931, and *Von dem besten Staate* in 1932. There was a delay until 1935 before the next volume *Der Briefwechsel B Bolzano's mit F Exner* was published. This contained letters between Bolzano and the philosopher Franz Exner (1802-1853). The fifth and final publication by the Committee was *Memoires géométriques* which did not appear until 1948. However, in 1948, Communists took control of the country and in 1952 the Czech Academy of Sciences was disbanded and at the same time the Bolzano Committee was also disbanded. The Czechoslovak Academy of Sciences was founded in 1952 but the Bolzano Committee was not at this stage re-established under the new academy. This did happen in 1958 but three years later the new Committee was disbanded with no further volumes of Bolzano's manuscripts being published during these three years.

The Czechoslovak Academy of Sciences began a new push to publish Bolzano's manuscripts in the early 1960s. Before the first volume in the series appeared in 1969 there were a number of related publications. Kazimír Vecerka published Bolzano's *Anti-Euclid* in 1967. It contained Bolzano's ideas concerning the reform and improvement of Euclidean geometry. Bob van Rootselaar published Bolzano's corrections to his *Functionenlehre* in 1969. It is, perhaps one of, Bolzano's attempts to correct certain errors in *Functionenlehre.* The first volume in the new series *Bernard Bolzano-Gesamtausgabe* published by *Friedrich Frommann Verlag* and edited by Eduard Winter, Jan Berg, Friedrich Kambartel, Jaromir Louzil, and Bob van Rootselaar, contains a biography of Bolzano together with details of the topics on which he worked: mathematics, logic, theology, philosophy and aesthetics. The second volume, which set the scene for the whole series, appeared in 1972. Jan Berg describes its contents:-

Further details of this series of volumes is given inThis is a detailed catalogue of that part of Bolzano's manuscripts which was donated in1892to the then existing Court Library in Vienna by his former disciple Robert Zimmermann(1824-1898). Moreover, in the supplement,19letters from Bolzano to the philosopher Franz Exner(1802-1853), dealing inter alia with questions of logical semantics, are described. ... The bibliography lists(1)in chronological order all works of Bolzano published in any language until March1971and(2)literature on Bolzano in alphabetic order ... The section on the division of the Bernard Bolzano-Gesamtaus-gabe does not correspond to the present conception of the editors. In particular, the plan for part II(the posthumous works of Bolzano)has been extensively revised during the last decade. For example,[the]subsection ... containing Bolzano's scientific diaries is now scheduled to embrace20volumes in about35instalments.

Bernard Bolzano's publications

**References (72 books/articles)**

**Article by:** *J J O'Connor* and *E F Robertson*