Bolzano, Bernard. 1996. "Contributions to a Better-Grounded Presentation of Mathematics (1810)." In From Kant to Hilbert: a
Source Book in the Foundations of Mathematics. Vol. I, edited by Ewald, William, 174-224. Oxford: Clarendon Press.
Translation by Steve Russ, (abbreviated: BD) revised reprint in The Mathematical Works of Bernard Bolzano, pp. 87-137.
"The work BD must be one of the first books devoted to what we would now call foundations of mathematics, or philosophy of
mathematics. (Indeed, this latter phrase was the title given to it by Fels for the second edition.) After a short Preface the first main part is devoted to the
nature of mathematics and its proper classification. The second part deals with definitions, axioms, proofs, and theorems. It is here that the
ground-consequence relation is introduced:
in the realm of truth . . . a certain objective connection prevails . . . . some of these judgements are the grounds of others and the latter
are the consequences of the former. (BD II § 2)
He goes on to explain that the proper purpose to pursue in a scientific exposition is to arrange the judgements so as to reflect this
objective connection. (p. 21-22)
An important contribution Bolzano makes here is a solution to the problem of how to define, or come to agreement, on the basic or simple
concepts of a system. He says we should do so in the same way as we first learn terms in our mother tongue by considering several propositions containing the
term (see BD II § 8). This is akin to what we would call implicit definition and, of course, is closely related to axiomatic systems. However, this is
hindsight; there is no suggestion that Bolzano did, or could, consider axiom systems with the degree of formality familiar since the work of Hilbert.
Any such book as BD could hardly ignore Kant and his thinking aboutmathematics and accordingly there is an Appendix devoted to a criticism of
Kant's theory of the construction of mathematical concepts through pure intuition. The central point of Bolzano's criticism was that the very notion of pure
intuition was incoherent containing, he believed, internal contradictions. An excellent account of this, and indeed the whole methodology outlined in
BD, is contained in Rusnock Bolzano's Philosophy and the Emergence of Modern Mathematics (2000), Ch. 2. For a more extended and philosophical
discussion of Bolzano's views of proof and their relationship to Kant see Lapointe (forthcoming) [Bolzano's Semantics and his Criticism of the
Decompositional Conception of Analysis" in The Analytic Turn, Michael Beaney (Ed.), London, Routledge, 2007, pp.219-234].
Steve Russ, from the reprint in The Mathematical Works of Bernard Bolzano, pp. 22-23.
———. 1972. Theory of Science, Attempt at a Detailed and in the Main Novel Exposition of Logic with Constant Attention to Earlier
Authors. Berkeley: University of California Press.
Translation of selected section of Wissenschaftslehre edited by Rolf George.
The Index lists the complete contents of the first three books of the Wissenschaftslehre.
Cited as: George 1972.
Contents: Acknowledgments VII; Editor's Introduction XXIII; Bibliography XLVIII; Introduction 1; Book One: Theory of Fundamentals 19; Book
Two: Theory of Elements 59; Book Three: Theory of Knowledge 303; Book Four: Heuretic 373; Book Five: Theory of Science Proper 385; Index of Special Symbols,
phrases and Sentence Forms 393; Index of Subjects 393; Index of Persons 396; (*) Names Omitted (list of persons to whom reference had to be omitted in the
present edition) 398; Translation of Key Terms 399.
"During the earliest stages of my work on this translation, I enjoyed the co-operation and advice of my teacher, the late Henry S.
Leonard. A preliminary draft of the first volume was finished in 1958, but at that time I was still thinking of a complete translation of all four volumes. I
was eventually persuaded that early complaints about the unnecessary bulk of the work had their point. Kambartel's very successful attempt at shortening the
first two volumes (Bernard Bolzano's Grundlegung der Logik, Hamburg, 1963) finally convinced me that an abbreviated version was not only feasible, but
desirable." (p. VII).
(*) Bolzano's Wissenschaftslehre is an admirable source book for, and commentary upon, the history of logical theory. The following
is the list of persons to whom references had to be omitted in the present edition." (p. 398)
———. 1973. Theory of Science. Dordrecht: Reidel.
Edited, with an introduction, by Jan Berg. Translated from the German by Burnham Terrell.
Part A: Selections from the Wissenschaftslehre pp. 35-367; Part B: Excerpts from Bolzano's Correspondence pp. 371-383; Bibliography
Cited as: Berg 1973.
Table of Contents: Preface XV; Editor's Intreoduction 1; Part A. A selection from the Wissenschaftslehre (Sulzbach 1837, Leipzig
1914-31) Volume One 35; Volume Two: 167; Volume Three 305; Volume Four 357; Part B. Excerpts from Bolzano's Correspondence 371; Bibliography 385; Name Index
391; Subject Index 393-398.
"The present selection from the Wissenschaftslehre of Bernard Bolzano aims at giving a compact view of his main ideas in logic,
semantics, epistemology and the methodology of science. These ideas are analyzed from a modern point of view in the Introduction. Furthermore, excerpts from
Bolzano's correspondence are included which yield important remarks on his own work.
The translation of the sections from the Wissenschaftslehre are based on a German text, which I have located in the Manuscript
Department of the University Library in Prague (signature: 75 B 459). It was one of Bolzano's own copies of his printed work and contains a vast number of
corrections made by Bolzano himself, thus representing the final stage of his thought, which has gone unnoticed in previous editions." (from the
———. 2014. Theory of Science. New York: Oxford University Press.
First complete translation by Rolf George and Paul Rusnock.
Volume One: Theory of Fundamentals and Theory of Elements (part I): Introduction; Book One: Theory Of Fundamentals; Part I: Of The Existence
of Truths in Themselves; Part II: Of the Recognizability of Truth; Book Two: Theory of Elements: Part I: Of Ideas in Themselves.
Volume Two: Theory of Elements (part II): Book Two: Theory of Elements (continued); Part II: Of Propositions in Themselves; Part III: Of True
Propositions; Part IV: Of Inferences.
Volume Three: Theory of Knowledge and the Art of Discovery; Book Three: Theory of Knowledge; Part I: Of Ideas; Part II: Of Judgements; Part
III: Of the Relation Between our Judgements and Truth; Part IV: Of Certainty, Probability, and Confidence in Judgements; Book Four: The Art of Discovery; Part
I: General Rules; Part II: Particular Rules.
Volume Four: Theory Of Science Proper; Book Five: Theory Of Science Proper; Part I: General Rules; Part II: On the Determination of the
Extensions of the Sciences; Part III: On the Choice of a Class of Readers for a Treatise; Part IV: On the Propositions Which Should Appear in a Treatise; Part
V: On the Divisions of a Treatise; Part VI: On the Order to Which the Propositions Belonging to a Treatise Should Appear; Part VII: Theory of Signs or, On the
Signs Used in Or Recommended by a Treatise; Part VIII: How the Author of a Treatise Should Behave; Part IX: On Scientific Books That Are Not Genuine
———. 2007. Selected Writings on Ethics and Politics. Amsterdam: Rodopi.
Translated by Paul Rusnock and Rolf Georg.
Contents: Introduction 1; I. Selected Exhortations 43; II. On Rights, Civil Disobedience, and Resistance to Authority 141; III. Ethics and
Philosophy of Religion 169; IV. Political Philosophy 241; V. Index 359-368.
"In his own day, few appreciated Bolzano's contributions to theoretical philosophy and mathematics: only a small number were
even aware that he had done this work. He was renowned. rather, for his work as "catechist". professor of religious science
(Religionswissenschaft) at the Charles University in Prague from 1805 to 1819. In this highly visible position, Bolzano had become one of the most
prominent advocates of social justice and reform in his homeland, a national philosopher who was the "social and political conscience of Bohemia"
(4) W. Künne, "Bernard Bolzano über Nationalismus und Rassismus in Böhmen," p. 97-139 in E. Morscher and O. Neumaier ed.
Bolzanos Kampf gegen Nationalismus und Rassismus. Beitrage zur Bolzano-Forschung 4, Sankt Augustin: Academia Verlag. 1996, p. 97.
———. 2015. "On the Concept of the Beautiful: A Philosophical Essay (§§1–25)." Estetika: The Central European Journal of
Aesthetics no. 52:229-266.
Partial translation by Adam Bresnahan of Über den Begriff des Schönen. Eine philosophische Abhandlung [On the concept of the
beautiful. A philosophical treatise] (Prague, 1843).
The fact that I have decided to fill so many pages with the analysis of a single concept may for some seem to demand explanation. I can only
reply that this concept seems to me to be of particular importance; and further, that the analysis of concepts is a matter that always demands expansive
inquiries if one is to go beyond merely saying that the concept is reducible to its parts and actually convince the reader, thus also taking care to
demonstrate that the attempts at explicating the concept that have been made thus far are lacking in one way or another. After I have completed this essay on
the fundamental concept of aesthetics, I will not deem it necessary to proceed with such thoroughness in the essays that follow." (p. 229)
———. 2004. The Mathematical Works of Bernard Bolzano. Oxford: Oxford University Press.
Contents: Preface XII, Introduction 1; Part I: Geometry and Foundations 11. 1.1: Elementary Geometry (1804) 25; 1.2: Contribution to a
Better-Grounded Presentation of Mathematics (1810 83); Part II: Early Analysis 139. 2.1: The Binomial Theorem (1816) 155; 2.2: A Purely Analytic Proof (1817)
255; 2.3: Three Problems of Rectification, Complanation and Cubature (1817) 279; Part III: Later Analysis and the Infinite 345. 3.1: Infinite Quantity Concepts
(1830s) 355; 3.2: Theory of Functions (1830s) 429; 3.3: Improvements and Additionsd to the Theory of Functions (F+) 573; 3.4: Paradoxes of the Infinite
(posthumous 1851) 591; Selected Works of Bernard Bolzano 679; Bibliography 685; Name Index 691; Subject Index 693.
"The main goal of this volume is to present a representative selection of the mathematical work and thought of Bolzano to those who read
English much better than they could read the original German sources. It is my hope that the publication of these translations may encourage potential research
students, and supervisors, to see that there are numerous significant and interesting research problems, issues, and themes in the work of Bolzano and his
contemporaries that would reward further study. Such research would be no small undertaking.
Bolzano's thoughtwas all of a piece and to understand his mathematical achievements properly it is necessary to study his work on logic and
philosophy, as well as, to some extent, on theology and ethics. Of course, it would also be necessary to acquire the linguistic, historical, and technical
skills fit for the purpose. But the period of Bolzano's work is one of the most exciting periods in the history of Europe, from intellectual, political, and
cultural points of view. And with over half of the projected 120 volumes of Bolzano's complete works (BGA) available, the resources for such research
have never been better. Thework on mathematics and logic has been particularly well-served through the volumes already published." (p. XII).
Ewald, William. 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Vol. I. Oxford: Clarendon Press.
Contents: 6. Bernard Bolzano (1781-1848): A. Preface to Considerations on some objects of elementary geometry 172 (Bolzano 1804); B.
Contributions to a better-grounded presentation of mathematics 174 (Bolzano 1810); C. 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 225 (Bolzano 1817a); D. From Paradoxes of the infinite 249-292 (Bolzano
Bolzano, Bernard. 1950. Paradoxes of the Infinite. London: Routledge & Kegan Paul.
The Paradoxien originally appeared in 1851, were reprinted in facsimile in 1889 and edited afresh in 1921 by A. Höfler, with
annotations by H. Hahn.
Translated from the German of the posthumous edition by Franz Prihonský and furnished with a historical introduction by Donald A. Steele.
New translation in The Mathematical Works of Bernard Bolzano, pp. 591-678.
Contents: Short title key to Bolzano references IX; Donald A. Steele: Historical Introduction 1; Translation 59; Selected bibliography 176;
Index of Persons and Place 185; Index of Topics 188-189.
"The Paradoxien are the work of Bolzano's old age. Indeed, the modern mathematical reader who takes the text as it stands will
be occasionally disappointed, and may misjudge Bolzano if he has not also read his earlier work. The qualification of taking the text 'as it stands' is not
otiose. Our received version, here translated from the 1851 Leipzig edition, is a posthumous one by a friend with whom Bolzano discussed the topics in question
for the last few years of his life. The competence of Prihosnky, as an editor of mathematical matter and the trustworthiness of the received version of the
Paradoxien have recently been placed in doubt by Martin Jasek, the discoverer of the counter-example.
The antecedents of the Paradoxien go back at least as far as 9 June 1842, when Bolzano read 'that part of his paper on the march of
ideas to be followed in a truly scientific exposition of mathematics which deals with the finite and infinite.' The next traces are three instalments of papers
read as follows: on 9 January 1845 about 'a solution of sundry paradoxes occurring in the mathematical sciences' ; on to December 1846 about 'calculations with
infinite numerical expressions'; and on 24 February 1848 about 'the paradoxes occasioned in mathematics by the idea of the infinite.' The gradual
crystallisation of the eventual title is manifest. Between the first and second paper, on 3 February 1845, Bolzano wrote to Prihonsky that he was busy with
sundry paradoxes in mathematics; between the second and third, he wrote twice to Fesl: on 26 February 1848 that he now realised more than ever the importance
of the topics of the Paradoxien for mathematics and its philosophy, and on 24 June 1848 that the matter for the Paradoxien is constantly
expanding under his pen. Finally, on 30 November 1848, only eighteen days before his death, Bolzano read a final instalment under the final title of
Paradoxien des Unendlichen.
Unlike the manuscript of the Funktionenlehre, that of the Paradoxien was never made ready for the press by the author
himself. That task fell to his devoted but none too mathematical friend Prihonsky. In fact, Bolzano himself had entertained doubts about the posthumous
treatment of his mathematical papers, for he wrote to Fesl as early as 12 June 1842 that he was convinced 'that his mathematical ideas, in their present form,
could not be really well edited by any of his friends.' This premonition on Bolzano's part comes to corroborate the suspicions of Jasek.
Those suspicions were aroused by a palpable contradiction between the Funktionenlehre and the Paradoxien.
The received text of §37, pages 65-66, as distinguished from the footnote, makes no actual assertion as to the universal existence of
derivates of continuous functions; Bolzano simply intends to choose such as are differentiable for a certain purpose: 'Ich begehre nichts anderes, als
dass.' But a footnote says it can be shown that 'all well defined functions' are bound to be differentiable 'save possibly for a set of arguments which
may indeed be infinite, but whose members must be individually isolated.' No proof is attempted. Jasek ('Aus dem handschriften Nachlass Bernhard Bolzanos',
Vestnik Kralovske Ceske Spolecnosti Nau, Trida matematicko-prirodovedeckd (1923), pp. 29-32) claims to possess evidence that the footnote is
interpolated. He suspects Slivka von Slivitz, another pupil and friend of Bolzano, of being Prihonsky necessary but not sufficient mathematical counsellor, and
of having timorously desired to shield Bolzano from the appearance of mathematical heresy which his counterexample may well have borne to contemporaries.
A fresh critical study of the manuscript is indicated, and external obstacles stand in the way for the present. The difficulty does not,
however, reside solely in the passage noticed by Jasek. It recurs in §45, page 88. Strictly speaking, again, there is even here no actual assertion that all
continuous functions are differentiable, and the question is further complicated by the admitted fact that the posthumous editor found the manuscript in places
rather illegible, in spite of his acquaintance with Bolzano's peculiar abbreviation practices. The presumption of guilt on the part of von Slivitz is slightly
enhanced, in Jasek's eyes, by a study of his marginal annotations to a copy of the Funktionenlehre. With baffling contrariety, it is also slightly
alleviated by the fact that the Zusammensetzung der Krafte, published in 1842 by Bolzano himself -- and embodying with acknowledgement some
suggestions by von Slivitz -- proceeds (§52, page 29) to differentiate a function of which only the continuity is known, together with its satisfaction of
conditions which are not quoted as if they established the otherwise uncertain differentiability. The hypothesis that Bolzano's mathematical discrimination had
become dulled between 1830 and 1848 on this point at least is simple and not to be rejected a priori; but neither is it to be accepted easily until
renewed and exact archivial research compels us to do so." (pp. 53-55).
———. 2004. On the Mathematical Method and Correspondence with Exner. Amsterdam: Rodopi.
Translated by Paul Rusnock and Rolf George.
Contents: Acknowledgements 4; Introduction 5; A note on the translation 37; I. On the Mathematical Method 39; II. Selections from the
Bolzano-Exner Correspondence 83; III. Bolzano and Exner on Ideas and their objects: an exchange from 1843 175; IV. Indices 185; Index of Names 186; Index of
There are thee version of the essay on mathematical method; the translation if from the latest version (pp. 23-78 of the original
"It is clear that Bolzano was not satisfied with the logic set out in the Contributions . Already in 1812 he had resolved
to write another treatise on logic,(4) a project he worked on for close to a decade following his dismissal. The result was one of the great works of
nineteenth-century philosophy, the Theory of Science.(5) The Theory of Science was ready for the press by 1830, at which time Bolzano started
writing the Theory of Magnitudes [Grossenlehre], a treatise intended to supply a unified foundation for all of contemporary mathematics.(6)
Although he wrote several hundred pages, many of them all but ready for the press, but died before finishing his work. Some elements of his mathematical system
were published after his death in the Paradoxes of the Infinite,(7) edited by his friend and student F. Prihonsky, but the bulk remained all but
unknown until well into the twentieth century. The essay "On the Mathematical Method", translated here, formed part of the introductory matter of the
Theory of Magnitudes, and was intended to present the essentials of Bolzano's logic to a mathematical audience.
Due in large part to his troubles with the Austrian authorities, the Theory of Science remained unpublished for almost a decade (it
was finally published outside Austria, in Bavaria, in 1837). This did not mean, however, that Bolzano's logic remained unknown. Long accustomed to an invasive
and often arbitrary censorship, Bohemian intellectuals had developed unofficial channels for communicating their ideas, a precursor of the Samizdat
system which was later to flourish in that land. Bolzano's mature logic received its first airing in this way, when, in 1833, he had a copy made of the essay
on mathematical method and sent it to Franz Exner, the newly appointed professor of philosophy in Prague.(8)
Exner (1802-1853) was born and educated in Vienna, where he studied philosophy with Rembold, who like Bolzano (and for similar political
reasons) had been removed from his university chair in 1825. In 1830, Exner was put in the uncomfortable position of taking his teacher's place, being called
upon to fill the vacant chair on a temporary basis. In 1832 he moved to Prague, where he was named to the chair of philosophy. Outside of his official duties,
in good Austrian fashion, he organized a "circle" of intellectuals which met regularly at his house. Although a born and bred Viennese, he was
sensitive to the special circumstances of Bohemia, particularly to the disadvantaged situation of the Czech majority. Politically, though not always
philosophically, he was very much on Bolzano's side: with the Bohemian enlightenment and opposed to the conservative reaction in both church and state. Exner
was a follower of Herbart, who had a substantial following in Austria at the time, and whose doctrines were to become in effect the official philosophy of the
Empire, in part due to Exner's influence when he worked for the Ministry of Education from 1845 until his death.(9)
Exner responded to Bolzano in June of 1833, beginning a correspondence that would continue for the rest of Bolzano's life. The most intense
philosophical exchanges occurred during 1833 and 1834, when the letters translated here were written. The two continued their discussion in person in 1834,
when Bolzano returned to Prague from June to November. There would also be a later exchange of views in a pair of papers read at the Royal Bohemian Academy of
Sciences in the early 1840s. We have translated Bolzano's contribution, which contains the relevant passages from Exner's, in this volume." (pp. 6-8)
(4) Philosophische Tagebucher 1811-1817, in J. Berg, F. Kambartel, J. Louzil, B. van Rootselaar, and E. Winter ed., Bernard
Bolzano-Gesamtausgabe (hereafter BBGA) (Stuttgart-Bad Cannstatt 1969-) Series 2B Vol. 16/1, p. 34-36.
(5) Wissenschaftslehre (Sulzbach, 1837). New edition by Jan Berg in the BBGA. Hereafter WL.
(6) BBGA IIA, Vols. 7-10; Volumes 7, 8 and 10/1 have already been published.
(7) Paradoxien des Unendlichen (Leipzig, 1851), English translation by D. Steele (London: Routledge and Kegan Paul, 1950).
(8) Three versions of the essay on mathematical method survive. Our translation is based upon the latest version. Most likely, however, the
version that Exner received differed somewhat from this one. A Czech translation of an earlier work on logic, "O logice" (= "Etwas über
Logik" BBGA 2A5, p. 139-168), was actually published somewhat earlier (1831).
(9) Cf. Biographisches Lexicon des Kaiserthums Oesterreich, part 4 (Vienna, I 858); Allgemeine deutsche Biographie, vol. 6
Russ, S. B. 1980. "A translation of Bolzano’s paper on the intermediate value theorem." Historia Mathematica no.
Summary: "This is the first English translation of Bolzano's paper, Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey
Werthen, die ein entgegengesetzes Resultat gewahren, wenigstens eine reelle Wurzel der Gleichung liege (Prague 1817). It has already appeared in French,
Russian, and Czechoslovakian translations.
The paper represents an important stage in the rigorous foundation of analysis and is one of the earliest occasions when the continuity of a
function and the convergence of an infinite series are both defined and used correctly."