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References
B. Bolzano,Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege. Prague: Haase (1817). A new edition, with P. E. B. Jourdain as editor, appeared inOstwald’s Klassiker der exakten Wissenschaften, No. 153, Leipzig (1905). An English translation also appears by the present author inHistoria Mathematica 7 (1980), 156–185.
For what is still the best survey in English of Bolzano’s life and mathematical work, see the article “Bolzano, Bernard“ by Professor Bob van Rootselaar in theDictionary of Scientific Biography New York: Scribners (1973). Although now out of date in some respects (e.g., the bibliographic details and the remarks on the real number work), this article also gives a useful summary of Bolzano’s contributions to logic in [4].
For an excellent survey of Bolzano’s topological contributions see the article, Dale M. Johnson, Prelude to dimension theory: The geometrical investigations of Bernard Bolzano,Arch. History Exact Sci. 17 (1976), 275-296.
B. Bolzano,Wissenschaftslehre. Versuch einer ausführlichen und grösstentheils neuen Darstellung der Logik mit steter Rücksicht auf deren bisherigen Bearbeiter, 1–4. Sulzbach: Seidel (1837). Substantial parts of this work are translated in R. George (ed.),Theory of Science, Oxford: Basil Blackwell (1972), and in J. Berg (ed.),Theory of Science, Dordrecht: Reidel (1973).
B. Bolzano,Betrachtungen über einige Gegenstände der Elementargeometrie, Prague: Barth (1804).
M. J. Fesl (ed.),Lebensbeschreibung des Dr Bernard Bolzano mit einigen seiner ungedruckten Aufsätze und dem Bildnisse des Verfassers. Sulzbach: Seidel (1836). The page numbers quoted refer, however, to the more easily available extract from this work which appears in E. Winter (ed.),Bernard Bolzano: Ausgewählte Schriften. Berlin: Union Verlag (1976).
E. Winter, J. Berg, et al. (eds.),B. Bolzano-Gesamtausgabe. Stuttgart: Frommann Holzboog (1969). Well over half of the proposed 56 volumes of this magnificent and scholarly edition have already been published. The bibliographic series of this edition, starting with Vol. E2/1 (1972), is now the most authoritative source for the extensive literature on Bolzano and the numerous editions of his works.
A useful survey in English of the contents of theGrössenlehre is given in Jan Berg,Bolzano’s Logic, Stockholm: Almquist and Wiksell (1962). Some volumes of the original manuscripts, enjoying the meticulous and helpful editorship of Jan Berg and Bob van Rootselaar, have already appeared in [7]. All the early works [1,5,9,11,12] and important parts of the work on real numbers and the theory of functions also have English translations forthcoming in S. B. Russ,The Mathematical Works of Bernard Bolzano. Oxford: Oxford University Press (to appear).
B. Bolzano,Beyträge zu einer begründeteren Darstellung der Mathematik. Erste Lieferung. Prague: Widtmann (1810).
In the late 1960s Lakatos analysed scientific theories in terms of the notion of a “research programme“, a theoretical structure which provides guidance for future research. See I. Lakatos and A. Musgrave (eds.),Criticism and the Growth of Knowledge, Cambridge: Cambridge University Press (1970), 91–195. The paper “Cauchy and the continuum“ in J. Worrall and G. Currie (eds.),Imre Lakatos,Philosophical Papers. Vol. 2,Mathematics, Science and Epistemology, Cambridge: Cambridge University Press (1978), 43-60, applies such principles in a particular case. The recent work, T. Koetsier,Lakatos’s Philosophy of Mathematics: A Historical Approach, Amsterdam: North-Holland (1991), is a study of these principles generally in mathematics. Because of the explicit accounts of his methodology, and its fruitfulness, Bolzano’s work would make a particularly interesting case study to analyse within the Lakatos’ framework.
B. Bolzano,Der binomische Lehrsatz und als Folgerung aus ihm der polynomische und die Reihen, die zur Berechnung der Logarithmen und Exponentialgrössen dienen, genauer als bisher erwiesen. Prague: Enders (1816).
B. Bolzano,Die drey Probleme der Rectification, der Complanation und der Cubirung, ohne Betrachtung des Unendlich Kleinen, ohne die Annahmen des Archimedes, und ohne irgend eine nicht streng erweisliche Voraussetzung gelöst;zugleich als Probe einer gänzlichen Umstaltung der Raumwissenschaft, allen Mathematikern zur Prüfung vorgelegt, Leipzig: Kummer (1817).
Bolzano actually uses superscripts centred over the F for the partial sums, but they are rendered here as subscripts for convenience.
See Ivor Grattan-Guinness, Bolzano, Cauchy and the ’New Analysis’ of the nineteenth century,Arch. History Exact Sci. 6 (1970), 372-400.
Philip Kitcher, Bolzano’s ideal of algebraic analysis,Stud. History Philos. Sci. 6 (1975), 229–269.
D. A. Steele,Paradoxes of the Infinite by Dr Bernard Bolzano, London: Routledge and Kegan Paul (1950). This is a translation of the work of Bolzano, F. Pfihonsky (ed.),Paradoxien des Unendlichen, Leipzig: Reclam (1851).
This appears in a section headedUnendliche Grössenbegriffe and is in Vol. 2A8 of [7].
This is found in 107 of theFunctionenlehre which is a part of theGrössenlehre that appears in Vol. 2A10 of [7]. Berg dates this part of the manuscript as being completed in 1834. It was also published as Vol. 1 of K. Rychlik,Bernard Bolzanos Schriften. Prague (1930).
Detlef Laugwitz, Bolzano’s infinitesimal numbers,Czechoslovak Math. J. 32(107) (1982), 667–670.
Detlef Spalt, Bolzanos Lehre von den messbaren Zahlen 1830-1989,Arch. History Exact Sci. 42(1) (1991), 15–70.
Bolzano uses superscript notation herey n,y 1, etc. We have used corresponding subscripts to avoid confusion with powers.
Vojtěch Jarník,Bolzano and the Foundations of Mathematical Analysis. Prague: Society of Czechoslovak Mathematicians and Physicists (1981). See the section, “On Bolzano’s Function,” 67–81.
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Russ, S. Years Ago. The Mathematical Intelligencer 14, 45–53 (1992). https://doi.org/10.1007/BF03025869
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DOI: https://doi.org/10.1007/BF03025869