Abstract
During the early 1830s Bernard Bolzano, working in Prague, wrote a manuscript giving a foundational account of numbers and their properties. In the final section of his work he described what he called ‘infinite number expressions’ and ‘measurable numbers’. This work was evidently an attempt to provide an improved proof of the sufficiency of the criterion usually known as the ‘Cauchy criterion’ for the convergence of an infinite sequence. Bolzano had in fact published this criterion four years earlier than Cauchy who, in his work of 1821, made no attempt at a proof. Any such proof required the construction or definition of real numbers and this, in essence, was what Bolzano achieved in his work on measurable numbers. It therefore pre-dates the well-known constructions of Dedekind, Cantor and many others by several decades. Bolzano’s manuscript was partially published in 1962 and more fully published in 1976. We give an account of measurable numbers, the properties Bolzano proved about them and the controversial reception they have prompted since their publication.
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- 1.
This is a slightly modified form of the theorem, the exact version is in Russ (2004, 266).
- 2.
This is an unpublished dissertation which we have not seen but rely on the report of it in Spalt (1991).
- 3.
Another interesting possibility for repairing the definition of measurable numbers is in Rusnock (2000, 185–186).
References
Bachmann P (1892) Vorlesungen über die Theorie der Irrationalzahlen, Leipzig
Bair J, Blaszczyk P, Ely R et al (2013) Is mathematical history written by the victors? Not Am Math Soc 60(7):886–904
Becker HP (1988) Dr B Bolzanos meßbaren Zahlen. Unpublished Diplomarbeit, Fachbereich Mathematik, Technische Hochschule, Darmstadt
Bolzano B (1816/2004) Binomische Lehrsatz und aus Folgerung aus ihm der polynomische, und die Reihen, die zur Berechnung der Logarithmen und Exponentialgrößen dienen, genauer als bisher erwiesen (trans: Russ (2004)). C.W. Enders, Prague
Bolzano B (1817/2004) Rein analytischer Beweis des Lehrsatzes, daß zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege (trans: Russ (2004)). Gottlieb Haase, Prague
Bolzano B (1837/2014) Wissenschaftslehre. Theory of science (trans: Rusnock P, Rolf G (2014)). Oxford University Press, Oxford
Bolzano B (1851/2004) Paradoxien des Unendlichen (trans: Russ (2004)). CH Reclam, Leipzig
Bolzano B (1976) Reine Zahlenlehre. In: Berg J (ed) Bernard Bolzano Gesamtausgabe Bd. 2A8, Frommann-Holzboog, Stuttgart-Bad Canstatt
Burn RP (1992) Irrational numbers in English language textbooks, 1890–1915: constructions and postulates for the completeness of the real numbers. Hist Math 19(2):158–176
Cauchy A (1821/2009) Cauchy’s Cours d’analyse. An annotated translation (trans: Bradley RE, Sandifer CE). Springer, Berlin
Dedekind R (1872/1963) Essays on the theory of numbers (trans: Beman WW). Dover, New York
Ferreirós J (2016) Mathematical knowledge and the interplay of practices. Princeton University Press, Princeton
Havil J (2012) The irrationals. Princeton University Press, Princeton
Ide J (1803) Anfangsgründe der reinen Mathematik. Heinrich Frölich, Berlin
Jordan C (1887) Cours d’analyse de l’Ecole Polytechnique Vol. 3, 1st ed. Gauthier-Villars, Paris
Kitcher P (1975) Bolzano’s ideal of algebraic analysis. Stud Hist Philos Sci 6(3):229–269
Klein F (1932) Elementary mathematics from an advanced standpoint. Macmillan, New York
Laugwitz D (1965) Bemerkungen zu Bolzanos Größenlehre. Arch Hist Exact Sci 2(5):398–409
Laugwitz D (1982) Bolzano’s infinitesimal numbers. Czechoslov Math J 32(4):667–670
Mainzer K (1990) Real numbers. In: Ebbinghaus, H-D et al. (ed) Numbers. Springer, Berlin
Morscher E (2008) Bernard Bolzano’s life and works. Academia Verlag, Sankt Augustin
Russ S (2004) The mathematical works of Bernard Bolzano. Oxford University Press, Oxford
Rusnock P (2000) Bolzano’s Philosophy and the Emergence of Modern Mathematics. Rodopi, Amsterdam
Rychlík K (1962) Theorie der reellen Zahlen im Bolzanos handschriftlichen Nachlasse. Czechoslovakian Academy of Sciences, Prague
Sebestik J (1992) Logique et mathématique chez Bolzano. Vrin, Paris
Simons P (2003) Bolzano on Quantities. In: Morscher E (ed) Bernard Bolzanos Leistungen in Logik, Mathematik und Physik. Academia Verlag, Sankt Augustin
Spalt D (1991) Bolzanos Lehre von den meßbaren Zahlen 1830–1989. Arch Hist Exact Sci 42(1):15–70
Stedall J (2006) Mathematics emerging. Oxford University Press, Oxford
van Rootselaar B (1963) Bolzano’s theory of real numbers. Arch Hist Exact Sci 2(2):168–180
van Rootselaar B (2003) Bolzanos Mathematik. In: Morscher E (ed) Bernard Bolzanos Leistungen in Logik, Mathematik und Physik. Academia Verlag, Sankt Augustin
Vopěnka P (1979) Mathematics in the alternative set theory. Teubner Verlagsgesellschaft, Leipzig
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Russ, S., Trlifajová, K. (2016). Bolzano’s measurable numbers: are they real?. In: Zack, M., Landry, E. (eds) Research in History and Philosophy of Mathematics. Proceedings of the Canadian Society for History and Philosophy of Mathematics/La Société Canadienne d’Histoire et de Philosophie des Mathématiques. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-46615-6_4
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