Foundations of Science

, Volume 23, Issue 4, pp 681–704 | Cite as

Bolzano’s Infinite Quantities

  • Kateřina Trlifajová


In his Foundations of a General Theory of Manifolds, Georg Cantor praised Bernard Bolzano as a clear defender of actual infinity who had the courage to work with infinite numbers. At the same time, he sharply criticized the way Bolzano dealt with them. Cantor’s concept was based on the existence of a one-to-one correspondence, while Bolzano insisted on Euclid’s Axiom of the whole being greater than a part. Cantor’s set theory has eventually prevailed, and became a formal basis of contemporary mathematics, while Bolzano’s approach is generally considered a step in the wrong direction. In the present paper, we demonstrate that a fragment of Bolzano’s theory of infinite quantities retaining the part-whole principle can be extended to a consistent mathematical structure. It can be interpreted in several possible ways. We obtain either a linearly ordered ring of finite and infinitely great quantities, or a partially ordered ring containing infinitely small, finite and infinitely great quantities. These structures can be used as a basis of the infinitesimal calculus similarly as in non-standard analysis, whether in its full version employing ultrafilters due to Abraham Robinson, or in the recent “cheap version” avoiding ultrafilters due to Terence Tao.


Bernard Bolzano Paradoxes of the infinite Measurable numbers Cantor’s Set Theory Infinite quantities Non-standard analysis 



I am grateful to Mikhail Katz, Jan Šebestík and Petr Kůrka for helpful comments on an earlier version of the manuscript. Many thanks to my colleague Jan Starý for fruitfull discussions about ultrafilters and to my husband Jan Trlifaj for notes on the final version. I am also very grateful to the two anonymous referees for their thorough reading of the manuscript and valuable comments.


  1. Albeverio, S., Hoegh-Kron, R., Febstad, J. E., & Lindstrom, T. (1986). Nonstandard methods in stochastic analysis and mathematical physics. Orlando, Florida: Academic Press.Google Scholar
  2. Bair, J., Blaszczyk, P., Ely, R., Henry, V., Kanovei, V., Katz, K., Katz, M. et al. (2013). Is mathematical history written by the victors? Notices of the American Mathematical Society, 60(7), 886–904.Google Scholar
  3. Bair, J., Baszczyk, P., Ely, R., Henry, V., Kanovei, V., Katz, K., Katz, M. et al. (2017). Interpreting the infinitesimal mathematics of Leibniz and Euler. Journal for General Philosophy of Science, 48(2), 195–238.CrossRefGoogle Scholar
  4. Bascelli, T., Błaszczyk, P., Kanovei, V., Katz, K., Katz, M., Kutateladze, S., Nowik, T., Schaps, D. & Sherry, D. (2018), Gregory’s sixth operation. Foundations of Science, 23(1), 1–12.CrossRefGoogle Scholar
  5. Behboud, A. (1998). Remarks on Bolzano’s collections. In W. Kűnne, M. Siebel, & M. Textor (Eds.), Bolzano and analytic philosophy, Amsterdam: Rodopi.Google Scholar
  6. Blaszczyk, P., Kanovei, V., Katz, K., Katz, M., Kutateladze, S., & Sherry, D. (2017). Toward a history of mathematics focused on procedures. Foundations of Science, 22(4), 763–783.CrossRefGoogle Scholar
  7. Bolzano, B. (1837/2014). Wissenschaftslehre. Seidel, Sulzbach. Translation Theory of Science (Rusnock, P. & Rolf, G. 2014). Google Scholar
  8. Bolzano, B., (1851/2004). Paradoxien des Unendlichen. CH Reclam, Leipzig. Translation Paradoxes of the infinite (Russ 2004).Google Scholar
  9. Bolzano, B., (1976). Reine Zahlenlehre. In Berg, J., (Ed.) Bernard Bolzano Gesamtausgabe Bd. 2A8, Frommann-Holzboog, Stuttgart-Bad Canstatt. Translation (Russ 2004).Google Scholar
  10. Cantor, G. (1883/1976). Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Leipzig: B. G. Teubner. Translation Foundations of a General Theory of Manifolds by Georg Cantor. The Campaigner. Journal of the National Caucus of Labor Committees, 9(1–2), 69–97.Google Scholar
  11. Chang, C. C., & Keisler, H. J. (1992). Model theory. Amsterdam: North-Holland Publishing Co.Google Scholar
  12. Dauben, J. W. (1979). Georg Cantor: His mathematics and philosophy of the infinite. Cambridge: Harvard University Press.Google Scholar
  13. Fraenkel, A., Bar-Hillel, Y., & Ley, A. (1973). Foundations of set theory. Elsevier: Studies in Logic.Google Scholar
  14. Ferreiros, J. (1999). Labyrinth of thought.  A history of Set theory and its role in modern mathematics. Basel: Birkhauser. Google Scholar
  15. Hallett, M. (1986). Cantorian set theory and limitation of size. Oxford: Clarendon Press.Google Scholar
  16. Krickel, F. (1995). Teil und Inbegriff: Bernard Bolzanos Mereologie. St. Augustin: Academia.Google Scholar
  17. Mancosu, P. (2009). Measuring the size of infinite collections of natural numbers: was cantor’s set theory inevitable? The Review of Symbolic Logic, 2(4), 612–646.CrossRefGoogle Scholar
  18. Pascal, B. (1866). De l’ésprit géometrique. Libraires-éditeurs, Paris: Ch. Delagrave et C.Google Scholar
  19. Robinson, A. (1966). Non-standard analysis. Amsterdam: North-Holland Publishing Co.Google Scholar
  20. Rusnock, P. (2000). Bolzano’s philosophy and the emergence of modern mathematics. Amsterdam: Rodopi.Google Scholar
  21. Rusnock, P. (2013). On Bolzano’s concept of sum. History and Philosophy of Logic, 34(2), 155–169.CrossRefGoogle Scholar
  22. Rusnock, P., & Rolf, G. (2014). Theory of science. Oxford: Oxford University Press.Google Scholar
  23. Russ, S. (2004). The mathematical works of Bernard Bolzano. Oxford: Oxford University Press.Google Scholar
  24. Russ, S., & Trlifajová, K. (2016). Bolzano’s measurable numbers: Are they real? In  M. Zack, & E. Landry (Eds.), Research in history and philosophy of mathematics. Basel: Birkhauser. Google Scholar
  25. Šebestík, J. (1992). Logique et mathématique chez Bernard Bolzano. Paris: Vrin.Google Scholar
  26. Simons, P. (1998). Bolzano on collections. In W. Kűnne, M. Siebel, & M. Textor (Eds.), Bolzano and analytic philosophy. Amsterdam: Rodopi.Google Scholar
  27. Simons, P. (2004). Bolzano on quantities. In E. Morscher (Ed.), Bernard Bolzanos Leistungen in Logik,  Mathematik und Physik. Wien: Academia. Google Scholar
  28. Tao, T., (2012a). A cheap version of non-standard analysis.
  29. Tao, T. (2012b). Compactness and contradiction. Providence: American Mathematical Society.Google Scholar
  30. Waldegg, G. (2005). Bolzano’s approach to the paradoxes of infinity: Implications for teaching. Science and Education, 14(6), 559–577.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Information TechnologyPraha 6Czech Republic

Personalised recommendations