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3 results

  • BOLZANO’S MATHEMATICAL INFINITE

  • Part of
  • ANNA BELLOMO, GUILLAUME MASSAS
  • Journal:
    The Review of Symbolic Logic / Volume 16 / Issue 1 / March 2023
    Published online by Cambridge University Press:
    22 February 2021, pp. 59-113
    Print publication:
    March 2023
    • Article
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  • Bernard Bolzano (1781–1848) is commonly thought to have attempted to develop a theory of size for infinite collections that follows the so-called part–whole principle, according to which the whole is always greater than any of its proper parts. In this paper, we develop a novel interpretation of Bolzano’s mature theory of the infinite and show that, contrary to mainstream interpretations, it is best understood as a theory of infinite sums. Our formal results show that Bolzano’s infinite sums can be equipped with the rich and original structure of a non-commutative ordered ring, and that Bolzano’s views on the mathematical infinite are, after all, consistent.

  • 2 - Formalism Freeness and Entanglement: Definitions

  • Juliette Kennedy, University of Helsinki
  • Book:
    Gödel, Tarski and the Lure of Natural Language
    Published online:
    03 December 2020
    Print publication:
    17 December 2020, pp 19-38

  • Summary

    Formalism freeness and logical entanglement are precisely defined. Anticipations of these concepts are cited in the work of Post and Brouwer, as well as in early work of Gödel. Varieties of entanglement and formalism freeness are given, with specific examples taken from set theory: extended constructibility and games; and model theory. Semantic characterisations of metamathematical concepts are discussed at length.

  • Chapter 14 - The metaphysical interpretation of logical truth

  • from Part III - Specific Issues
  • Edited by Penelope Rush, University of Tasmania
  • Book:
    The Metaphysics of Logic
    Published online:
    05 October 2014
    Print publication:
    16 October 2014, pp 233-248

  • Summary

    Bolzano's Theory of Science presents the first explicit and methodical espousal of internal logical realism. It also contains a formidable number of theoretical innovations. They include: the first account of the distinction between sense and reference; definitions of analyticity and consequence, i.e. deducibility based on a new substitutional procedure that anticipates Quine's and Tarski's, respectively; and an account of mathematical knowledge that excludes, contra Kant. In Bolzano's case, one of the main purposes in introducing propositions in themselves is to achieve precise and satisfactory definitions. By way of consequence, on Bolzano's own account the success of the endeavour depends on whether his commitment to propositions allows him to deliver a good theory of logic, or at least one that is preferable to its rivals. Bolzano did have views on epistemic modality, though unfortunately, there is no place for a discussion of the latter here.