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abstract bibtex

abstract bibtex

Bolzano's `measurable numbers' are the first attempt ever at an arithmetical definition of the reals. Despite its intrinsic interest as a key step in the history of the so-called arithmetization of analysis, Bolzano's construction has gone largely unnoticed in the literature on the foundations of mathematics. A major reason for this is that Bolzano's construction still awaits proper account - one that can do justice to the remarkable intellectual significance of Bolzano's efforts. In this paper we provide such an account. Interpreting Bolzano's construction of the reals is challenging because the proofs are contained in an unfinished manuscript from the 1830s/1840s, and dotted with inconsistencies and mistakes. Existing interpretations of Bolzano's work on the reals typically concentrate on fixing the construction, and doing so by resorting on present-day set theory; in particular, they identify Bolzano's reals with sequences. We contrast the *sequence*-account just mentioned with what we call a *face-value* account, and side with the latter in providing a careful textual examination of Bolzano's proofs and the theoretical resources he actually employs. We show that it is doubtful that Bolzano's construction requires any set theoretical resources, that Bolzano's theory of measurable numbers is still ascribable to the science-of-quantity conception of mathematics, and that studying Bolzano's proposal only through the lenses of a set-theoretic reconstruction is what has hindered previous commentators in their philosophical appraisal of Bolzano's measurable numbers. Our analysis backs up a characterization of Bolzano's construction as between what Epple (2003) calls the traditional conception of mathematics exhibited by Frege and the more modern approach of Cantor and Dedekind.

@article{bellomo_bolzano_0, title = {Bolzano on the {Reals} [ongoing]}, abstract = {Bolzano's `measurable numbers' are the first attempt ever at an arithmetical definition of the reals. Despite its intrinsic interest as a key step in the history of the so-called arithmetization of analysis, Bolzano's construction has gone largely unnoticed in the literature on the foundations of mathematics. A major reason for this is that Bolzano's construction still awaits proper account - one that can do justice to the remarkable intellectual significance of Bolzano's efforts. In this paper we provide such an account. Interpreting Bolzano's construction of the reals is challenging because the proofs are contained in an unfinished manuscript from the 1830s/1840s, and dotted with inconsistencies and mistakes. Existing interpretations of Bolzano's work on the reals typically concentrate on fixing the construction, and doing so by resorting on present-day set theory; in particular, they identify Bolzano's reals with sequences. We contrast the *sequence*-account just mentioned with what we call a *face-value* account, and side with the latter in providing a careful textual examination of Bolzano's proofs and the theoretical resources he actually employs. We show that it is doubtful that Bolzano's construction requires any set theoretical resources, that Bolzano's theory of measurable numbers is still ascribable to the science-of-quantity conception of mathematics, and that studying Bolzano's proposal only through the lenses of a set-theoretic reconstruction is what has hindered previous commentators in their philosophical appraisal of Bolzano's measurable numbers. Our analysis backs up a characterization of Bolzano's construction as between what Epple (2003) calls the traditional conception of mathematics exhibited by Frege and the more modern approach of Cantor and Dedekind.}, author = {Bellomo, Anna and Betti, Arianna}, year = {0}, }

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