{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,24]],"date-time":"2024-08-24T17:40:09Z","timestamp":1724521209396},"reference-count":39,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2021,2,22]],"date-time":"2021-02-22T00:00:00Z","timestamp":1613952000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2023,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Bernard Bolzano (1781\u20131848) is commonly thought to have attempted to develop a theory of size for infinite collections that follows the so-called part\u2013whole 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\u2019s 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\u2019s infinite sums can be equipped with the rich and original structure of a non-commutative ordered ring, and that Bolzano\u2019s views on the mathematical infinite are, after all, consistent.<\/jats:p>","DOI":"10.1017\/s1755020321000010","type":"journal-article","created":{"date-parts":[[2021,2,22]],"date-time":"2021-02-22T07:49:42Z","timestamp":1613980182000},"page":"59-113","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["BOLZANO\u2019S MATHEMATICAL INFINITE"],"prefix":"10.1017","volume":"16","author":[{"given":"ANNA","family":"BELLOMO","sequence":"first","affiliation":[]},{"given":"GUILLAUME","family":"MASSAS","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2021,2,22]]},"reference":[{"key":"S1755020321000010_r9","first-page":"591","volume-title":"Paradoxien des Unendlichen","author":"Bolzano","year":"1851"},{"key":"S1755020321000010_r32","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198539308.001.0001","volume-title":"The Mathematical Works of Bernard Bolzano","author":"Russ","year":"2004"},{"doi-asserted-by":"publisher","key":"S1755020321000010_r38","DOI":"10.1007\/BF00384332"},{"doi-asserted-by":"publisher","key":"S1755020321000010_r40","DOI":"10.1007\/BF00357654"},{"key":"S1755020321000010_r12","volume-title":"Einleitung zur Gr\u00f6ssenlehre. 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Published by Cambridge University Press on behalf of The Association for Symbolic Logic","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}