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Logic"],"published-print":{"date-parts":[[2021,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Formalism in the philosophy of mathematics has taken a variety of forms and has been advocated for widely divergent reasons. In Sects.\u00a01 and 2, I briefly introduce the major formalist doctrines of the late nineteenth and early twentieth centuries. These are what I call empirico-semantic formalism (advocated by Heine), game formalism (advocated by Thomae) and instrumental formalism (advocated by Hilbert). After describing these views, I note some basic points of similarity and difference between them. In the remainder of the paper, I turn my attention to Hilbert\u2019s instrumental formalism. My primary aim there will be to develop its formalist elements more fully. These are, in the main, (i) its rejection of the axiom-centric focus of traditional model-construction approaches to consistency problems, (ii) its departure from the traditional understanding of the basic nature of proof and (iii) its distinctively descriptive or observational orientation with regard to the consistency problem for arithmetic. More specifically, I will highlight what I see as the salient points of connection between Hilbert\u2019s formalist attitude and his finitist standard for the consistency proof for arithmetic. I will also note what I see as a significant tension between Hilbert\u2019s observational approach to the consistency problem for arithmetic and his expressed hope that his solution of that problem would dispense with certain epistemological concerns regarding arithmetic once and for all.<\/jats:p>","DOI":"10.1007\/s00153-021-00782-7","type":"journal-article","created":{"date-parts":[[2021,6,15]],"date-time":"2021-06-15T15:04:11Z","timestamp":1623769451000},"page":"529-546","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Formalism and Hilbert\u2019s understanding of consistency problems"],"prefix":"10.1007","volume":"60","author":[{"given":"Michael","family":"Detlefsen","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,6,15]]},"reference":[{"key":"782_CR1","unstructured":"Bernays, P.: \u00dcber Hilbert\u2019s Gedanken zur Grundlegung der Arithmetik. Jahresberichte der deutsche Mathematiker-Verienigung 30, 10\u201319 (1922). English trans. in [26]"},{"key":"782_CR2","unstructured":"Bolzano, B.: The Mathematical Works of Bernard Bolzano, trans. and ed. by Steve Russ. Oxford University Press, Oxford (2004)"},{"key":"782_CR3","unstructured":"Bolzano, B.: Beitr\u00e4ge zu einer begr\u00fcndeteren Darstellung der Mathematik, Caspar Widtmann, Prague, 1810. English trans. in [2]"},{"key":"782_CR4","doi-asserted-by":"crossref","unstructured":"Brouwer, L.E.J.: \u00dcber die Bedeutung des Satzes vom ausgescholessenen Dritten in der Mathematik, insbesondere in der Funktiontheorie. Journal f\u00fcr die reine und angewandte Mathematik 154, 1\u20137 (1923) English translation in [15], 334\u2013345. Page references are to this translation","DOI":"10.1515\/crll.1925.154.1"},{"key":"782_CR5","unstructured":"Brouwer, L.E.J.: Intuitionistische Betrachtungen \u00fcber den Formalismus, Koninklijke Akademie van wetenschappen de Amsterdam. Proc. Sect. 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