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This requires abstracting the notion of a proof as a set of sentences ordered by provability. We can then distinguish between discrete<jats:italic>steps<\/jats:italic>of a proof and possibly continuous<jats:italic>stages<\/jats:italic>, defining indexing functions to pick these out. Proof stages can be associated with the application of continuously variable rules, connecting continuity in lines of reasoning with continuously variable reasons. Some examples of continuous proofs are provided. We conclude by presenting some fundamental facts about continuous proofs, analogous to continuous structural rules and composition. We take this to be a development on its own, as well as lending support to non-finitistic constructionism.<\/jats:p>","DOI":"10.1007\/s11225-019-09892-z","type":"journal-article","created":{"date-parts":[[2020,1,22]],"date-time":"2020-01-22T15:02:40Z","timestamp":1579705360000},"page":"1145-1160","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Reasoning Continuously: A Formal Construction of Continuous Proofs"],"prefix":"10.1007","volume":"108","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7609-7254","authenticated-orcid":false,"given":"T. D. P.","family":"Brunet","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"E.","family":"Fisher","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,1,22]]},"reference":[{"key":"9892_CR1","unstructured":"Bell, J. 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