Pragmatism and category theory
Identifieur interne : 001A15 ( Istex/Corpus ); précédent : 001A14; suivant : 001A16Pragmatism and category theory
Auteurs : Ralf KrömerSource :
- Science Networks. Historical Studies ; 2007.
Abstract
Abstract: In the introduction, I said that the way mathematicians work with categories reveals interesting insights into their implicit philosophy (how they interpret mathematical objects, methods, and the fact that these methods work). On the grounds of the evidence presented, we can now observe that the history of CT shows a switch in this interpretation: at first, objects of categories were always interpreted as sets (as in the case of the representations of Eilenberg and Mac Lane; see section 5.4.4.2); the purely formal character of categorial concepts was acknowledged but not consequently stressed. What was stressed positively is that concerning the categories themselves, the “all” is to be taken seriously <#20 p.237>. One was not aware of the fact that the difference between set theory and formal CT allowed for an interpretation of CT beyond sets (as far as the objects are concerned). This changed with Grothendieck on the one hand and Buchsbaum on the other. Grothendieck was interested in infinitistic argumentation and tried to extend the scope of the (formal) concept of set. Buchsbaum was interested in formal purity. The result of this development is a new technical intuition. This paradigm change took a different shape in the American and the French community, respectively.
Url:
DOI: 10.1007/978-3-7643-7524-9_8
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ID="Aff12"><OrgAddress><City>Liège</City></OrgAddress></Affiliation><Affiliation ID="Aff13"><OrgAddress><City>Bonn</City></OrgAddress></Affiliation><Affiliation ID="Aff14"><OrgAddress><City>Regensburg</City></OrgAddress></Affiliation><Affiliation ID="Aff15"><OrgAddress><City>Leipzig</City></OrgAddress></Affiliation><Affiliation ID="Aff16"><OrgAddress><City>Mainz</City></OrgAddress></Affiliation><Affiliation ID="Aff17"><OrgAddress><City>Tokyo</City></OrgAddress></Affiliation><Affiliation ID="Aff18"><OrgAddress><City>Minneapolis</City></OrgAddress></Affiliation></EditorGroup></SeriesHeader><Book Language="En"><BookInfo BookProductType="Monograph" ContainsESM="No" Language="En" MediaType="eBook" NumberingStyle="Outline" TocLevels="0"><BookID>978-3-7643-7524-9</BookID><BookTitle>Tool and Object</BookTitle><BookSubTitle>A History and Philosophy of Category Theory</BookSubTitle><BookVolumeNumber>32</BookVolumeNumber><BookSequenceNumber>35</BookSequenceNumber><BookDOI>10.1007/978-3-7643-7524-9</BookDOI><BookTitleID>127100</BookTitleID><BookPrintISBN>978-3-7643-7523-2</BookPrintISBN><BookElectronicISBN>978-3-7643-7524-9</BookElectronicISBN><BookChapterCount>8</BookChapterCount><BookCopyright><CopyrightHolderName>Birkhäuser Verlag AG</CopyrightHolderName><CopyrightYear>2007</CopyrightYear></BookCopyright><BookSubjectGroup><BookSubject Code="SCM" Type="Primary">Mathematics</BookSubject><BookSubject Code="SCM23009" Priority="1" Type="Secondary">History of Mathematical Sciences</BookSubject><BookSubject Code="SCM11035" Priority="2" Type="Secondary">Category Theory, Homological Algebra</BookSubject><SubjectCollection Code="SUCO11649">Mathematics and Statistics</SubjectCollection></BookSubjectGroup><BookContext><SeriesID>4883</SeriesID></BookContext></BookInfo><BookHeader><AuthorGroup><Author AffiliationIDS="Aff19"><AuthorName DisplayOrder="Western"><GivenName>Ralf</GivenName><FamilyName>Krömer</FamilyName></AuthorName><Contact><Email>kromer@univ-nancy2.fr</Email></Contact></Author><Affiliation ID="Aff19"><OrgDivision>LPHS-Archives Poincaré (UMR7117 CNRS)</OrgDivision><OrgName>Université Nancy 2</OrgName><OrgAddress><Street>Campus Lettres 23, Bd Albert 1er</Street><Postcode>54015</Postcode><City>Nancy Cedex</City><Country>France</Country></OrgAddress></Affiliation></AuthorGroup></BookHeader><Chapter ID="Chap8" Language="En"><ChapterInfo ChapterType="OriginalPaper" ContainsESM="No" Language="En" NumberingStyle="Outline" TocLevels="0"><ChapterID>8</ChapterID><ChapterNumber>Chapter 8</ChapterNumber><ChapterDOI>10.1007/978-3-7643-7524-9_8</ChapterDOI><ChapterSequenceNumber>8</ChapterSequenceNumber><ChapterTitle Language="En">Pragmatism and category theory</ChapterTitle><ChapterFirstPage>303</ChapterFirstPage><ChapterLastPage>316</ChapterLastPage><ChapterCopyright><CopyrightHolderName>Birkhäuser Verlag AG</CopyrightHolderName><CopyrightYear>2007</CopyrightYear></ChapterCopyright><ChapterGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ChapterGrants><ChapterContext><SeriesID>4883</SeriesID><BookID>978-3-7643-7524-9</BookID><BookTitle>Tool and Object</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para TextBreak="No">In the introduction, I said that the way mathematicians work with categories reveals interesting insights into their implicit philosophy (how they interpret mathematical objects, methods, and the fact that these methods work). On the grounds of the evidence presented, we can now observe that the history of CT shows a switch in this interpretation: at first, objects of categories were always interpreted as sets (as in the case of the representations of Eilenberg and Mac Lane; see section 5.4.4.2); the purely formal character of categorial concepts was acknowledged but not consequently stressed. What was stressed positively is that concerning the categories themselves, the “all” is to be taken seriously <#20 p.237>. One was not aware of the fact that the difference between set theory and formal CT allowed for an interpretation of CT beyond sets (as far as the objects are concerned). This changed with Grothendieck on the one hand and Buchsbaum on the other. Grothendieck was interested in infinitistic argumentation and tried to extend the scope of the (formal) concept of set. Buchsbaum was interested in formal purity. The result of this development is a new technical intuition. This paradigm change took a different shape in the American and the French community, respectively.</Para></Abstract></ChapterHeader><NoBody></NoBody></Chapter></Book></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Pragmatism and category theory</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>Pragmatism and category theory</title></titleInfo><name type="personal"><namePart type="given">Ralf</namePart><namePart type="family">Krömer</namePart><affiliation>LPHS-Archives Poincaré (UMR7117 CNRS), Université Nancy 2, Campus Lettres 23, Bd Albert 1er, 54015, Nancy Cedex, France</affiliation><affiliation>E-mail: kromer@univ-nancy2.fr</affiliation></name><typeOfResource>text</typeOfResource><genre displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" type="research-article" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>Birkhäuser Basel</publisher><place><placeTerm type="text">Basel</placeTerm></place><dateIssued encoding="w3cdtf">2007</dateIssued><dateIssued encoding="w3cdtf">2007</dateIssued><copyrightDate encoding="w3cdtf">2007</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: In the introduction, I said that the way mathematicians work with categories reveals interesting insights into their implicit philosophy (how they interpret mathematical objects, methods, and the fact that these methods work). On the grounds of the evidence presented, we can now observe that the history of CT shows a switch in this interpretation: at first, objects of categories were always interpreted as sets (as in the case of the representations of Eilenberg and Mac Lane; see section 5.4.4.2); the purely formal character of categorial concepts was acknowledged but not consequently stressed. What was stressed positively is that concerning the categories themselves, the “all” is to be taken seriously <#20 p.237>. One was not aware of the fact that the difference between set theory and formal CT allowed for an interpretation of CT beyond sets (as far as the objects are concerned). This changed with Grothendieck on the one hand and Buchsbaum on the other. Grothendieck was interested in infinitistic argumentation and tried to extend the scope of the (formal) concept of set. Buchsbaum was interested in formal purity. The result of this development is a new technical intuition. 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