{"title":"Challenges to Predicative Foundations of Arithmetic","authors":"G. Hellman, S. Feferman","doi":"10.1017/CBO9780511570681.017","DOIUrl":"https://doi.org/10.1017/CBO9780511570681.017","url":null,"abstract":"This is a sequel to our article \"Predicative foundations of arithmetic\" (Feferman and Hellman, 1995), referred to in the following as PFA; here we review and clarify what was accomplished in PFA, present some improvements and extensions, and respond to several challenges. The classic challenge to a program of the sort exemplified by PFA was issued by Charles Parsons in a 1983 paper, subsequently revised and expanded as Parsons (1992). Another critique is due to Daniel Isaacson (1987). Most recently, Alexander George and Daniel Velleman (1996) have examined PFA closely in the context of a general discussion of different philosophical approaches to the foundations of arithmetic. The plan of the present paper is as follows: Section I reviews the notions and results of PFA, in a bit less formal terms than there and without the supporting proofs, and presents an improvement communicated to us by Peter Aczel. Then, Section II elaborates on the structuralist perspective that guided PFA. It is in Section III that we take up the challenge of Parsons. Finally, Section IV deals with the challenges of George and Velleman, and thereby, that of Isaacson as well. The paper concludes with an Appendix by Geoffrey Hellman, which verifies the predicativity, in the sense of PFA, of a suggestion credited to Michael Dummett for another definition of the natural number concept.","PeriodicalId":448297,"journal":{"name":"Mathematics and Its Logics","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116785080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Maoist Mathematics?","authors":"","doi":"10.1017/9781108657419.007","DOIUrl":"https://doi.org/10.1017/9781108657419.007","url":null,"abstract":"","PeriodicalId":448297,"journal":{"name":"Mathematics and Its Logics","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134601023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"What Is Categorical Structuralism?","authors":"","doi":"10.1017/9781108657419.003","DOIUrl":"https://doi.org/10.1017/9781108657419.003","url":null,"abstract":"","PeriodicalId":448297,"journal":{"name":"Mathematics and Its Logics","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116920089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On Nominalism","authors":"","doi":"10.1017/9781108657419.006","DOIUrl":"https://doi.org/10.1017/9781108657419.006","url":null,"abstract":"","PeriodicalId":448297,"journal":{"name":"Mathematics and Its Logics","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116661533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"If “If-Then” Then What?","authors":"","doi":"10.1017/9781108657419.015","DOIUrl":"https://doi.org/10.1017/9781108657419.015","url":null,"abstract":"","PeriodicalId":448297,"journal":{"name":"Mathematics and Its Logics","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128084696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Index","authors":"","doi":"10.1017/9781108657419.017","DOIUrl":"https://doi.org/10.1017/9781108657419.017","url":null,"abstract":"","PeriodicalId":448297,"journal":{"name":"Mathematics and Its Logics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126044691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Significance of the Burali-Forti Paradox","authors":"G. Hellman","doi":"10.1093/ANALYS/ANR091","DOIUrl":"https://doi.org/10.1093/ANALYS/ANR091","url":null,"abstract":"","PeriodicalId":448297,"journal":{"name":"Mathematics and Its Logics","volume":"39 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125874098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Predicativism as a Philosophical Position","authors":"G. Hellman","doi":"10.1017/9781108657419.010","DOIUrl":"https://doi.org/10.1017/9781108657419.010","url":null,"abstract":"Les exigences qui definissent la predicativite, a savoir que les objets soient explicitement presentables et les preuves acceptables du point de vue predicatif sont distinguees des theses predicativistes de caractere philosophique. Celles de ces theses qui expriment un scepticisme a Vegard de I'objectivite de ['ensemble de toutes les parties d'un ensemble infini sont familieres. Cependant, Varticulation de theses limitatives fortes se revele problematique: des modes de pensee impredicatifs se glissent dans les formulations mimes de ces theses, par exemple quart d on afftrme que la « definissabilite predicative » marque la limite de l'« intelligibility ». Une experience de pensee estproposee pour ebranler I'idee que les parties arbitraires d'un tout infini d'atomes « dependent de Vesprit » ou « dependent du langage ». D'un autre cote, des theses plus faibles, comme, par exemple, celle selon laquelle les mathematiques predicatives sont « plus sures » que les mathematiques impredicatives, sont presque des platitudes. L'influence philosophique interessante du predicativistne semble etre de caractere negatif dans sa contestation des arguments d'indispensabilite a la Godel-Friedman en faveur du transfini en mathematiques pures, ou a la Quine-Putnam enfaveur des mathematiques abstraites dans les sciences. II y a de plus en plus de raisons qui plaident enfaveur de Godel-Friedman, par exemple Vimpredicativite des formulations sans variables de theoremes comme ceux de Kruskal ou du « Graph Minor », et, d'une portee plus grande, les travaux recents en theorie des relations booleennes. On pent etre ainsi conduit a realiser Videe de Godel, qui etait de justifier les axiomes de grands cardinaux par leur role explicatif unificateur en mathematiques, en analogie avec la facon dont on justifie certaines hypotheses en physique theorique.","PeriodicalId":448297,"journal":{"name":"Mathematics and Its Logics","volume":"81 S361","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132226818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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