Gödel argued that the incompleteness theorems entail that the mind is not a machine, or that certain arithmetical propositions are absolutely undecidable. His view was that the mind is not a machine, and that no arithmetical propositions are absolutely undecidable. I argue that his position presupposes that the idealized mathematician has an ability which I call the recursive-ordinal recognition ability. I show that we have this ability if, and only if, there are no absolutely undecidable arithmetical propositions. I argue that there are such propositions, but that no recognizable example of one can be identified, even in principle.
{"title":"Gödel’s Disjunctive Argument","authors":"Wesley Wrigley","doi":"10.1093/philmat/nkac013","DOIUrl":"https://doi.org/10.1093/philmat/nkac013","url":null,"abstract":"\u0000 Gödel argued that the incompleteness theorems entail that the mind is not a machine, or that certain arithmetical propositions are absolutely undecidable. His view was that the mind is not a machine, and that no arithmetical propositions are absolutely undecidable. I argue that his position presupposes that the idealized mathematician has an ability which I call the recursive-ordinal recognition ability. I show that we have this ability if, and only if, there are no absolutely undecidable arithmetical propositions. I argue that there are such propositions, but that no recognizable example of one can be identified, even in principle.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48118638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Frederick Kroon, Jonathan McKeown-Green, and Stuart Brock. A Critical Introduction to Fictionalism","authors":"M. Leng","doi":"10.1093/philmat/nkac012","DOIUrl":"https://doi.org/10.1093/philmat/nkac012","url":null,"abstract":"","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45409091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Salvatore Florio and Øystein Linnebo. The Many and the One. A Philosophical Study of Plural Logic","authors":"F. Boccuni","doi":"10.1093/philmat/nkac009","DOIUrl":"https://doi.org/10.1093/philmat/nkac009","url":null,"abstract":"","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49451448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Bradley Armour-Garb and Frederick Kroon, eds. Fictionalism in Philosophy","authors":"F. Kroon","doi":"10.1093/philmat/nkac010","DOIUrl":"https://doi.org/10.1093/philmat/nkac010","url":null,"abstract":"","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45344363","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Most philosophers take Benacerraf's argument in ‘What numbers could not be’ to rebut successfully the reductionist view that numbers are sets. This philosophical consensus jars with mathematical practice, in which reductionism continues to thrive. In this note, we develop a new challenge to Benacerraf's argument by contesting a central premise which is almost unanimously accepted in the literature. Namely, we argue that — contra orthodoxy — there are metaphysically relevant reasons to prefer von Neumann ordinals over other set-theoretic reductions of arithmetic. In doing so, we provide set-theoretical facts which, we believe, are crucial for informed assessment of reductionism.
{"title":"Breaking the Tie: Benacerraf's Identification Argument Revisited","authors":"Arnon Avron;Balthasar Grabmayr","doi":"10.1093/philmat/nkac022","DOIUrl":"10.1093/philmat/nkac022","url":null,"abstract":"Most philosophers take Benacerraf's argument in ‘What numbers could not be’ to rebut successfully the reductionist view that numbers are sets. This philosophical consensus jars with mathematical practice, in which reductionism continues to thrive. In this note, we develop a new challenge to Benacerraf's argument by contesting a central premise which is almost unanimously accepted in the literature. Namely, we argue that — contra orthodoxy — there are metaphysically relevant reasons to prefer von Neumann ordinals over other set-theoretic reductions of arithmetic. In doing so, we provide set-theoretical facts which, we believe, are crucial for informed assessment of reductionism.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"31 1","pages":"81-103"},"PeriodicalIF":1.1,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46615070","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Penelope Maddy. A Plea for Natural Philosophy: And Other Essays","authors":"","doi":"10.1093/philmat/nkac030","DOIUrl":"10.1093/philmat/nkac030","url":null,"abstract":"","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"31 1","pages":"143-143"},"PeriodicalIF":1.1,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44021473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The mathematician G.F.C. Griss is known for his program of negationless intuitionistic mathematics. Although Griss's rejection of negation is regarded as characteristic of his philosophy, this is a consequence of an executability requirement that mental constructions presuppose agents’ executing corresponding mental activity. Restoring Griss's executability requirement to a central role permits a more subtle characterization of the rejection of negation, according to which D. Nelson's strong constructible negation is compatible with Griss's principles. This exposes a ‘holographic’ theory of negation in negationless mathematics, in which a full theory of negation is ‘flattened’ in a putatively negationless setting.
{"title":"Negation in Negationless Intuitionistic Mathematics","authors":"Thomas Macaulay Ferguson","doi":"10.1093/philmat/nkac026","DOIUrl":"10.1093/philmat/nkac026","url":null,"abstract":"The mathematician G.F.C. Griss is known for his program of negationless intuitionistic mathematics. Although Griss's rejection of negation is regarded as characteristic of his philosophy, this is a consequence of an executability requirement that mental constructions presuppose agents’ executing corresponding mental activity. Restoring Griss's executability requirement to a central role permits a more subtle characterization of the rejection of negation, according to which D. Nelson's strong constructible negation is compatible with Griss's principles. This exposes a ‘holographic’ theory of negation in negationless mathematics, in which a full theory of negation is ‘flattened’ in a putatively negationless setting.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"31 1","pages":"29-55"},"PeriodicalIF":1.1,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47393855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Frédéric Patras. The Essence of Numbers","authors":"Bonnie Gold","doi":"10.1093/philmat/nkac028","DOIUrl":"https://doi.org/10.1093/philmat/nkac028","url":null,"abstract":"","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"31 1","pages":"132-142"},"PeriodicalIF":1.1,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68133728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Andrew Arana and Marco Panza, eds. Précis de philosophie de la logique et des mathématiques. Vol. 2. Philosophie des mathématiques","authors":"","doi":"10.1093/philmat/nkac031","DOIUrl":"10.1093/philmat/nkac031","url":null,"abstract":"","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"31 1","pages":"146-146"},"PeriodicalIF":1.1,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42268848","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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