While sets are combinatorial collections, defined by their elements, classes are logical collections, defined by their membership conditions. We develop, in a potentialist setting, a predicative approach to (logical) classes of (combinatorial) sets. Some reasons emerge to adopt a stricter form of potentialism, which insists, not only that each object is generated at some stage of an incompletable process, but also that each truth is “made true” at some such stage. The natural logic of this strict form of potentialism is semi-intuitionistic: where each set-sized domain is classical, the domain of all sets or all classes is intuitionistic.
{"title":"Predicative Classes and Strict Potentialism","authors":"Øystein Linnebo, Stewart Shapiro","doi":"10.1093/philmat/nkae020","DOIUrl":"https://doi.org/10.1093/philmat/nkae020","url":null,"abstract":"While sets are combinatorial collections, defined by their elements, classes are logical collections, defined by their membership conditions. We develop, in a potentialist setting, a predicative approach to (logical) classes of (combinatorial) sets. Some reasons emerge to adopt a stricter form of potentialism, which insists, not only that each object is generated at some stage of an incompletable process, but also that each truth is “made true” at some such stage. The natural logic of this strict form of potentialism is semi-intuitionistic: where each set-sized domain is classical, the domain of all sets or all classes is intuitionistic.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"40 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142601934","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}
In ‘Intuition, iteration, induction’, Mark van Atten argues that iteration is an object of intuition for Brouwer and explains the intuitive character of the act of iteration drawing from Husserl’s phenomenology. I find the arguments for this reading of Brouwer unconvincing. In this note I set out some issues with his claim that iteration is an object of intuition and his Husserlian explication of iteration. In particular, I argue that van Atten does not accomplish his goals due to tensions with Brouwer’s comments on second-order mathematics and because Husserl does not understand the experience of succession as Brouwer does.
{"title":"Is Iteration an Object of Intuition?","authors":"Bruno Bentzen","doi":"10.1093/philmat/nkae019","DOIUrl":"https://doi.org/10.1093/philmat/nkae019","url":null,"abstract":"In ‘Intuition, iteration, induction’, Mark van Atten argues that iteration is an object of intuition for Brouwer and explains the intuitive character of the act of iteration drawing from Husserl’s phenomenology. I find the arguments for this reading of Brouwer unconvincing. In this note I set out some issues with his claim that iteration is an object of intuition and his Husserlian explication of iteration. In particular, I argue that van Atten does not accomplish his goals due to tensions with Brouwer’s comments on second-order mathematics and because Husserl does not understand the experience of succession as Brouwer does.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"38 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142325606","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}
Set-theoretic potentialism is one of the most lively trends in the philosophy of mathematics. Modal accounts of sets have been developed in two different ways. The first, initiated by Charles Parsons, focuses on sets as objects. The second, dating back to Hilary Putnam and Geoffrey Hellman, investigates set-theoretic structures. The paper identifies two strands of open issues, technical and conceptual, to clarify these two different, yet often conflated, views and categorize the potentialist approaches that have emerged in the contemporary debate. The final outcome is a taxonomy that should help researchers navigate the rich landscape of modal set theories.
{"title":"A Taxonomy for Set-Theoretic Potentialism","authors":"Davide Sutto","doi":"10.1093/philmat/nkae016","DOIUrl":"https://doi.org/10.1093/philmat/nkae016","url":null,"abstract":"Set-theoretic potentialism is one of the most lively trends in the philosophy of mathematics. Modal accounts of sets have been developed in two different ways. The first, initiated by Charles Parsons, focuses on sets as objects. The second, dating back to Hilary Putnam and Geoffrey Hellman, investigates set-theoretic structures. The paper identifies two strands of open issues, technical and conceptual, to clarify these two different, yet often conflated, views and categorize the potentialist approaches that have emerged in the contemporary debate. The final outcome is a taxonomy that should help researchers navigate the rich landscape of modal set theories.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"4 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084654","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}
Practical approaches to the notions of subsets and extension sets are compared, coming from broadly set-theoretic and category-theoretic traditions of mathematics. I argue that the set-theoretic approach is the most practical for ‘looking down’ or ‘in’ at subsets and the category-theoretic approach is the most practical for ‘looking up’ or ‘out’ at extensions, and suggest some guiding principles for using these approaches without recourse to either category theory or axiomatic set theory.
{"title":"Up with Categories, Down with Sets; Out with Categories, In with Sets!","authors":"Jonathan Kirby","doi":"10.1093/philmat/nkae010","DOIUrl":"https://doi.org/10.1093/philmat/nkae010","url":null,"abstract":"Practical approaches to the notions of subsets and extension sets are compared, coming from broadly set-theoretic and category-theoretic traditions of mathematics. I argue that the set-theoretic approach is the most practical for ‘looking down’ or ‘in’ at subsets and the category-theoretic approach is the most practical for ‘looking up’ or ‘out’ at extensions, and suggest some guiding principles for using these approaches without recourse to either category theory or axiomatic set theory.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"108 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140551916","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}
Boffa non-well-founded set theory allows for several distinct sets equal to their respective singletons, the so-called ‘Quine atoms’. Rieger contends that this theory cannot be a faithful description of set-theoretic reality. He argues that, even after granting that there are non-well-founded sets, ‘the extensional nature of sets’ precludes numerically distinct Quine atoms. In this paper we uncover important similarities between Rieger’s argument and how non-rigid structures are conceived within mathematical structuralism. This opens the way for an objection against Rieger, whilst affording the theoretical resources for a defence of Boffa set theory as a faithful description of set-theoretic reality.
{"title":"Identity and Extensionality in Boffa Set Theory","authors":"Nuno Maia, Matteo Nizzardo","doi":"10.1093/philmat/nkad025","DOIUrl":"https://doi.org/10.1093/philmat/nkad025","url":null,"abstract":"Boffa non-well-founded set theory allows for several distinct sets equal to their respective singletons, the so-called ‘Quine atoms’. Rieger contends that this theory cannot be a faithful description of set-theoretic reality. He argues that, even after granting that there are non-well-founded sets, ‘the extensional nature of sets’ precludes numerically distinct Quine atoms. In this paper we uncover important similarities between Rieger’s argument and how non-rigid structures are conceived within mathematical structuralism. This opens the way for an objection against Rieger, whilst affording the theoretical resources for a defence of Boffa set theory as a faithful description of set-theoretic reality.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"29 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139750353","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}
This paper studies internal (or intra-)mathematical explanations, namely those proofs of mathematical theorems that seem to explain the theorem they prove. The goal of the paper is a rigorous analysis of these explanations. This will be done in two steps. First, we will show how to move from informal proofs of mathematical theorems to a formal presentation that involves proof trees, together with a decomposition of their elements; secondly we will show that those mathematical proofs that are regarded as having explanatory power all display an increase of conceptual complexity from the assumptions to the conclusion.
{"title":"Mathematical Explanations: An Analysis Via Formal Proofs and Conceptual Complexity","authors":"Francesca Poggiolesi","doi":"10.1093/philmat/nkad023","DOIUrl":"https://doi.org/10.1093/philmat/nkad023","url":null,"abstract":"This paper studies internal (or intra-)mathematical explanations, namely those proofs of mathematical theorems that seem to explain the theorem they prove. The goal of the paper is a rigorous analysis of these explanations. This will be done in two steps. First, we will show how to move from informal proofs of mathematical theorems to a formal presentation that involves proof trees, together with a decomposition of their elements; secondly we will show that those mathematical proofs that are regarded as having explanatory power all display an increase of conceptual complexity from the assumptions to the conclusion.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"43 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138559460","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":"Luciano Boi and Carlos Lobo, eds. When Form Becomes Substance: Power of Gestures, Diagrammatical Intuition and Phenomenology of Space","authors":"","doi":"10.1093/philmat/nkad024","DOIUrl":"https://doi.org/10.1093/philmat/nkad024","url":null,"abstract":"","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"35 13","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138598119","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":"Joel D. Hamkins.Lectures on the Philosophy of Mathematics","authors":"J. Ferreirós","doi":"10.1093/philmat/nkad022","DOIUrl":"https://doi.org/10.1093/philmat/nkad022","url":null,"abstract":"","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"29 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138595463","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}
Brouwer’s view on induction has relatively recently been characterised as one on which it is not only intuitive (as expected) but functional, by van Dalen. He claims that Brouwer’s ‘Ur-intuition’ also yields the recursor. Appealing to Husserl’s phenomenology, I offer an analysis of Brouwer’s view that supports this characterisation and claim, even if assigning the primary role to the iterator instead. Contrasts are drawn to accounts of induction by Poincaré, Heyting, and Kreisel. On the phenomenological side, the analysis provides an alternative to that in Tieszen’s Mathematical Intuition, and confirms a view of Gödel on his Dialectica Interpretation.
{"title":"Intuition, Iteration, Induction","authors":"Mark van Atten","doi":"10.1093/philmat/nkad017","DOIUrl":"https://doi.org/10.1093/philmat/nkad017","url":null,"abstract":"Brouwer’s view on induction has relatively recently been characterised as one on which it is not only intuitive (as expected) but functional, by van Dalen. He claims that Brouwer’s ‘Ur-intuition’ also yields the recursor. Appealing to Husserl’s phenomenology, I offer an analysis of Brouwer’s view that supports this characterisation and claim, even if assigning the primary role to the iterator instead. Contrasts are drawn to accounts of induction by Poincaré, Heyting, and Kreisel. On the phenomenological side, the analysis provides an alternative to that in Tieszen’s Mathematical Intuition, and confirms a view of Gödel on his Dialectica Interpretation.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"74 4","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"110423257","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}
Journal Article Dominique Pradelle.Être et genèse des idéalités. Un ciel sans éternité Get access Dominique Pradelle.*Être et genèse des idéalités. Un ciel sans éternité, [Being and genesis of ideal elements: A heaven without eternity.] Collection Épiméthée. Paris: PUF [Presses universitaires de France], 2023. Pp. 544. ISBN: 978-2-13-083587-5 (pbk); 978-2-13-083588-2 (epub); 978-2-13-085194-3 (pdf). Bruno Leclercq Bruno Leclercq Philosophy Department, Université de Liège, 4000 Liège, Belgium E-mail: b.leclercq@uliege.be https://orcid.org/0000-0002-3322-943X Search for other works by this author on: Oxford Academic Google Scholar Philosophia Mathematica, nkad020, https://doi.org/10.1093/philmat/nkad020 Published: 08 November 2023
{"title":"Dominique Pradelle.<i>Être et genèse des idéalités. Un ciel sans éternité</i>","authors":"Bruno Leclercq","doi":"10.1093/philmat/nkad020","DOIUrl":"https://doi.org/10.1093/philmat/nkad020","url":null,"abstract":"Journal Article Dominique Pradelle.Être et genèse des idéalités. Un ciel sans éternité Get access Dominique Pradelle.*Être et genèse des idéalités. Un ciel sans éternité, [Being and genesis of ideal elements: A heaven without eternity.] Collection Épiméthée. Paris: PUF [Presses universitaires de France], 2023. Pp. 544. ISBN: 978-2-13-083587-5 (pbk); 978-2-13-083588-2 (epub); 978-2-13-085194-3 (pdf). Bruno Leclercq Bruno Leclercq Philosophy Department, Université de Liège, 4000 Liège, Belgium E-mail: b.leclercq@uliege.be https://orcid.org/0000-0002-3322-943X Search for other works by this author on: Oxford Academic Google Scholar Philosophia Mathematica, nkad020, https://doi.org/10.1093/philmat/nkad020 Published: 08 November 2023","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"28 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135430051","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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