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}
Abstract Just as mathematics helps us to represent and reason about the natural world, in its internal applications one branch of mathematics helps us to represent and reason about the subject matter of another. Recognition of the close analogy between internal and external applications of mathematics can help resolve two persistent philosophical puzzles concerning its applicability: a platonist puzzle arising from the abstractness of mathematical objects; and an empiricist puzzle arising from mathematical propositions’ lack of empirical factual content. In order to see how this is the case, we will examine what it is to apply mathematics internally and describe examples.
{"title":"Internal Applications and Puzzles of the Applicability of Mathematics","authors":"Douglas Bertrand Marshall","doi":"10.1093/philmat/nkad019","DOIUrl":"https://doi.org/10.1093/philmat/nkad019","url":null,"abstract":"Abstract Just as mathematics helps us to represent and reason about the natural world, in its internal applications one branch of mathematics helps us to represent and reason about the subject matter of another. Recognition of the close analogy between internal and external applications of mathematics can help resolve two persistent philosophical puzzles concerning its applicability: a platonist puzzle arising from the abstractness of mathematical objects; and an empiricist puzzle arising from mathematical propositions’ lack of empirical factual content. In order to see how this is the case, we will examine what it is to apply mathematics internally and describe examples.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"198 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135168582","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":"Sorin Bangu, Emiliano Ippoliti, and Marianna Antonutti Marfori, eds. <i>Explanatory and Heuristic Power of Mathematics</i>","authors":"","doi":"10.1093/philmat/nkad018","DOIUrl":"https://doi.org/10.1093/philmat/nkad018","url":null,"abstract":"","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"73 4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136079969","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}
Abstract The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper I question this claim. I show that there is a kind of maximality (namely absoluteness) on which large cardinal axioms come out as restrictive relative to a formal notion of restrictiveness. Within this framework, I argue that large cardinal axioms can still play many of their usual foundational roles.
{"title":"Are Large Cardinal Axioms Restrictive?","authors":"Neil Barton","doi":"10.1093/philmat/nkad014","DOIUrl":"https://doi.org/10.1093/philmat/nkad014","url":null,"abstract":"Abstract The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper I question this claim. I show that there is a kind of maximality (namely absoluteness) on which large cardinal axioms come out as restrictive relative to a formal notion of restrictiveness. Within this framework, I argue that large cardinal axioms can still play many of their usual foundational roles.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135925160","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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