Neologicism (NL) invokes a “syntactic-priority thesis” (SPT) to derive existence of numbers, etc., from abstraction principles. Innumerable counterexamples to the SPT, however, are seen to arise from fiction, e.g., “Pegasus is (entirely) fictive”. Examination of possible defenses of the SPT leads to just one viable option, based on quasi-modal “in-fiction” operators. This, however, applies just as well to abstraction principles themselves, thereby undermining NL’s case for countenancing mathematical abstracta. Appeal to the linguistics principle of compositionality is seen not to help NL.
{"title":"Neologicism Meets Fiction","authors":"Geoffrey Hellman","doi":"10.1093/philmat/nkaf009","DOIUrl":"https://doi.org/10.1093/philmat/nkaf009","url":null,"abstract":"Neologicism (NL) invokes a “syntactic-priority thesis” (SPT) to derive existence of numbers, etc., from abstraction principles. Innumerable counterexamples to the SPT, however, are seen to arise from fiction, e.g., “Pegasus is (entirely) fictive”. Examination of possible defenses of the SPT leads to just one viable option, based on quasi-modal “in-fiction” operators. This, however, applies just as well to abstraction principles themselves, thereby undermining NL’s case for countenancing mathematical abstracta. Appeal to the linguistics principle of compositionality is seen not to help NL.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"38 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143940197","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 International Humanities Council has established a new international research network Diversity of Mathematical Research Cultures & Practices (DMRCP) at the Universität Hamburg. In a tripartite contribution, we outline and discuss the specific philosophical approach that DMRCP seeks to promote for which we use the term ‘empirical philosophy of mathematics’: the contribution is therefore programmatic and methodological, rather than a contribution to a specific philosophical research question. This article forms Part III of the triptych.
{"title":"A Triptych on Empirical Philosophy of Mathematics. Part III: How?","authors":"Deborah Kant, Benedikt Löwe","doi":"10.1093/philmat/nkaf004","DOIUrl":"https://doi.org/10.1093/philmat/nkaf004","url":null,"abstract":"The International Humanities Council has established a new international research network Diversity of Mathematical Research Cultures & Practices (DMRCP) at the Universität Hamburg. In a tripartite contribution, we outline and discuss the specific philosophical approach that DMRCP seeks to promote for which we use the term ‘empirical philosophy of mathematics’: the contribution is therefore programmatic and methodological, rather than a contribution to a specific philosophical research question. This article forms Part III of the triptych.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"119 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143920101","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}
Modal potentialism argues that mathematics has a generative nature, and aims to formalise mathematics accordingly using quantified modal logic. This paper shows that Øystein Linnebo’s approach to modal potentialism in his book Thin Objects is incoherent. In particular, he is committed to the legitimacy of introducing a primitive modal predicate of formulae. However, as with the semantic paradoxes, natural principles for such a predicate are inconsistent; no such predicate can underpin an account of modal potentialism. Hence, Linnebo’s intended interpretation of the primitive modality and his formal framework do not match up.
{"title":"How Should We Understand the Modal Potentialist’s Modality?","authors":"Boaz D Laan","doi":"10.1093/philmat/nkaf007","DOIUrl":"https://doi.org/10.1093/philmat/nkaf007","url":null,"abstract":"Modal potentialism argues that mathematics has a generative nature, and aims to formalise mathematics accordingly using quantified modal logic. This paper shows that Øystein Linnebo’s approach to modal potentialism in his book Thin Objects is incoherent. In particular, he is committed to the legitimacy of introducing a primitive modal predicate of formulae. However, as with the semantic paradoxes, natural principles for such a predicate are inconsistent; no such predicate can underpin an account of modal potentialism. Hence, Linnebo’s intended interpretation of the primitive modality and his formal framework do not match up.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"13 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143827659","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 proposes that mathematicians routinely use inference to the best explanation (IBE) to confirm their conjectures. Mathematicians can justly reason that the ‘best explanation’ of some mathematical evidence they possess would be a proof of it that likewise proves a given conjecture. By IBE, the evidence thereby confirms that such an as-yet-undiscovered proof exists and that the conjecture holds. This reasoning can be expressed in Bayesian terms once Bayesianism’s logical omniscience has been circumvented. A Bayesian analysis identifies considerations affecting a mathematical IBE’s strength and helps to unify mathematical IBEs with scientific IBEs.
{"title":"Inference to the Best Explanation as a Form of Non-Deductive Reasoning in Mathematics","authors":"Marc Lange","doi":"10.1093/philmat/nkae024","DOIUrl":"https://doi.org/10.1093/philmat/nkae024","url":null,"abstract":"This paper proposes that mathematicians routinely use inference to the best explanation (IBE) to confirm their conjectures. Mathematicians can justly reason that the ‘best explanation’ of some mathematical evidence they possess would be a proof of it that likewise proves a given conjecture. By IBE, the evidence thereby confirms that such an as-yet-undiscovered proof exists and that the conjecture holds. This reasoning can be expressed in Bayesian terms once Bayesianism’s logical omniscience has been circumvented. A Bayesian analysis identifies considerations affecting a mathematical IBE’s strength and helps to unify mathematical IBEs with scientific IBEs.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"3 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143775500","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}
Free choice sequences play a key role in the Brouwerian continuum. Using recent modal analysis of potential infinity, we can make sense of free choice sequences as potentially infinite sequences of natural numbers without adopting Brouwer’s distinctive idealistic metaphysics. This provides classicists with a means to make sense of intuitionistic ideas from their own classical perspective. I develop a modal-potentialist theory of real numbers that suffices to capture the most distinctive features of intuitionistic analysis, such as Brouwer’s continuity theorem, the existence of a sequence that is monotone, bounded, and non-convergent, and the inability to decompose the continuum non-trivially.
{"title":"A Potentialist Perspective on Intuitionistic Analysis","authors":"Ethan Brauer","doi":"10.1093/philmat/nkae025","DOIUrl":"https://doi.org/10.1093/philmat/nkae025","url":null,"abstract":"Free choice sequences play a key role in the Brouwerian continuum. Using recent modal analysis of potential infinity, we can make sense of free choice sequences as potentially infinite sequences of natural numbers without adopting Brouwer’s distinctive idealistic metaphysics. This provides classicists with a means to make sense of intuitionistic ideas from their own classical perspective. I develop a modal-potentialist theory of real numbers that suffices to capture the most distinctive features of intuitionistic analysis, such as Brouwer’s continuity theorem, the existence of a sequence that is monotone, bounded, and non-convergent, and the inability to decompose the continuum non-trivially.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"75 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143744947","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}
Hume’s Principle (HP) does not determine the truth values of ‘mixed’ identity statements like ‘$ #F $ = Caesar’. This is the Caesar Problem (CP). Still, neologicists such as Hale and Wright argue that (1) HP is a priori, and (2) HP introduces the pure sortal concept Number. We argue that Neologicism faces a Caesar-problem Problem (CPP): if neologicists solve CP by establishing that ‘$ #Fneq $ Caesar’ is true, (1) and (2) cannot be retained simultaneously. We examine various responses neologicists might provide and show that they do not address CPP. We conclude that CP uncovers a fatal tension in Neologicism.
{"title":"The Caesar-problem Problem","authors":"Francesca Boccuni, Luca Zanetti","doi":"10.1093/philmat/nkaf002","DOIUrl":"https://doi.org/10.1093/philmat/nkaf002","url":null,"abstract":"Hume’s Principle (HP) does not determine the truth values of ‘mixed’ identity statements like ‘$ #F $ = Caesar’. This is the Caesar Problem (CP). Still, neologicists such as Hale and Wright argue that (1) HP is a priori, and (2) HP introduces the pure sortal concept Number. We argue that Neologicism faces a Caesar-problem Problem (CPP): if neologicists solve CP by establishing that ‘$ #Fneq $ Caesar’ is true, (1) and (2) cannot be retained simultaneously. We examine various responses neologicists might provide and show that they do not address CPP. We conclude that CP uncovers a fatal tension in Neologicism.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"28 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143546334","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}
Michael Dummett argues that acceptance of potentially infinite collections requires that we abandon classical logic and restrict ourselves to intuitionistic logic. In this paper we examine whether Dummett is correct. After developing two detailed accounts of what, exactly, it means for a concept to be potentially infinite (based on ideas due to Charles McCarty and Øystein Linnebo, respectively), we construct a Kripke structure that contains a natural number structure that satisfies both accounts. This model supports a logic much stronger than intuitionistic logic, demonstrating that Dummett was wrong. We conclude by briefly examining ways to extend the account(s) in question to indefinitely extensible concepts such as Cardinal, Ordinal, and Set.
Michael Dummett认为，接受潜在的无限集合要求我们放弃经典逻辑，并将自己限制在直觉逻辑中。本文将检验达米特的观点是否正确。在发展了两个详细的描述之后，确切地说，一个概念是潜在无限的意思（分别基于查尔斯·麦卡蒂和Øystein Linnebo的想法），我们构建了一个Kripke结构，它包含一个满足两种解释的自然数结构。这个模型支持的逻辑比直觉逻辑强得多，证明达米特错了。最后，我们简要地考察了将所讨论的帐户扩展到无限可扩展的概念（如基数、序数和集合）的方法。
{"title":"The Logic of Potential Infinity","authors":"Roy T Cook","doi":"10.1093/philmat/nkae022","DOIUrl":"https://doi.org/10.1093/philmat/nkae022","url":null,"abstract":"Michael Dummett argues that acceptance of potentially infinite collections requires that we abandon classical logic and restrict ourselves to intuitionistic logic. In this paper we examine whether Dummett is correct. After developing two detailed accounts of what, exactly, it means for a concept to be potentially infinite (based on ideas due to Charles McCarty and Øystein Linnebo, respectively), we construct a Kripke structure that contains a natural number structure that satisfies both accounts. This model supports a logic much stronger than intuitionistic logic, demonstrating that Dummett was wrong. We conclude by briefly examining ways to extend the account(s) in question to indefinitely extensible concepts such as Cardinal, Ordinal, and Set.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"116 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142825480","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}
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}
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