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What Numbers Really Cannot Be and What They Plausibly Are 数字不可能是什么，它们似乎是什么
IF 1.1 1区 哲学
Philosophia Mathematica Pub Date : 2025-07-02 DOI: 10.1093/philmat/nkaf014
Arnon Avron
{"title":"What Numbers Really Cannot Be and What They Plausibly Are","authors":"Arnon Avron","doi":"10.1093/philmat/nkaf014","DOIUrl":"https://doi.org/10.1093/philmat/nkaf014","url":null,"abstract":"We show that structuralism has the very serious defect of having no satisfactory notion of identity which can be associated with its central notion: structure. We also refute the structural thesis about the nature of the natural numbers by showing that there are at least two completely different structures that are entitled to be taken as ‘the structure of the natural numbers’, and any choice between them would arbitrarily favor one of them over the equally legitimate other. Finally, we argue for the high plausibility of the identification of the natural numbers with the finite von Neumann ordinals.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"36 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144547095","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}
引用次数: 0
Choice in the Iterative Conception of Set 集合迭代概念中的选择
IF 1.1 1区 哲学
Philosophia Mathematica Pub Date : 2025-07-02 DOI: 10.1093/philmat/nkaf010
Bruno Jacinto, Beatriz Souza
{"title":"Choice in the Iterative Conception of Set","authors":"Bruno Jacinto, Beatriz Souza","doi":"10.1093/philmat/nkaf010","DOIUrl":"https://doi.org/10.1093/philmat/nkaf010","url":null,"abstract":"The iterative conception (IC) is arguably the best worked out conception of set available. What is the status of the axiom of choice under this conception? Boolos argues that it is not justified by IC. We show that Boolos’s influential argument overgenerates. For, if cogent, it would imply that none of the axioms of ZFC which Boolos took to be justified by IC is so justified. We furthermore show that, to the extent that they are consequences of a plural formulation of stage theory, all those axioms are justified by IC — axiom of choice included.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"4 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144547094","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}
引用次数: 0
Artificial Intelligence and Inherent Mathematical Difficulty 人工智能与固有数学难度
IF 1.1 1区 哲学
Philosophia Mathematica Pub Date : 2025-07-02 DOI: 10.1093/philmat/nkaf005
Walter Dean, Alberto Naibo
{"title":"Artificial Intelligence and Inherent Mathematical Difficulty","authors":"Walter Dean, Alberto Naibo","doi":"10.1093/philmat/nkaf005","DOIUrl":"https://doi.org/10.1093/philmat/nkaf005","url":null,"abstract":"This paper explores the relationship of artificial intelligence to resolving open questions in mathematics. We first argue that limitative results from computability and complexity theory retain their significance in illustrating that proof discovery is an inherently difficult problem. We next consider how applications of automated theorem proving, Sat-solvers, and large language models raise underexplored questions about the nature of mathematical proof — e.g., about the status of brute force and the relationship between logical and discovermental complexity. Nevertheless, we finally suggest that the results obtained thus far by automated methods do not tell against the inherent difficulty of proof discovery.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"15 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144547096","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}
引用次数: 0
Fregean Metasemantics Fregean Metasemantics
IF 1.1 1区 哲学
Philosophia Mathematica Pub Date : 2025-05-23 DOI: 10.1093/philmat/nkaf003
Ori Simchen
{"title":"Fregean Metasemantics","authors":"Ori Simchen","doi":"10.1093/philmat/nkaf003","DOIUrl":"https://doi.org/10.1093/philmat/nkaf003","url":null,"abstract":"How the semantic significance of numerical discourse gets determined is a metasemantic issue par excellence. At the sub-sentential level, the issue is riddled with difficulties on account of the contested metaphysical status of the subject matter of numerical discourse, i.e., numbers and numerical properties and relations. Setting those difficulties aside, I focus instead on the sentential level, specifically, on obvious affinities between whole numerical and non-numerical sentences and how their significance is determined. From such a perspective, Frege’s 1884 construction of number, while famously mathematically untenable, fares better metasemantically than extant alternatives in the philosophy of mathematics.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"24 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144133725","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}
引用次数: 0
Intrinsic Justification for Large Cardinals and Structural Reflection 大基数和结构反射的内在理由
IF 1.1 1区 哲学
Philosophia Mathematica Pub Date : 2025-05-13 DOI: 10.1093/philmat/nkaf006
Joan Bagaria, Claudio Ternullo
{"title":"Intrinsic Justification for Large Cardinals and Structural Reflection","authors":"Joan Bagaria, Claudio Ternullo","doi":"10.1093/philmat/nkaf006","DOIUrl":"https://doi.org/10.1093/philmat/nkaf006","url":null,"abstract":"We deal with the issue of whether large cardinals are intrinsically justified set-theoretic principles (Intrinsicness Issue). To this end, we review, in a systematic fashion, the abstract principles that have been formulated to motivate them and their mathematical expressions, and assess their intrinsic justifiability. A parallel, but closely linked, issue is whether there exist mathematical principles that yield all large cardinals (Universality Issue), and we also test principles for their ability to respond to this issue. Finally, we discuss Structural Reflection Principles and their responses to Intrinsicness and Universality, and also make some further considerations on their general justifiability.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"27 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143945668","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}
引用次数: 0
Neologicism Meets Fiction 新词与小说相遇
IF 1.1 1区 哲学
Philosophia Mathematica Pub Date : 2025-05-12 DOI: 10.1093/philmat/nkaf009
Geoffrey Hellman
{"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}
引用次数: 0
A Triptych on Empirical Philosophy of Mathematics. Part III: How? 实证数学哲学三联画。第三部分：怎么做？
IF 1.1 1区 哲学
Philosophia Mathematica Pub Date : 2025-05-07 DOI: 10.1093/philmat/nkaf004
Deborah Kant, Benedikt Löwe
{"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}
引用次数: 0
How Should We Understand the Modal Potentialist’s Modality? 我们应该如何理解情态潜在者的情态？
IF 1.1 1区 哲学
Philosophia Mathematica Pub Date : 2025-04-13 DOI: 10.1093/philmat/nkaf007
Boaz D Laan
{"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}
引用次数: 0
Inference to the Best Explanation as a Form of Non-Deductive Reasoning in Mathematics 作为数学非演绎推理形式的最佳解释推理
IF 1.1 1区 哲学
Philosophia Mathematica Pub Date : 2025-04-02 DOI: 10.1093/philmat/nkae024
Marc Lange
{"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}
引用次数: 0
A Potentialist Perspective on Intuitionistic Analysis 直觉分析的潜在主义视角
IF 1.1 1区 哲学
Philosophia Mathematica Pub Date : 2025-03-31 DOI: 10.1093/philmat/nkae025
Ethan Brauer
{"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}
引用次数: 0
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