In non-well-founded set theory, which anti-foundation axiom is philosophically justified, BAFA, FAFA, SAFA, AFA, or some other one? In this paper, we investigate a general approach to answering this question: first, considering which identity condition for sets is justified; second, considering which anti-foundation axiom it justifies. Specifically, we study in detail two plausible identity conditions, $ text{IC}_{1} $ and $ text{IC}_{2} $: we show that $ text{IC}_{2} $ justifies $ text{FAFA}_{2} $ and $ text{IC}_{1} $ justifies AFA, and argue for AFA by offering an argument for $ text{IC}_{1} $.
{"title":"Comparing Anti-foundation Axioms by Comparing Identity Conditions for Sets","authors":"Daheng Ju, Hang Qi Jing","doi":"10.1093/philmat/nkaf022","DOIUrl":"https://doi.org/10.1093/philmat/nkaf022","url":null,"abstract":"In non-well-founded set theory, which anti-foundation axiom is philosophically justified, BAFA, FAFA, SAFA, AFA, or some other one? In this paper, we investigate a general approach to answering this question: first, considering which identity condition for sets is justified; second, considering which anti-foundation axiom it justifies. Specifically, we study in detail two plausible identity conditions, $ text{IC}_{1} $ and $ text{IC}_{2} $: we show that $ text{IC}_{2} $ justifies $ text{FAFA}_{2} $ and $ text{IC}_{1} $ justifies AFA, and argue for AFA by offering an argument for $ text{IC}_{1} $.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"18 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145525130","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}
Liesbeth De Mol, Yuri V Matiyasevich, Eugenio G Omodeo, Alberto Policriti, Wilfried Sieg, Elaine J Weyuker
In his autobiographical essay written in 1999, ‘From logic to computer science and back’, Martin David Davis (1928 3 8–2023 1 1) indicated that he viewed himself as a logician and a computer scientist. He expanded the essay in 2016 and expressed a new perspective through a changed title, ‘My life as a logician’. He points out that logic was the unifying theme underlying his scientific career. Our paper attempts to provide a consistent vision that illuminates Davis’s successive contributions leading to his landmark writings on computability, unsolvable problems, automated reasoning, as well as the history and philosophy of computing.
马丁·大卫·戴维斯（Martin David Davis， 1928年8月- 2023年11月）在1999年写的自传体文章《从逻辑到计算机科学再回来》中表示，他认为自己既是逻辑学家又是计算机科学家。2016年，他扩展了这篇文章，并通过更改标题“我作为逻辑学家的生活”表达了新的视角。他指出，逻辑是他科学生涯的统一主题。我们的论文试图提供一个一致的愿景，阐明戴维斯的连续贡献，导致他在可计算性，不可解决的问题，自动推理，以及计算的历史和哲学方面的里程碑式的著作。
{"title":"Martin Davis: An Overview of his Work in Logic, Computer Science, and Philosophy","authors":"Liesbeth De Mol, Yuri V Matiyasevich, Eugenio G Omodeo, Alberto Policriti, Wilfried Sieg, Elaine J Weyuker","doi":"10.1093/philmat/nkaf016","DOIUrl":"https://doi.org/10.1093/philmat/nkaf016","url":null,"abstract":"In his autobiographical essay written in 1999, ‘From logic to computer science and back’, Martin David Davis (1928 3 8–2023 1 1) indicated that he viewed himself as a logician and a computer scientist. He expanded the essay in 2016 and expressed a new perspective through a changed title, ‘My life as a logician’. He points out that logic was the unifying theme underlying his scientific career. Our paper attempts to provide a consistent vision that illuminates Davis’s successive contributions leading to his landmark writings on computability, unsolvable problems, automated reasoning, as well as the history and philosophy of computing.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"94 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145295616","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}
Emily Carson, Øystein Linnebo, Gila Sher, Wilfried Sieg, Mark van Atten
Charles Dacre Parsons passed away on April 19, 2024, aged 91. In this obituary, four of his PhD students and one colleague and collaborator discuss, in an order (roughly) determined by the development of Parsons’s career, his engagement with proof theory; Quine; Kant; Brouwer and Gödel; and mathematical structuralism.
{"title":"Charles Parsons April 13, 1933 – April 19, 2024","authors":"Emily Carson, Øystein Linnebo, Gila Sher, Wilfried Sieg, Mark van Atten","doi":"10.1093/philmat/nkaf018","DOIUrl":"https://doi.org/10.1093/philmat/nkaf018","url":null,"abstract":"Charles Dacre Parsons passed away on April 19, 2024, aged 91. In this obituary, four of his PhD students and one colleague and collaborator discuss, in an order (roughly) determined by the development of Parsons’s career, his engagement with proof theory; Quine; Kant; Brouwer and Gödel; and mathematical structuralism.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"90 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145246622","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}
Extremal axioms impose a condition of either minimality or maximality on the admissible models of an axiomatic theory. In this paper, I propose an alternative formulation of arithmetic and real analysis based on extremal axioms. Once properly formulated, the second-order extremal axiom restricts the quantifiers of the theory to the minimal or maximal domain of discourse. It is proved that extremal axioms are logically equivalent to standard assumptions of, respectively, second-order Induction and Archimedean Completeness. Finally, I distinguish between internalist and externalist accounts of mathematical structures as characterized by extremal axioms and their corresponding axiomatic theories.
{"title":"What are Extremal Axioms?","authors":"Nicola Bonatti","doi":"10.1093/philmat/nkaf020","DOIUrl":"https://doi.org/10.1093/philmat/nkaf020","url":null,"abstract":"Extremal axioms impose a condition of either minimality or maximality on the admissible models of an axiomatic theory. In this paper, I propose an alternative formulation of arithmetic and real analysis based on extremal axioms. Once properly formulated, the second-order extremal axiom restricts the quantifiers of the theory to the minimal or maximal domain of discourse. It is proved that extremal axioms are logically equivalent to standard assumptions of, respectively, second-order Induction and Archimedean Completeness. Finally, I distinguish between internalist and externalist accounts of mathematical structures as characterized by extremal axioms and their corresponding axiomatic theories.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"4 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145246620","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}
I defend two theses concerning the semantics of number words, such as ‘two’. First, as nouns, they have taxonomic meanings whereby they describe or refer to kinds. Secondly, since numerical singular terms refer to numbers, if anything, numbers are kinds. Jointly, these two theses have several significant implications for the philosophy of mathematics. For example, they provide a natural and independently motivated resolution to a revenge version of Benacerraf’s Identification Problem.
{"title":"Numbers, Kinds, and the Identification Problem","authors":"Eric Snyder","doi":"10.1093/philmat/nkaf015","DOIUrl":"https://doi.org/10.1093/philmat/nkaf015","url":null,"abstract":"I defend two theses concerning the semantics of number words, such as ‘two’. First, as nouns, they have taxonomic meanings whereby they describe or refer to kinds. Secondly, since numerical singular terms refer to numbers, if anything, numbers are kinds. Jointly, these two theses have several significant implications for the philosophy of mathematics. For example, they provide a natural and independently motivated resolution to a revenge version of Benacerraf’s Identification Problem.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"19 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144920671","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}
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.
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
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