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Did Gentzen Prove the Consistency of Arithmetic? 根岑证明了算术的一致性吗？
IF 1.1 1区 哲学
Philosophia Mathematica Pub Date : 2026-10-13 DOI: 10.1093/philmat/nkag018
Daniel Waxman
{"title":"Did Gentzen Prove the Consistency of Arithmetic?","authors":"Daniel Waxman","doi":"10.1093/philmat/nkag018","DOIUrl":"https://doi.org/10.1093/philmat/nkag018","url":null,"abstract":"Abstract In 1936 Gentzen gave a proof of the consistency of Peano Arithmetic. While its validity is not in doubt, this paper asks whether it is a proof in the epistemic sense — an argument that can generate or enhance justification for its conclusion. Gentzen claimed it was a ‘real vindication’ of arithmetic, while many later critics judged it circular or inert. I argue that the truth lies in between: the proof is non-trivial, yet falls short of a real vindication of the consistency of arithmetic. Along the way, I provide a framework for assessing the epistemic status of consistency proofs.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"31 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149210629","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
The Mathematical Infinite as an Illusion 数学上的无限是一种幻觉
IF 1.1 1区 哲学
Philosophia Mathematica Pub Date : 2026-10-06 DOI: 10.1093/philmat/nkag016
Graham Priest
{"title":"The Mathematical Infinite as an Illusion","authors":"Graham Priest","doi":"10.1093/philmat/nkag016","DOIUrl":"https://doi.org/10.1093/philmat/nkag016","url":null,"abstract":"Abstract The paper describes and explains (though it does not endorse) a view of mathematical reality to the effect that: (i) there is only a finite number of things; (ii) that there is an infinite number of things is, in a sense to be made clear, an illusion; (ii) paradox is an integral aspect of the illusion. The view is made possible by an important technical result in the model-theory of paraconsistent logic: the Collapsing Lemma. The paper ends with a few comments on the relevance of this view to the possibility of a finitistic view of the physical universe.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"10 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149092921","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
Vopěnka’s Alternative Set Theory as a Framework for Feasible Numbers 作为可行数框架的voponnka的可选集理论
IF 1.1 1区 哲学
Philosophia Mathematica Pub Date : 2026-10-06 DOI: 10.1093/philmat/nkag015
Zuzana Haniková
{"title":"Vopěnka’s Alternative Set Theory as a Framework for Feasible Numbers","authors":"Zuzana Haniková","doi":"10.1093/philmat/nkag015","DOIUrl":"https://doi.org/10.1093/philmat/nkag015","url":null,"abstract":"Abstract This paper takes feasible numbers in Yessenin-Volpin’s work and assesses how this notion could be modelled in Vopěnka’s Alternative Set Theory. The route explored also makes an attempt to model Dummett’s weakly infinite, weakly finite totalities, the coherence of which he took to be decisive of Yessenin-Volpin’s foundational conception. The outcome turns out to be negative: Alternative Set Theory can be taken as a model of some, but not quite all features of Yessenin-Volpin’s feasible numbers. Given that only one particular approach is explored here, the outcome has limited bearing on general questions surrounding ultrafinitism or modelling feasible numbers.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"27 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149092786","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 Road to Mathematical Content 通往数学内容之路
IF 1.1 1区 哲学
Philosophia Mathematica Pub Date : 2026-08-11 DOI: 10.1093/philmat/nkag014
Zeynep Soysal
{"title":"A Road to Mathematical Content","authors":"Zeynep Soysal","doi":"10.1093/philmat/nkag014","DOIUrl":"https://doi.org/10.1093/philmat/nkag014","url":null,"abstract":"Abstract Kennedy develops formalism freeness and logical entanglement to study the relation between informal mathematical contents and the formalisms that attempt to capture them, suggesting that these notions can lay down a road to content. I reconstruct Kennedy’s outlook and argue that whether logical entanglement signals a breakdown in the adequacy of formalisms depends in part on the role of formalization. I then use a language-first perspective to clarify this dependence, vindicate Kennedy’s claim that some mathematics must be done in informal language, and show how her notions can contribute to an account of content.","PeriodicalId":49004,"journal":{"name":"Philosophia Mathematica","volume":"29 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148899045","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
Comparing Anti-foundation Axioms by Comparing Identity Conditions for Sets 通过比较集合的单位条件比较反基础公理
IF 1.1 1区 哲学
Philosophia Mathematica Pub Date : 2025-11-16 DOI: 10.1093/philmat/nkaf022
Daheng Ju, Hang Qi Jing
{"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}
引用次数: 0
Martin Davis: An Overview of his Work in Logic, Computer Science, and Philosophy Martin Davis：概述他在逻辑、计算机科学和哲学方面的工作
IF 1.1 1区 哲学
Philosophia Mathematica Pub Date : 2025-10-15 DOI: 10.1093/philmat/nkaf016
Liesbeth De Mol, Yuri V Matiyasevich, Eugenio G Omodeo, Alberto Policriti, Wilfried Sieg, Elaine J Weyuker
{"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}
引用次数: 0
Charles Parsons April 13, 1933 – April 19, 2024 查尔斯·帕森斯1933年4月13日- 2024年4月19日
IF 1.1 1区 哲学
Philosophia Mathematica Pub Date : 2025-10-02 DOI: 10.1093/philmat/nkaf018
Emily Carson, Øystein Linnebo, Gila Sher, Wilfried Sieg, Mark van Atten
{"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}
引用次数: 0
What are Extremal Axioms? 什么是极值公理？
IF 1.1 1区 哲学
Philosophia Mathematica Pub Date : 2025-10-02 DOI: 10.1093/philmat/nkaf020
Nicola Bonatti
{"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}
引用次数: 0
Numbers, Kinds, and the Identification Problem 数量、种类和识别问题
IF 1.1 1区 哲学
Philosophia Mathematica Pub Date : 2025-08-30 DOI: 10.1093/philmat/nkaf015
Eric Snyder
{"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}
引用次数: 0
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
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