On the one hand, I show that the later Wittgenstein's practice‐based approach to meaning, including the idea that the meaningfulness of mathematics ultimately is rooted in the everyday ‘applications’ it emerged from, as well as his insistence on the variability in and contingency of mathematical and mathematics‐like practices, foreshadows more recent work in Philosophy of Mathematical Practice (PMP), although Wittgenstein's approach was more radically practice‐based than what is prevalent in present‐day PMP. On the other hand, I also show that Wittgenstein's work on mathematics is driven by a pervasive critical attitude towards some key characteristics (monism, foundationalism and exceptionalism) of mainstream philosophical discourse about mathematics (then and now), which is mostly absent from present‐day PMP, and I argue that this aspect of Wittgenstein's work on mathematics could inspire present‐day practitioners of PMP to rethink the identity of their discipline.
{"title":"Wittgenstein's critical Philosophy of Mathematical Practice","authors":"Frank Scheppers","doi":"10.1111/phin.12442","DOIUrl":"https://doi.org/10.1111/phin.12442","url":null,"abstract":"On the one hand, I show that the later Wittgenstein's practice‐based approach to meaning, including the idea that the meaningfulness of mathematics ultimately is rooted in the everyday ‘applications’ it emerged from, as well as his insistence on the variability in and contingency of mathematical and mathematics‐like practices, foreshadows more recent work in Philosophy of Mathematical Practice (PMP), although Wittgenstein's approach was more radically practice‐based than what is prevalent in present‐day PMP. On the other hand, I also show that Wittgenstein's work on mathematics is driven by a pervasive critical attitude towards some key characteristics (monism, foundationalism and exceptionalism) of mainstream philosophical discourse about mathematics (then and now), which is mostly absent from present‐day PMP, and I argue that this aspect of Wittgenstein's work on mathematics could inspire present‐day practitioners of PMP to rethink the identity of their discipline.","PeriodicalId":47112,"journal":{"name":"PHILOSOPHICAL INVESTIGATIONS","volume":"877 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142251618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In what way does a mathematical proof depend on the notation used in its presentation? This paper examines this question by analysing the computational differences, in the sense of Larkin and Simon's ‘Why a diagram is (sometimes) worth 10,000 words’, between diagrammatic and sentential notations as a means for presenting geometric proofs. Wittgenstein takes up the question of mathematical notation and proof in Section III of Remarks on the Foundations of Mathematics. After discussing his observations on a proof's ‘characteristic visual shape’ in Section III with respect to arithmetical proofs, the paper shows how the notion of a characteristic visual shape illuminates the special effectiveness of diagrammatic notation in geometry.
数学证明在哪些方面取决于它的表达方式？本文通过分析拉金和西蒙的 "为什么一张图（有时）胜过一万字"（Why a diagram is (sometimes) worth 10,000 words）一文中的计算差异，来探讨这个问题。维特根斯坦在《关于数学基础的评论》第三节中讨论了数学符号和证明的问题。本文讨论了他在第三节中针对算术证明对证明的 "特征视觉形状 "的观察，然后说明了特征视觉形状的概念如何揭示了图解符号在几何中的特殊功效。
{"title":"Geometric diagrams as an effective notation","authors":"John Mumma","doi":"10.1111/phin.12440","DOIUrl":"https://doi.org/10.1111/phin.12440","url":null,"abstract":"In what way does a mathematical proof depend on the notation used in its presentation? This paper examines this question by analysing the computational differences, in the sense of Larkin and Simon's ‘Why a diagram is (sometimes) worth 10,000 words’, between diagrammatic and sentential notations as a means for presenting geometric proofs. Wittgenstein takes up the question of mathematical notation and proof in Section III of <jats:italic>Remarks on the Foundations of Mathematics</jats:italic>. After discussing his observations on a proof's ‘characteristic visual shape’ in Section III with respect to arithmetical proofs, the paper shows how the notion of a characteristic visual shape illuminates the special effectiveness of diagrammatic notation in geometry.","PeriodicalId":47112,"journal":{"name":"PHILOSOPHICAL INVESTIGATIONS","volume":"209 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142251619","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Language, Mind and Value By SeverinSchroeder, London: Anthem. 2024","authors":"Reshef Agam‐Segal","doi":"10.1111/phin.12443","DOIUrl":"https://doi.org/10.1111/phin.12443","url":null,"abstract":"","PeriodicalId":47112,"journal":{"name":"PHILOSOPHICAL INVESTIGATIONS","volume":"36 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The mature Wittgenstein's groundbreaking analyses of sense and the logical must—and the powerful new method that made them possible—were the result of a multi‐year process of writing, re‐arranging, re‐writing and one large‐scale revision that eventually produced the Philosophical Investigations and RFM I. In contrast, his struggles during the same period with questions of arithmetic and higher mathematics remained largely in first‐draft form, and he drops the topic entirely after 1945. In this paper, I argue that Wittgenstein's new method can be applied to the cases of arithmetic and set theory and that the result is innovative, recognizably Wittgensteinian, and independently appealing. I conclude by acknowledging the reasons Wittgenstein himself might have had to resist applying his own proven method to the case of mathematics—particularly to set theory—and by indicating why I think those reasons are ultimately unsound.
{"title":"Wittgenstein on mathematics","authors":"Penelope Maddy","doi":"10.1111/phin.12441","DOIUrl":"https://doi.org/10.1111/phin.12441","url":null,"abstract":"The mature Wittgenstein's groundbreaking analyses of sense and the logical must—and the powerful new method that made them possible—were the result of a multi‐year process of writing, re‐arranging, re‐writing and one large‐scale revision that eventually produced the <jats:italic>Philosophical Investigations</jats:italic> and <jats:italic>RFM</jats:italic> I. In contrast, his struggles during the same period with questions of arithmetic and higher mathematics remained largely in first‐draft form, and he drops the topic entirely after 1945. In this paper, I argue that Wittgenstein's new method can be applied to the cases of arithmetic and set theory and that the result is innovative, recognizably Wittgensteinian, and independently appealing. I conclude by acknowledging the reasons Wittgenstein himself might have had to resist applying his own proven method to the case of mathematics—particularly to set theory—and by indicating why I think those reasons are ultimately unsound.","PeriodicalId":47112,"journal":{"name":"PHILOSOPHICAL INVESTIGATIONS","volume":"54 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper challenges a dualistic picture popularised by Nigel Pleasants at the centre of influential investigations into the possibility of Wittgensteinian forms of moral certainty. The dualistic picture takes it for granted that moral certainty concerns both a series of hinge propositions that are beyond doubt, make no sense to justify and cannot be expressed in ordinary discourse and a phenomenon that is only ever instantiated in our ways of acting. I consider tensions in this account as they relate to ‘Murder is wrong’ as a moral hinge proposition by drawing on the life and lyrics of US hip‐hop artist Kendrick Lamar. My claim is that the Lamar example highlights important tensions in our relation to the wrongness of murder that many would understand to occur within the moral realm, but which are conceptually ruled out from dualistic accounts of moral certainty. The plausibility of the dualistic picture must therefore be reassessed.
{"title":"Is the wrongness of murder a universal moral hinge?","authors":"Ryan Manhire","doi":"10.1111/phin.12439","DOIUrl":"https://doi.org/10.1111/phin.12439","url":null,"abstract":"This paper challenges a dualistic picture popularised by Nigel Pleasants at the centre of influential investigations into the possibility of Wittgensteinian forms of moral certainty. The dualistic picture takes it for granted that moral certainty concerns both a series of hinge propositions that are beyond doubt, make no sense to justify and cannot be expressed in ordinary discourse and a phenomenon that is only ever instantiated in our ways of acting. I consider tensions in this account as they relate to ‘Murder is wrong’ as a moral hinge proposition by drawing on the life and lyrics of US hip‐hop artist Kendrick Lamar. My claim is that the Lamar example highlights important tensions in our relation to the wrongness of murder that many would understand to occur within the moral realm, but which are conceptually ruled out from dualistic accounts of moral certainty. The plausibility of the dualistic picture must therefore be reassessed.","PeriodicalId":47112,"journal":{"name":"PHILOSOPHICAL INVESTIGATIONS","volume":"57 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183329","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Wittgenstein and the logical positivists attempted to explain logical necessity in terms of linguistic conventions. It is often thought that their respective accounts have been conclusively refuted by objections from Quine, Dummett and others. We argue that this verdict is premature. Several of the most popular anti‐conventionalist arguments fail, partly because they misconstrue the idea of truth by convention in Wittgenstein and/or logical positivism. Correctly understood, conventionalism claims that, given certain linguistic conventions, some sentences are unconditionally true, that is true irrespective of how the world happens to be. This claim is difficult to deny, and the corresponding conventionalism about logical necessity remains a viable position.
{"title":"Logic and conventions","authors":"Kai Michael Büttner, Hans‐Johann Glock","doi":"10.1111/phin.12437","DOIUrl":"https://doi.org/10.1111/phin.12437","url":null,"abstract":"Wittgenstein and the logical positivists attempted to explain logical necessity in terms of linguistic conventions. It is often thought that their respective accounts have been conclusively refuted by objections from Quine, Dummett and others. We argue that this verdict is premature. Several of the most popular anti‐conventionalist arguments fail, partly because they misconstrue the idea of truth by convention in Wittgenstein and/or logical positivism. Correctly understood, conventionalism claims that, given certain linguistic conventions, some sentences are unconditionally true, that is true irrespective of how the world happens to be. This claim is difficult to deny, and the corresponding conventionalism about logical necessity remains a viable position.","PeriodicalId":47112,"journal":{"name":"PHILOSOPHICAL INVESTIGATIONS","volume":"36 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183334","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Wittgenstein was hostile towards set theory; see his remark in §22 of RFM II: ‘I believe and hope that a future generation will laugh at this hocus pocus’. At the same time, he says that what philosophy owes set theory is ‘tremendous’ and that this is something ‘deep’. I want to clarify these two statements and to reconcile them. The hocus‐pocus remark is mainly directed at the temptation to talk about an own world of sets, focussing on extensions and thereby forgetting the actual mathematical manoeuvres leading to them. Wittgenstein does not shy away from using set‐theoretical symbols and expressions, but he regularly deprives them of their genuine set‐theoretical character. When speaking of the ‘depth’ we encounter with regard to set theory, what he means is the depth explained in PI §111: It arises ‘through a misinterpretation of our forms of language’, and this is particularly so in set theory.
{"title":"Wittgenstein and set theory","authors":"Felix Mühlhölzer","doi":"10.1111/phin.12435","DOIUrl":"https://doi.org/10.1111/phin.12435","url":null,"abstract":"Wittgenstein was hostile towards set theory; see his remark in §22 of RFM II: ‘I believe and hope that a future generation will laugh at this hocus pocus’. At the same time, he says that what philosophy owes set theory is ‘tremendous’ and that this is something ‘deep’. I want to clarify these two statements and to reconcile them. The hocus‐pocus remark is mainly directed at the temptation to talk about an own world of sets, focussing on extensions and thereby forgetting the actual mathematical manoeuvres leading to them. Wittgenstein does not shy away from using set‐theoretical symbols and expressions, but he regularly deprives them of their genuine set‐theoretical character. When speaking of the ‘depth’ we encounter with regard to set theory, what he means is the depth explained in PI §111: It arises ‘through a misinterpretation of our forms of language’, and this is particularly so in set theory.","PeriodicalId":47112,"journal":{"name":"PHILOSOPHICAL INVESTIGATIONS","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The status of mathematical facts has long been taken to be unclear in Wittgenstein's philosophy of mathematics, and often, it seems that he wants to eliminate mathematical facts in favour of facts about our beliefs or behaviour. In this paper, I argue that by reading Wittgenstein as a radical conventionalist, we can give a reading of the relevant passages according to which Wittgenstein doesn't deny that there are mathematical facts, but rather denies that one needs a metaphysical account of what mathematical facts are and how they relate to the world that goes beyond the minimal claims of radical conventionalism—that empirical facts about how we would find natural to project our training into new cases and the constitution of concepts by our agreement are enough. At the end of the paper, I discuss the implications of this reading on how to understand a rule's determination of its own application.
{"title":"Wittgenstein on mathematical facts","authors":"Ásgeir Berg","doi":"10.1111/phin.12438","DOIUrl":"https://doi.org/10.1111/phin.12438","url":null,"abstract":"The status of mathematical facts has long been taken to be unclear in Wittgenstein's philosophy of mathematics, and often, it seems that he wants to eliminate mathematical facts in favour of facts about our beliefs or behaviour. In this paper, I argue that by reading Wittgenstein as a radical conventionalist, we can give a reading of the relevant passages according to which Wittgenstein doesn't deny that there are mathematical facts, but rather denies that one needs a metaphysical account of what mathematical facts are and how they relate to the world that goes beyond the minimal claims of radical conventionalism—that empirical facts about how we would find natural to project our training into new cases and the constitution of concepts by our agreement are enough. At the end of the paper, I discuss the implications of this reading on how to understand a rule's determination of its own application.","PeriodicalId":47112,"journal":{"name":"PHILOSOPHICAL INVESTIGATIONS","volume":"33 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper offers an evaluation of Wittgenstein's critique of Cantorian set theory, illustrating his broader philosophical stance on mathematics. By emphasizing the constructed nature of mathematical theories, Wittgenstein encourages a reflective approach to mathematics that acknowledges human agency in its development. His engagement with Cantorian set theory provides valuable insights into the philosophical and practical dimensions of mathematics, urging a reconsideration of its foundations and the nature of mathematical proofs. This perspective aligns closely with the philosophy of mathematical practice, which emphasizes the human side of mathematics by its focus on mathematical practice.
{"title":"Wittgenstein in Cantor's paradise","authors":"Karim Zahidi","doi":"10.1111/phin.12436","DOIUrl":"https://doi.org/10.1111/phin.12436","url":null,"abstract":"This paper offers an evaluation of Wittgenstein's critique of Cantorian set theory, illustrating his broader philosophical stance on mathematics. By emphasizing the constructed nature of mathematical theories, Wittgenstein encourages a reflective approach to mathematics that acknowledges human agency in its development. His engagement with Cantorian set theory provides valuable insights into the philosophical and practical dimensions of mathematics, urging a reconsideration of its foundations and the nature of mathematical proofs. This perspective aligns closely with the philosophy of mathematical practice, which emphasizes the human side of mathematics by its focus on mathematical practice.","PeriodicalId":47112,"journal":{"name":"PHILOSOPHICAL INVESTIGATIONS","volume":"52 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Anxiety and wonder: On being human By MariaBalaska, London, UK: Bloomsbury. 2024","authors":"Oscar Davis","doi":"10.1111/phin.12434","DOIUrl":"https://doi.org/10.1111/phin.12434","url":null,"abstract":"","PeriodicalId":47112,"journal":{"name":"PHILOSOPHICAL INVESTIGATIONS","volume":"6 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183387","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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