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Added by: Fenner Stanley TanswellAbstract:
Over a period of more than 30 years, more than 100 mathematicians worked on a project to classify mathematical objects known as finite simple groups. The Classification, when officially declared completed in 1981, ranged between 300 and 500 articles and ran somewhere between 5,000 and 10,000 journal pages. Mathematicians have hailed the project as one of the greatest mathematical achievements of the 20th century, and it surpasses, both in scale and scope, any other mathematical proof of the 20th century. The history of the Classification points to the importance of face-to-face interaction and close teaching relationships in the production and transformation of theoretical knowledge. The techniques and methods that governed much of the work in finite simple group theory circulated via personal, often informal, communication, rather than in published proofs. Consequently, the printed proofs that would constitute the Classification Theorem functioned as a sort of shorthand for and formalization of proofs that had already been established during personal interactions among mathematicians. The proof of the Classification was at once both a material artifact and a crystallization of one community’s shared practices, values, histories, and expertise. However, beginning in the 1980s, the original proof of the Classification faced the threat of ‘uninvention’. The papers that constituted it could still be found scattered throughout the mathematical literature, but no one other than the dwindling community of group theorists would know how to find them or how to piece them together. Faced with this problem, finite group theorists resolved to produce a ‘second-generation proof’ to streamline and centralize the Classification. This project highlights that the proof and the community of finite simple groups theorists who produced it were co-constitutive–one formed and reformed by the other.Martin, Ursula, Pease, Alison. Mathematical Practice, Crowdsourcing, and Social Machines2013, in Intelligent Computer Mathematics. CICM 2013. Lecture Notes in Computer Sciences, Carette, J. et al. (eds.). Springer.-
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Added by: Fenner Stanley TanswellAbstract:
The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. Crowdsourcing pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers check proofs too long and complicated for humans to comprehend. The Study of Mathematical Practice is an emerging interdisciplinary field which draws on philosophy and social science to understand how mathematics is produced. Online mathematical activity provides a novel and rich source of data for empirical investigation of mathematical practice - for example the community question-answering system mathoverflow contains around 40,000 mathematical conversations, and polymath collaborations provide transcripts of the process of discovering proofs. Our preliminary investigations have demonstrated the importance of “soft” aspects such as analogy and creativity, alongside deduction and proof, in the production of mathematics, and have given us new ways to think about the roles of people and machines in creating new mathematical knowledge. We discuss further investigation of these resources and what it might reveal. Crowdsourced mathematical activity is an example of a “social machine”, a new paradigm, identified by Berners-Lee, for viewing a combination of people and computers as a single problem-solving entity, and the subject of major international research endeavours. We outline a future research agenda for mathematics social machines, a combination of people, computers, and mathematical archives to create and apply mathematics, with the potential to change the way people do mathematics, and to transform the reach, pace, and impact of mathematics research.Comment (from this Blueprint): In this paper, Martin and Pease look at how mathematics happens online, emphasising how this embodies the picture of mathematics given by Polya and Lakatos, two central figures in philosophy of mathematical practice. They look at multiple venues of online mathematics, including the polymath projects of collaborative problem-solving, and mathoverflow, which is a question-and-answer forum. By looking at the discussions that take place when people are doing maths online, they argue that you can get rich new kinds of data about the processes of mathematical discovery and understanding. They discuss how online mathematics can become a “social machine”, and how this can open up new ways of doing mathematics.
Francois, Karen, Vandendriessche, Eric. Reassembling Mathematical Practices: a Philosophical-Anthropological Approach2016, Revista Latinoamericana de Etnomatemática Perspectivas Socioculturales de la Educación Matemática, 9(2): 144-167.-
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Added by: Fenner Stanley TanswellAbstract:
In this paper we first explore how Wittgenstein’s philosophy provides a conceptual tools to discuss the possibility of the simultaneous existence of culturally different mathematical practices. We will argue that Wittgenstein’s later work will be a fruitful framework to serve as a philosophical background to investigate ethnomathematics (Wittgenstein 1973). We will give an overview of Wittgenstein’s later work which is referred to by many researchers in the field of ethnomathematics. The central philosophical investigation concerns Wittgenstein’s shift to abandoning the essentialist concept of language and therefore denying the existence of a universal language. Languages—or ‘language games’ as Wittgenstein calls them—are immersed in a form of life, in a cultural or social formation and are embedded in the totality of communal activities. This gives rise to the idea of rationality as an invention or as a construct that emerges in specific local contexts. In the second part of the paper we introduce, analyse and compare the mathematical aspects of two activities known as string figure-making and sand drawing, to illustrate Wittgenstein’s ideas. Based on an ethnomathematical comparative analysis, we will argue that there is evidence of invariant and distinguishing features of a mathematical rationality, as expressed in both string figure-making and sand drawing practices, from one society to another. Finally, we suggest that a philosophical-anthropological approach to mathematical practices may allow us to better understand the interrelations between mathematics and cultures. Philosophical investigations may help the reflection on the possibility of culturally determined ethnomathematics, while an anthropological approach, using ethnographical methods, may afford new materials for the analysis of ethnomathematics and its links to the cultural context. This combined approach will help us to better characterize mathematical practices in both sociological and epistemological terms.Comment (from this Blueprint): Francois and Vandendriessche here present a later Wittgensteinian approach to “ethnomathematics”: mathematics practiced outside of mainstream Western contexts, often focused on indigenous or tribal groups. They focus on two case studies, string-figure making and sand-drawing, in different geographic and cultural contexts, looking at how these practices are mathematical.
Carter, Jessica. Diagrams and Proofs in Analysis2010, International Studies in the Philosophy of Science, 24(1): 1-14.-
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Added by: Fenner Stanley TanswellAbstract:
This article discusses the role of diagrams in mathematical reasoning in the light of a case study in analysis. In the example presented certain combinatorial expressions were first found by using diagrams. In the published proofs the pictures were replaced by reasoning about permutation groups. This article argues that, even though the diagrams are not present in the published papers, they still play a role in the formulation of the proofs. It is shown that they play a role in concept formation as well as representations of proofs. In addition we note that 'visualization' is used in two different ways. In the first sense 'visualization' denotes our inner mental pictures, which enable us to see that a certain fact holds, whereas in the other sense 'visualization' denotes a diagram or representation of something.Comment (from this Blueprint): In this paper, Carter discusses a case study from free probability theory in which diagrams were used to inspire definitions and proof strategies. Interestingly, the diagrams were not present in the published results making them dispensable in one sense, but Carter argues that they are essential in the sense that their discovery relied on the visualisation supplied by the diagrams.
McCallum, Kate. Untangling Knots: Embodied Diagramming Practices in Knot Theory2019, Journal of Humanistic Mathematics, 9(1): 178-199.-
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Added by: Fenner Stanley TanswellAbstract:
The low visibility and specialised languages of mathematical work pose challenges for the ethnographic study of communication in mathematics, but observation-based study can offer a real-world grounding to questions about the nature of its methods. This paper uses theoretical ideas from linguistic pragmatics to examine how mutual understandings of diagrams are achieved in the course of conference presentations. Presenters use shared knowledge to train others to interpret diagrams in the ways favoured by the community of experts, directing an audience’s attention so as to develop a shared understanding of a diagram’s features and possible manipulations. In this way, expectations about the intentions of others and appeals to knowledge about the manipulation of objects play a part in the development and communication of concepts in mathematical discourse.Comment (from this Blueprint): McCallum is an ethnographer and artist, who in this piece explores the way in which mathematicians use diagrams in conference presentations, especially in knot theory. She emphasises that there are a large number of ways that diagrams can facilitate communication and understanding. The diagrams are dynamic in many way, and she shows how the way in which a speaker interacts with the diagram (through drawing, erasing, labelling, positioning, emphasising etc.) is part of explaining the mathematics it represents.
De Toffoli, Silvia, Giardino, Valeria. An Inquiry into the Practice of Proving in Low-Dimensional Topology2015, in From Logic to Practice, Gabriele Lolli, Giorgio Venturi and Marco Panza (eds.). Springer International Publishing.-
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Added by: Fenner Stanley TanswellAbstract:
The aim of this article is to investigate specific aspects connected with visualization in the practice of a mathematical subfield: low-dimensional topology. Through a case study, it will be established that visualization can play an epistemic role. The background assumption is that the consideration of the actual practice of mathematics is relevant to address epistemological issues. It will be shown that in low-dimensional topology, justifications can be based on sequences of pictures. Three theses will be defended. First, the representations used in the practice are an integral part of the mathematical reasoning. As a matter of fact, they convey in a material form the relevant transitions and thus allow experts to draw inferential connections. Second, in low-dimensional topology experts exploit a particular type of manipulative imagination which is connected to intuition of two- and three-dimensional space and motor agency. This imagination allows recognizing the transformations which connect different pictures in an argument. Third, the epistemic—and inferential—actions performed are permissible only within a specific practice: this form of reasoning is subject-matter dependent. Local criteria of validity are established to assure the soundness of representationally heterogeneous arguments in low-dimensional topology.Comment (from this Blueprint): De Toffoli and Giardino look at proof practices in low-dimensional topology, and especially a proof by Rolfsen that relies on epistemic actions on a diagrammatic representation. They make the case that the many diagrams are used to trigger our manipulative imagination to make inferential moves which cannot be reduced to formal statements without loss of intuition.
Secco, Gisele Dalva, Pereira, Luiz Carlos. Proofs Versus Experiments: Wittgensteinian Themes Surrounding the Four-Color Theorem2017, in How Colours Matter to Philosophy, Marcos Silva (ed.). Springer, Cham.-
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Added by: Fenner Stanley TanswellAbstract:
The Four-Colour Theorem (4CT) proof, presented to the mathematical community in a pair of papers by Appel and Haken in the late 1970's, provoked a series of philosophical debates. Many conceptual points of these disputes still require some elucidation. After a brief presentation of the main ideas of Appel and Haken’s procedure for the proof and a reconstruction of Thomas Tymoczko’s argument for the novelty of 4CT’s proof, we shall formulate some questions regarding the connections between the points raised by Tymoczko and some Wittgensteinian topics in the philosophy of mathematics such as the importance of the surveyability as a criterion for distinguishing mathematical proofs from empirical experiments. Our aim is to show that the “characteristic Wittgensteinian invention” (Mühlhölzer 2006) – the strong distinction between proofs and experiments – can shed some light in the conceptual confusions surrounding the Four-Colour Theorem.Comment (from this Blueprint): Secco and Pereira discuss the famous proof of the Four Colour Theorem, which involved the essential use of a computer to check a huge number of combinations. They look at whether this constitutes a real proof or whether it is more akin to a mathematical experiment, a distinction that they draw from Wittgenstein.
Mihaljević, Helena, Santamaría, Lucía. Authorship in top-ranked mathematical and physical journals: Role of gender on self-perceptions and bibliographic evidence2020, Quantitative Science Studies, 1(4): 1468-1492.-
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Added by: Fenner Stanley TanswellAbstract:
Despite increasing rates of women researching in math-intensive fields, publications by female authors remain underrepresented. By analyzing millions of records from the dedicated bibliographic databases zbMATH, arXiv, and ADS, we unveil the chronological evolution of authorships by women in mathematics, physics, and astronomy. We observe a pronounced shortage of female authors in top-ranked journals, with quasistagnant figures in various distinguished periodicals in the first two disciplines and a significantly more equitable situation in the latter. Additionally, we provide an interactive open-access web interface to further examine the data. To address whether female scholars submit fewer articles for publication to relevant journals or whether they are consciously or unconsciously disadvantaged by the peer review system, we also study authors’ perceptions of their submission practices and analyze around 10,000 responses, collected as part of a recent global survey of scientists. Our analysis indicates that men and women perceive their submission practices to be similar, with no evidence that a significantly lower number of submissions by women is responsible for their underrepresentation in top-ranked journals. According to the self-reported responses, a larger number of articles submitted to prestigious venues correlates rather with aspects associated with pronounced research activity, a well-established network, and academic seniority.Comment (from this Blueprint): Mihaljević and Santamaría here use large-scale quantitative research methods to investigate the gender gap in contemporary mathematics. I’ve recommended reading the introduction and conclusion in order to see what they were doing and what they found out, but the rest of the paper is worth looking at if you want more detailed methods and results.
Barrow-Green, June. Historical Context of the Gender Gap in Mathematics2019, in World Women in Mathematics 2018: Proceedings of the First World Meeting for Women in Mathematics, Carolina Araujo et al. (eds.). Springer, Cham.-
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Added by: Fenner Stanley TanswellAbstract:
This chapter is based on the talk that I gave in August 2018 at the ICM in Rio de Janeiro at the panel on The Gender Gap in Mathematical and Natural Sciences from a Historical Perspective. It provides some examples of the challenges and prejudices faced by women mathematicians during last two hundred and fifty years. I make no claim for completeness but hope that the examples will help to shed light on some of the problems many women mathematicians still face today.Comment (from this Blueprint): Barrow-Green is a historian of mathematics. In this paper she documents some of the challenges that women faced in mathematics over the last 250 years, discussing many famous women mathematicians and the prejudices and injustices they faced.
Schattschneider, Doris. Marjorie Rice (16 February 1923–2 July 2017)2018, Journal of Mathematics and the Arts, 12(1): 51-54.-
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Added by: Fenner Stanley TanswellAbstract:
Marjorie Jeuck Rice, a most unlikely mathematician, died on 2 July 2017 at the age of 94. She was born on 16 February 1923 in St. Petersburg, Florida, and raised on a tiny farm near Roseburg in southern Oregon. There she attended a one-room country school, and there her scientific interests were awakened and nourished by two excellent teachers who recognized her talent. She later wrote, ‘Arithmetic was easy and I liked to discover the reasons behind the methods we used.… I was interested in the colors, patterns, and designs of nature and dreamed of becoming an artist’?Comment (from this Blueprint): Easwaran discusses the case of Marjorie Rice, an amateur mathematician who discovered new pentagon tilings. This obituary gives some details of her life and the discovery.
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Steingart, Alma. A Group Theory of Group Theory: Collaborative Mathematics and the ‘Uninvention’ of a 1000-page Proof
2012, Social Studies of Science, 42(2): 185-213.
Comment (from this Blueprint): Steingart is a sociologist who charts the history and sociology of the development of the extremely large and highly collaborative Classification Theorem. She shows that the proof involved a community deciding on shared values, standards of reliability, expertise, and ways of communicating. For example, the community became tolerant of so-called “local errors” so long as these did not put the main result at risk. Furthermore, Steingart discusses how the proof’s text is distributed across a wide number of places and requires expertise to navigate, leaving the proof in danger of uninvention if the experts retire from mathematics.