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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.

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Added by: Fenner Stanley TanswellAbstract:
A source of tension between Philosophers of Mathematics and Mathematicians is the fact that each group feels ignored by the other; daily mathematical practice seems barely affected by the questions the Philosophers are considering. In this talk I will describe an issue that does have an impact on mathematical practice, and a philosophical stance on mathematics that is detectable in the work of practising mathematicians. No doubt controversially, I will call this issue ‘morality’, but the term is not of my coining: there are mathematicians across the world who use the word ‘morally’ to great effect in private, and I propose that there should be a public theory of what they mean by this. The issue arises because proofs, despite being revered as the backbone of mathematical truth, often contribute very little to a mathematician’s understanding. ‘Moral’ considerations, however, contribute a great deal. I will first describe what these ‘moral’ considerations might be, and why mathematicians have appropriated the word ‘morality’ for this notion. However, not all mathematicians are concerned with such notions, and I will give a characterisation of ‘moralist’ mathematics and ‘moralist’ mathematicians, and discuss the development of ‘morality’ in individuals and in mathematics as a whole. Finally, I will propose a theory for standardising or universalising a system of mathematical morality, and discuss how this might help in the development of good mathematics.
Comment (from this Blueprint): Cheng is a mathematician working in Category Theory. In this article she complains about traditional philosophy of mathematics that it has no bearing on real mathematics. Instead, she proposes a system of “mathematical morality” about the normative intuitions mathematicians have about how it ought to be.

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Added by: Fenner Stanley TanswellAbstract: During the 1970s and 1980s, a team of Automated Theorem Proving researchers at the Argonne National Laboratory near Chicago developed the Automated Reasoning Assistant, or AURA, to assist human users in the search for mathematical proofs. The resulting hybrid humans+AURA system developed the capacity to make novel contributions to pure mathematics by very untraditional means. This essay traces how these unconventional contributions were made and made possible through negotiations between the humans and the AURA at Argonne and the transformation in mathematical intuition they produced. At play in these negotiations were experimental practices, nonhumans, and nonmathematical modes of knowing. This story invites an earnest engagement between historians of mathematics and scholars in the history of science and science studies interested in experimental practice, material culture, and the roles of nonhumans in knowledge making.
Comment (from this Blueprint): Dick traces the history of the AURA automated reasoning assistant in the 1970s and 80s, arguing that the introduction of the computer system led to novel contributions to mathematics by unconventional means. Dick’s emphasis is on the AURA system as changing the material culture of mathematics, and thereby leading to collaboration and even negotiations between the mathematicians and the computer system.

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Added by: Fenner Stanley TanswellPublisher’s Note: This comprehensive account of the concept and practices of deduction is the first to bring together perspectives from philosophy, history, psychology and cognitive science, and mathematical practice. Catarina Dutilh Novaes draws on all of these perspectives to argue for an overarching conceptualization of deduction as a dialogical practice: deduction has dialogical roots, and these dialogical roots are still largely present both in theories and in practices of deduction. Dutilh Novaes' account also highlights the deeply human and in fact social nature of deduction, as embedded in actual human practices; as such, it presents a highly innovative account of deduction. The book will be of interest to a wide range of readers, from advanced students to senior scholars, and from philosophers to mathematicians and cognitive scientists.
Comment (from this Blueprint): This book by Dutilh Novaes recently won the coveted Lakatos Award. In it, she develops a dialogical account of deduction, where she argues that deduction is implicitly dialogical. Proofs represent dialogues between Prover, who is aiming to establish the theorem, and Skeptic, who is trying to block the theorem. However, the dialogue is both partially adversarial (the two characters have opposite goals) and partially cooperative: the Skeptic’s objections make sure that the Prover must make their proof clear, convincing, and correct. In this chapter, Dutilh Novaes applies her model to mathematical practice, and looks at the way social features of maths embody the ProverSkeptic dialogical model.

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Added by: Fenner Stanley TanswellAbstract: Despite increasing rates of women researching in mathintensive 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 topranked 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 openaccess 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 topranked journals. According to the selfreported responses, a larger number of articles submitted to prestigious venues correlates rather with aspects associated with pronounced research activity, a wellestablished network, and academic seniority.
Comment (from this Blueprint): Mihaljević and Santamaría here use largescale 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.

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Added by: Fenner Stanley TanswellAbstract: Prominent mathematician William Thurston was praised by other mathematicians for his intellectual generosity. But what does it mean to say Thurston was intellectually generous? And is being intellectually generous beneficial? To answer these questions I turn to virtue epistemology and, in particular, Roberts and Wood's (2007) analysis of intellectual generosity. By appealing to Thurston's own writings and interviewing mathematicians who knew and worked with him, I argue that Roberts and Wood's analysis nicely captures the sense in which he was intellectually generous. I then argue that intellectual generosity is beneficial because it counteracts negative effects of the reward structure of mathematics that can stymie mathematical progress.
Comment (from this Blueprint): In this paper, Morris looks at ascriptions of intellectual generosity in mathematics, focusing on the mathematician William Thurston. She looks at how generosity should be characterised, and argues that it is beneficial in counteract some of the negative effects of the reward structure of mathematics.

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Added by: Fenner Stanley TanswellAbstract:
We investigate the truth conditions of knowledge ascriptions for the case of mathematical knowledge. The availability of a formalizable mathematical proof appears to be a natural criterion:
(*) X knows that p is true iff X has available a formalizable proof of p.
Yet, formalizability plays no major role in actual mathematical practice. We present results of an empirical study, which suggest that certain readings of (*) are not necessarily employed by mathematicians when ascribing knowledge. Further, we argue that the concept of mathematical knowledge underlying the actual use of “to know” in mathematical practice is compatible with certain philosophical intuitions, but seems to differ from philosophical knowledge conceptions underlying (*).
Comment (from this Blueprint): MüllerHill is interested in the question of when mathematicians have mathematical knowledge and to what extent it relies on the formalisability of proofs. In this paper, she undertakes an empirical investigation of mathematicians’ views of when mathematicians know a theorem is true. Amazingly, while they say that they believe proofs have an exact definition and that the standards of knowledge are invariant, when presented with various toy scenarios, their judgements seem to suggest systematic contextsensitivity of a number of factors.

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Added by: Fenner Stanley TanswellAbstract: The FourColour 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 FourColour 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.

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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 facetoface 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 ‘secondgeneration 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 coconstitutive–one formed and reformed by the other.
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 socalled “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.

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Added by: Fenner Stanley TanswellAbstract: Some personal thoughts and opinions on what “good quality mathematics” is and whether one should try to define this term rigorously. As a case study, the story of Szemer´edi’s theorem is presented.
Comment (from this Blueprint): Tao is a mathematician who has written extensively about mathematics as a discipline. In this piece he considers what counts as “good mathematics”. The opening section that I’ve recommended has a long list of possible meanings of “good mathematics” and considers what this plurality means for mathematics. (The remainder details the history of Szemerédi’s theorem, and argues that good mathematics also involves contributing to a great story of mathematics. However, it gets a bit technical, so only look into it if you’re particularly interested in the details of the case.)
Comment (from this Blueprint): BarrowGreen 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.