Open-access writing : An investigation into the online drafting and revision of a research article in pure mathematics

Despite significant improvements over the last few generations, the discipline of mathematics still counts a disproportionately small number of women among its practitioners. These women are underrepresented as conference speakers, even more so than the underrepresentation of women among PhD-earners as a whole. This underrepresentation is the result of implicit biases present within all of us, which cause us (on average) to perceive and treat women and men differently and unfairly. These mutually reinforcing biases begin in primary school, remain active through university study, and continue to oppose women's careers through their effects on hiring, evaluation, awarding of prizes, and inclusion in journal editorial boards and conference organization committees. Underrepresentation of women as conference speakers is a symptom of these biases, but it also serves to perpetuate them; therefore, addressing the inequity at conferences is valuable and necessary for countering this underrepresentation. We describe in detail the biases against women in mathematics, knowing that greater awareness of them leads to a better ability to mitigate them. Finally, we make explicit suggestions for organizing conferences in ways that are equitable for female mathematicians.

/pdf/addressing-the-underrepresentation-of-women-in-mathematics-1eslc4ersn.pdf

Addressing the underrepresentation of women in mathematics conferences

Analysis of online mathematics forums can help reveal how explanation is used by mathematicians; we contend that this use of explanation may help to provide an informal conceptualization of simplicity. We extracted six conjectures from recent philosophical work on the occurrence and characteristics of explanation in mathematics. We then tested these conjectures against a corpus derived from online mathematical discussions. To this end, we employed two techniques, one based on indicator terms, the other on a random sample of comments lacking such indicators. Our findings suggest that explanation is widespread in mathematical practice and that it occurs not only in proofs but also in other mathematical contexts. Our work also provides further evidence for the utility of empirical methods in addressing philosophical problems. This article is part of the theme issue 'The notion of 'simple proof' - Hilbert's 24th problem'.

https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2018.0159

Explanation in mathematical conversations: An empirical investigation

Beginning at the end of the nineteenth century, systematic scientific abstracting played a crucial role in reconfiguring the sciences on an international scale. For mathematicians, the 1931 launch of the Zentralblatt fur Mathematik and 1940 launch of Mathematical Reviews marked and intensified a fundamental transformation, not just to the geographic scale of professional mathematics but to the very nature of mathematicians’ research and theories. It was not an accident that mathematical abstracting in this period coincided with an embrace across mathematical research fields of a distinctive form of symbolic and conceptual abstraction. This essay examines the historical, institutional, embodied, and conceptual bases of mathematical abstracting and abstraction in the mid-twentieth century, placing them in historical context within the first half of the twentieth century and then examining their consequences and legacies for the second half of the twentieth century and beyond. Focused on scale, media, and the relationship between mathematical knowledge and its forms of articulation, my analysis connects the changing social structure of modern mathematical research communities to their changing domains of investigation and resources for representation and collective understanding.

/pdf/abstract-relations-bibliography-and-the-infra-structures-of-4l0zbvjck3.pdf

Abstract relations: bibliography and the infra-structures of modern mathematics

The growing use of the internet for collaboration, and of numeric and symbolic software to perform calculations it is impossible to do by hand, not only augment the capabilities of mathematicians, but also afford new ways of observing what they do. In this essay we look at four case studies to see what we can learn about the everyday practice of mathematics: the polymath experiments for the collaborative production of mathematics, which tell us about mathematicians attitudes to working together in public; the minipolymath experiments in the same vein, from which we can examine in finer grained detail the kinds of activities that go on in developing a proof; the mathematical questions and answers in math overflow, which tell us about mathematical-research-in-the-small; and finally the role of computer algebra, in particular the GAP system, in the production of mathematics. We conclude with perspectives on the role of computational logic.

https://www.springer.com/cda/content/document/cda_downloaddocument/9783319214009-c2.pdf?SGWID=0-0-45-1519886-p177560026

Stumbling Around in the Dark: Lessons from Everyday Mathematics

For any $m \geq 1$, let $H_m$ denote the quantity $H_m := \liminf_{n \to \infty} (p_{n+m}-p_n)$, where $p_n$ denotes the $n^{\operatorname{th}}$ prime; thus for instance the twin prime conjecture is equivalent to the assertion that $H_1$ is equal to two. In a recent breakthrough paper of Zhang, a finite upper bound was obtained for the first time on $H_1$; more specifically, Zhang showed that $H_1 \leq 70000000$. 
Almost immediately after the appearance of Zhang's paper, improvements to the upper bound on $H_1$ were made. In order to pool together these various efforts, a \emph{Polymath project} was formed to collectively examine all aspects of Zhang's arguments, and to optimize the resulting bound on $H_1$ as much as possible. After several months of intensive activity, conducted online in blogs and wiki pages, the upper bound was improved to $H_1 \leq 4680$. As these results were being written up, a further breakthrough was introduced by Maynard, who found a simpler sieve-theoretic argument that gave the improved bound $H_1 \leq 600$, and also showed for the first time that $H_m$ was finite for all $m$. The polymath project, now with Maynard's assistance, then began work on improving these bounds, eventually obtaining the bound $H_1 \leq 246$, as well as a number of additional results, both conditional and unconditional, on $H_m$. 
In this article, we collect the perspectives of several of the participants to these Polymath projects, in order to form a case study of online collaborative mathematical activity, and to speculate on the suitability of such an online model for other mathematical research projects.

The "bounded gaps between primes" Polymath project - a retrospective

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D. H. J. Polymath

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The "bounded gaps between primes" Polymath project - a retrospective