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We present a case study of how mathematicians write for mathematicians. We have conducted interviews with two research mathematicians, the talented PhD student Adam and his experienced supervisor Thomas, about a research paper they wrote together. Over the course of 2 years, Adam and Thomas revised Adam’s very detailed first draft. At the beginning...
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... Another strand in the philosophy of mathematical practice has focused on how experts and learners come to establish conviction about and trust in mathematical results (Andersen, 2020;Andersen et al., 2021;Weber et al., 2014). Using a variety of methods, such research has shown that experiments, examples, authority and other non-deductive forms of knowledge play much larger roles in establishing conviction than a purely formalist, autonomous (individual-centred) epistemology would accept. ...
From a purely formalist viewpoint on the philosophy of mathematics, experiments cannot (and should not) play a role in warranting mathematical statements but must be confined to heuristics. Yet, due to the incorporation of new mathematical methods such as computer-assisted experimentation in mathematical practice, experiments are now conducted and used in a much broader range of epistemic practices such as concept formation, validation, and communication. In this article, we combine corpus studies and qualitative analyses to assess and categorize the epistemic roles experiments are seen—by mathematicians—to have in actual mathematical practice. We do so by text-mining a corpus of reviews from the Mathematical Reviews, which include the indicator word “experiment”. Our qualitative, grounded classification of samples from this corpus allows us to explore the various roles played by experiments. We thus identify instances where experiments function as references to established knowledge, as tools for heuristics or exploration, as epistemic warrants, as communication or pedagogy, and instances simply proposing experiments. Focusing on the role of experiments as epistemic warrants, we show through additional sampling that in some fields of mathematics, experiments can warrant theorems as well as methods. We also show that the expressed lack of experiments by reviewers suggests concordant views that experiments could have provided epistemic warrants. Thus, our combination of corpus studies and qualitative analyses has added a typology of roles of experiments in mathematical practice and shown that experiments can and do play roles as epistemic warrants depending on the mathematical field.
... Second, in this article we observed several structural patterns regarding the way definitions are presented. Andersen et al. (2021) has conducted an investigation of how mathematicians write research papers. Identifying these structural patterns contributes to this investigation. ...
In the philosophy of mathematical practice, the aim is to understand the various aspects of this practice. Even though definitions are a central element of mathematical practice, the study of this aspect of mathematical practice is still in its infancy. In particular, there is little empirical evidence to substantiate claims about definitions in practice. In this article, we address this gap by reporting on an empirical investigation on how mathematicians create definitions and which roles and properties they attribute to them. On the basis of interviews with thirteen research mathematicians, we provide a broad range of relevant aspects of definitions. In particular, we address various roles of definitions and show that definitions are not just a product of mathematical factors, but also of social and contingent factors. Furthermore, we provide concrete examples of how mathematicians interact and think about definition. This broad empirical basis with a variety of examples provides an optimal starting point for future investigations into definitions in mathematical practice.
... But Line Edslev Andersen's interviews with working mathematicians indicate that audience consideration does feature into how papers are revised. Andersen's interviews provide insight into how mathematicians write for mathematicians (Andersen et al., 2021), how peer reviewers receive and evaluate papers (Andersen, 2017), and how mathematicians choose which gaps to include in their papers (Andersen, 2020). Additionally, mathematical results must be translated to students in a pedagogical setting. ...
Formal logic has often been seen as uniquely placed to analyze mathematical argumentation. While formal logic is certainly necessary for a complete understanding of mathematical practice, it is not sufficient. Important aspects of mathematical reasoning closely resemble patterns of reasoning in nonmathematical domains. Hence the tools developed to understand informal reasoning, collectively known as argumentation theory, are also applicable to much mathematical argumentation. This chapter investigates some of the details of that application. Consideration is given to the many contrasting meanings of the word ``argument''; to some of the specific argumentation-theoretic tools that have been applied to mathematics, notably Toulmin layouts and argumentation schemes; to some of the different ways that argumentation is implicated in mathematical practices; and to the social aspects of mathematical argumentation.
... Similarly, Raman (2003) distinguishes between private and public aspects of proof to stress that mathematicians' arguments for self-understanding are usually different but connected to those they generate for a particular mathematics community. A proof may also transform in mathematicians' interaction, such as a collaboration or a peer-review (e.g., Andersen et al., 2021). To resort to a metaphor, we associate a proof with an elastic object that mathematicians purposefully manipulate to fit the organizational structure of a target social situation. ...
... 247-248) In regards to (1), mathematics education research often builds on conceptualizations of proof in mathematics (e.g., Davis, 1986;De Villiers, 1990;Rav, 1999) and investigates how individual mathematicians engage with proof (e.g., Inglis et al., 2013;Lew & Mejía-Ramos, 2020;Weber et al., 2022). Less attention is paid to the social mechanisms through which proof functions in the mathematics community and how mathematicians interact with proof and with each other within broader communal activities (e.g., Andersen, 2020;Andersen et al., 2021). In this paper, we elaborated on De Millo et al.'s (1993) view on proofs as successive social processes and argued that these processes unfold in situations that are socially organized, proof-transformative, and sequential. ...
Coming from a social perspective, we introduce a classroom organizational frame, where students’ proofs progress from collaborative construction in small groups, through whole-class presentation at the board by one of the constructors, to a posteriori reflection. This design is informed by a view on proofs as successive social processes in the mathematics community. To illustrate opportunities for mathematics learning of proof progressions, we present a commognitive analysis of a single proof from a small course in topology. The analysis illuminates the processes through which students’ proof was restructured, developed previously unarticulated elements, and became more formal and elaborate. Within this progression, the provers developed their mathematical discourses and the course teacher seized valuable teachable moments. The findings are discussed in relation to key themes within the social perspective on proof.
... Thus, the case study we present in this paper stands out by being an account of a discovery, from the initial conception of the mathematical idea to the publication of a mathematical article, for a case where the agent is not a professional mathematician and where the novel geometric construction is relatively simple (in fact, it can be done with straightedge and compass alone). 1 By supplying personal context to the discovery process, this paper gives insight into a single episode in the development of mathematics, showing real mathematical practice beyond the purely technical results. We believe that adding this particular case study to the literature contributes to displaying the diversity of the processes underlying the development of new mathematics and encourages further investigations into mathematical creativity. ...
... The move from the back to the front, facilitated by an experienced researcher and two anonymous referees, involved a radical transformation: the initial order of presentation, the appeal to examples, the use of concepts and results from other areas of mathematics, the terminology for new concepts, and even the title changed. Similar observations are made in a detailed study of the writing style in professional mathematics [1] and in Ashton's discussion of the role that intended audiences play in mathematical proofs [2]. ...
We present some aspects of the genesis of a geometric construction, which can
be carried out with compass and straightedge, from the original idea to the
published version (Fernández González 2016). The Midpoint Path Construction
makes it possible to multiply the length of a line segment by a rational number
between 0 and 1 by constructing only midpoints and a straight line. In the form
of an interview, we explore the context and narrative behind the discovery, with
first-hand insights by its author. Finally, we discuss some general aspects of this
case study in the context of philosophy of mathematical practice.
... So accepting this particular audience is not impossible or contradictory but it falls short of the reasons discussed above. 18 Some evidence for this can be found in (Andersen et al. 2019) which describes the process of turning a PhD student's proof into a publishable research paper. The PhD student who is experienced in mathematics but inexperienced in publishing, produces a proof which has little concern for what the journal reviewers will be convinced by. ...
The role of audiences in mathematical proof has largely been neglected, in part due to misconceptions like those in Perelman & Olbrechts-Tyteca (1969) which bar mathematical proofs from bearing reflections of audience consideration. In this paper, I argue that mathematical proof is typically argumentation and that a mathematician develops a proof with his universal audience in mind. In so doing, he creates a proof which reflects the standards of reasonableness embodied in his universal audience. Given this framework, we can better understand the introduction of proof methods based on the mathematician's likely universal audience. I examine a case study from Alexander and Briggs's work on knot invariants to show that we can fruitfully reconstruct mathematical methods in terms of audiences.
... The respondents would often rely on professional experience and their assessment of where they could gain a 'head start' by drawing on their work with similar problems or similar promising methods. The third criterion was perceived as being both of utmost importance and difficulty, as essentially it focuses on carving a niche for your work among your intended audience (Andersen et al., 2019;Ashton, 2020). Therefore, recognizability is a powerful value in problem choice, as peers would have to be able to identify and learn from your work. ...
In this article we consider important developments in artificial intelligence within automated and interactive theorem provers (ATP/ITP). Our focus is to describe and analyze key challenges for interactive theorem provers in mainstream mathematical practice. Our broader research program is motivated by studying the functions of visual internal and external representations in human mathematicians and the role of epistemic emotions. These aspects remain gorges to bridge in developing ITPs. But by seeing ITPs as augmenting the human mathematician, we stand to gain the best of two epistemic practices in the form of a hybrid — a centaur.
In this chapter, I chart out a topology of the epistemic environment in which policy debates about social justice in mathematics are being had. This topology reveals a dilemma for those of us who hold that social injustices have partially shaped contemporary mainstream mathematical reasoning. Proposing policies to rectify these injustices requires arguments in an epistemic environment which is hostile towards the conclusion of these arguments and may result in backlash and hinder social justice efforts more generally. Not proposing such policies, however, fails a moral imperative to call out and seek to rectify injustices where one is able. I suggest that this dilemma is tragic in that it has no higher-order resolution.
