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Mathematical explanation: Epistemic aims and diverging assessments Vrije Universiteit Brussel
Mathematicians suggest that some proofs are valued for their explanatory value. This has led to a philosophical debate about the distinction between explanatory and non-explanatory proofs. In this paper, we explore whether contrasting views about the explanatory value of proof are possible and how to understand these diverging assessments. By considering an epistemic and contextual conception of explanation, we can make sense of disagreements ...
The mark of understanding: In defense of an ability account Vrije Universiteit Brussel
Understanding is a valued trait in any epistemic practice, scientific or not. Yet, when it comes to characterizing its nature, the notion has not received the philosophical attention it deserves. We have set ourselves three tasks in this paper. First, we defend the importance of this endeavor. Second, we consider and criticize a number of proposals to this effect. Third, we defend an alternative account, focusing on abilities as the proper mark ...
Polymath as an epistemic community Vrije Universiteit Brussel
The Polymath Project is an online collaborative enterprise that was initiated in 2009, when Timothy Gowers asked whether and how groups could work together to solve mathematical problems that “do not naturally split up into a vast number of subtasks.” Gowers proposed to answer this question himself by actually trying to set up such a collaboration, based on interactions taking place in the comment-threads of a series of posts on a WordPress ...
Collective Understanding Vrije Universiteit Brussel
Could groups ever be an understanding subject (an epistemic agent ascribed with understanding) or should we keep our focus exclusively on the individuals that make up the group? The way this paper will shape an answer to this question is by starting from a case we are most willing to accept as group understanding, then mark out the crucial differences with an unconvincing case, and, ultimately, explain why these differences matter. In order to ...
Characterising Understanding and the Understanding Subject Vrije Universiteit Brussel
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Studying mathematical practices: the dilemma of case studies Vrije Universiteit Brussel
In this paper we argue that the choice of research methods reflects the theoretical framework even before these methods have been put to use in case studies. We understand the term ‘case study’ broadly in this paper and argue that neither thinking of them as cherry-picked cases to support preconceived ideas about mathematical practices nor thinking of them as inductive leaps from (too) few cases to general features is suitable. By realising the ...
The “Artificial Mathematician” Objection Vrije Universiteit Brussel
Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer....
Mathematical Aims Beyond Justification Vrije Universiteit Brussel
The past decades, there has been an increased philosophical interest in mathematical practice. These philosophers intend to answer questions related to the activities of working mathematicians. One of these questions is what aims direct mathematical research. After all, mathematicians often express the desire for good or beautiful mathematics. These types of reflections indicate the need for an understanding of aims that go beyond ...
Model-based reasoning in mathematical practice Vrije Universiteit Brussel
The nature of mathematical reasoning has been the scope of many discussions in philosophy of mathematics. This chapter addresses how mathematicians engage in specific modelling practices. We show, by making only minor alterations to accounts of scientific modelling, that these are also suitable for analyzing mathematical reasoning.
In order to defend such a claim, we take a closer look at three specific cases from diverse mathematical ...
In order to defend such a claim, we take a closer look at three specific cases from diverse mathematical ...