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The polymath project: lessons from a successful online collaboration in mathematics
 In Proc. CHI ’11. ACM
, 2011
"... Although science is becoming increasingly collaborative, there are remarkably few success stories of online collaborations between professional scientists that actually result in real discoveries. A notable exception is the Polymath Project, a group of mathematicians who collaborate online to solve ..."
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Although science is becoming increasingly collaborative, there are remarkably few success stories of online collaborations between professional scientists that actually result in real discoveries. A notable exception is the Polymath Project, a group of mathematicians who collaborate online to solve open mathematics problems. We provide an indepth descriptive history of Polymath, using data analysis and visualization to elucidate the principles that led to its success, and the difficulties that must be addressed before the project can be scaled up. We find that although a small percentage of users created most of the content, almost all users nevertheless contributed some content that was highly influential to the task at hand. We also find that leadership played an important role in the success of the project. Based on our analysis, we present a set of design suggestions for how future collaborative mathematics sites can encourage and foster newcomer participation. Author Keywords largescale collaboration, online collaborative mathematics, online collaborative science, online communities
The informal logic of mathematical proof
 ASPIC2 Argumentation Service Platform with Integrated Components http://www.argumentation.org
, 2007
"... Paul Erdős famously remarked that ‘a mathematician is a machine for turning coffee into theorems ’ [9, p. 7]. The proof of mathematical theorems is central to mathematical practice and to much recent debate about the nature of mathematics. This paper is an attempt to introduce a new perspective on t ..."
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Paul Erdős famously remarked that ‘a mathematician is a machine for turning coffee into theorems ’ [9, p. 7]. The proof of mathematical theorems is central to mathematical practice and to much recent debate about the nature of mathematics. This paper is an attempt to introduce a new perspective on the argumentation characteristic of mathematical proof. I shall argue that this account, an application of informal logic to mathematics, helps to clarify and resolve several important philosophical difficulties. It might be objected that formal, deductive logic tells us everything we need to know about mathematical argumentation. I shall leave it to others [14, for example] to address this concern in detail. However, even the protagonists of explicit reductionist programmes—such as logicists in the philosophy of mathematics and the formal theorem proving community in computer science—would readily concede that their work is not an attempt to capture actual mathematical practice. Having said that, mathematical argumentation is certainly not inductive either. Mathematical proofs do not involve inference from particular
Mathematical knowledge
, 2007
"... Abstract The original proof of the fourcolor theorem by Appel and Haken sparked a controversy when Tymoczko used it to argue that the justification provided by unsurveyable proofs carried out by computers cannot be a priori. It also created a lingering impression to the effect that such proofs depe ..."
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Abstract The original proof of the fourcolor theorem by Appel and Haken sparked a controversy when Tymoczko used it to argue that the justification provided by unsurveyable proofs carried out by computers cannot be a priori. It also created a lingering impression to the effect that such proofs depend heavily for their soundness on large amounts of computationintensive custombuilt software. Contra Tymoczko, we argue that the justification provided by certain computerized mathematical proofs is not fundamentally different from that provided by surveyable proofs, and can be sensibly regarded as a priori. We also show that the aforementioned impression is mistaken because it fails to distinguish between proof search (the context of discovery) and proof checking (the context of justification). By using mechanized proof assistants capable of producing certificates that can be independently checked, it is possible to carry out complex proofs without the need to trust arbitrary customwritten code. We only need to trust one fixed, small, and simple piece of software: the proof checker. This is not only possible in principle, but is in fact becoming a viable methodology for performing complicated mathematical reasoning. This is evinced by a new proof of the fourcolor theorem that appeared in 2005, and which was developed and checked in its entirety by a mechanical proof system.