Polymath Project

Abstract: Alan Turing proposed to consider the question, “Can machines think?” in his famous article [40]. We consider the question, “Can machines do mathematics, and how?” Turing suggested that intelligence be tested by comparing computer behaviour to human behaviour in an online discussion. We hold that this approach could be useful for assessing computational logic systems which, despite having produced formal proofs of the Four Colour Theorem, the Robbins Conjecture and the Kepler Conjecture, have not achieved widespread take up by mathematicians. It has been suggested that this is because computer proofs are perceived as ungainly, brute-force searches which lack elegance, beauty or mathematical insight. One response to this is to build such systems which perform in a more human-like manner, which raises the question of what a “human-like manner” may be. Timothy Gowers recently initiated Polymath [4], a series of experiments in online collaborative mathematics, in which problems are posted online, and an open invitation issued for people to try to solve them collaboratively, documenting every step of the ensuing discussion. The resulting record provides an unusual example of fully documented mathematical activity leading to a proof, in contrast to typical research papers which record proofs, but not how they were obtained. We consider the third Mini-Polymath project [3], started by Terence Tao and published online on July 19, 2011. We examine the resulting discussion from the perspective: what would it take for a machine to contribute, in a human-like manner, to this online discussion? We present an account of the mathematical reasoning behind the online collaboration, which involved about 150 informal mathematical comments and led to a proof of the result. We distinguish four types of comment, which focus on mathematical concepts, examples, conjectures and proof strategies, and further categorise ways in which each aspect developed. Where relevant, we relate the discussion to theories of mathematical practice, such as that described by Pólya [36] and Lakatos [24], and consider how their theories stand up in the light of this documented record of informal mathematical collaboration.

Abstract: In formal education, learning mathematics is typically done by receiving direct instruction within the confines of a classroom. From first grade through graduate school, students are expected to learn mathematics primarily by being taught by instructors with previous knowledge of the subject. Research mathematicians, on the other hand, must rely on other methods; the mathematics they are trying to understand may not, as yet, be known to anyone else. Hence, they learn primarily through experimentation, self-directed study, and collaboration with peers. In recent years, these methods have been expanded to use modern tools and ideas. Research mathematicians initiated several successful large-scale online collaboration projects, such as the Polymath project and the MathOverflow website. In this chapter, we discuss these two projects, along with various other examples of online collaborative learning of mathematics. Our primary motivation is captured in the following question: why aren’t we all learning math this way? While a complete answer is beyond the scope of this work, we hope to at least stimulate a debate among a wide audience. The major part of our discussion is thus informal; we defer the contextualization of these examples within modern education research until the end of the chapter.

Abstract: REINVENTING DISCOVERY: THE NEW ERA OF NETWORKED SCIENCE by Michael Nielsen, Princeton University Press, 2012 ISBN: 978-0-691-14890-8This book describes the modern initiatives of open science and collective intelligence. The author believes that through new methodology and management of science, "we have an opportunity to change the way knowledge is constructed", (p. 206) This new collaborative, internet-enabled, completely open, scientific research and problem-solving paradigm is called networked science. Clear descriptions of both a foundational framework with detailed analysis as well as several inspiring applied examples of how networked science has worked are provided. Neilsen begins by explaining the Tim Growers' mathematical challenge problem posted on Growers' blog on the internet. Here the collective skills of the group (27 collaborators contributed) solved an exacting open problem in just 37 days. This was called the Polymath Project. Nielsen goes on to describe GenBank, an online database of genetic information used by numerous collaborators, as well as the Galaxy Zoo project, where 200,000 online volunteers help classify galaxy images. Neilsen comments on the evolution and success of Wikipedia and Linux as the most massive and open of these kinds of collective efforts, although he also reflects on scientists as a steadfast conservative group that primarily values writing rarely read scientific papers rather than contributing to the more influential, frequently read Wikipedia entries. So it is usually when collaboration and credit for promotion and recognition are both present that modern scientists do their best work. Nielsen concludes his first chapter by explaining that this new way of doing science is important because "it means speeding up things such as curing cancer, solving the climate-change problem, launching humanity permanently into space." (p. 11)My favorite example of collaboration in the book is the chess match of world champion Gary Kasparov against the World Team (actually over 50,000 people who voted online for the daily moves against the champion). It took 62 moves for Kasparov to prevail. The World Team's strategy was far stronger than any one individual who participated and the collective group played extremely well against a far-superior opponent. The technical chess advice for the collaboration was assembled through an online forum to discuss options as the open internet participants voted each day for the next move of the World Team. The advice forum was written by teenaged chess players who were good players but nowhere near the level of Kasparov. It is amazing to me that this cooperative voting methodology was able to do so well. …