Abstract
Because the conclusion of a correct proof follows by necessity from its premises, and is thus independent of the mathematician’s beliefs about that conclusion, understanding how different pieces of mathematical knowledge can be distributed within a larger community is rarely considered an issue in the epistemology of mathematical proofs. In the present chapter, we set out to question the received view expressed by the previous sentence. To that end, we study a prime example of collaborative mathematics, namely the Polymath Project, and propose a simple formal model based on epistemic logics to bring out some of the core features of this case-study.
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Notes
- 1.
- 2.
What is the formal nature of proofs? Are proofs out there (in “The Book”) or constructed by us? What is a surveyable proof and what not? How does one grasp (the content of) a proof? Are there more and less beautiful proofs? (etc.)
- 3.
For a first connection between abstract argumentation and modal logic, see (Grossi, 2010).
- 4.
- 5.
A more complete and highly accessible version of events can be found in (Nielsen, 2011).
- 6.
This highly compact formulation of the DHJ can be complemented by, especially for the non-mathematician, “The gentle introduction to the Polymath project” to be found at: http://numberwarrior.wordpress.com/2009/03/25/a-gentle-introduction-to-the-polymath-project/.
- 7.
Actually, a proof already existed. It was formulated in (Furstenberg and Katznelson, 1991). However, this original proof relied on methods and techniques from domains far away from combinatorics, such as ergodic theory. So, as often happens in mathematical research, although one has a proof of the theorem, nevertheless this does not prevent mathematicians from searching for an alternative and, more importantly, an elementary proof, i.e., a proof using the concepts, proof methods and techniques of the domain itself. For more on this topic, see, e.g., (Rav, 1999).
- 8.
One of the authors of this chapter is himself the editor of a logic journal, Logique et Analyse, and has a nice collection of, e.g., disproofs of Gödel’s theorems.
- 9.
This type of research, much loved by some of the aforementioned amateurs can indeed be seen as “outsourcing drudge work, comparable to Amazon’s Mechanical Turk” (https://www.mturk.com/mturk/welcome), as remarked by one of the referees.
- 10.
- 11.
See http://polymathprojects.org/category/discussion/, for more specific comments and on a related blog, http://polymathprojects.org/general-discussion/, for more general comments.
- 12.
Again, this is something that the standard formalism does not allow for since the notions of general and particular knowledge are simply definable in terms of, respectively, the conjunction and disjunction of individual knowledge claims for each member of the group. Nevertheless, such restrictions on the available groups can be introduced syntactically with, what amounts to, awareness filters. We do not pursue the details of this additional modification, but stick to a rather informal approach.
- 13.
An intuitively plausible way to think about such levels of abstraction is this: Sometimes we like to consider the (or a) mathematical community as a monolithic bloc (in terms of consensus in that community), but equally often we prefer to take a more refined look that allows us to focus on more local phenomena.
- 14.
This approach trivialises the idea of distributed computing, but the comments we make equally apply to more refined protocols for inference networks.
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Acknowledgements
The first author is a postdoctoral fellow of the Research Foundation–Flanders, which through project G.0431.09 also supported research for this chapter by the third author.
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Allo, P., Van Bendegem, J.P., Van Kerkhove, B. (2013). Mathematical Arguments and Distributed Knowledge. In: Aberdein, A., Dove, I. (eds) The Argument of Mathematics. Logic, Epistemology, and the Unity of Science, vol 30. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6534-4_17
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