- “Massive” online collaboration in mathematics initiated by Timothy Gowers in 2009.
- Collaborative mathematical problem-solving (proving a theorem) via interaction in blog discussion-threads.
- 12 Polymath-projects started in the last 8 years.

In [3]:

```
plot_overview(PM_FRAME, annotate=True)
```

In [5]:

```
plot_discussion_tree(PM_FRAME, **PM4_FOCUS)
```

In [6]:

```
plot_discussion_tree_radial(PM_FRAME, **PM4_FOCUS)
```

- Source-material to answer questions about real-life mathematical inquiry.
- Polymath as a massive repository of mathematics in action.
- Mathematics “as we know it,” but now on the front-stage.

- Example of e-research, crowd-science, and citizen-science.
- Polymath as an example of how to organise scientific collaboration online.
- A “new kind of mathematics”.

In [7]:

```
plot_scatter_authors_hits(PM_FRAME, "Polymath 4", thread_type="research threads", fontsize=10)
```

- Source-material to answer questions about real-life mathematical inquiry.
- Polymath as a massive repository of mathematics in action.
- Mathematics “as we know it,” but now on the front-stage.

- Example of e-research, crowd-science, and citizen-science.
- Polymath as an example of how to organise scientific collaboration online.
- A “new kind of mathematics”.

How do we combine both perspectives, and use insights and methods from the study of online collaboration to contribute to the philosophy of mathematical practices, and more generally to our understanding of scientific problem-solving processes?

Explain how effective collaboration in Polymath is possible?

- Inference to the best explanation.
- Answering “how possible” questions.
- Which ”general principles” or “theoretical insights” are needed to explain, with the available data, how succesful collaboration is possible?

- Problem of idealisation in simulation-models of scientific communities (Martini & Pinto, 2016).
- Explain how epistemically succesuful collaboration is possible by building on a specific data-set.

Successful collaboration and strong group-level attitudes

- Effective participation requires a detailed shared understanding of the problem-solving situation, because aggregation is a social/interactive process:
- Cannot be reduced to outsourcing small tasks and aggregating the results.
- Poorly defined epistemic tasks.
- No hard-coded aggregation or error-correction procedure.

- Epistemic goals and epistemic standards are more diverse:
- Increased efficiency is not the only goal of collaborative proving.
- Showing mathematics is action is an explicit goal.
- Improving and reflecting on the process is part of the projects.

- Additional features we associate with scientific communities:
- No strict separation between leaders and participants.
- Research-agenda is also crowd-sourced.

- Permanent and publicly accessible repository of contributions.
- Regular summaries, and in some cases wiki-based knowledge-base.

- Small contributions rule to avoid unncecessary creation of information-asymmetries:
- Keep contributions small.
- Share ideas quickly.

In [8]:

```
plot_comment_sizes(PM_FRAME, **PM4_FOCUS, resample="weekly")
```

Source: Dunin-Keplicz, B., & Verbrugge, R. (2010). Wiley Series in Agent Technology: Teamwork in multi-agent systems. A formal approach. Chichester: Wiley.

- Role of group-level attitudes
- General, mutual, and collective intentions.
- Collective intentions require
**common belief of mutual intentions**.

- This is much harder to study and keep track of!

Common beliefs are possible because interactions take place in informationally transparent shared contexts.

- Common beliefs require reliable communication (no uncertainty about message and about receipt of message).

- But: asynchronous communication is a text-book example of
*un*reliable communication.

- And: merely claiming that access is universal isn't enough.

(Everybody*can*read, but who*does*read?)

- Common belief within
*G*requires public announcement to all members of*G*. - But: strict interaction-patterns only reveal one-to-one communication.

In [9]:

```
draw_network(PM_FRAME, project="Polymath 4", thread_type="research threads", stage=-1, fontsize=10)
```

- Affiliation-networks relate individuals to something else (events, places, interests).
- Can be used to construct a network based on common affiliation.

- Use affiliation to certain
*communicative events*or*contexts*to model co-presence. - Treat co-presence as a sufficient condition to make public-announcements.

- But which events or contexts?

- “Centre of discussion” as a metaphor for the dynamic context in which participants interact.
- Intended to capture idea that joint presence is epistemically transparent.

- Contributing instantly puts one in the centre of discussion.
- Moving out of the centre of discussion happens gradually.

In [11]:

```
plot_centre_crowd(PM_FRAME, "Polymath 4", stage=2, thread_type="research threads")
```

- Closeness to the centre of discussion signals presence in the centre of discussion, and
- is sufficient to know who is also close.

- Common information of who is present is necessary and sufficient for making public announcements.

- Ability to know who is also close is sufficient for common knowledge of who's close.

**(TC)**$\mathrm{Close}_i \to (\mathrm{Close}_j \to \mathsf{K}_i \mathrm{Close}_j)$

- If
**(TC)**is an axiom-scheme (or is common knowledge) we can prove $\bigwedge_{i \in G} \mathrm{Close}_i \to \mathsf{C}_G \bigwedge_{i \in G} \mathrm{Close}_i$

- $\mathrm{Close}_i \to (\mathrm{Close}_j \to \mathsf{K}_i \mathrm{Close}_j)$ is sufficient to prove $\bigwedge_{i \in G} \mathrm{Close}_i \to \mathsf{C}_G \bigwedge_{i \in G} \mathrm{Close}_i$,
- but is also susceptible to
*imprecise knowledge*-style counterexamples.

- Episodes of intense interaction can be identified through clustering, and participants can be
*affiliated*to events.

In [12]:

```
plot_activity_thread(PM_FRAME, "Polymath 4", stage=3, thread_type="all threads", last="2009-09-01")
```

In [13]:

```
draw_network(PM_FRAME, project="Polymath 4", thread_type="research threads", stage=-1, graph_type="cluster", fontsize=10)
```