Workshop, University of Oxford, 8th & 9th April 2017

Organisers: Fenner Stanley Tanswell, Lorenzo Lane, Josh Habgood-Coote & Sarah Baldwin

Part of the “Social Machine of Mathematics” project led by Prof Ursula Martin CBE, Department of Computer Science, University of Oxford.

Speakers include:-

  • Patrick Allo (Oxford)
  • Line E. Andersen (Aarhus)
  • Catarina Dutilh Novaes (Groningen)
  • S. Orestis Palermos (Edinburgh)
  • Jeroen de Ridder (VU Amsterdam)

Location: Room 278 in the Oxford e-Research Centre, 7 Keble Road, Oxford ,OX1 3QG

Mathematics is a deeply social discipline. The stereotype of the “lone genius” is one which does not fit the breadth and depth of mathematical work, which also features everything from one-on-one collaborations to massive collective efforts. Indeed, it is common for proofs of significant theorems to rely on work by many mathematicians working together and in parallel. The most famous example is the proof of the classification of finite simple groups, which relied on the work of scores of mathematicians. Likewise, the polymath project has shown the power of massive collaboration in mathematics at settling open problems and discovering mathematical proofs.   

Nonetheless, the new questions that arise for specifically collaborative work on mathematics have received insufficient philosophical attention, unlike parallel questions about the importance of collaboration in science which have received significant attention. Social epistemology and work on group knowledge have in recent years made a great deal of progress on general issues surrounding epistemic cooperation. Similarly, sociological research on mathematical collaboration has revealed much of the everyday workings of mathematics in practice. In this workshop we aim to explore the social dimensions of mathematics, connecting new work in social epistemology, mathematical practice and sociology, in order to gain a better understanding of how collaboration in mathematics produces knowledge, proofs and understanding.

We will address the following questions:

  • Is there distinctively social knowledge in mathematics?
  • How can massive mathematical collaboration produce novel proofs and novel theorems?
  • What are the epistemic benefits of collaboration and the division of cognitive labour in mathematical work?
  • Can crowdsourcing and online collaboration produce genuine proofs of interesting theorems?
  • Does it help to view massive collaboration as part of a “social machine” of mathematics?


Registration is free, but must be done in advance to have the lunches. There will also be a workshop dinner at Al-Shami on Saturday evening, at around £20. To register, please email Fenner.Tanswell [at] including whether you would like to attend the workshop dinner and any dietary restrictions or further needs. Registration for lunch and the dinner will close on the 1st of April.

Any further enquiries can also be directed to Fenner by email:  Fenner.Tanswell [at]



SATURDAY 8th April
12:00-13:00 LUNCH
13:00-14:15 S. Orestis Palermos “Is Polymath a Social Machine?”
14:15-15:30 Fenner Stanley Tanswell “Proving Activities and Collaborative Mathematics”
15:30-16:00 TEA/COFFEE
16:00-17:15 Patrick Allo “Tracking common information and public announcements in Polymath: A data-driven and logically informed method”
17:15-18:30 Catarina Dutilh Novaes “Adversariality and Cooperation in Mathematical Proofs’
19:30 DINNER
SUNDAY 9th April
9:00-10:15 Josh Habgood-Coote “Collective Mathematical Knowledge”
10:15-10:30 TEA/COFFEE
10:30-11:45 Line E. Andersen “Acceptable Gaps in Mathematical Proofs”
11:45-13:00 Lorenzo Lane “Coordinating Visions: Documenting collaborative knowledge production at mathematics institutes”
13:00-14:00 LUNCH
14:00-15:15 Jeroen de Ridder “Beyond social knowledge”

The workshop will begin on Saturday at midday with lunch and registration, and the first talk will commence at 13:00. We will be done on Sunday by mid-to-late afternoon, currently planned for 15:30. This will allow people from nearby to travel on those days and only require one night of accommodation. A full programme will be added soon.

For college accommodation (£50+/night), we recommend using:

There are a number of rooms currently available in Keble college, which is just next door to the workshop. Most of the accommodation they offer will be within easy walking distance of us.

For budget accommodation (£25/night), we recommend the Oxford YHA youth hostel:

Again, this is not far from the workshop location.

Panel 1


Panel 2


Sunday 28th April

Pre-workshop informal gathering for those already in Bristol. Fenner arrives early afternoon, and is keen to head to a café/pub with people as they arrive. We’re intending to pick a pub somewhere central for dinner and socialising Sunday evening.

Monday 29th April, Wills Memorial Building room 3.33

9 -9.30 Coffee+Welcome

9.30-10.30 Jacob Ward “Communicating a Theory of Mathematical Communication”

10.30-10.50 Coffee

10.50-11.50 Philip Welch “In Cahoots: The Cabal Seminar 1976 – 1981 – a case study”

11.50-12.00 Break

12-1 Adam Dunn “The Secret Life of Statistics: How the evolution of eighteenth-century statistical thought depended on networks, communities and collaborations”

1-2 Lunch + coffee

2-3 Ursula Martin “Journeys in mathematical landscapes: genius or craft?”

3-3.30 Coffee

3.30-4.30 Fenner Tanswell & Josh Habgood-Coote “Group Knowledge and Mathematical Collaboration”

4.30-5 Coffee

5-6 Stephen Crowley “What does the mathematical community do? Some initial speculations”

7:30 Conference dinner at Krishna’s Inn

Tuesday 30th April, Wills Memorial Building room 1.5



9 -9.30 Coffee+Welcome

9.30-10.30 Line Edslev Andersen “The High Bar for Relying on Testimony in Mathematics”


10.30-10.50 Coffee


10.50-11.50 Catarina Dutilh Novaes “Adversarial collaboration and transferability in mathematical proofs”

11.50-12.00 Break

12-1 Brendan Larvor “The Gatekeepers’ Last Sigh: reflections on the Jaffe-Quinn episode in the age of arXiv”

1-2 Lunch + coffee

2-3 Haixin Dang “Peer Review and the Social Structure of Mathematics”

3-3.30 Coffee


3.30-4.30 Elizabeth de Freitas “Bruno Latour’s Gaia: The mathematical aesthetic and dis/trust in science”

4.30-5 Coffee

5-6 James Ladyman TBC

Informal dinner, organised on the night.



Jacob Ward “Communicating a Mathematical Theory of Communication: Academia,
Industry, and Bureaucracy at the London Symposia of Information Theory”
From 1950 to 1960, four editions of the London Symposia on Information Theory took place. Information theory was a new research area, first formulated as a ‘mathematical theory of communication’ by the mathematician Claude Shannon at Bell Telephone Laboratories in 1948. Notable mathematicians and scientists, such as Alan Turing, Denis Gabor, Donald Mackay, and Warren McCulloch attended the London Symposia, which became a key site for the international exchange of ideas on information theory, particularly between the USA and the UK.
These symposia, however, were also sponsored and attended by groups and researchers
from outside the academy. The Ministry of Supply, Britain’s wartime supply industry for the armed forces, sponsored the first London Symposia, and communications engineers from the Post Office, which ran Britain’s telephone network, also participated, going on to collaborate with British information theorists such as Colin Cherry and Denis Gabor through the 1950s and 1960s.
This paper explores the London Symposia of Information Theory as a mathematical
community composed not only of academics, but also of bureaucrats and industry
researchers, and will show how the institutional structures and collaborative practices in
this community generated a distinctive field of information theory that had immediate and immeasurable impact in British government and telecommunications.

Philip Welch “In Cahoots: The Cabal Seminar 1976 – 1981 – a case study”
A somewhat Whiggish version of history sees a group of mathematicians come together in the southern California region (principally UCLA, Caltech) to solve a foundational problem in set theory – a problem that increased with significance as attempts to disentangle the implications of various concepts were made.

The problem concerned the relationship between assumptions concerning the winning strategies of two person perfect information games and the universe of sets of mathematical discourse. Postulates concerning such ‘determinacy’ of games had been in the air for over two decades, as they implied the analysis of certain subsets of the real line became much more regular. However they contradicted the axiom of choice. The trigger for new endeavour came unexpectedly from a statistician David Blackwell (who also has an interesting history) at Berkeley.It is a story very much of its pre-LaTex typesetting, pre-Internet time. The name ‘Cabal Seminar’ was a slightly pejorative one: the dynamics of collaboration was intricate, but the excitement at times was high.

However it was difficult for outsiders to actually discover what was going on, let alone be given access to proofs.  The set theorists at those institutions could seem to be involved in some esoteric struggle with a hermeneutics of texts that were, worse, unseen by the outside world. One had to go there to find out.  Nevertheless 4 volumes of Springer Lecture Notes appeared over these years of the Seminar’s deliberations.

Adam Dunn “The Secret Life of Statistics: How the evolution of eighteenth-century statistical thought depended on networks, communities and collaborations”
Statistics is not a solitary practice. It is and has always been a collaborative effort. However, historians of statistical thought often focus either on individual case studies or national case studies. This paper aims to challenge this view and illustrate the importance of collaboration, community and the development of networks in the evolution of statistics during the eighteenth century. It will argue that a crucial component in the development of statistics during the Enlightenment was the
foundation of ‘epistemic communities’, that is, communities which centre on individuals or institutions whose focus is a similar area of knowledge/knowledge production, and the networks that surrounded them. The paper will focus on two case-studies: August Ludwig von Schlözer and Sir John Sinclair. Through a broad comparison of the statistical networks and community that both men established the paper will illustrate their importance in the evolution of statistical thought in the eighteenth century from a descriptive model to a more mathematical and scientifically minded discipline. Further, it will demonstrate that the wide scope of these networks and communities was a crucial feature in this development. Statistics cannot be seen as product of a nation or a few
individuals, but was, instead, based on collaboration between a wide variety of people from disparate backgrounds. This paper will explore how these collaborative networks and communities functioned and what their role was in the evolution of statistical thought. Through the comparison of these two epistemic communities it will analyse the importance of collaborative effort both in the collection of information as well as the spread of ideas and influence. It shall demonstrate how Schlözer and Sir John formed the foundation of modern statistical practice and how this would never have been possible without massive collaborative effort and the underlying foundation of an ‘epistemic community’ on which such collaboration could be based.

Ursula Martin “Journeys in mathematical landscapes: genius or craft?”

We look at how Anglophone mathematicians have, over the last hundred years or so, presented their activities using metaphors of landscape and journey. We contrast romanticised self-presentations of the isolated genius with ethnographic studies of mathematicians at work, both alone, and in collaboration, looking particularly at on-line collaborations in the “polymath” format. The latter provide more realistic evidence of mathematicians daily practice, consistent with the the “growth mindset” notion of mathematical educators, that mathematical abilities are skills to be developed, rather than fixed traits.

We place our observations in a broader literature on landscape, social space, craft and wayfaring, which combine in the notion of the production of mathematics as crafting the exploration of an unknown landscape. We indicate how “polymath” has a two-fold educational role, enabling participants to develop their skills, and providing a public demonstration of the craft of mathematics in action.


Fenner Tanswell & Josh Habgood-Coote “Group Knowledge and Mathematical Collaboration”

While the epistemic inaccessibility of the computer proof of the Four Colour Theorem led to a major re-examination of the role of proofs in mathematics, a similar reckoning has not come about in response to individually inaccessible “big” proofs. These proofs, such as the Classification of Finite Simple Groups, which require large numbers of mathematicians working in collaboration, present us with several epistemic difficulties. Unlike traditional proofs, they cannot easily be checked and verified by an individual mathematician, so seemingly cannot become known in the usual way. With bigger proofs, the probability of errors and gaps also increases; so what kinds of errors and gaps are tolerable without undermining the status of the proof? We draw on the tools of social epistemology, seeing a group of mathematicians as the relevant object of study, and use this to reject the potential negative conclusions about the epistemic status of big proofs.

Stephen Crowley “What does the mathematical community do? Some initial speculations”

Talking to mathematicians makes it clear that mathematics is a surprisingly communal activity. Why might that be? I identify two stories about the epistemic value of communal work (common knowledge and Longino’s knowledge making account). I ague that while mathematical practice has elements of both ‘common knowledge’ and ‘knowledge making’ these two stories do not exhaust the value of the communal in mathematical work. I conclude by offering a speculation (based on the work of Tanswell and colleagues) concerning the ‘meaningfulness’ of mathematical work which I suggest helps make sense of communal practices not otherwise accounted for.


Line Edslev Andersen “The High Bar for Relying on Testimony in Mathematics”

Mathematicians appear to have quite high standards for when they will rely on the results of others without checking their proofs.  We examine why this is.  We argue that for each expert who testifies that she has checked the proof of a result p and found no errors, the likelihood that the proof contains a substantial error decreases and can become extremely small.  In particular, when a number of experts testify that they have checked the proof of p and found no errors, we have good reason to believe they are being truthful.  For this reason, it is relatively easy for a mathematician to have a high bar for relying on testimony, and by having a high bar for relying on testimony, she can protect her own work and the work of her community from errors.  Our argument thus provides an explanation for why many mathematicians have high standards for when they will rely on the results of others without checking their proofs.

Catarina Dutilh Novaes “Adversarial collaboration and transferability in mathematical proofs”

The idea of adversarial collaboration has been introduced and practiced (albeit modestly) in the social sciences in the last decades. It consists in a protocol whereby scientists who disagree with each other on a particular issue can collaborate in investigating it. Now, that there is something like adversarial collaboration in mathematics has been known for a long time, at least since Lakatos’ Proofs and Refutations; according to him, it is through the ‘dialectic of proofs and refutations’ that mathematical knowledge is produced. (One can accept this general point even while disagreeing with some aspects of Lakatos’ account.) However, for adversarial collaboration in mathematics (thus understood) to be successful, proofs must be formulated in such a way that members of the relevant mathematical community who critically scrutinize a proof can understand it. In other words, proofs must have the property of *transferability*, a concept introduced by Easwaran (2009). In my talk, I develop the concept of adversarial collaboration in mathematics in terms of a rational reconstruction of mathematical proofs as dialogues involving two fictive participants: Prover and Skeptic, and the notion of transferability. I then discuss three examples: the reception of Gödel’s incompleteness proofs in the 1930s, E. Nelson’s failed proof of the inconsistency of Peano arithmetic in 2011, and the still ongoing saga of Mochizuki’s purported proof of the ABC conjecture.

Brendan Larvor “The Gatekeepers’ Last Sigh: reflections on the Jaffe-Quinn episode in the age of arXiv”

In this talk, I re-examine the arguments made in the debate in 1993-4 between Jaffe & Quinn and Thurston, with particular attention to the reflections and anxieties they express about the effect of digital communications on mathematical research.  I then contrast these notes from the infancy of the internet with some more recent cases and offer some reflections about the arguments from the early 1990s in the light of more recent experience.

Haixin Dang “Peer Review and the Social Structure of Mathematics”

Many social epistemologists of science have been concerned with how to best organize the scientific community so that the community would be more reliable and better at truth seeking or truth tracking. For example, social epistemologists have examined how the priority rule encourages successful scientific research, how increased access to information could cause scientists’ methods to become more or less reliable, and how collaboration among scientists can be incentivized. In this talk, I review some results from the existing literature on the social structure of science and ask whether these results can be applied to mathematical practice. In particular, I focus on recent philosophical work on peer review and the credit economy of science. I argue that pre-publication peer review should be massively restructured in mathematics. If our goal is to produce and curate an accurate body of mathematical knowledge, then pre-publication peer-review is not the most reliable mechanism because the process often lets in proofs with errors. I propose that our focus should shift to post-publication peer review, and that the reward structure of mathematics should be reformed to reflect this.

Elizabeth de Freitas “Bruno Latour’s Gaia: The mathematical aesthetic and dis/trust in science”

This paper focuses on recent work of Bruno Latour, and the issue of dis/trust in science. I explore the materialist ontology at the heart of Latour’s work, and discuss implications for how we conceive the relationship between mathematics and the physical sciences.

Confirmed speakers:

Philip Welch (University of Bristol) “In Cahoots: The Cabal Seminar 1976 – 1981 – a case study”

Jacob Ward (University of Oxford) “Communicating a Theory of Mathematical Communication”

Ursula Martin (University of Oxford) “Journeys in mathematical landscapes: genius or craft?”

Brendan Larvor (University of Hertfordshire) “The Gatekeepers’ Last Sigh: reflections on the Jaffe-Quinn episode in the age of arXiv”

James Ladyman (University of Bristol)

Catarina Dutilh Novaes (Vrije Universiteit Amsterdam) “Adversarial collaboration and transferability in mathematical proofs”

Adam Dunn  (University of St Andrews) “The Secret Life of Statistics: How the evolution of eighteenth-century statistical thought depended on networks, communities and collaborations”

Elizabeth de Freitas (Manchester Metropolitan University)

Haixin Dang (University of Pittsburgh)

Stephen Crowley (Boise State University) “Is mathematical knowledge common knowledge?”

Line Edslev Andersen (Aarhus Universitet) “The high bar for relying on testimony in mathematics”