Mathematical collaboration

I have just spent an entire weekend at a workshop on mathematical collaboration. It blew a huge hole in the time that I had to get on with urgent work that needs to be done, but it was such a good experience that I don’t really mind. I had been thinking of titling this post “I’m a mathematician – get me out of here!”, but in fact the mathematicians were not treated just as performing seals or experimental rats but were made to feel like collaborators in the enterprise.

The meeting was in the philosophy department, which stands on the edge of the cliff with a stunning view over the sea, the West Sands, and beyond. I am ashamed to say that after five years in St Andrews, this was the first time I had been into the building, though I had been told one of its features. It has two doors, labelled (if I recall correctly) Metaphysics and Moral Philosophy, but you can enter either, they lead into the same space.

The meeting was organised by Fenner Tanswell (St Andrews) and Josh Habgood-Coote (now in Bristol), and was associated in some way with Ursula Martin’s grant. Ursula showed up on both days, but was visibly coming down with some illness (possibly related to the ones that struck me down three times this winter) and didn’t stay long.

The first talk was by Josh, who spoke about authorship, a somewhat vexed question in these days of REF. He began by showing us a 12-page physical paper (on a measurement of the mass of the Higgs boson) which had 5135 authors, whose names covered 16 pages. His proposal was to split the functions of authorship, so that a paper has contributors (who did the work), a spokesperson (responsible for the write-up), and a guarantor (who takes responsibility for the project). The mathematicians in the audience were not entirely happy with this. Our default position is that every author is a guarantor; we expect every author to have read and approved all of the paper. But clearly not all is well in the academy. There are guest authors (the head of the institution whose name must be put on all papers) and ghost authors (a problem in the pharmaceutical world, and also in economics, where the research is done and the paper written entirely in the commercial organisation and an academic author is added just to make it look respectable). Read Ben Goldacre on this!

Next was Benedikt Loewe, who teaches in Amsterdam, Hamburg, and Cambridge. He described a course he put on for potential researchers in the “philosophy of mathematical practice”. Here we first heard the term “experimental philosophy”. He used the analogy of buying clothes. If you want clothes, you probably go to a shop. But perhaps nothing in the shop fits very well. You might consider a bespoke tailor. But this is more expensive, and the tailor might not make exactly what you want because of some kind of artistic pride. So as a last resort, you might try to make your own clothes, knowing that they may end up not terribly well made. In the same way, “experimental philosophy” tries to use the services of neuropsychologists (for example), but many end up doing the experiments themselves. Philosophy of scientific practice has existed for some years, but these people regard mathematics with “a mixture of awe and lack of interest”.

Kamilla Rekvenyi, a mathematics undergraduate in St Andrews, presented material which had grown out of her History of Mathematics project, on “Paul Erdős’ mathematics as a social activity”. She told about how he encouraged young mathematicians by asking interesting questions (exemplified by letters he wrote to the current Regius Professor of Mathematics in St Andrews, who was in the audience), and about the row between Erdős and Atle Selberg about the elementary proof of the Prime Number Theorem (where one could say that Erdős thought they were collaborating but Selberg didn’t).

I was very impressed by Stephen Crowley, the last speaker of the day. His topic was “Does collaboration make mathematicians virtuous?” His answer was “no”; but since he so clearly had come to learn, and since he himself showed so clearly one of the virtues he mentioned (humility – I hope he doesn’t mind me saying this), I was not entirely convinced. He made a good argument that mathematics is an ideal test case for questions of this sort. In any case, this depends on something called “virtue theory” which several people mentioned, but which I have been unable to get a clear grasp of. Stephen managed to confuse us by wearing a T-shirt with a map of Idaho labelled “Iowa”; whether this was a subtle dig at the standard of geographic knowledge of many Americans, or something quite different, I was not really sure.

Then there was a panel discussion, which I won’t even attempt to capture, and dinner, where we were given a good example of the opposite of virtue by a large party of golfers at the next table who drank lots of beer and spoke so loudly that conversation on our table was severely curtailed.

The next day began with a joint presentation by Fenner and Colin Rittberg on “Epistemic injustice in mathematics”. One of his examples concerned a mathematician who claimed to have been unjustly treated because her papers were rejected by journals with the comment that “that is not difficult to prove”, even though no published proof could be cited. There is a real issue here, but the mathematicians were confused by the fact that the authors called such results “folk theorems”, more or less defining this term to mean results which “the experts” know even though there is no proof in the literature. My understanding, confirmed by others present, is that a “folk theorem” is one where we do not know who first proved it; if we use it, we have either to refer to a textbook proof or to give our own proof, so there are often several proofs in the literature of such results (perhaps also a problem but hardly an injustice). We suggested that another term be used for the first concept, possibly “ghost theorem” – suggestions for better terminology welcome!

Chris Kelp talked about “Inquiry, Knowledge and Understanding”. This was a more traditional philosophy talk. He believes that the aim of an inquiry is to gain knowledge, and the inquiry is successful if the knowledge is gained. He reminded us that, as Edmund Gettier pointed out, knowledge is not the same as “justified true belief”, as philosophers had once thought. It seemed to me that perhaps a mathematical example of such a situation was the moment when Andrew Wiles, at the Newton Institute, announced Fermat’s Last Theorem. It is true, and he believed at that moment (with justification) that he had proved it. But subsequent events showed that he didn’t know that because the proof had a hole.

The final presentation was done remotely, via video and Skype, by Katie MacCallum from Brighton. This remarkable individual is an artist and a philosopher and is also knowledgeable about mathematics; she is working closely with nine mathematicians at various career stages, attempting to understand what they are doing whey they work, and to translate this understanding into works of art. The video combined her presentation with animated examples of knot diagrams drawn on a blackboard. Remarkably, a long segment of her talk consisted of a defence of the mathematicians’ customary use of blackboard and talk to communicate. She gave three reasons: briefly,

• this method slows the presenter down to the audience’s speed, which often doesn’t happen with a computer presentation;
• writing is quite hard work, and the presenter often pauses at the end of a line, and fills the silence with “meta-information” about the material, which aids understanding;
• use of different parts of the board (the top, a box to one side) gives the audience clues to the structure of the material presented.

And so the meeting ended, since Ursula was not well enough to give the last talk. She encouraged us to walk on West Sands, but I slipped off home, to try to catch up on urgent tasks …