Sunday, 20 May 2018

Mathematical discoveries

I was lucky enough to attend a meeting organised by the British Society for the History of Mathematics on "The History of Cryptography and Coding".  It was a quite exceptional meeting - six excellent talks.  As one of the other audience members said, I learned something from every talk, and a lot from several talks.  (Anyone who goes to these events will know that this isn't always the case.)

The final talk, by Clifford Cocks on the discoveries of the public key cryptography, was fascinating in many respects.  (Cocks was one of the people at GCHQ who discovered both the Diffie-Hellman key exchange method and the Rivest-Shamir-Adleman (RSA) algorithm before those after whom the ideas are named, but this wasn't known until GCHQ made it public over 25 years after the event.)  Cocks told us about the (different) reasons why the British and American discoverers were looking for these methods.  I was particularly struck by his insights into the creative processes that led to the discoveries.

In 1970 James Ellis at GCHQ had the idea of public-key cryptography.  Many people at GCHQ tried to find a way to implement it, without success.  Cocks suggested that this was because of "tunnel vision" - because Ellis's paper suggested using look-up tables, everyone was focused on that idea. Cocks had just arrived at GCHQ from university, and his mentor mentioned Ellis's problem to him, but described it in general terms without mentioning look-up tables.  Without having been led in a wrong direction, Cocks quickly came up with the idea of using factorisation, and the problem was solved.  (When Cocks told his colleague and housemate Malcolm Williamson about his paper, Williamson overnight worked up the idea of using the discrete logarithm problem, anticipating Diffie and Hellman.)

Cocks also told us about how Diffie was working on these discoveries having left his academic job, supporting himself on his savings - something which I don't recall knowing.

Then Cocks told us about Rivest, Shamir and Adleman's discovery of RSA.  They had tried about 30 ideas, none of which worked.  Then after a Passover meal at which alcohol flowed freely, Rivest had the big idea, wrote it down, and checked the next morning to see if it still worked.

I think these stories shed some light on mathematical creativity.  It needs hard work, of course, but it also needs flexibility.  Cocks (by his modest account) had the advantage over his colleagues that his mind wasn't conditioned by an unproductive idea.  Rivest's solution came after a break from thinking about it.  Of course, there are many other examples - PoincarĂ©'s inspiration as he was getting on a bus is the standard one - but it is always interesting to hear how great mathematical discoveries came about, and to hear this story from Cocks himself was a wonderful privilege.

Sunday, 28 January 2018

London buses, and the use of the mean as an estimate

A couple of weeks ago, I got onto my morning bus, climbed the stairs, holding on as the bus lurched forward, and sat down, to hear for the first time the new announcement "Please hold on: the bus is about to move", introduced by TfL (Transport for London).  Over the next few days this announcement was widely ridiculed. It was broadcast after every stop, but often - in my experience almost always - AFTER the bus had started moving, and sometimes when it was slowing down for the next stop, making the announcement appear ridiculous.  Occasionally, at busy stops like the railway station, it was broadcast while large numbers of people were still waiting to board, presumably causing consternation to prospective passengers who took it seriously.  And on one occasion, while the bus was stationary, I heard "The bus is about to move" followed immediately by the announcement "The driver has been instructed to wait here for a few minutes", flatly contradicting the previous words.

What was happening? TfL explained that they were piloting the announcement for four weeks, to try to reduce the number of injuries sustained by passengers on moving buses - apparently of the order of 5000 each year.  The timing of the announcement was based on the average time buses spent at each stop - I suspect by "average" they meant the mean.

The intention is laudable.  But the problem with using a mean in situations like this is that it doesn't really tell you how long a particular bus will wait at a given stop.  My bus home probably spends longer stopped at the railway station than at all the other stops put together.  Just as most people earn less than the mean national salary, which is heavily influenced by the very small number of people earning millions each year, so I imagine most of the time a bus spends less time at a stop than the mean.  So a system based on the mean time spent at a stop will result in the announcement usually being played after the us has left the stop, leading to ridicule.

Now, TfL are pretty good at maths - their planning of the transport around London during the 2012 Olympics was a very successful example of operational research in action.  So did they really get this wrong?  After all, one would think that a few tests would have shown the problem.  

Certainly one result of the announcements was  a great deal of publicity, which perhaps has made people more aware of the need for care when standing and moving on a bus. The announcements themselves may have a short-term effect, but in fact one very quickly ceases to notice them (or at least I have found that they very rarely impinged on my attention, after the first few instances on the first day).  But perhaps the press coverage, and people talking about the announcements, had more impact than the announcements themselves.

But if the announcements are to continue, how can TfL avoid the absurdity of an announcement that the bus is about to move being broadcast after it has moved?  The solution TfL have adopted (as well as apparently changing the timing) is simple.  The wording of the announcement is now "Please hold on while the bus is moving".  The timing no longer offers the possibility of absurdity.  The solution to this problem was not mathematical modelling, but thoughtful use of language.