Monday, 20 April 2020

Memories of John Conway

The mathematician John Horton Conway died just over a week ago. (I recommend Siobhan Roberts's biography Genius at Play.)  I' intend to write a post later about what his book On Numbers and Games meant to me, but in this post I will share a couple of memories of Conway.  I emphasise - these are memories and their accuracy is not guaranteed.

First,a personal story of how I was caught out by his exam question.  When I studied Part III of the Mathematical Tripos at Cambridge I took Conway's course on Sporadic Simple Groups (this at a time when the Classification of the Finite Simple Groups was not complete and indeed before it had been proved that the Monster group actually existed).  In the exam we had to do any three questions out of six (or similar).  The last question was something like "Write an essay on anything in the course you haven't covered in your previous answers."  Now, I had three good topics I knew pretty well, and they all came up in other questions.  So I thought to myself, "Well, I'll do the essay question first, while I think about which two of the other three questions to do."  As a result I wrote the essay on my fourth-best topic, and didn't use in the exam one of my three strongest topics.  Not good exam technique!  (To be honest, I feel that's why it wasn't a very fair question to set in an exam!)  I have sometimes wondered: if I had answered that question first, and written about my best topic, and then afterwards answered the question on that topic, since that wouldn't have been a previous answer, could I have got credit for the same material twice?)

My other memories are of an email mailing list for those interested in the history of mathematics, which ran for some years from the late 1990s before splitting and dissolving in acrimony as these things tended to do at the time.  Anyone could discuss or ask any questions, and Conway was one of the regulars.

One one occasion two schoolboys posted an email saying they were in primary school and they wanted to know more about some mathematical topic.  Conway sent a lovely long reply, at an appropriate mathematical level, saying probably how nice it was to hear from such enthusiastic students.  The boys sent a thank-you letter, adding at the end that they weren't actually at all interested in mathematics, but their teacher had told them to ask a question.

On another occasion a university student (I think a PhD student) asked about a result of Lagrange (I think), noting that there were several proofs of the result and asking if anyone knew which was Lagrange's original proof.  Conway wrote a long reply, saying that he didn't know, and going on to explain why he didn't think it mattered: there were several simple proofs and it would have been a matter of chance which one Lagrange came up with first, since the proof of the theorem was trivial. Immediately someone jumped in, accusing Conway of appalling rudeness in describing the question as "trivial" (he hadn't) and suggesting that such treatment from an established mathematician would likely deter the student from further mathematical studies.  Others argued about this (not all members of the list were native English speakers and the word "trivial" did seem to have very negative connotations for some: which surprised me as I was very familiar with its use in this mathematical context and didn't think it at all offensive).  Conway said he hadn't intended to be rude or to suggest the question was trivial - indeed, by writing a long and detailed answer, he had shown his respect for the questioner - and he apologised profusely.  But the criticism continued until the student who had asked the original question replied.  He said that far from feeling upset by Conway's response, not only did he not interpret it as critical, but he was absolutely thrilled that the great Conway himself had replied to his question: he had been walking on air ever since!

I admired Conway's willingness to engage with anybody on that mailing list, regardless of their age or experience.  I hope I have learned from that.

Sunday, 29 March 2020

Self-referential humour

I' am one of these annoying people who loves self-reference.  I had a sign saying "Why can't anyone these days do anything properyl" in my student room: now I wear a T-shirt labelled "Prefectionist", so my sense of humour hasn't developed.  Books which I loved as a student (and still do) are Patrick Hughes and George Brecht's Vicious Circles and Infinity and Douglas Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid.  The first of these books pushed my interest in art towards Fluxus, and then later I discovered Hughes's brain-baffling perspective paintings.   And through Hughes I found the wonderful, amaxingly modern logical paradoxes of the fourteenth-century philosopher Jean Buridan.

And so I'm one of these annoying people who think it's witty to include "Self-reference" in the index of a software user-manual, referring to the page number for that index page. (Yes, I did that, but it was a long time ago.)

When did this enjoyment of self-reference start?  Possibly with my father, although I can't immediately think of any examples.  Certainly when I was a teenager.  For me, the funniest Monty Python sketch was the one where we see a group of intrepid explorers reaching their destination, never before visited by man, and duly celebrate with handshakes all round.  But while they are celebrating, one of them wonders, "Who is filming us?" and we see them looking in all directions before pointing, rushing straight towards the camera and then introducing themselves to the camera crew, who obviously got there before them.  And as we cut to another angle, they realise that there must be another camera crew too ...  (This is my memory of the sketch and it may not be very accurate.)

I had wonderful maths teachers at school - Jimmy Cowan and Ivan Wells - but I don't remember them particularly encouraging this kind of humour.  But recently (ie when I first planned writing this blog post, so not very recently) I was amused by this gag from Moray Hunter's radio comedy "Alone":  .

"Remind me, Morris, never to ask you to do anything … at all  … ever"
"Does that include reminding you never to ask me to do anything at all ever?"
"Probably – I lost interest halfway through the question"

(The first speaker is the Angus Deayton character, and the concluding line is not only ideal for his world-weary delivery, but really perfects the joke.)

Why do I mention this?  Because Moray Hunter was at my old school and shared the same maths teachers as I had.  So perhaps there is a connection...

Saturday, 22 February 2020

A brilliant solution to the Tower of Hanoi

I have always underestimated the interest value of the Tower of Hanoi problem.  If you don;t remember the puzzle, there are three positions for piles of discs of different sizes, starting with all discs piled from smallest to largest on one pile (in this case the leftmost).
The Tower of Hanoi
The problem is to move disks so that the whole pile is now in the rightmost position, but one can only move one disc at a time and one can never place a larger disc on top of a smaller one.

The surprise is how many moves are necessary - to move a pile of n discs takes 2^(n-1) operations.  (it's a good example showing various ideas about algorithms, such as recursion.)  There is a legend that monks in India are engaged in moving a tower of 64 discs, one operation per day, and that when the move is complete, the world will end.  How long would that take?

Although I won a T-shirt by solving the puzzle in Lewisham Shopping Centre on a Saturday morning in Maths Year 2000, I had never felt that the puzzle has mass appeal: the procedure for solution is too laborious.  But recently I took the puzzle in the photograph to the Green STEM Fest for children at the University of Greenwich (along with things which I thought were more likely to attract interest) and this was the exhibit the kids enjoyed most.  A few nine-year-olds (or so) started racing each other against the clock, recording times of around 35 seconds, which seemed pretty quick to me.

Then one of them produced the brilliant solution in the video below (which is a reconstruction)!  This is the work of a real mathematician who has recognised the symmetry in the situation.

Sunday, 12 January 2020

Thoughts on Hannah Fry's Royal Institution Christmas Lectures

Another New Year, another resolution to write more blog posts.  We'll see.

A mathematician delivering the Royal Institution Christmas Lectures is always something special.  Hannah Fry's three lectures this year were rather different from Christopher Zeeman's classics.  Rather than one person directly talking to the audience for the full hour each time, she brought in guests for short interviews, showed activities outside the lecture room, and presented a huge range of activities and apparatus with volunteers from the audience (and in one case, a plant, when the audience member asked to solve Rubik's cube turned out to be the nation's champion speedcuber).  This was a team effort and everyone who took part performed splendidly: Matt Parker's many contributions deserve special mention. 

And it was wonderful.  The excitement was palpable.  The enthusiasm of the audience, the rush to put hands up whenever a volunteer was wanted - even allowing for possibly selective editing, it was clear that all the students were having a whale of a time.  Did one ever expect to see young people so excited by a maths lecture?  (Sure, Zeeman was also exciting, but in a very different way.)

I have seen some comments to the effect that there wasn't very much maths in the lectures.  I think that is misguided.  There was plenty of maths, with the applications shown but without the technical details.  I don't have any problem with that.  As  a kid I was always motivated by the abstract mathematics rather than the applications, but I'm in a minority.  And today, a kid wanting to know  the details of anything Hannah talked about can just get out their phone.  And what a wonderful panorama Hannah presented of the power of mathematics in today's world of data and machine intelligence.  (It was nice to see MENACE, the match-box game player, taking its rightful place in the show!)

(The one unfortunate thing was that the first lecture included an upbeat segment about using maths to judge when it was safe to explore volcanoes - although it must have been filmed before the tragedy in New Zealand which could not have been foreseen, that bit should have been edited out or reshot for the broadcast.)

So - Hannah Fry's lectures have inspired schoolkids to take maths seriously.  Hopefully some of them will be motivated to study maths at University.  And what will happen when they attend their first lecture?  If these Christmas Lectures are their first experience of mathematics lectures, they will be expecting wildly interactive sessions with guest speakers introduced every few minutes, lots of demonstrations and fast-moving material. How will they react to a lecturer spending an hour going through a complex pure mathematical proof line by line?

Have the Royal Institution misled their audience by presenting as a lecture something so far removed from a traditional lecture?  Possibly, but the Christmas Lectures is their brand, so they cannot be blamed for doing that.

If we as university lecturers are to avoid disappointing our future students, perhaps we need to rethink our lectures.  Rather than go through detailed mathematics at a pace which cannot be right for everyone in the room, perhaps we should try to emulate Hannah's RI Lectures.  We could (as I'm sure some already do) present many voices (on video if not live), lots of ideas, and as much interactivity as we can manage to keep the audience enthusiastic, leaving the technical details for students to study in their own time.  We can provide lecture notes (or, better, screencasts) that they can go through at their own pace, pausing when their brain is full and returning to them later, and going to Youtube or similar when they get stuck, just as we ourselves study from books and papers.  We can use our large-class time to build enthusiasm and give the big picture rather than getting lost in detail.

If Hannah's "lectures" help speed up the move to more useful use of students' time than the traditional lecture, that will be another benefit from these remarkable Christmas Lectures.

(As always, I am presenting my own personal views - that is what a blog is for!  I don't expect everyone to agree with me.)

Saturday, 15 June 2019

Looking forward to the Festival of Mathematics

So in ten days time the University is hosting Greenwich Maths Time, the 2019 IMA Festival of Mathematics and its Applications.  The Festival takes place on Tuesday 25 and Wednesday 26 June.  Everything is free!

The weather forecast is currently fine and we are looking forward to a feast of mathematics.  The programme and booking details can be found at the Festival website the Twitter hashtag is #IMAFest19.

The Festival has attracted a wonderful collections of performers and activities.  Visitors will have the opportunity to learn about noon-Newtonian fluids by walking on custard!  We're grateful to everyone who is taking part and who has worked to make the Festival possible, and especially to the Festival sponsors, the University of Greenwich, the Institute of Mathematics and its Applications, the Advanced Mathematics Support Programme, The OR Society, FDM, the Mathematical Association, and an anonymous individual donor.

Sunday, 8 July 2018

The Big Internet MathOff

The Big Internet MathOff is now taking place.  Sixteen mathematicians are competing in a knock-out tournament to present the most interesting piece of mathematics.  With the final two matches of the first round now taking place, there has already been a wealth of fascinating mathematics presented.  Christian Lawson-Perfect deserves huge thanks for creating this wonderful showcase of exciting maths.

But not only are the contestants' pitches fascinating.   There is also interesting maths involved in the background.

First, Each match runs for 48 hours.  As a contestant, my most pressing question today is whether I will win my first round match against Jo Morgan, in which case I will need to work urgently on my pitch for round 2.  So the question is, how soon can I extrapolate from the early votes to be reasonably sure of the outcome?

I am sure that this has been studied in the context of political elections.  But this is harder because the electorate is everybody with access to the internet - several hundred million people (though not all will actually vote). 

Now, when I first looked at the score ten minutes after the match started, I was leading by 2 votes to 1, but I think that was a bit too early to be confident of the result.  As I write this, four hours into the match, Jo is leading by 58 votes to 29, and my share of the vote has consistently been around 30% for some time.  Although time differences may mean that America has yet to vote, my intuition is that I can safely abandon my pitch for round 2 and turn to other things like writing blog posts.

I've already had to find my answer to my second problem.  Each contestant has to provide a different piece of fun mathematics for each round so we all had to submit four topics before the tournament started.  It's unlikely that each set of four topics were all equally good. So what order do you choose for your topics?  Do you save your best one for the final?  In that case, you might not get to the final and your best idea is wasted.  Or do you use your best ideas in the early rounds, to improve your chances of progress, but risk putting your weakest idea forward in the final?

There may be a similarity with World Cup penalty shoot-outs.  With five penalties to take, it's clear that a team should select their best five penalty takers.  But in which order should they take the penalties?  One suggestion is that you should save your best penalty-takers for the end, when the pressure will be greatest.  But that could mean that your team has lost 5-3 and your best taker hasn't taken one of the penalties.  The great Ally McCoist, commenting on last night's shoot-out between Russia and Croatia, argued for putting your best takers first, guaranteeing that at least they will take penalties.  I'm not convinced that's a sound argument, though if you score your early penalties then the pressure on your opponents will increase.

So what did I do for the MathOff? Did I save my best idea for the final, or bring it out for the first round?  Well, there is a further complication.  I have no idea how to judge which of my topics would win most votes in an internet poll. Ranking them in order would have been pure guesswork.  So, shamefully, I didn't use any game theory or simulation to decide the order of my pitches - I just entered them in the order in which I thought of them.  Which may be as good a strategy as any!

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.