Richard Kilvington (ca.1302–1361) was a Franciscan who worked in Oxford for a time in the early 14th century. He was associated with the Merton mathematicians, though it seems that he was not actually at Merton himself (despite some later reports).

He worked in terminist logic, mathematical physics, and the new theology. In particular, he refuted Aristotle’s claim that metabasis (using arguments or methods from one branch of science in a different branch) was illicit.

On infinity, he says that integers are potentially infinite because one can always find a larger integer, but not actually infinite since there is no single infinite number.

His best-known work is his Sophismata, a discussion of paradoxes or questionable truths. These have been translated, edited and commented by Norman Kretzmann and Barbara Ensign Kretzmann, The Sophismata of Richard Kilvington, Cambridge University Press, 1990. (But beware; there is another book with the same title and editors, published by Oxford University Press in the same year, which is a collation and transcription of the existing Latin texts; unless you are expert in mediaeval Latin, this may not be what you want.)

Here, courtesy of Google Books, is the list of sophismata which Kilvington discusses. I think they throw an interesting light on the philosophy of the time. Traditionally ridiculed as being about angels dancing on pinheads, the subject was really much more. Simply reading this list, you will spot connections with Zeno’s paradox, infinity, infinite divisibility, change, the basis of empirical knowledge, and one of the obsessions of the Merton mathematicians, quantification. These are things which still perturb philosophers and others. Don’t be put off by all the stuff about whiteness; this is just an arbitrary attribute which can change and might be quantifiable.

1. Socrates is whiter than Plato begins to be white.
2. Socrates is infinitely whiter than Plato begins to be white.
3. Socrates begins to be whiter than Plato begins to be white.
4. Socrates begins to be whiter than he himself begins to be white.
5. Socrates will begin to be as white as he himself will be white.
6. Socrates will begin to be as white as Plato will be white.
7. Socrates will be whiter than Plato will be white in any of these.
8. Socrates will be precisely as Plato will be white in any of these.
9. Socrates will be as white as Plato will cease to be white.
10. Socrates will be two times whiter than Plato will be white at instant A.
11. Something has produced degree B of whiteness.
12. Socrates has traversed distance A.
13. Socrates will traverse distance A.
14. Socrates will begin to traverse distance A, and Socrates will begin to have traversed distance A, and he will not begin to traverse distance A before he will begin to have traversed distance A.
15. Distance A begins to have been traversed.
16. A begins to be true.
17. A and B will be true.
18. A was moving continuously during some time after B, and A is not moving.
19. Socrates will as quickly cease to move as he will move.
20. Socrates will as quickly have been destroyed as he will have been generated.
21. A bgins to intensify whiteness in some part of B, and each proportional part in B will without interval be diminished.
22. A begins to whiten some part in B, and no part in B will be whiter than it now is white.
23. A will generate whiteness up to point C, and no whiteness will be immediate to point C.
24. D will begin at the same time to have been divided and not divided.
25. A will begin to have been divided from B.
26. A will begin to be per se whiter than B.
27. Socrates will begin to be able to traverse distance A.
28. Distance A will begin to have been traversed by Socrates.
29. Socrates will move over some distance when he will not have the power to move over that distance.
30. Socrates moves two times faster than Plato.
31. Socrates and Plato will begin to move equally fast.
32. Socrates does not move faster than Plato.
33. Socrates will move faster than Socrates now moves.
34. Plato can move uniformly during some time and as fast as Socrates now moves.
35. Socrates will begin to be able to move at degree A of speed.
36. Socrates will begin to be able to move stone A.
37. Socrates can as quickly have the power to move stone A as Plato will have the power to traverse distance C.
38. Plato can begin to be the strongest of the men who are in here.
39. A and half of A begin at the same time to be destroyed by agent B.
40. It is infinitely easier to make C be true than to make D be true.
41. B will make C true.
42. It is infinitely easier for B to make it be the case that the proposition “Infinitely many parts of A have been traversed” is true than to make it the case that the proposition “All of A has been traversed” is true.
43. Infinitely sooner will A be true than B will be true.
44. As many proportional parts in A Socrates will traverse as Plato.
45. You know this to be everything that is this.
46. You know this to be Socrates.
47. You know that the King is seated.
48. A is known by you.

The LMS–Gresham Lecture

Last week I gave the annual LMS–Gresham lecture. When they asked me to do it, a year ago, I was writing the course material for Mathematical Structures, and couldn’t think of anything else, so I said I would talk about that. In fact, I believe that a cohort of talented and passionate mathematicians is important – why else would I be doing this job?

Anyway, the lecture went well. You can see the slides here; the slides don’t have all the ad libs, or the questions afterwards. After the lecture, we went for a pleasant meal in the nearby Indian restaurant.

Incidentally, I went to the LMS–Gresham lecture last year to see how someone else (in this case Bernard Silverman) did it. But this year I have no idea who my successor is.

Marking the exam

The good news here is that the students seem to have done rather well in the exam. Of course there is lots of monitoring and standardization to be done yet, but it looks as if both the median and the average mark are just above the B/C borderline. The students have done me proud. (I did wish, though, that I had a little stamp saying “An example is not a proof!”)

However, I noticed a couple of curious tendencies:

• Having given a counterexample to one statement, people tend to hunt for a different counterexample to the next, even if the same one works (for example, if the second statement is the contrapositive of the first). It’s as if the counterexample has exhausted its potency on one statement.
• When asked whether several properties hold (e.g. is a relation reflexive, symmetric, transitive?), there is a tendency to mention only the ones that do hold. We all have a mindset that inclines towards the positive, but to test a hypothesis you have to look at possible negatives.

Here is a selection of things written by students which caught my eye. These are not in any sense a representative selection. Some of them show a lack of understanding which you might find worrying; I would excuse a lot of this on the grounds that in the stress of an exam you will almost certainly write things that with calm consideration you wouldn’t. Some contain good sense hidden under poor expression. Some show original and creative thought. Some of them show that the students have picked up my passion for mathematics! I have sometimes lightly edited these.

• √2 must be irrational since it falls between √1 and √3.
• “Clearly” cannot be used in a proof since nothing is clear without a proof of it. [This was a common reaction. One candidate said "You cannot be lazy and write "Clearly P(0) is true", you must prove it. It may be clear to you, not always clear to the reader". Bravo!]
• A set with no elements contains a single element which is the empty set.
• [After working out the cases n = 1,2,3] So it seems the formula is correct. However the above is not a proof.
• m² is also even, since anything squared becomes even.
• If “less than” is related to ℕ, then ℕ is related to “less than”.
• Hence it is shown that P(n) is true for P(n+1).
• The proof does not include a conclusion box ☐ to indicate the end of proof [and is therefore invalid].
• Therefore A is infinitely countable. [This quite common.]
• The relation “less than” on ℕ is not reflexive since a is not related to a for most natural numbers a.
• We need to prove by rejection.
• Suppose √2 is a rational number. Then it can be written in the form m/n. But we cannot write √2 in this form, so contradiction.
• The statement is not true as this cannot be proven. The contrapositive is not true as the statement is often true.

Previous

It was a bit more tightly focussed than I would have liked. We had a lot of references to Szemerédi’s Regularity Lemma; even Ben Green was using a version of the Regularity Lemma for integers, proved by him and Terry Tao, in his talk. (Curiously, Szemerédi was giving a new proof of a known result, avoiding the use of the Regularity Lemma in order to obtain better bounds.) No finite geometry, no enumeration …

I’ll just talk briefly about three of the highlights.

An old result of Erdős asserts that, if A is any finite set of natural numbers of size n, then A contains a sum-free subset of size at least n/3. The proof is simple and beautiful, so here it is. Pick a random real number a from the uniform distribution on [0,1], and let S_a be the set of those n in A for which the fractional part of an is between 1/3 and 2/3. Clearly S_a is sum-free. The average size of S_a is obviously n/3. So there is some choice of a for which the size of S_a is at least n/3.

Ben, with his students Sean Eberhard and Freddie Manners, has proved that the constant 1/3 here is best possible. He gave us a clear outline of the proof. It is necessary to avoid two kinds of “large” sum-free sets. These were very familiar to me from my own work on sum-free sets. The first are periodic sets, such as the set of odd numbers, the set of numbers congruent to 2 or 3 (mod 5), and so on, which occur with positive probability in the choice of a random sum-free set; the second consists of the interval [x, 2x) which comes up (along with the odd numbers) in the Cameron–Erdős conjecture (which was proved by Ben in his PhD thesis).

He described the method of doing this in a couple of simplified cases. To me, it had an adèlic flavour about it; the real completion of the rationals is involved with avoiding the intervals, and the p-adic completions in avoiding the periodic sets.

Gábor Kun, as usual, had a very interesting project to talk about. This is a “finitization” of the concept of amenability. A bit technical, so I won’t attempt a description; but it involved a conjecture of von Neumann, a conjecture of Thomassen, and an algorithmic the Lovász Local Lemma.

The final talk on the second day, the Norman Biggs lecture, was given by Noga Alon, who always gives a good talk, and this was no exception. He was talking about random Cayley graphs, and asking in particular what can be said about the girth or the chromatic number of a random Cayley graph for a given group. He started off by asking: if the order of the group is 10^(10^(10^10)) (I won’t attempt that in HTML), and we choose a generating set of size 10^10, then the chromatic number is with high probability 2 if the group is an elementary abelian 2-group, 3 if it is cyclic of prime order, and bigger than 10 if it is PSL(2,p).

There were lots of technical results, but there was only time to give us a brief taste of the methods used.

Given that these techniques are now available, is it time to revisit Babai’s problem:

Is it true that, if G is a group which is neither abelian nor generalized dicyclic, then a random Cayley graph for G has automorphism group precisely G with high probability?

This is partly, I think, because of the law of unintended consequences. The people who evaluate our research have the idée fixe that conference proceedings are of less value than refereed journals, so anyone who submits one to the REF is committing academic suicide. This despite the fact that many conference proceedings are refereed to the same standard as journal articles; but the box-ticking of research evaluation cannot cope with this subtlety.

There are all sorts of reasons why I might want to have a good paper in a conference proceedings. Perhaps the people I want to read the paper belong to the community that puts on that conference. Perhaps the conference celebrates a mathematician I respect. And so on.

It wasn’t always so. I want to review briefly the history of the proceedings of the British Combinatorial Conference as an example. (Full details are kept by Keith Edwards.) The proceedings appeared intermittently until 1983, when regular publication in Ars Combinatoria began. (In some of the earlier volumes, the invited and contributed papers appeared together; but from 1977, the practice of publishing the invited talks in advance of the conference was adopted.) In 1991, the contributed papers volume moved to Discrete Mathematics, where it has been since; the proceedings of the 2011 Exeter conference appeared quite recently. The papers were refereed to the usual standard of the journal by a guest editorial board appointed by the BCC, and everything was vetted by the editor-in-chief.

The proceedings of the 1997 conference, for example, had 60 papers covering 795 pages in volumes 197/198, together with the problems presented at the conference. Later, when the publisher used to publicise the 25 most-downloaded papers from the journal every three months, papers from the BCC proceedings were usually high on the list.

It is clear that publication in the proceedings was highly valued by many conference participants.

But no more. Discrete Mathematics has been moving away from conference proceedings, and has given notice to the BCC. The committee have tried but failed to find an alternative forum for publication. We required a journal with a good reputation, since this is one of the things that delegates appreciated; putting the papers on a web page would be easy enough but would not really fill the gap. So, for this year’s conference, there will not be a volume of contributed papers.

Academic publishing is changing very rapidly, and it may well be that the committee will decide to try out a new option, perhaps an epijournal in the future. But, for what it’s worth, a tradition of some standing will be broken.

So I got a bit of a shock recently when it started nagging me to say that my current version of Ubuntu is no longer supported, and inviting me to upgrade. I was a bit nervous; what if the upgrade went disastrously wrong? (I’ve told the story of PC World technical support trying to remove the Recycler virus from my Windows XP laptop a couple of years ago, ending up removing everything except the virus.)

I am now in the middle of marking close to 300 exam scripts. Let your troubles all come together, I thought, so I went ahead with the upgrade while I sat in my armchair marking.

A little bit of research showed that you should only upgrade one step at a time or disaster will certainly strike. So, with some trepidation, I clicked the button to upgrade from 11.10 (“oneiric ocelot”) to 12.04 LTS (“precise pangolin”) at lunchtime yesterday. There were some nervous moments, such as when it rebooted and couldn’t find the /etc directory; but finally, after several hours, it was working.

Everything seems fine. I haven’t lost any of my shell script commands or aliases; my desktop looks exactly the same, apart from the fact that Skype has disappeared (which I don’t really mind); and all the programs I’ve tried seem to run as they should.

As its name might suggest, oneiric ocelot was a bit dozy; invoking various programs from the command line gave a cascade of error messages (but the programs still worked); and it seemed to take a long time to shut down. Precise pangolin has fixed both of these glitches.

Best of all, the magic letters LTS (“long-term support”) suggest that it won’t start nagging me again for a while, which is a good thing as far as I am concerned!

The post announces an initiative, backed by the Sloan Foundation, to look beyond current issues of open access and copyright and ask for

how a World Digital Mathematical Library could be so much more than just a collection of digitally available mathematical documents.

All ideas welcome: read the article, and think creatively!

Ingrid particularly asks for help from “internet-savvy, imaginative, social-networking young mathematicians”, and also “bloggers and blog-readers”. These two groups are not the same; I belong to the second but not the first. But let’s imagine that she really means anyone who has a good idea about how we can use new technology to enrich the publishing and reading of mathematics, and particular, would capture the imagination of the mathematical community.

Marcin Krzywkowski has asked for the walks to be swapped, i.e. the Chilterns walk on the Sunday before the conference. Is there any support for this? It would be an all-day walk; we might upgrade the Thursday walk to something a little more substantial. Let me know what you think!

I hope to see many old and new friends at the conference this summer (8–10 July).

So I propose a couple of walks, one on the Sunday before, and one on the Thursday after. Please come on at least one of these, if you can.

On Sunday, if you are arriving by lunchtime, I propose an afternoon walk somewhere in London. Tentative suggestion: High Barnet to Cockfosters (both easily accessible by tube), with a stop at the Cock and Dragon in Cockfosters (which doesn’t serve Fosters, at least it didn’t last time I was there). This would be about 6km altogether. More details later.

For Thursday, I propose something just a little more serious, a walk in the Chilterns, including lunch at the Woody Nook at Woodcote, a restaurant whose wines come from its own boutique vineyard on the Margaret River in Western Australia. We will go by train to Goring, from which it is about 5km to Woodcote (up a hill); after lunch, the most likely option is a slightly longer route (maybe 10km), down the hill to the Thames, and back beside the river to Goring. Variants are possible as well.

I will need to have a good estimate of numbers, to make a booking at the Woody Nook.

If you are interested in one or both of these walks, please either email me (p.j.cameron AT qmul.ac.uk) or leave a comment here. If you would like to propose a big alteration (such as having the Chilterns walk on Sunday instead), please do so; everything is changeable at the moment.

Perhaps the first crack in the wall came with the work of Lászlo Babai in the late 1970s. Permutation groups had struggled for a century to find good bounds for the order of a primitive groups of degree n not containing the symmetric and alternating group. Wielandt found an exponential bound for groups which are not 2-transitive; this was extended and improved by Praeger and Saxl to give a bound 4ⁿ for all primitive groups. Then Babai, using ideas from the combinatorics of coherent configurations and the probabilistic method, produced a bound for the orders of primitive but not 2-transitive groups which is much better for large n, namely n^{4√n log n}. This is best possible apart from the logarithm in the exponent.

Babai just managed to save this problem from the Classification of Finite Simple Groups, which bulldozes away some very attractive problems and results, and enables us to prove much stronger bounds with known exceptions.

One of my papers with Laci also contains a novel application of the probabilistic method. There are a number of results of the form “every [finite] group is the automorphism group of a [finite] X”; for example, Frucht in the case where X=graph (or cubic graph). Sometimes there is a simple reason why such a result cannot hold. The automorphism group of a tournament must have odd order; but every finite group of odd order is the automorphism group of a finite tournament.

The result that Laci and I proved concerns automorphism groups of switching classes of tournaments. Such a group must have a double cover with a unique involution; so first we need to understand such groups. A cohomological argument due to George Glauberman, described here, shows that the groups are precisely those whose Sylow 2-subgroups are cyclic or dihedral. Then an application of the probabilistic method shows that, given such a group, we can construct a switching class of tournaments for which it is the full automorphism group. The novelty is the use of the probabilistic method rather than explicit construction.

The person who did most to bring together the “Hungarian” probabilistic combinatorics with finite geometry was another Hungarian, Tamas Szőnyi, who combined a deep knowledge of finite and algebraic geometry with a thorough understanding of random techniques. The British Combinatorial Conference always has to struggle a bit to find speakers who can interest researchers in the most diverse areas of combinatorics. Tamas fitted the bill very well, when he spoke at the conference at Queen Mary and Westfield College (as we then were) in 1997.

These things are on my mind now because of a lovely talk last Friday by Jeroen Schillewaert, on small maximal partial ovoids in generalized quadrangles. Briefly, then, the definitions:

• a generalized quadrangle is a point-line geometry, with at least three points on a line and at least three lines through a point, having the properties that two points lie on at most one line, and if a point p is not on a line l then it is collinear with a unique point of l.
• Such a geometry has a constant number (say s+1) of points on a line and a constant number (say t+1) of lines through a point; the pair (s,t) is called the order of the GQ.
• A partial ovoid is a set of points, no two collinear; it is maximal if no point can be added without violating this condition.

It is known that √s ≤ t ≤ s². There are obvious upper and lower bounds for the size of a maximal partial ovoid, of size about s and st respectively. In the case where t = s², the best explicit construction of a small maximal partial ovoid produces one of size about s²/2.

Jeroen Schillewaert and his co-author Jacques Verstraëte have given a randomized construction procedure that succeeds, asymptotically almost surely, in finding a maximal partial ovoid of size s times a power of log s, a dramatic improvement and not far from the trivial lower bound. There is not too much clever finite geometry (this has the advantage that it works for any GQ with these parameters, not just a classical GQ), but some fairly sophisticated probability theory. They hope to extend the result to wider classes of GQs.

And it was a very good talk, using blackboard and chalk.

It is a consultation document, released on 25 February this year, and requiring responses by 25 March (so it is already way too late). The title is Open Access and Submissions to the Research Excellence Framework post-2014.

The first thing to note is that it begs the question of whether there will be, or should be, a REF post-2014. When the RAE (which, as I like to point out, was correctly named: it was an exercise to assess research) was introduced in the late 1980s, there is general agreement that it had an effect: it concentrated people’s minds on the need to focus on doing good research, and was a wake-up call to some. But having done that, it became less useful; and the change from dual support to overheads on grants took away another plank, since the money it had to distribute was much reduced. After 2001, many voices were raised to say it has outlived its usefulness. Indeed, the reason for the seven-year gap until the next one (previous gaps had been four or five years) was the need to win the battle to have another RAE before it could be set up. Then came the present absurdity, the REF: meaningless title, universally unpopular emphasis on impact, etc.

I really wish that academics could all stand together and say, enough is enough. But I know this won’t happen; however bad it is, there are some people who profit from it (or imagine that they will), who will let down the united front, and HEFCE will listen to them.

Anyway, end of rant. As far as HEFCE are concerned, there will be another REF, and in order for papers to be eligible, they must be published under some version of open access. Although it is a consultation document, many of its conclusions are already said to be irrevocable. So, for example, in their words,

• “work which has been originally published in an ineligible form then retrospectively made available in time for the post-2014 REF submission date should not be eligible”,
• “all submitted outputs … shall be available through a repository of the submitting institution”.

The second point is particularly worrying. It says quite clearly that putting your papers on the arXiv will not qualify. I have struggled with the absurd software used by repositories in two universities; this requirement makes no sense whatsoever. They do, however, envisage the possibility that your institutional repository simply has a link to the arXiv; so why bother going round the houses?

They admit that there are serious problems for which no solution is yet in sight. For example, in some disciplines, a common form for output is the monograph. We don’t yet have a model for open-access monographs; do they expect publishers to give them away free? who will pay the postage?

As a final, general comment, a lot of the trouble comes from HEFCE’s desire to treat all subjects in exactly the same way. Perhaps this counts as “transparency”, and is thought to be a good thing by people who have not thought about it.

On a related subject, there was an article about epijournals in the current issue of the European Mathematical Society Newsletter. Most of it makes sense, but one proposal leaves me with a bit of unease. The organisation supporting epijournals will make available software for direct communication between author and referee, preserving the anonymity of the referee.

During my time as an academic, I have seen a big shift of responsibility from editor to referee. It used to be the case that the referee’s job was to advise the editor primarily on whether the paper was worth publishing, and second on any improvements which could/should be made. I have on more than one occasion had a disagreement with a referee and taken my case to the editor. (On one occasion the referee said, “page 7, line 23: this is wrong”, giving no indication of why. I told the editor that it was not wrong, and the editor believed me.)

However, there is a tendency now for editors simply to require that authors obey all the referees’ demands. This development in communications seems to take us some way further in this direction.

By coincidence, this issue of the EMS Newsletter also has an article about the British Society for the History of Mathematics.

I was involved in a lot of the department’s activity (though obviously not teaching). Thus, I designed a new module which (if approved) I will teach next year; I marked some student presentations; I gave a Pure Mathematics Colloquium on Derangements; I was on the victorious Pure Maths team in the staff/student charity quiz; I struggled with the truly awful University software for recording research details and publications (which thinks I am a statistician, and cannot be persuaded to change its mind); more happily, I have begun producing my own personal web page. I chaired a public lecture by a colleague, and had a few meetings with my first St Andrews PhD student. I went on an LDWA Heart of Scotland Group walk near Crieff with Kenneth Falconer; I had lunch or dinner with several colleagues; and, most important, I have started what I hope will develop into several new collaborations.

One of my colleagues announced at coffee, “I now have Erdős number 3, and all three people are in this building.”

The student newspaper is no better or worse than many, except for a startling example of innumeracy on the front page of one issue: The proportion of students gaining first or upper second class degrees rose from 68% in 2002/03 to 85% in 2011/12, “representing an almost 152% increase over the past ten years” (!!)

St Andrews is an interesting town; more than almost any other, people and cars inhabit parallel universes. Lade Braes is a path through the town and then along the Kinness and Cairnsmill Burns for some distance. But my favourite is a route which starts a little to the west of the ruined Blackfriars Chapel on South Street. Duck under a lintel below an old cottage, and keep on in a straight line, or as near to straight as you can in St Andrews. After about 2km you are out of the town, climbing Pipeland Hill with spectacular views back over the town and Eden estuary. After 6km you are on the bank of Cameron Burn, where the path ends. All this with only one very short stretch on the road.

We were told that St Andrews never has five fine days in a row; if there are four, then for sure the weather will change on the fifth. After the first couple of weeks, I was prepared to revise “five” and “four” to “one” and “half”; but the weather improved later, even if it could still come up with a cloudless sky and a heavy hailstorm within a few hours of each other. Spring was late coming, but by the end of the month the hawthorn and horse chestnut trees were in leaf. The evenings were drawing out, so it was still light walking home after dinner, and with that incomparable northern light.

Rosemary and I walked two longish stretches of the Fife Coastal Path, from Newburgh to the Tay Road Bridge and from there back to St Andrews. (We couldn’t resist walking over the bridge and back!) On another outing we went by train to Stonehaven and walked to the spectacular Dunnottar Castle on the Aberdeenshire coast.