After infinity

After the Mass Interaction debate on the nature of infinity, I talked to quite a few people. Among them were Ania and George, whose ears pricked up when I waid I was going to Lisbon soon. They said they were also going to Lisbon, and hoped that we could meet up.

The chance of people randomly meeting up in Lisbon, when two are downtown and the other commuting between Benfica and the University, are pretty remote. But Ania and George found my email and contacted me, suggesting a meeting. So on Thursday, after lecturing on group theory, I went over to the Cidade Universitaria metro to meet them. Right on time, there they were.

After taking them to the mathematics department to introduce them to João (who is my boss at the moment, I suppose, having just taken over as head of department the week before I arrived), we went and had a coffee and a chat, ranging widely from the Polish cryptographers who first broke the Enigma cipher (from Ania’s home town of Poznan) to the architectural style encouraged by the Portuguese dictator Antonio Salazar (much in evidence around the university). Then we went and sat on a seat outside, and Ania interviewed me for her blog. (I will post a link when the interview goes up.)

At the end they were polite enough to say that the conversation had been a highlight of their visit to Lisbon. I feel enormously privileged to be able, not only to do mathematics in such a beautiful city, but to take a break from the mathematics and talk to lively and interesting people.

PS It was George and Ania who came up with the lovely title. In some ways even better than that of the Horizon program “To infinity and beyond”.

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According to Astronomy Picture of the Day, this month’s full moon is the largest since 1948, and there won’t be another as large until 2034. (This means that this month, the point in the moon’s orbit where the full moon occurs is very close to the perigee, or point of closest approach to the earth.)

Anyway, this is what it looked like, rising over Portugal:


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Lisbon 2016


As the picture (medronho fruit and eucalyptus leaves) might suggest, I am in Portugal: this was taken in Monsanto (not the wicked chemical company but the forest park on a hill just west of Lisbon). I was meant to be here in July but shingles put paid to that.

Anyway, I am (as usual in Lisbon) very busy, and should have some mathematics to report on very soon. Also, I will be giving a “crash course” on group theory, and there may be notes …

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Looking back

I grew up in the 1950s and 1960s. Many things are written about the 1960s now, some by people who weren’t there, some by people who don’t remember, so I probably don’t have much to add. Things were not a paradise in those days: there were many problems. But there was a feeling in the air then that things could get better: by our own efforts, we could build a fairer, more tolerant, world, one in which artistic and scientific progress for the good of everyone was really possible.

I don’t find that sense any more. One of the main changes is that we are not all humans, all in the same predicament, and all helping each other. Rather, almost everyone has some flag they can rally round, to attack those under a different flag. And I do not only mean political flags, of course. In a world in which male writers are criticised for writing about female characters (and similarly for other pairs of conflicting opposites), people would rather have a safe space among others like them than the adventure of celebrating their common humanity. This also extends to rich and poor; everywhere, the state is retreating from its commitment to offer health services, education, public transport, and social security to all, and the rich (with a few honorable exceptions) have no intention of picking up the burden. Repressive and intolerant regimes force millions of people to live as refugees, and now the US has a president who will build a wall and expel many of his own people.

I think I am not the only one to notice this. Bob Dylan, unlike his creation Mr Jones, knows what is happening. In the 1960s, he sang

When you got nothin’
You got nothin’ to lose

But thirty years later, almost the same lines had a very different feel:

When you think that you’ve lost everything
You find out you can always lose a little more

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The Institute of Combinatorics and its Applications keeps a low profile. There is no on-line version of its Bulletin. When I was forced to leave my office in Queen Mary at two weeks notice, old issues of the Bulletin (along with many other journals) had to be put in the recycling, and now I have no access to that material.

For the last few years, the Institute has been relatively inactive, failing (for example) to award its annual medals.

However, I am glad to say that the sleeping giant seems to be stirring. There is, as of March this year, an ICA blog, which I have put on the blogroll so you can take a look. (I have only just been notified about this, eight months after the event.) There is even a promise to look into the possibility of an ICA website and even on-line publication of the Bulletin.

We shall see!

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The nature of infinity, 2

On Thursday, the first in a series of public discussions on scientific topics was put on by an organisation called Mass Interaction (the name comes from a statement by Richard Feynman that “all mass is interaction”). John Barrow and I discussed the nature of infinity, moderated by the driving force behind the event, Max Sanderson.

The event was held in a new venue called AirSpace, in Oxford Street near Tottenham Court Road station. This area is a huge building site at the moment because of Crossrail works, so I had a bit of trouble finding my way to the venue; but I did make it, only a little late.

I will write here something which is not a summary of the discussion but was certainly inspired by it.

We agreed, more or less, that there are three kinds of infinity:

  • The mathematicians’ infinity. This included the paradise opened to us by the work of Georg Cantor, with its dizzying vision of infinitely many different sizes of infinity, but is not just that: it also includes the infinitesimals of Newton’s calculus, over which he was criticized by Berkeley (who called them “the ghosts of departed quantities”), and more besides. Do these infinities exist? In mathematics, existence does not necessarily mean correspondence with a physical object in the universe; self-consistency is enough for mathematical existence.
  • The physicists’ infinity. This has several subdivisions, which I will indicate by questions: is the Universe infinitely large? Has it existed for infinitely long? Will it continue for infinitely long? Is space infinitely divisible? Is time infinitely divisible? Is it possible to have a point within our finite part of the universe where a physical quantity such as density or temperature is infinite? These infinities may potentially exist; it is reasonable to believe that the question of whether the universe has existed for an infinite time already has a definite answer, even if we cannot be certain what the answer is.
  • The theologians’ or mystics’ infinity. This is much harder to discuss. Many people have had a mystical experience of union with the infinite, but it is notoriously difficult to put the experience into words afterwards. If we cannot agree on what infinity is, it is unlikely that we can decide whether it really exists or not.

Of course, the different types are not sharply separated. The Buddha’s disciple Mālunkyaputta went to his master with a number of questions, to which he desperately wanted answers:

  • whether the world is eternal or not eternal,
  • whether the world is finite or not,
  • whether the soul (life) is the same as the body, or whether the soul is one thing and the body another,
  • whether a Buddha (Tathagata) exists after death or does not exist after death, whether a Buddha both exists and does not exist after death, and whether a Buddha is non-existent and not non-existent after death.

Are these questions about the physical universe, or are they transcendental? It doesn’t really matter, they are surely the great questions that have troubled humans for as long as there have been humans.

(The Buddha refused to answer these questions. He said, “Whether the view is held that the world is eternal, or that the world is not eternal, there is still re-birth, there is old age, there is death, and grief, lamentation, suffering, sorrow, and despair.”)

Indeed, I would add here that our first exposure to infinity comes at the point in our childhood where it suddenly comes home to us that, at a certain point, the world will carry on but we will cease to exist. We naturally wonder whether the world is infinite in time, and whether in some other sense we are also infinite. I think it takes great courage to face these questions squarely. This is perhaps why most of us put them aside and don’t think about them any more.

The great civilisations of the first millennium BCE thought about infinity, and left us with records of their thought. The Greeks had the greatest influence on subsequent European thought. Partly as a result of problems such as Zeno’s paradox, Aristotle forbade consideration of actual infinity and permitted only “potential infinity” (such as the progression from a natural number to the next). This damped down European speculation for a long time, abetted by the Catholic Church, which taught that humans were uniquely creatures of God and an infinite universe would have no distinguished centre for us to inhabit. (Giordano Bruno was burnt at the stake for maintaining an infinite universe.) Albert of Saxony and Galileo both considered the problem of infinity but decided that it was too difficult, and it was only faced in a constructive way by Cantor in the late nineteenth century. The Indians were much more bold, aided perhaps by their invention of zero (the counterpart and “inverse” of infinity).

However, as I outlined above, I think that our ancestors thought about infinity long before this. One member of the audience objected that, prior to the present, people were too preoccupied with the problem of getting enough to eat to engage in speculation. I disagree, on several grounds.

  • Even if young children are put to work, they can still think. I grew up on a farm, and I vividly remember summing geometric series while chasing the cows in to be milked. If anything, the present age of screens and light pollution discourages children from looking up at the stars and wondering if they are infinite.
  • There is evidence that there was actually much more leisure in hunter-gatherer societies than in farming societies. Also, signs in the sky are important to both. Moreover, the Ishango bone (an artefact from the Congo basin now in a museum in Brussels) shows that our ancestors were mathematicians long before they were farmers.

Could we do without infinity? Our present civilization rests on mathematics to such a great extent that it is clear we could not do without mathematics. I believe that we cannot have mathematics without infinity. (A large majority of mathematicians would agree, though not all: there are constructivists who insist on restricting to the finite.) A vast amount of engineering design depends on differential equations, which depends on real numbers. Electronics depends on quantum theory, which depends on complex numbers. Real and complex numbers are essentially infinite objects: although we can only ever use real numbers which have a finite description, we need the entire set in order for properties of functions such as continuity and differentiability to work properly. My students struggle with the construction of the real numbers from the natural numbers; they would struggle much harder if I tried to construct them from finite fragments of the natural numbers. So infinity in mathematics is here to stay!

On the other hand, as John mentioned, physics has the Cosmic Censorship hypothesis, according to which a singularity within the finite part of the universe we inhabit can only occur behind an event horizon, and so can have no effect on us. This has been shown (except in a few cases) to be a consequence of general relativity. Physicists still abhor infinity, and where it appears (e.g. in studying point charges) they try to develop a theory (such as string theory) which will smooth it out.

If you enjoyed this, you can find more in my short course on The Infinite Quest at the Institute of Art and Ideas. (You have to work: the lecture is interrupted by questions, so you have to turn on your brain from time to time!) Also, we are promised that there will be a podcast and more on-line material about the Mass Interaction event at some point: I will publicise it when this happens.

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Old Codgers 2016

Yesterday, a beautiful sunny autumn day, I spent at the Old Codger’s Combinatorics Colloquium in Reading, where the campus with its lake and its many mature trees were looking glorious.

Tree at University of Reading

There is some controversy about the position of the apostrophe in the title. Before the s, it suggests that there is only one old codger involved, presumably the organiser Anthony Hilton, even though several regular attendees (myself included) could be so described; after the s, it suggests that everyone involved is an old codger, which was not the case, as several young people attended. In my slides, I fudged the issue by leaving the apostrophe out.

The talks were full of interest. First Terry Griggs told us about pentagonal geometries, joint work with Klara Stokes. He reminded us that a generalized n-gon is an incidence structure of points and lines whose bipartite incidence graph (or Levi graph) has diameter n and girth 2n, according to the definition of Jacques Tits; the Feit–Higman theorem tells us that no finite generalized pentagon exists (apart from the ordinary pentagon). So how should we define a pentagonal geometry? The idea of Ball, Bamberg, Devillers and Stokes was to observe that, in a pentagon, the points not collinear with a given point p comprise a line, the “opposite” of p; they take this (and the condition that two points lie on at most one line) to define a pentagonal geometry. The added regularity condition that any line has k points and any point lies on r lines is assumed. Terry took us through a number of properties and constructions of these objects, with links to Moore graphs, projective planes, and orthogonal Latin squares, among other things.

Then Steve Noble gave us a clear introduction to delta-matroids, the gadgets which fill the blank in his title “Graphs are to matroids as embedded graphs are to ???”. They are motivated by an abstraction of embedded graphs called ribbon graphs, where the vertices are discs and the edges are ribbons. Delta-matroids admit a richer collection of operations than do ordinary matroids (which occur as a special case). The last part of the talk was devoted to the enumeration of delta-matroids. It is not hard to see that the number dn of delta-matroids on n points is bigger than the square root of the number 22n of families of subsets, so that log log dn is at least n−1; the difference is a decreasing function of n which might tend to zero.

After lunch, two Johnsons (Matthew and Robert) spoke. Matthew considered the notions of Kempe chain in a properly vertex-coloured graph (a connected component of a 2-coloured subgraph) and a Kempe change (interchanging the colours in a Kempe chain). These notions were invented by Kempe in his failed proof of the four-colour theorem, and made precise by Heawood who salvaged the proof of the five-colour theorem from Kempe’s work. Bojan Mohar conjectured that, for k ≥ 3, if G is a k-regular graph which is not complete, then any proper k-colouring of G can be transformed into any other by a sequence of Kempe changes. This was refuted by Jan van den Heuvel, who pointed out that it is false for the triangular prism. Matthew and his collaborators have shown that the triangular prism is the only exception. He gave us an outline of the proof, and left us with two problems, one of which seems extremely natural: what is the best upper bound for the number of Kempe changes required?

Robert told us about a nice result he and his student Nick Day have proved. There is an easy argument that shows that a complete graph on 2k vertices cannot be covered by k bipartite graphs; so any edge-colouring of this graph with k colours contains a monochromatic odd cycle. Erdős and Graham asked about the smallest such cycle that could be guaranteed; they forgot to ask, and left to Chung, the question of whether the length of such a cycle is bounded. They have shown that it is not. Robert gave us the elegant proof. As a by-product of the argument, they have found new lower bounds for the Ramsey number for odd cycles, refuting a conjecture of Bondy and Erdős.

Then, after the tea break, David Conlon gave a beautiful board talk about quasirandomness in graphs. It has been known for a while that, for dense graphs, several “quasirandomness” conditions are equivalent: the three he talked about were the condition that the number of edges between two disjoint subsets is close to the expected number, that the non-principal eigenvalues of the adjacency matrix are bounded, and (surprisingly) that the number of 4-cycles is close to the expected number. He discussed two themes. First, what happens for sparse graphs? The equivalence of the first two conditions fails, but David and his student Zhao have succeeded in giving substitute results in some cases, including Cayley graphs. Then he talked about his work with Jacob Fox and Benny Sudakov on replacing the 4-cycle in the third condition by other graphs. This is closely related to the famous Sidorenko conjecture. If we replace a 4-cycle by a 3-cycle, the result is false, but can be restored by assuming the triangles are nicely distributed (any large induced subgraph has about the right number). This result of Simonovits and Sós used the Regularity Lemma, and so gave tower bounds for the inequalities involved; these have been reduced to polynomial, and the right answer could be linear.

I concluded the meeting by talking about three problems that I would like to see solved: cliques in graphs in the Johnson scheme, derangements and Latin squares, and sum-free sets. The slides are in the usual place.

After an enjoyable dinner at the Forbury restaurant, we headed home.

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