A plane, three trains, various buses and tubes, and I have been translated from Lisbon to Prague. Time to celebrate with some eye candy perhaps: from here

to here:

Prague is beautiful too, and full of magic:

A plane, three trains, various buses and tubes, and I have been translated from Lisbon to Prague. Time to celebrate with some eye candy perhaps: from here

to here:

Prague is beautiful too, and full of magic:

Michael Kinyon reminded me of Edgar Allan Poe’s comments on algebraists in his story “The Purloined Letter”. Here they are in full.

“But is this really a poet?” I asked. “There are two brothers, I know; and both have attained reputation in letters. The minister I believe has written learnedly on the Differential Calculus. He is a mathematician, and no poet.”

“You are mistaken; I know him well; he is both. As poet *and* mathematician, he could reason well; as mere mathematician, he could not have reasoned at all, and thus would have been at the mercy of the Prefect.”

“You surprise me,” I said, “by these opinions, which have been contradicted by the voice of the world. You do not mean to set at naught the well-digested idea of centuries. The mathematical reason has long been regarded as *the* reason *par excellence*.”

“*‘Il y a à parièr,’*” replied Dupin, quoting from Chamfort, “‘*que toute idée publique, toute convention reçue, est une sottise, car elle a convenue au plus grand nombre.’* The mathematicians, I grant you, have done their best to promulgate the popular error to which you allude, and which is none the less an error for its promulgation as truth. With an art worthy of a better cause, for example, they have insinuated the term ‘analysis’ into application to algebra. The French are the originators of this particular deception; but if a term is of any importance—if words derive any value from applicability—then ‘analysis’ conveys ‘algebra’ about as much as, in Latin, *‘ambitus’* implies ‘ambition’, *‘religio’* religion, or *‘homines honesti’* a set of *honorable* men.”

“You have a quarrel on hand, I see,” said I, “with some of the algebraists of Paris; but proceed.”

“I dispute the availability, and thus the value, of that reason which is cultivated in any especial form other than the abstractly logical. I dispute, in particular, the reason educed by mathematical study. The mathematics are the science of form and quantity; mathematical reasoning is merely logic applied to observation upon form and quantity. The great error lies in supposing that even the truths of what is called *pure* algebra are abstract or general truths. And this error is so egregious that I am confounded at the universality with which it has been received. Mathematial axioms are *not* axioms of general truth. What is true of *relation*—of form and quantity— is often grossly false in regard to morals, for example. In this latter science it is very usually *untrue* that the aggregated parts are equal to the whole. In chemistry also the axiom fails. In the consideration of motive it fails; for two motives, each of a given value, have not, necessarily, a value when united, equal to the sum of their values apart. There are numerous other mathematical truths which are only truths within the limits of *relation*. But the mathematician argues from his *finite* truths, through habit, as if they were of an absolutely general applicability—as the world indeed imagines them to be. Bryant, in his very learned ‘Mythology’, mentions an analogous source of error, when he says that ‘Although the pagan fables are not believed, yet we forget ourselves continually, and make inferences from them as existing realities.’ With the algebraists, however, who are pagans themselves, the ‘pagan fables’ *are* believed, and the inferences are made, not so much through lapse of memory as through an unaccountable addling of the brains. In short, I never yet encountered the mere mathematician who could be trusted out of equal roots, of one who did not clandestinely hold it as a point of his faith that *x*^{2}+*px* was absolutely and unconditionally equal to *q*. Say to one of these gentlemen, by way of experiment, if you please, that you believe occasions may occur when *x*^{2}+*px* is *not* altogether equal to *q*, and, having made him understand what you mean, get out of his reach as speedily as convenient, for, beyond doubt, he will endeavour to knock you down.”

I must admit I would quite like to knock him down for this amazing farrago. But it is more interesting to refute him point by point. A good exercise for students?

And yet there is something in what he says. Julian Jaynes gives an example of valid poetic reasoning, which goes something like this: “Man dies; grass dies; so man is grass”. Could it be that this kind of reasoning is closer to what makes Dupin, Holmes, and their ilk great detectives than Boolean logic? Indeed, is it closer to how we *do* mathematics, rather than how we write it up afterwards?

I actually don’t think that mathematicians are much better than other people in applying logical reasoning to things outside mathematics; and I think that we teachers should feel a bit ashamed of that fact.

The University of Coimbra is the oldest in Portugal, having been founded in 1290 (younger than Oxford, older than St Andrews), but after bouncing back and forth between Coimbra and Lisboa for a while, it finally settled in Coimbra in 1537. Dom João III gave the University his royal palace at the top of the hill, and this is now the heart of the University, with an ancient library, chapel, meeting room, examination room, and so on. The wonderful University palace quadrangle and surrounding buildings are since 2013 a World Heritage site.

Rosemary and I spent two quite extraordinary days in and around Coimbra. I thought the purpose of the trip was for me to give a colloquium talk – I did that, speaking on “The Random Graph”, and drew an enthusiastic audience of over thirty, quite remarkable for this time of year – but it seems that the real purpose was for me to be shown some of the wonders of this part of Portugal, and to receive Portuguese hospitality from my hosts Jorge Picado and Maria Clementino.

Jorge met us at the station, gave us a tour of the University palace and courtyard, and then with Maria took us to lunch. After my talk, we were delivered to the hotel with instructions about where to find Coimbra fado that evening.

Coimbra fado is different from the Lisboa variety, and its performance is jealously guarded. It seems to me that it gives the musicians much more interesting things to play. We heard a singer and two guitarists (one playing a Portuguese guitar, whose tuning pegs radiate out like a peacock’s tail, the other a regular Spanish guitar) at the Santa Cruz café. Fado, beer, and ham and cheese to pick at. After the fado, we walked through the park on the banks of the river Mondego, the largest river entirely within Portugal (the Tejo and Douro both rise in Spain), where we sat and watched swallows, kites, and wispy clouds.

The next day was all sightseeing, first to the Roman town of Conimbriga, and then to the forest of Bussaco.

The guidebook says that the name Conimbriga gave rise to Coimbra when it was transferred to the town formerly known as Aeminium, though according to Jorge, not all authorities agree. There was a sizeable settlement here from about 900BC, on a plateau at the edge of a steep river gorge. The town flourished in Roman times; only 15% of it has been excavated. Some of the most extensive houses were demolished to build a defensive wall against the barbarians at the end of the Roman empire. Much of the underfloor heating systems and many fine mosaic floors remain, and many artefacts of Roman life have been found.

Many of the mosaic patterns are geometric, including a couple of vertex-transitive tesselations (one with squares and octagons, one with triangles, squares and hexagons).

Bussaco has an alternative spelling Buçaco, sometimes in the same document. It was owned by a monastery of Carmelite friars, to whom its remoteness was a great benefit. They built a via sacra and hermitages as well as a monastery. In the Napoleonic Wars it was the scene of a fierce battle, when Portuguese and British forces inflicted the first defeat on Napoleon’s troops. Later a hotel was built overshadowing the old monastery, in a flamboyant neo-baroque style in which the stone almost seems to be a living and growing organism.

The monks left much of the forest in its natural state, but also planted a wide variety of trees, so that now everything from pines to tree-ferns flourishes there. Many trees were blown down in a severe storm in January 2013, but there are so many trees in the forest, and so much conservation work has been done already, that the scars are not too noticeable. We had a pleasant hike down the stream and back up beside an artificial cascade, and then on the via sacra leading to the Coimbra Gate giving fine views over the countryside below.

At the end of the day, a fast train took us back to Lisbon, in half an hour less than it had taken us getting there the day before.

Posted in events, geography, history
Tagged Bussaco forest, Coimbra fado, Conimbriga Roman town, the random graph
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Pedro Nunes was a Portuguese mathematician of the sixteenth century, perhaps the greatest mathematician of his time in Europe.

Yesterday I was treated to a very informative short presentation about Nunes and his work by the historian of science Henrique Leitão. Here are three things I learned.

First, one of Nunes’ five books was a book on Algebra. What is remarkable about it for its time is the philosophy. Nunes believed that algebra is not just a growth from the root of geometry, but an entirely new subject. His proof was that some results in geometry are more easily proved by means of algebra than by geometric methods.

Second was his discussion of the *rhumb line* (now called the *loxodromic curve*), the line traced by a ship which sails always on a bearing making a constant angle with the meridian. Such a line is not a great circle, since it spirals in to the north and south poles. (This fact was already a great novelty at the time, a curve having a finite limit point.) The mathematical tools of the time did not permit finding its equation, but Nunes proposed a “finite difference method”. The navigator sets a bearing making the given constant angle with the meridian, and sails straight (i.e. alon a great circle) until his bearing deviates from the required value by more than a given fixed amount (say one degree); then he corrects the bearing and continues. This gives a practical method for calculating rhumb lines. Nunes’ method was used by many others, and tables were produced.

There has been a lot of interest in the question of how Mercator calculated his map projection. Leitão and a colleague propose a new answer to this. Since rhumb lines appear as straight lines in Mercator’s projection, he could simply use existing tables based on Nunes’ method to derive the spacings of the parallels. This hypothesis appears to fit the data better than any other suggestion.

Nunes’ third remarkable achievement was the following. Suppose that you place a vertical stick in the ground, and watch the movement of its shadow as the day progresses. Almost everyone would say that the movement of the shadow was monotonic. However, Nunes did the calculations and showed that retrograde motion of the shadow was possible under some conditions, and worked out exactly what the conditions were. He admitted that he had never seen the phenomenon, despite knowing what to look for, and nobody he had spoken to had seen it either; yet he had sufficient confidence in his mathematics that he could confidently assert its existence. This caused a certain amount of religious controversy; the fact of a shadow standing still is described in the Bible as a miracle, and yet Nunes was proposing that standing still or even reversing can happen strictly in accordance with natural laws. (I believe that this phenomenon, though small, has now been observed.)

Anyway, the reason for this was an extraordinary event yesterday. I mentioned in March that I was teaching a group theory course to PhD students in compuational algebra at the Universidade Aberta (the Portuguese Open University). There was a celebration of the successful completion of the first year of the course, at which two Pedro Nunes awards (voted by the students on the course) were presented, to Michael Kinyon and me, by the Chancellor of the University. The ceremony began with a short presentation by João Araújo (the driving force behind the course) of how the computational algebra course was set up, and how it had run.

All this in a morning out from the Portuguese Mathematical Society summer meeting, at which I lectured. I will probably say something about this meeting later.

Things have been a little quieter for the last couple of weeks. Tomorrow, off to Portugal for a week and a half, then the Czech Republic for a week, then a couple of weeks to catch my breath before New Zealand …

I have managed a bit of rushing round to see family.

Neill’s second book, *How to make Awesome Comics*, is out next month from David Fickling Books. For the young person in your life, make sure you get hold of one! He also did some artwork for the Cowley Road Carnival last weekend, and said it was quite spooky to see it plastered on every bus stop in Oxford!

I learned a little bit more about Hester’s working life too. She was surprised to find a recent presentation she gave about decommissioning in the oil industry available on the web. To my somewhat biased view, she has done a good job, presenting hard facts and mostly avoiding management-speak.

Apart from all that, I have been doing some work: Pablo Spiga and I have a first draft of the paper on the theorem I discussed in the last post, and Sebastian Cioabă and I have nearly finished the paper on edge partitions of complete graphs I discussed last year.

Oh, and I set a resit exam for Mathematical Structures (but I don’t have to mark it :) )

Posted in Uncategorized
Tagged awesome comics, decommissioning, graph partitions, switching classes
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Last year I wrote here about switching classes of graphs for which the switching class has a primitive automorphism group. (I repeat the definitions briefly below.) I conjectured that, except for the trivial switching classes of the complete and null graphs and finitely many others, such a class must contain a graph with trivial automorphism group. This is an example of the point of view of Laci Babai: from the Classification of Finite Simple Groups we learn that conditions which seem to imply a great deal of symmetry (such as primitive automorphism group) actually severely restrict the amount of symmetry.

The operation of *Seidel switching* a graph with respect to a set *S* of vertices involves replacing edges between *S* and its complement with non-edges, and non-edges with edges, leaving edges and non-edges inside and outside *S* unchanged. This is an equivalence relation on the class of graphs on a given vertex set, whose classes are called *switching classes*. An *automorphism* of a switching class is a permutation which permutes among themselves the graphs in the class. An equivalent combinatorial concept is a *two-graph*, a collection of 3-element subsets of the vertex set with the property that any 4-element subset contains an even number of members of the collection. I talked about two-graphs at the Villanova conference, and you can find the slides here.

Soon after posting that, I was able to prove the conjecture. Now Pablo Spiga and I have worked out the finite list of exceptions. Up to complementation, there is just one such switching class on 5, 6, 9, 10, 14 and 16 vertices, and no others. The automorphism groups are *D*_{10}, PSL(2,5), 3^{2}:*D*_{8}, PΣL(2,9), PSL(2,13), and 2^{4}:*S*_{6}. All these are well-known. For an odd number of vertices, these are the switching classes of the finite homogeneous graphs with primitive automorphism groups (I don’t think this is more than a small-number coincidence). On an even number of vertices, they all have doubly transitive automorphism groups, and are among the list of such things given by Don Taylor a long time ago.

The proof that the list is finite comes by confronting upper bounds for orders of primitive groups derived from CFSG with lower bounds from the assumption that every graph in the switching class possesses a non-trivial automorphism. By pushing these arguments as hard as possible, Pablo was able to show that there were no examples on more than 32 points, and I was able to search all the primitive groups with degree up to 32 and come up with the list.

I hope we will have a paper available quite soon.

One interesting thing emerged from the investigation, which is probably worth a further look. For reasons I won’t go into here, it suffices to consider the case where the number of vertices is even. (The odd case is covered by the results of Ákos Seress that I discussed in the earlier post.) The switching classes with primitive automorphism group on *n* vertices, with *n* even and *n* ≤ 32 fall into two types:

- those with doubly transitive groups, which are in Taylor’s list; and
- some with very small groups: two on 10 vertices with group
*A*_{5}, six on 28 vertices with group PGL(2,7), and six more on 28 vertices with group PSL(2,8).

I’d never seen anything like the second type before, so I looked at the first two examples, on 10 points.

The action of *A*_{5} is on the 2-element subsets of the domain {1,…,5}, which we can think of as edges of a graph. The orbits of the symmetric group *S*_{5} are isomorphism classes, of which there are just four with 3 edges, namely *K*_{3}, *K*_{2}∪*P*_{3}, *K*_{1,3} and *P*_{4}, where *K* means complete (or complete bipartite) graph and *P* means path, the subscript being the number of vertices. You can check that every 4-edge graph contains an even number of copies of *P*_{4}, so these form a two-graph containing *S*_{5} in its automorphism group. The full automorphism group is larger; it is the group PΣL(2,9) (aka *S*_{6}) and appears on our list.

However, the automorphism group of *P*_{4} consists of even permutations, so under the action of *A*_{5}, the 60 copies of this graph fall into two orbits of 30. The table below shows the numbers of graphs in the various *A*_{5}-orbits which are contained in each of the four-edge graphs on five vertices.

You can see from the table that taking one of the *A*_{5}-orbits on *P*_{4}s, together with the orbit on *K*_{1,3}s, form a two-graph. So here are the mysterious two switching classes with automorphism group *A*_{5}. We see that their symmetric difference is the much more symmetric two-graph consisting of all the *P*_{4}s.

Surely the examples on 28 points also have some nice structure! And what happens beyond?

Posted in exposition, mathematics
Tagged Akos Seress, Don Taylor, Pablo Spiga, primitive group, switching class, two-graph
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