Remarkable forgeries?

In my pigeonhole at Queen Mary recently, I found an unsolicited copy of a book with the title Meetings with Remarkable Forgeries, by M. J. Harper. According to the cover blurb, it is a debunking of Christopher de Hamel’s Meetings with Remarkable Manuscripts, about which I said something here. Perhaps the publishers sent it to me because of my post about the earlier book.

In fact, it is not quite what the cover says. The blurb ends up by saying “The author has bigger fish he wants to fry”, and indeed on the first page of the text he mentions “my long-term goal of getting the universities abolished”, which hardly endears him to me. I must say that, on the strength of this book, universities needn’t start quaking in their boots just yet.

De Hamel describes twelve manuscripts, of which only two are claimed to be forgeries by Harper, the Gospels of St Augustine and the Book of Kells. In particular, the Leiden Aratea, on which I commented, doesn’t draw his fire, even though de Hamel describes its production by what might count as an earlier version (at the court of Charlemagne) of the production lines Harper regards as being responsible for forging other early manuscripts: he starts with an admission that mediaeval monks copied manuscripts, and while we find something wrong with this, they clearly did not. (Before the printing press, copying was essential to ensure wide circulation.) Indeed, it often happened that the copy was a finer manuscript than the original, in which case it appears they felt no compunction to carefully preserve the original.

Anyway, Harper claims that the two manuscripts I mentioned are both “forgeries”, produced in Durham in the twelfth century. He describes the Benedictine order as the “Thomas Cook of their day”, arranging itineraries for pilgrims, and as a profitable sideline, producing “ancient” manuscripts for the pilgrims to see on their journeys. He claims that Durham was an important centre of this manuscript production. The secular clergy of Durham cathedral were replaced by Benedictine monks in the twelfth century, and the Bishop of Durham ensured that he was master of the County Palatine, and that the Sheriff of Northumberland had no authority in Durham.

The other reason for the monks to forge “ancient” gospel books was to record the charters documenting their claim to various properties. Certainly, mediaeval monks transcribed legal documents of this sort into spare pages in gospel books. I used to have a lovely book on the history of Eynsham Abbey (alas, I can no longer find it) which describes such practices. Presumably something written in a gospel book was less likely to be challenged than if it was on a loose piece of vellum in the abbot’s study.

Harper claims that there is no archaeological evidence of early monasteries on either Iona or Holy Island. I have never been to Iona. It is true that there is nothing old to see on Holy Island. I am not sure what this proves. Harper’s claim is that St Cuthbert’s Gospels and the Lindisfarne Gospels could not have been produced there.

Another manuscript in his sights is the Llandeilo Gospels, claimed to be an ancient Welsh book, produced in Llandeilo, but according to Harper produced in another forgery factory, this time in Lichfield. He takes issue with the usual derivation of the name “Llandeilo” as the place of St Teilo, and instead interprets it as “an enclosure where dung was spread”.

The Wikipedia article for “llan” confirms that “[t]he various forms of the word are cognate with English land and lawn and presumably initially denoted a specially cleared and enclosed area of land.” But it adds, “In late antiquity, it came to be applied particularly to the sanctified land occupied by communities of Christian converts”, and goes on to add that nearly all of the 630 placenames in Wales containing this element “have some connection with a local patron saint.”

Perhaps Harper wants to close down Wikipedia as well as the universities?

Harper grumbles at the Welsh writing Llanfair for “St Mary’s parish” as being unable to spell the name of the second most important person in the world, but I believe that this is actually a correct translation. Welsh declines the beginning, rather than the end, of a word: Merthyr Mawr, but Fforest Fawr. I wonder what experts on mediaeval Welsh make of all this?

As befits one with such a hatred of scholars, his book has no table of contents and no bibliography. He quotes various things from other books, but with the exception of de Hamel’s book, he doesn’t tell us what these are. (Fortunately he does have an index, without which I would have had a much harder job writing this account.)

By way of light relief, here is a crossword clue which you should easily be able to solve:

Giggling troll follows Clancy, Larry, Billy and Peggy who howl, wrongly
disturbing a place in Wales (58)

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Summer school at Marienheide

Summer school

Last week, I was lecturing at a summer school in Franz Dohrmann Haus, a very pleasant conference centre in the small town of Marienheide, not far from Köln. Apart from a walk on Wednesday afternoon, I didn’t get much exercise, and the food was excellent and plentiful (including fresh figs for breakfast), so I am sure I have come back a kilogram or two heavier. (Table tennis was available, but the students preferred to play bizarre card games, occasionally involving physical assaults of various kinds.)

There were various jobs I was hoping to do, but the wi-fi in the place was totally inadequate, so that it was impossible to upload or send anything bigger than a small textfile. This had the effect of keeping us focussed on the mathematics.

The topic of the summer school was permutation groups. The main speakers, apart from me, were Cheryl Praeger, Csaba Schneider, and Pablo Spiga. As you might expect, we were very much focussed on finite permutation groups, and in particular, on groups which were at least quasiprimitive but not 2-transitive; and the O’Nan–Scott Theorem played a big part in the proceedings.

Cheryl talked about quasiprimitive permutation groups. For example, she explained how, for s ≥ 2, a connected non-bipartite s-arc transitive graph has a quotient which is also s-arc transitive with the same valency, and its automorphism group is quasiprimitive on vertices. (This means that every non-identity normal subgroup is transitive. This condition is implied by primitivity, but is strictly weaker; for example, every transitive action of a simple group is quasiprimitive.)

Cheryl then described her extension of the O’Nan–Scott Theorem to quasiprimitive groups. Of the eight cases in Kovács’ version of the theorem, in several of them the group is necessarily primitive, and in most of the others the conclusions are almost identical with those in the usual version. The exception is the product action case, where the group may have an invariant partition so that it acts on the parts as a primitive group of product action type, but the intersction of the socle with the point stabiliser is a subdirect product (rather than a direct product) of factors coming from the base group.

Purely by chance, she was explaining the pioneering work of Tutte on arc-transitivity; I happened to be wearing a t-shirt with Adrian Bondy’s photo of Tutte on the back, so I stood up and exhibited it (to a round of applause).

Following this, she described progress (or lack of it) on dealing with these cases for s arc transitive graphs.

Previously, she had described distance-transitive graphs, and given Derek Smith’s description of the ways in which the automorphism group of such a graph can be imprimitive, and how to reduce to the primitive case.

Csaba and Pablo both talked about aspects of the O’Nan–Scott Theorem. Pablo had to leave after three days, so only gave four lectures. He took us through the theorem, and described some consequences. One of these was an impressive new theorem which asserts that, with known exceptions (basically the large groups, subgroups of the wreath product of the symmetric group of degree n on k-sets with some transitive group of degree l containing the socle (An)l), every element of such a group is a “regular” permutation, one which has a cycle of length equal to its order.

Pablo’s other application was to a conjecture of Richard Weiss made nearly 40 years ago. It asserts that if a finite graph is connected and vertex-transitive, and the vertex stabiliser acts primitively on its neighbours, then the order of the vertex stabiliser is bounded by a function of the valency. (This is related to the Sims conjecture.) He proved it for twisted wreath products as an illustration of the techniques.

Csaba’s focus was on what I call “non-basic groups”, those which preserve a Cartesian decomposition of the domain. Such a decomposition is a collection of partitions with the property that, given an arbitrary choice of one part from each partition, there is a unique point in the intersection of those parts. (The points can be identified with all words of length l over an alphabet A, and the ith partition divides the words according to their entries in the ith coordinate. He and Cheryl are writing a book about this; a prelimiary version is available on his website.

He spoke about how to recognise when a group preserves a Cartesian decomposition, and how to find the identification if it is. He showed us that the only almost simple groups which preserve a Cartesian decomposition have socle A6, M12, or Sp(4,2a).

(In fact, I have GAP code for checking this, which I wrote in connection to the road closure problem, which (to my great surprise) seems to be competitive with other methods, but which is still capable of significant improvement.)

In his final lecture, he spoke about twisted wreath products. Primitive groups with a unique minimal normal subgroup which is non-abelian and acts regularly are necessarily of this form; this case was omitted from the first versions of the O’Nan–Scott Theorem, but Aschbacher and Scott, and Kovács, pointed out that examples do exist. The twisted wreath product construction was already known; it was probably discovered by B. H. Neumann, and is dealt with in Suzuki’s book. Csaba explained the necessary and sufficient conditions for a twisted wreath product to be a primitive permutation group. (The smallest example is a semi-direct product of (A5)6 by A6, and is a permutation group of degree 606; so it is unlikely to arise in a computational problem for a few years yet.

My lectures were about permutation group problems arising from transformation semigroups; in particular, what are the conditions on a permutation group G if the semigroup generated by G and f has some nice property (such as synchronizing, regular, or idempotent-generated apart from the group G) for all non-permutations f, or all in some restricted class (such as all of rank k, or all with image a prescribed set of size k). The resulting conditions are typically equivalent to or closely related to primitivity if k = 2, or to higher degrees of homogeneity for larger values. So I was interested in groups slightly higher in the hierarchy than my fellow lecturers, for the most part.

In addition, there was a lecture by Tomasz Popiel, on results he found with his Perth colleagues on generalized polygons with point-primitive groups. They use the O’Nan–Scott methodology, together with some nice new results on fixed point ratios of primitive groups of various types. Tomasz asked for help in obtaining similar results in the twisted wreath product case (which has not yet succumbed to their analysis).

The lecturers also posed exercises for the students to work (and gave solutions to them), and there was also a problem session, which will be written up in due course.

Misty morning

The weather at the start of the week was not so good. When we started out on the walk (round two local lakes, about 17km) on Wednesday afternoon, we could hardly see where we were going at the start; but by the time we returned, it had become a beautiful afternoon. The next day was so nice that we moved outside for the problem session. The lecture room had two large blackboards, each a triptych which could be folded out to give lots of board space, or closed to give some additional space for results that would be needed later; they were on rollers, and so it was possible to move one, and enough chairs, outside.

Problem session

Course material (including exercises) is available here.

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London Mathematical Society Newsletter

The first issue of the revamped LMS newsletter, edited by Iain Moffatt, reached me a little while ago.

The first obvious difference is that it has gone from A5 to A4 format, the same as the newsletters of the European Mathematical Society and the Institute for Mathematics and its Applications.

A lot of content hasn’t changed, though now on the larger pages it is laid out more spaciously: news, reports of meetings, Council diary, forthcoming Society meetings and other events, book reviews, obituaries. But along with that there are a couple of longer features: Béla Bollobás on Bill Tutte’s centenary, Andrew Granville on a “panopoly” of proofs that there are infinitely many primes (which has non-empty intersection with the first chapter of Proofs from the Book, not surprisingly), and an article by Elizabeth Quaglia on how to ensure efficient performance of networks to download requests while keeping the requests confidential.

A new feature is “micro/nano-theses”, designed to highlight new results by young researchers. In this issue, Matthew Burfitt tells us about “Free loop cohomology of homogeneous spaces”. We also have the first two in a series of “Success stories in mathematics”, which will hopefully showcase what great careers are available to mathematicians.

The Newsletter is not entirely free of misprints. It corrects several in the last issue (including a garbling of Ramanujan’s famous equation 103+93 = 123+13), but manages to miss out completely the accented letter in the name Erdős.

But definitely a change for the better. Well done Iain and his team!

The Newsletter is available on-line at https://www.lms.ac.uk/publications/lms-newsletter .

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p-values, 2

In this sequel, I want to tell a story which perhaps indicates that the philosophical positions of scientists and statisticians have less influence on what really goes on than much more practical things like publication policies of journals. This story concerns a paper I saw nearly 50 years ago. I hope that this sort of thing couldn’t happen now, but I wish I felt more confident about that.

The paper was in a journal of animal behaviour, I really don’t remember which one but I think it was a “reputable” journal. I haven’t tried to locate it, so everything I say is from memory and thus unreliable. But back then, you needed to have found something statistically significant at the 5% level at least (that is, p ≤ 0.05) to have any hope of getting your work published.

Anyway, the scientist had observed baby birds for the first fifteen days of their life, and had measured some attribute (I don’t recall what) each day. To analyse the results, the value of the attribute on the i-th day was compared with the value on the j-th day, for all pairs {i,j}. One of these pairs of values was found to be significantly different, and on the strength of this significant result, the paper was accepted for publication.

Now I think it hardly needs saying, but apparently it escaped the authors, that the real discovery they had made is that only one pair of results are significantly different, when I would have expected five!

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p-values

To Glasgow on Monday for the 36th Fisher Memorial Lecture, Stephen Senn talking on “And thereby hangs a tail (the strange history of p-values)”.

p-values have taken a bit of a hammering lately. I understand that they are being blamed for the “crisis in irreproducibility” of scientific results; at least one journal has banned them; and even that fine ranter David Colquhoun has weighed in against them. In this debate, R. A. Fisher has been cast as the villain. Robert Matthews wrote in the popular press that “The plain fact is that 70 years ago Ronald Fisher gave scientists a mathematical machine for turning baloney into breakthroughs”. Some biographers delight in pointing out his feet of clay; we are all encouraged to reject his methods and become pure and honest Bayesians.

As you might expect (at least, for me, my prior probability on this would be quite high), things are not so straightforward.

Now I am not a statistician, and the lecture, though a delight as a presentation, did go too fast for me to be able to take notes, and I have not been able to find the PowerPoint on the web. So I apologise in advance if I have got things wrong.

What is the probability that the sun will rise tomorrow? Believe it or not, Bayesian statistics can calculate this. The argument might go back to Laplace. Starting with a prior that it is equally as likely or not, if I see the sun rise every day for m days, then I calculate the posterior probability that it will rise tomorrow to be (m+1)/(m+2), a satisfyingly high value.

However, the Cambridge philosopher C. D. Broad pointed out the flaw in this argument as a proof of a scientific theory. A very slight extension of this argument shows that the probability that the sun will rise every day for the next n days is (m+1)/(m+n+1), which is small if n is large, and indeed tends to zero as n goes to infinity. A five-year-old Bayesian (if there is such a thing) will be fairly sure that the sun will fail to rise one day before the end of his life.

[As a paranthetical remark, I read Broad’s book The Mind and its Place in Nature when I was much younger; it influenced my thinking, though at this remove I cannot really say how much.]

Anyway, Stephen Senn showed us extracts from the work of W. S. Gosset (“Student”) who clearly regarded the numbers that came out of his calculation as “the probability that the hypothesis is correct”.

Stephen showed us a very simple diagrammatic argument to show that the number that comes out of Gosset’s calculation (under a reasonable assumption on the prior, in Bayesian terms) is identical with Fisher’s p-value. To be slightly more precise, under assumptions of normality, if the prior assumption is that treatment B is better than treatment A, and an experiment shows that A beats B by two standard deviations, the probability that our assumption is correct drops to 5%. [I don’t really know what probability is, and I have absolutely no idea how you can assign probabilities to things like this.]

But Fisher’s interpretation is considerably more nuanced. Rather than talk about these meaningless(?) probabilities, he would say that, if it is true that there is no difference between the treatments, only one time in 20 would random experimental variation give a result as extreme as this. This is a much more realistic thing to say, in my opinion. (Fisher’s method also gives an added safety factor, the possibility of using a two-tailed test.)

So where is the problem? Is it that scientists simply don’t understand what Fisher said? It seems to me, contra Matthews, that a computing technique which gives you the probability that a scientific theory is correct is the real baloney machine! But what would I, a mere mathematician, know? (Though you may recall that here I did report a talk by Bollobás in which he computed the probability that his theorem was true; I was not the only audience member a bit worried by this! Perhaps a Fisherian interpretation would be more honest.)

Stephen Senn also remarked that the result of the Bayesian calculation depends crucially on the prior assumption; if a non-zero probability is given to the event that treatments A and B have the same effect, then the answer will be different. But I can’t say I followed the hints about this calculation that he gave us.

The conclusion he drew is that the current fuss is really a turf war between two camps of Bayesians, with Fisher caught in the firing line.

As I said, it was a blistering performance, and finished in good time for us to get to the drinks reception in the wonderful Glasgow City Chambers, where we were two months ago for the BCC drinks reception.

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Initial decisions on REF2021

This document has just been released by HEFCE. If you haven’t read it, here are a few small things from my perusal of it.

Collaborative research

Item 18 of the document states:

There was broad support in the consultation for better recognising collaborative activity in the REF. We will therefore include in the revised environment template (see paragraphs 27–29) an explicit focus on the submitting unit’s approach to supporting collaboration with organisations beyond higher education.

It seems that collaboration between, say, mathematicians and biologists doesn’t count. As for collaboration between group theorists and analysts, forget it.

Impact

The definitions of impact and their interpretation have not yet been figured out, despite the fact that it is such a crucial part of the assessment. But the “excellent” (i.e. at least two-star) research underpinning impact must have been conducted since 2000, and the impact itself must have occurred since 2013. The link between impact case studies and number of staff submitted will be maintained, though they appear to have no idea how it will work.

Impact may be rolled in with environment (there are some noises about this but it is not clear to me what is being said). They are working with an organisation called “Forum for Responsible Research Metrics”. You know my views, this is something like the Forum for Flat Earth Studies. And impact will rise from 20% to 25% of the overall assessment (no surprise there, they always get their way in the end).

Next steps

A provisional timetable is enclosed (assuming it will be REF2021 and not REF2022 as has been suggested). The only thing imminent is consultation with stakeholders about the composition of subpanels and self-nomination for sub-panel chairs (this month and next, respectively).

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Thanks to South West Trains

South-West Trains has rebranded itself as “The South-West Railway”, at least in its station announcements.

Usually I only mention train companies to grumble about them; I think this is true for others too. But a pleasant change: yesterday this train company did me a huge favour, saving me from the potentially quite serious consequences of a senior moment.

I was heading for a family reunion and barbecue in Camberley, not an easy place to reach by train, with no direct service from London. In addition, media coverage of the engineering works at Waterloo suggested that the chaos there is not yet over. So I decided to take the tube to Richmond, buy a ticket there, and get the Reading train to Ascot where I would change for the branch line to Camberley.

Because of the timing, I left home without coffee. I was hoping for a leisurely coffee in Richmond, but when I had bought my ticket I saw that the Reading train was going in a few minutes from the other side of the station, and I hurried over to catch it. (After years of commuting, my instinct is to catch any train going in the right direction.) I guess that the lack of coffee partly explained what followed.

The train was tremendously crowded; not all the passengers were going to the rugby at Twickenham. With some difficulty I managed to find a seat, and put my bag on the rack. In the bag were beer and sausages for the barbecue, birthday presents, and my St Andrews house and office keys.

We pulled in to Ascot, and I leapt up and got off the train. As the doors shut and it pulled out, I realised what I had done: I left my bag on the rack.

I have left things on trains a couple of times before. Once the item came back days later; all the other times it vanished without trace. But before yielding to despair, I hurried to the ticket office, which fortunately was not busy. I told the ticket seller what I had done. She phoned through to Bracknell; I described the bag to them and told them where it was on the train.

Then I sat down on edge, to wait.

Twenty minutes later, I had my bag back. A SWT guard got off the train carrying a blue bag. I approached him. “What’s in the bag?” he asked. “Beer and sausages”, I replied. He handed it over with some banter, about these being the essential items for a barbecue, anything else is just frippery.

So when I got on the Camberley train, it was the one I would have caught had I had a coffee in Richmond and been less out of it in Ascot. The same man who had brought my bag was on the train, travelling to his next job, cue more banter; I invited him to the barbecue, which he regretfully declined.

So thank you, SWT staff, you saved me there!

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