http://thecostofknowledge.com

Many thanks to Tyler Neylon for designing a website where one can declare one’s unwillingness to work for Elsevier journals. Already, without any announcement apart from brief mentions quite some way into the comments on the last post, it has 31 signatures, many of them from France, where for various reasons they are particularly annoyed with Elsevier.

This post is primarily to give the site some visibility, which I’ll also do on Google+ (if you support the venture, then please spread the word). It is not necessarily to persuade you to sign. I well understand that we are all in different situations and signing is easier for some people than others. But one thing I would definitely say is that if you already have a private non-cooperation policy (as I myself have done for years) then you will have much more effect if you go public about it. As I said in my previous post, the more people who sign, the more morally and socially acceptable it becomes to sign too: a private protest is just a nuisance to other mathematicians, but larger and more public one may have a chance of achieving something. So I hope that each signature will beget several others, at least for a while.

Elsevier — my part in its downfall

The Dutch publisher Elsevier publishes many of the world’s best known mathematics journals, including Advances in Mathematics, Comptes Rendus, Discrete Mathematics, The European Journal of Combinatorics, Historia Mathematica, Journal of Algebra, Journal of Approximation Theory, Journal of Combinatorics Series A, Journal of Functional Analysis, Journal of Geometry and Physics, Journal of Mathematical Analysis and Applications, Journal of Number Theory, Topology, and Topology and its Applications. For many years, it has also been heavily criticized for its business practices. Let me briefly summarize these criticisms.

1. It charges very high prices — so far above the average that it seems quite extraordinary that they can get away with it.

2. One method that they have for getting away with it is a practice known as “bundling”, where instead of giving libraries the choice of which journals they want to subscribe to, they offer them the choice between a large collection of journals (chosen by them) or nothing at all. So if some Elsevier journals in the “bundle” are indispensable to a library, that library is forced to subscribe at very high subscription rates to a large number of journals, across all the sciences, many of which they do not want. (The journal Chaos, Solitons and Fractals is a notorious example of a journal that is regarded as a joke by many mathematicians, but which libraries all round the world must nevertheless subscribe to.) Given that libraries have limited budgets, this often means that they cannot subscribe to journals that they would much rather subscribe to, so it is not just libraries that are harmed, but other publishers, which is of course part of the motivation for the scheme.

3. If libraries attempt to negotiate better deals, Elsevier is ruthless about cutting off access to all their journals.

4. Elsevier supports many of the measures, such as the Research Works Act, that attempt to stop the move to open access. They also supported SOPA and PIPA and lobbied strongly for them.

SOPA — my part in its downfall

If you haven’t heard, SOPA, which stands for Stop Online Piracy Act, is a US bill that was proposed in order to do what its name suggests. Although it has been defeated for now, its proponents have not given up, so many websites, notably including Wikipedia, are going on strike tomorrow (January 18th) in order to show just how potentially damaging the bill could be to the internet. I haven’t looked in much detail into what the adverse consequences of SOPA would be, but I’ve read enough, from people whose opinions I trust, to believe that I should join this strike. My technical competence is insufficient to follow the instructions that have been offered for doing this (and the same applies to any instructions that anyone reading this might feel moved to offer so I suggest not bothering). Therefore, I plan to mark this blog as private (and therefore inaccessible) for the day, an operation that I will undo on Thursday.

If you’d like more details about what’s wrong with the bill, then Google “SOPA” and you’ll find all you could possibly want.

Edit: I was about to change the blog to private when I noticed that WordPress has a Protest SOPA/PIPA setting. I’ve gone for that. It results in the ribbon you see in the top right-hand corner of this page, and a total blackout, with a page explaining why, from 8am to 8pm EST. So that will kick in properly at 1pm UK time.

Farewell to a pen-friend

A few days ago I learnt from the Guardian of the death of the novelist and critic Gilbert Adair. I was saddened by this, partly because I have hugely enjoyed his writing (though I’m glad to say that I haven’t read his entire oeuvre, so there are still treats in store) and partly because I knew him. The title of this post is a pun of a kind I hope he would have approved of: our interactions were mostly by email, but one can also take the “pen” to mean “almost” (as in “peninsula”), which is why I used a hyphen. We met a couple of times, and might have become proper friends if I had been less socially lazy. It turns out that he had a stroke a year ago, but I didn’t hear about it, so his death just over a week ago came as a surprise and leaves me regretting that I didn’t see more of him while I had the chance.

Since there’s nothing I can do about that, I thought that I’d try to use this blog as an outlet for the resulting feeling of loss, which is out of proportion to the amount that I actually had to do with him. Or perhaps it isn’t, since the very fact that I didn’t see him much is part of what now bothers me. It is also why I had no idea that my last contact with him might be my last, and why his death now seems a bit unreal.

A maths blog is not a completely inappropriate place to write about him, because I met him through mathematics and it was because of mathematics, which fascinated him, that that initial meeting led to a couple of further meetings. A secondary purpose of this post is to recommend his books, which are extremely clever in a way that many mathematicians would like. I’ll describe some of them as I go along.

Group actions IV: intrinsic actions

I have a confession to make. When I was an undergraduate at Cambridge (hmm, that sounds as though it might be the beginning of quite an interesting confession, so I’d better forestall any disappointment by saying right now that it isn’t), there was a third-year course in group theory, taught by John Thompson no less, on which I did not do very well. For a few weeks it seemed to cover material that we’d done in our first year, and then suddenly it got serious, with things like the Sylow theorems. And at that point I got lost, and was unable to do the questions on the examples sheets. I can’t remember much about the questions, but I think my difficulty was that there was a slightly indirect style of proof that caused me to find arguments hard to remember and even harder to come up with. And I never got round to doing anything about it: I went into a different area of maths, and even now I don’t know the proofs of the Sylow theorems. In fact, I don’t even know the statements, though I know they’re about the existence of subgroups of various cardinalities, and I know that they are proved using cleverly defined group actions. I’ve skim-read the proofs, so I have a fairly good idea of their flavour, but I don’t know the details. In particular, I don’t know which action does the job.

A short post on countability and uncountability

There is plenty I could write about countability and uncountability, but much of what I have to say I have said already in written form, and I don’t see much reason to rewrite it. So here’s a link to two articles on the Tricki, which, if you don’t know, is a wiki for mathematical techniques. The Tricki hasn’t taken off, and probably never will, but it’s still got some useful material on it that you might enjoy looking at. The articles in question are one about how to tell almost instantly whether a set is countable and another about how to find neat proofs that sets are countable when they are.

Group actions III — what’s the point of them?

Somebody told me recently that a few years ago they had a supervision with a colleague of mine (who shall remain nameless, but he or she is an applied mathematician) and asked what the point of group actions was. “I have absolutely no idea,” was the response, and the implication that one might draw from it was apparently intended.

No pure mathematician could hold such a view. I’ve stated a few times that group actions tell you a lot about groups. In this post I want to try to explain why that is, though there is far more to say than I am capable of explaining, let alone fitting into one blog post.

Several proofs that use group actions seem to depend on almost magically coming up with an action that just happens, when you analyse it the right way, to tell you what you wanted to know. I am not an algebraist and do not have a good all-purpose method for finding actions to prove given statements. I don’t rule out that such a method might exist, at least for reasonably simple statements, and would be interested to hear from anybody who thinks they can usefully add to what I have to say.

Normal subgroups and quotient groups

The traditional presentation of normal subgroups and quotient groups goes something like this. First, you define a subgroup to be normal if it satisfies a certain funny condition. Then, given a group and a normal subgroup , you show that you can define an operation on the cosets of , and that that operation turns the set of all cosets into a group, called the quotient group. Ideally, you also show that one can’t give a natural group structure to the left cosets of an arbitrary subgroup: that justifies restricting attention to normal subgroups.

There’s nothing terribly wrong with this approach, but it does leave one question unanswered: why bother with all this stuff? The traditional approach to that question is to ignore it, confident that the answer will gradually reveal itself. The more group theory you do, the more normal subgroups and quotients will arise naturally and demonstrate their utility, so if you just diligently keep studying, you will (fairly soon) come to regard normal subgroups and quotient groups as natural concepts that were obviously worth introducing.

Proving the fundamental theorem of arithmetic

How much of the standard proof of the fundamental theorem of arithmetic follows from general tricks that can be applied all over the place and how much do you actually have to remember? At first it may seem as though you have to remember quite a bit: there is a non-obvious sequence of lemmas, starting with Bézout’s theorem, continuing with the clever proof that if then either or , bumping that up to a proof for bigger products, and eventually deducing the theorem itself.

But what if one were simply asked to come up with a proof? Would there be any chance of discovering that sequence of lemmas? I maintain that there would — if, that is, you are aware of certain general tricks.

Why isn’t the fundamental theorem of arithmetic obvious?

The fundamental theorem of arithmetic states that every positive integer can be factorized in one way as a product of prime numbers. This statement has to be appropriately interpreted: we count the factorizations and as the same, for instance. Note that it is essential not to count 1 as a prime, or else we could stick a product of 1s on to the end of any factorization to get a different one: . But doesn’t that mean that 1 itself cannot be written as a product of primes? No — we define the “empty product” (what you get when you take a bunch of … no numbers at all and multiply them together) to be 1. That is a sensible convention because we would like multiplying a product of numbers by the empty product not to make any change to the result.