Archive for the ‘General’ Category

Is AV better than FPTP?

April 20, 2011

On May 5th the UK will vote in a referendum for only the second time ever. (The first time was in 1975, when we voted on whether to remain in the EU, or the Common Market as it was then called.) Now we have a chance to decide whether to retain our current voting system, misleadingly known as First Past The Post, or whether to switch to the Alternative Vote. Let me come clean straight away. Although in this post I shall try to write dispassionately about these two voting systems, my actual attitude is anything but dispassionate: I have yearned for a better voting system ever since I have had any political awareness at all, and am steeling myself for what is probably going to be a huge disappointment when the country votes for the status quo. And I am writing this post in the genuine hope of making a difference. Since it is extremely hard to change anybody’s mind in politics, I think the best I can hope for is to persuade somebody to vote yes (the question will be phrased in such a way that “yes” means you want AV and “no” means you want FPTP) who might otherwise not have bothered to vote at all. This is a mathematics blog, so I will give this post a mildly mathematical slant, but all I really mean by this is that I know when I write that a typical reader of this post will be mathematically literate, which may make the post different in tone from how it would be if I were writing for a more general readership.


A little physics problem

October 13, 2010

A few months ago I accidentally spotted an amusing phenomenon during my son’s bathtime. He has some plastic cups that he likes playing with in the bath, and as the water was running out, I took one of them, turned it upside-down, and pushed it down so that it was full of air and just above the plughole, surrounding it. If you want the hard version of the problem it is to work out what happened next. The easier version of the problem is to read past the fold, where I will say what happened, and then to explain it. I am hopeless at this kind of problem, so I don’t know the answer myself, and I also don’t rule out that the answer is too easy to be interesting. If that is the case, then apologies in advance.

A mathematician watches tennis II

June 24, 2010

This has been a year to remember for anybody whose interest in tennis is more that of a nerd than that of a tennis player (which, given the uselessness of my serve, very much applies to me), in that it has given us two records that may well never be beaten. First we have Roger Federer’s record of 23 consecutive Grand Slam semi-finals (set at the Australian Open, and finally fixed at 23 when he lost in the quarter-finals at Roland Garros), and now, something I’ve been hoping for all my life: a seemingly endless match. At the time of writing, John Isner and Nicolas Mahut are waiting to resume a match that has gone into a third day. They will do so later today, with the score standing at 59-59 in the final set. This doesn’t just beat previous records — it utterly smashes them. This set is way more than twice as long as the previous longest set in a Grand Slam, it alone is far longer than the previous longest ever full match in professional tennis, both players have served far more aces in a single match (95 for Mahut, 98 for Isner) than anybody before, and so on. And if you also take account of the fact that the previous two sets had to be settled by tie-breaks, with no breaks of serve in either, then we have had 142 games in a row with no breaks of serve. (I can’t remember when the break occurred in the second set, but even this number 142 can probably be improved slightly.) [Update. The match is now over, with Isner winning 70-68, so the eventual number of consecutive unbroken service games was 137 in the final set, 161 if you include the previous two sets, and a few more still, I think, if you include the last few games of the second set. The number of aces for both players ended up well into triple figures.]

Isner said, with some justification, that nothing like this will ever happen again. But with how much justification? As ever, to answer this question involves choosing some kind of probabilistic model, and it is far from obvious how to choose an appropriate one. But it is possible to get some feel for the probabilities by looking at a crude model, while being fully aware that it is not realistic. (more…)

If politicians were mathematicians

May 9, 2010

Before I start, let me get one thing over and done with: I fully admit that professional mathematicians are as capable as anyone else of making stupid collective decisions.

But I don’t want to imagine what the world would be like if it were run by mathematical researchers. I just wonder how much difference it would make if politicians understood enough mathematics to be able to understand an argument of more than one sentence. Or to put it more accurately, what would it be like if the following rules of political life were no longer accepted?

1. An argument that is slightly complicated but correct is trumped by an argument that is punchy, amusing, and wrong.

2. If option A is better than B in some respects and worse in others, then instead of weighing up the pros and cons, you decide which side you are on and then just mention the pros of the option you prefer and the cons of the other option.

3. If option A is better than B in every respect, but your party supports B, then you support B.

4. If one of your political opponents points out a flaw in your argument, then count to ten and repeat the flawed argument.

If that were the case, then one consequence would be that one could advocate new ways of doing politics and have them discussed seriously. In this post, I would like to mention a few ideas that would be dismissed as utter lunacy by any politician. But perhaps people who read this blog would be prepared to engage with them properly and weigh up the pros and cons. I’m sure there are cons — but I don’t think the ideas are utter lunacy. (more…)

Is the British voting system fair?

May 3, 2010

At the last general election, the percentages of votes and numbers of seats in parliament for the three main parties in Britain were as follows: Labour, 35.3% of votes, 356 seats; Conservatives, 32.3% of votes, 198 seats; Liberal Democrats, 22.1% of votes, 62 seats. In the election coming up on the 6th May, there is a distinct possibility of some quite bizarre outcomes. For example, if some recent polls give a true picture of how people will vote (which is of course far from certain), then there is a good chance that the Liberal Democrats will get more votes than Labour, but well under half the number of seats. It is also a commonplace that the Conservatives will need a higher percentage of votes than Labour to become the party with the largest number of seats. In the past there have been occasions where the party with the largest number of votes has lost the election. (Much of what I am saying applies equally to the system for electing a US president, but I shall stick with the British system in this post.)

Supporters of the first-past-the-post system argue, correctly, that it makes it much more likely that one party will have an absolute majority. They also argue, much more controversially, that this is a good idea. However, regardless of outcome of that argument, there can be no doubt that it has the potential to lead to anomalous results, and this potential has been thrown into sharp focus in the last week or two because it has a good chance of being realized. Here I would like to discuss whether it is correct to describe these anomalies as unfair. (more…)

Swine flu and British public health policy

June 5, 2009

One of my children has just recovered from swine flu, as a result of which I now have a clearer idea of what British policy is towards outbreaks. Much of it was perfectly sensible, but not quite all. Since there’s a small amount of mathematics involved, and since I wanted to get this off my chest, I thought I’d blog about it.

The good part was that everyone who had been in close contact with the child who had swine flu was immediately put on Tamiflu, which seems to have stopped any of the rest of us getting it. (It’s now been long enough that we can be almost certain of this.) The less good part was the piece of advice that I mainly want to discuss. The main question I had was, of course, to what extent I and my family should avoid contact with other people. The advice I was given, which, it was made clear to me, was the official policy and not just the whim of the public health official I spoke to, was that we should continue to lead our lives as normal for as long as we did not show any symptoms. (more…)

When normality is abnormal

May 9, 2009

Suppose you were reading a novel, or watching a play or film, that included a fictional mathematician …

My guess is that the moment you read the two words “fictional mathematician” a second or two ago, your mind leapt ahead and you had a pretty good idea of what he—yes he, since even if there are female fictional mathematicians out there, femaleness is unlikely to be part of your instant and not fully conscious reaction to the phrase—was like: a social misfit who is prone to flashes of extraordinary insight that completely baffle everybody else, or perhaps a social misfit who would like to have those flashes but doesn’t and goes mad instead, or perhaps a social misfit who does have the insights but with madness the huge price he has to pay.

So here is a question: is there any example of a mathematician in literature, theatre or cinema who is a fairly normal person socially, and pretty good at maths but not astoundingly so? Some examples that do not work are Uncle Petros, from Uncle Petros and Goldbach’s Conjecture, both the father and the daughter in Proof, and Will from Good Will Hunting: they’re all either ridiculously good at maths (usually without having to do all that routine stuff like learning the proof of Schur’s lemma, or the open mapping theorem, or the Gram-Schmidt orthogonalization process etc.) or mad, or both. I also don’t count characters if they are colleagues of a crazy genius and their main role in the book/play/film is to marvel at how clever the crazy genius is. Let’s say that the character has to be the main one, or at least the main mathematical one. (more…)

General news—February 2008

February 1, 2008

Not entirely surprisingly, my hibernation is going to go on for longer than I had hoped. The reason for this is quite simple: there is a major push to get the Princeton Companion to Mathematics finished within the next couple of months or so, and if I spend time blogging then it won’t happen. There has also been a delay with the Tricks Wiki, but that may be less severe because I am not the main bottleneck for that (though there are a few things I need to do before it can be up and running). This term I am giving a first course on probability. I had planned a few blog entries on that too, and I hope I’ll have time for some in due course.

General news — December 2007

December 19, 2007

As I predicted in an earlier post, my rate of posting has (temporarily) gone right down. This is partly for the reasons I said—I am very busy with a final push to finish the Princeton Companion to Mathematics, and busy in general—but also for another reason. I was going to keep personal matters rigorously excluded from this blog but since my secret is out (see the comments on Examples First II on November 17th) I may as well also mention that I have a five-week-old son, Octave, who doesn’t leave much time for blogging given that my other commitments won’t go away. So this isn’t a proper post but just a way of saying that my blog hasn’t died: it’s just hibernating. Meanwhile, I can at least briefly mention that a “Tricki”—that is, a Wiki-style website devoted to theorem-proving techniques—will almost certainly exist in the near future. Remarkably, my earlier post on this idea led to an offer of technical help that will be enough to turn it from a fantasy into a reality. And that’s saying something, since my own technical ability in this area is basically zero. I’ve seen a prototype and it looks great. Probably we’ll get a small site up and running and I’ll then ask for comments about how it could be improved before we throw it open. (We still haven’t decided what policy to adopt about who can edit what, but we are actively thinking about it.) And that’s it from me until 2008.

About this blog

September 15, 2007

As I mentioned in my first post, I am very busy at the moment, so my rate of posting will soon go right down, probably until about February. However, I wanted to have at least one post that had mathematical content (as opposed to remarks about the presentation of mathematics) since I hope that in the end that will be the main focus of the blog. So the post about cubics is supposed to be more representative of what this blog is about. Thanks to all who have contributed so far. Although I had read about the power of blogs it nevertheless came as a surprise to see from the stats how many more visits a blog gets than a homepage, and also to discover how easy it is to get useful advice. I’ve even had a serious offer of technical help with setting up new wiki-style websites, so there’s a good chance that some of those ideas will actually be realized.