Archive for May, 2010

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…)

A little experiment IV

May 8, 2010

I have another experiment that may not add all that much to the previous one, but I’m posting it anyway because I don’t want to waste the (admittedly not huge) effort it has just taken me to follow Jason Dyer’s suggestion and create a presentation using Prezi. If you follow this link, you will be taken to it. If you hover near the bottom of the box, a left arrow and a right arrow will appear, which will allow you to move right or left in an expression that, again, needs to be simplified. I’ve made it a bit easier than the last one, and you also get to see slightly more than one character at a time. There is also a button that allows you to shrink the entire expression so that it fits into the box: obviously if you use that then it counts as giving up on the experiment, but it may be interesting to do the simplification in your head according to the strict rules first and then see what happens to the information in your head when you then click the pan-out button.

Incidentally, I’m not sure that Prezi supports mathematical symbols, so I’ll save you some potential irritation by pointing out in advance that every x is an x rather than a “times”. (There’s one that looks a bit like a “times” because of the unusual spacing.)

I haven’t bothered with a poll this time. If anyone has anything interesting to report, then by all means let me know. Otherwise, think of it as a weird form of entertainment.

If you haven’t done the previous experiment (or even if you have but just want to see the same idea in a different format), you might like to look at the following two pdf files, kindly sent to me by Olivier Gerard, both of which present one chunk per page. In the first file the definition of “chunk” is the same as it was for me, so e^x would have e on one page, ^ on the next and x on the next. In the second, exponents are attached to the previous symbol, so x^2 would be on a single page, and (x+1)^2 would be on five pages of which the fifth would be )^2.

Fully separated version

Version with exponents not separated

A little experiment III

May 7, 2010

This one is a little bit different, and the associated poll questions are rather vague. I am curious to know what effect it would have on our ability to do routine manipulations if we could look at only one symbol at a time. Maybe at some point after the experiment I will say what my motivation is for this, but for now I want to influence what happens as little as possible.

After the fold, you will find a mathematical expression that can be simplified. Usually, one would look at the expression and take in chunks of it at a time, but I have embedded it in a lot of junk, so that that will not be possible. I want it to be easy to find the individual characters that make up the expression, but not easy to look at more than one at a time, so the non-junk characters are quite widely separated, but they are signalled by being enclosed in dollar signs. For example, to convey the expression e^x I would write something like


Your task, if you feel like participating in the experiment, is to simplify the expression as much as you can in your head. (If you write it down, then obviously it makes it completely pointless for me to have written it in this strange form.) I would then like to hear, in as much detail as you can remember, what thoughts went through your head, and in what order. I am particularly interested in what your eyes were doing and how they interacted with these thoughts. It may not be easy to remember all that, but if you do the best you can then I’ll be happy. (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…)

A little experiment II

May 1, 2010

A few people felt that my experiment about solving the equation x^2+2x+1=x^2 would have been more interesting if the equation had been more interesting. I’m not sure I agree with that: part of my intention was to get some evidence about what we do in a very simple situation. However, I now have another example that I want to know about, and it happens to be a harder one. Not much harder, but harder nevertheless.

One of the byproducts of its being a more interesting equation is that it will be quite a lot harder to encode all possible experiences that people might have and run a vote as I did last time. So I’ve made the vote a little bit vaguer, and I will be interested not just in the numbers but also in people’s descriptions of what they did (as I was first time round). My main aim is to get a fairly complete picture of all the ways that different mathematicians find it natural to think about the problem.

As last time, if you want to take part, then try to keep a close eye on all your thought processes: I’m interested not just in the ones that got you to the answer, but also in the ones that you might have thought about and dismissed. So try not to look at the problem until you are ready to keep track of what your reactions are. The problem will appear immediately after the fold. For the benefit of those who read this using RSS, I’ve also left a bit of space, and I’ve left space between the problem and my discussion of it. (more…)