What I read on my holiday... Riddles in Mathematics

Riddles in Mathematics: A Book of Paradoxes is a popular mathematics book written in 1944, revised in 1961. I purchased a 1975 reprint at Hay, and finally got round to reading it on holiday.

It is good fun, if a little dry and could be accused of being a little like a text book: there is a discussion, and couple of examples and then some paradoxes for you to resolve (or exercises). To get the most out of the book you do need to engage with the maths: you'll need pencil and paper and a willingness to get your algebra out.

The dated bits are in geometry: sadly Euclidean proof has dropped off the school curriculum (in favour of more statistics). I think there is nothing in the section that isn't in GCSE Maths, but probably in a different style. Also the four colour theorem has now been proved, albeit in an aesthetically unsatisfactory way.

The chapter on the infinite is great, and the chapter on probability has provoked me to get Grimmett and Walsh out to revise my probability.

A couple of my favourite paradoxes. Firstly one from p169 that I've put into steps and changed slightly.

1. If something has probability 0 it is impossible.
2. It is known there are an infinite number of primes
3. We only know a finite number of them (the highest known prime has just under thirteen million digits). Call it P.
4. The probability of picking a known prime is P dividied by the number of primes.
5. Any finite number divided by infinity is 0.
6. Therefore the probability that an arbitrary prime is known is 0.
7. Therefore it is impossible that an arbitrary prime is known.
8. Therefore there are no known primes

A round of applause to anyone who can pick the (many) holes in that. It is the same basic construction as Douglas Adam's proof that the population of the universe is 0.

Also worth a look are the geometric probabilities (p172-3).

Some good ol' algebraic sleight of hand is included, like this from p86:

1. Let b and c be two different positive numbers.
2. Let a = b + c. Note a must be greater than b.
3. Multiply by (a-b) to get a^2 - ab = ab - b^2 + ac - bc
4. Take off ac: a^2 - ab - ac = ab - b^2 - bc
5. Factorise: a(a-b-c) = b (a-b-c)
6. Divide by a-b-c to get a = b, but this contradicts a being bigger than c. So any two different numbers are the same (chose to be c=a-b)
7. In fact as b+c=a we have b+c=b, which means c=0.
8. But c was an arbitrary positive numbers, so all arbitrary positive numbers are 0!

Again a round of applause to those who can hit that one into touch. A bit easier I think.

I think you'd need to be a bit odd to enjoy the book: you'd need solid secondary school mathematics, and a willingness to play with the maths. Like me.

Graduate tax?

I must confess I'm a little worried about how Britain will pay for Universities.

I'll start by coming clean. In some ways I am unashamedly elitist: for higher education to be worthwhile it has to exclude some people, whether through admissions processes or by people failing or dropping out. The Open University is open to all but a challenging route to a degree which some will not complete, at the other extreme the Oxford and Cambridge admissions lottery thins out many. In some sense it is unfair that not everyone is able to benefit from a university education but for it to be useful it has to be that way. I am a huge fan of the Open University and the work it does to provide access to those who can't by other means. We need to be sure that access is as fair as possible to people: those who'd benefit from it can access it and aren't put off by issues beyond that.

I take it as a given that both the state and individuals benefit from education. At the crudest level it keeps a number of 18-21 year olds off the street and off benefit (at my most cynical I have thought this might be the reason that the government was so keen on the 50% figure). At the other end it provides people with skills and expertise the state needs.

There is an argument that rather too many jobs demand a graduate when all they really need is someone who could get a degree, and they might be better recruiting based on A-Levels. But there are many benefits to a university education.

I do think there is another argument around whether as many as 50% would benefit fully from a university education, but that is for another day.

Given that there are benefits what is the best way to pay for it?

Vince Cable has recently reopened the idea of a graduate tax, and prior to that most expected the Browne report to suggest increasing fees and loans. The idea behind both is that students should repay the benefit they receive in part which seems attractive at face value. I've had a look at the full text of Vince's speech.

Firstly, Vince rightly points out that under the current system the amount of the education people pay for is the same for a teacher or care worker at one end, and at the other end a doctor or a merchant banker. This means someone earning £20k a year ultimately pays the same overall as someone earning £200. This is regressive. A graduate tax could redress this issue.

However the concern I have is with the argument behind it. In his speech Vince says

for our students there are on average good rates of return to HE qualifications,
which have held up despite large increases in participation: over a £100,000 net
of tax over a working life relative to a non-graduate. This suggests employers
continue to see additional value in graduate skills, knowledge and capability.

[...]

My generation had the remarkable privilege of being
educated free. There was an implicit assumption that we paid for the graduate
premiums in our income through higher income tax. But there was also a sense of
unfairness articulated by Alan Johnson when he was Minister: why should a young
postman contribute through his tax to pay for an already privileged group to
avoid earning a living for three years and then emerge with higher earnings
potential?

In any event, a model designed for 10% of the population
could not be applied to 40%: hence the move to a graduate contribution.

This is where the problem can be seen. Let us accept the £100,000 figure, then this means that over life time then the person is paying £20,000 in extra income tax if they only pay basic rate or more if they pay higher rate. Those who pay higher rate will be paying £40-50,000. This is, of course, not all the tax they pay. This is ignored in his example.


Let me turn this round as well. Suppose we have two postmen. Both earning the same. Should one pay more tax because he has completed a degree? Is education something we wish to discourage? Or consider two people, one a nursery teacher (a graduate, average of £18,875 pa) the other a car salesman (£26,584 pa). Both basic rate tax payers. Why do we want the nursery teacher paying a higher rate of tax? Or take Bill Gates who dropped out of his university to part found Microsoft: imagine a British version of Microsoft where two people set up a company at the end of the final year of one, and the second year of the other. Why would one be paying more tax than the other? Did one really benefit more than the other from their university experience? If so how?

Another argument is that it puts people off applying. However the poor economic climate means we have no idea if this is true. The overall numbers applying have risen steadily over the last five years. What we don't know is if this rise is in the brightest and best, or just the middle classes. I would like to see figures on applications from deprived areas, and if anyone could show me those I'd be grateful.

Here a graduate tax would continue to be problematic. Would we put off people who didn't know if they need a degree? Would some groups of people select careers to avoid the tax?


I can see the argument that people should pay the cost of their education as not all people get that education. I disagree as it ignores the wider benefits, and will put off people from disadvantaged backgrounds, but I can see a consistent argument.


I can see arguments for paying from general taxation: the same principle as with secondary and further education, we need as a society to have people educated. I can see the argument for providing the service people will benefit from free at point of use as in the health service.


The problem with the graduate tax is it is neither. It says you should pay for the service, but this payment should not be linked to the cost of it. It says that some should pay vastly more than the cost, some much less. In the election we, as Liberal Democrats, campaigned to say it was wrong to have high earners paying less of his income proportionally than low earners. Now we are saying that we want to introduce a new tax that can and will do the same.

Maybe the graduate tax is the way to bridge a gap between Liberal Democrats who want it to be from general taxation, and the Conservatives who want individuals to pay. If so then maybe it is a reasonable compromise, but I need a lot of convincing of this.

Executions and criminalising sexuality

A cheery topic I know and in the news because of the dreadful Ashtiani case in Iran. I am worried that Iran expects us to be satisified that death by stoning will instead be replaced by another method of execution.

To a huge extent the method is irrelevant. The issue is far simpler: why does the state feel the need to kill someone for a consensual sex act?

I am viscerally opposed to the death penalty in any form. I was really pleased when, as part of the Human Rights Act, the last Labour government scrapped the few remaining theoretical death penalties on the UK's statute books. A real Labour civil liberties win, albeit in a theoretical area. Even for murder I prefer life imprisonment, which in some cases will mean full life.

In opposing it I understand that the campaigners have to get emotional play to convince some people, but concentrating on the barbaric method seems an own goal. Iran can now say "we've banned stoning" (which they have) and claim to have improved their record. Which they haven't. The issue isn't that their hanging method is also a cruel, painful way to die (which it is), but that this killing is even being considered by the state.

To be honest there is even the question why is adultery even a crime, never mind a capital one? I'm going to invoke Mill here and look for harm.

The harm in adultery is to the other partner. Maybe in a medieval environment the harm of raising another's child might be significant, however in other circumstances this is encouraged by religions so it seems unlikely to be that. The betrayed trust is a huge emotional hurt (I really want to be clear that I'm not downplaying this) but we don't criminalise other emotional hurts.(To take a trivial example we don't criminalise giving someone the hard shoulder, or insulting them; at less trivial the damage done by poor exam results doesn't criminalise the teacher who set the test.) If there is no other partner (as in this case, the adultery emerged at the trial of his alleged murderers) is there a harm? Consequentially there isn't.

To move to a wider ethical view, is deceptive adultery morally wrong? Yes. Does that mean the state should legislate against it? In many cases we make deception a civil matter, even when substantial harm may be involved. Some cases we do make criminal, such as obtaining goods or services by deception.

The question whether this hurt is sufficient for state intervention is an interesting one. My gut reaction a clear no, however my arguments here are not strong. I think my argument comes down to seeing marriage as a contract, albeit a very special one, and thinking that violations of the contract should therefore be seen as a civil matter. In that I think I may be underplaying the potential emotional damage as equivalent to an insult. I could play a privacy argument: but this falls down by comparing it with domestic violence where most would like to see action taken even if the victim doesn't complain. I'd like a better argument.

On an easier area, even less defensible is criminalising homosexuality. There simply is no harm here. (I am very pleased that the UK Supreme Court has decided that homosexual asylum seekers shouldn't be asked to be more discreet, and should not be deported. It always seemed odd to accept political activists (who could also be discreet) but not people persecuted for something they have less choice over.)

Last year Iran executed more than 388 people according to Iran. That is at least 1 in 185000 people. (Src Amnetsy and World Bank, via Google). The second highest per capita rate in the world after Iraq.

Alan Johnson's arguments for Stop and Search

Read this BBC article on the Supreme Court ruling on the abuse of anti-terrorism law by the police. Good news (well a start anyway).

Then I get to Alan Johnson's bit. Alan Johnson shows Blair and Brown's Labour's attitude to Civil Liberties. They still don't understand why people got annoyed with police abusing powers.

The BBC gives two of Johnson's arguments against the Court ruling and May's position on asking for reasonable suspicion. These need to be looked at, as they are... interesting.

"The number of stop and searches under Section 44 has reduced considerably over the last two years"

This is an argument to keep the powers? Where? How? Did I miss something?

So it was wrong then and it is wrong now. It would actually suggest the power isn't much use: they used it more when it was new but didn't find it as useful as they'd like. Hardly an argument against the judgement or for the power. Barely even relevant.

Second the BBC says he "said the decision would restrict the powers of the police."

Let me take this to the ridiculous extreme. The police could stop a lot of crime if they could shoot to kill anyone outside after dark. This is a restriction on the power of the police. Just because it is a restriction has nothing to do with whether it is right or wrong.

I'm not a fan of (Liberal Democrat) Lord Carlile. I feel since he was given his job of reviewing terrorism law he has gone native. He was given this job by Charles Clarke in 2005, not the coalition. He points out that the powers were simply not effective: so the restriction is not significant. Pennie Quinton, who brought the case was interviewed on PM this evening, and she said that Section 44 had not led to a single charge.

I have had a quick browse round the web to see if he had better arguments, but when the Guardian only give the Labour view a paragraph then it may be seen as a bad sign.

I think the need for reasonable suspicion is hardly an insurmountable barrier for stopping someone the police reasonably suspect to be up to no good...

Extra-solar planets

When I was a child there were no extra-solar planets known about. People assumed they may exist but we had no evidence that they did. This allowed us to continue to believe that there may be something special about our solar system. Indeed I remember arguments of the form that there may be no other solar systems, and this was an argument for the Earth as a special case in a special solar system.

What boggles is me is that the since the first planet was confirmed in 1992 is the speed with which this argument has been effortly demolished!

There are now over 464 confirmed extrasolar planets according to Wikipedia, and more possibles. This boggles my mind. What is amazing is the variety that we are able to determine.

In December 2009 GJ 1214 b was confirmed as "the first of a new class of planets with small size and relatively low density", which could be a large rocky planet and seems likely to have water. A very significant fact as water means that life would be possible. The presence of liquid water remains one of the special features of Earth. Of course at the moment we don't have direct evidence of water yet, but it looks likely.

Given the speed at which different types of planets are being found and in different ways, including by visual observation (with the Hubbles space telescope).

This is a real revolution in our knowledge about our place in the universe since 1992, but it has gone by relatively quietly. The chances of our solar system being unique before the first planets was real, now it seems unlikely in the extreme.

It struck me that the appeal to exceptionality was a really appealing argument intuitively, it feels like a strong argument but has a big problem. If the exceptionality is lost the argument falls. However it was always a weak argument: This has convenient special properties does not mean there are reasons for the special properties.

The fact that "why is our planet capable of intelligent life" can only be asked by creatures living on planets capable of intelligent life, given the scale of our galaxy, and the number of planets found in a tiny fraction of it, it seems highly like that there is another planet with people making similar arguments to this. Better arguments are needed.

Goal line technology

Well that was pretty awful. England's second goal should have stood and should have hidden a woeful performance: I couldn't see us being so gung-ho at 2-2 as we were at 2-1. Germany would probably have still won though: they were much better than England.

So what could be done about the ridiculous goal line decision? I'm not convinced by the introduction of technology but I'm even less convinced by FIFA's arguments against.

According to the media there seem to be three basic issues and one lame one that they raise:

1. Fans enjoy debating the controversy
2. That it can't be done everywhere, so it shouldn't be done anywhere (so called "universality of the Laws of the Game")
3. It would interrupt the game too much
4. That it costs too much

Let's look at these one by one...

Controversy

It is clearly true to say fans enjoy discussing decisions, but irrelevant. We want games decided by footballers not random, poor or corrupt referees. The reason I mention the last is that Brian Clough's Derby team were robbed by a corrupt ref (who later admitted it). There is enough unavoidable controversy without allowing for a load of avoidable and unnecessary problems. Surely FIFA want to remove the really bad decisions: otherwise they can let me ref the world cup final.

This is not a sound argument.

Universality

They make a big issue of this and it simply isn't true. Competitive football is different all the way down the pecking order. At big games (internationals and premier league) there are a ref, two linesmen, and a fourth and fifth official. As you go down the levels you lose these step by step. At village football level you often have a ref and club linesmen, in pub leagues you just have a ref. At the level I played competitive football we had club refs (you'd nominate a player to ref the first half, and the other team the other).

We already accept different levels of officiating as we go down, so this is not a sound argument.

Interruptions

This one has some merit, but is not impossible to work round. Firstly these incidents are rare, I don't recall two in a game.

The ref can stop the game for various reasons and does (if a player is injured, to administer a booking, if an outside influence interferes). Just give the ref the power to stop the game for a check with an appropriate restart. Or give each team an appeal and if they win it they get it back, if they lose it they can't appeal again and the opposition have an indirect free kick.

Or just get the fifth official to check the video and radio the ref who can take the game back. The fifth official hasn't much to do anyway. No interruption at all, but how long can you give for it to be decided?

So there is an issue here, but it can be rationed (like substitutions) and would be rare. It isn't like a wicket in cricket.

Cost

Don't make me laugh. However these incidents are incredibly rare so is it worth it? That however is a different argument.

Conclusion?

FIFA's arguments aren't that good. There is a problem to be addressed concerning interruptions in the game but that could be resolved with creative thinking.

As I said I'm not convinced that the change should be made. I agree with IFAB that the way ahead could be for two extra assistants at big matches (as mentioned in this account of the IFAB meeting). (Remember it isn't FIFA who set the rules, though they have an effective veto.)Ironically, England and Scotland voted in favour of technology at that meeting, and were voted down by FIFA, Wales and Ireland.