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The Bizarre World of Mathematics

There's a few interesting things about Maths that you might not appreciate at first glance. Most people think of numbers as soon as they hear the word maths, but there's a lot more to it than first meets the eye.

The first thing to realise is that it is a completely made up subject and has no basis in reality whatsoever. Ok so that might be an over-statement, but really Maths is not as scientific as the other sciences. For example think about the idea of zero, one of the most basic and fundamental concepts in Maths. Where have you ever seen zero? "Aha" you say, "I've got nothing on my table. That's zero!" Well the thing about that is can you take that zero and put it in a bag and take it to work? Can you give someone else zero cars, can you touch zero elephants? Zero isn't an actual thing at all, it is a complete absence of things. And therefore doesn't exist in the real world, but can only exist in your mind. It is a purely imaginary concept. And to be honest while it seems quite ordinary and natural is has some unpleasant side-effects. For example you cannot divide by zero without getting into all sorts of trouble.

Take for example the following: 8/2 = 4 This says that eight divided into two piles give you four in each pile. If they are equal (which they are), then if I double both of them then they should remain equal. Now dividing eight by two then doubling it should leave me with eight, as half then double doesn't change anything, so I get

8 = 2 times 4 = 8

Perfect.

Now let's try and use the same logic with zero. Given that it's not obvious what dividing by zero should be, I'll just make a guess to start off with and see if I'm right. Let's suppose:

8 / 0 = 4

Then again, if they are equal, multiplying both by zero should keep them the same, and also diving 8 by zero then multiplying it by zero shouldn't change anything so we get:

8 = 4 times 0

Anything times zero is just zero, since you have zero lots of that amount, so we get:

8 = 0

Which is a bit of a snag. We could repeat the trick with any number, and get results like:

5 = 0

21 = 0

In fact it would result in all numbers being equal to zero. The entire system of maths would collapse in on itself like a black-hole. Not good. So we get a first peek into what maths is really about here, we have a certain amount of choice over how we do maths. See maths is just a collection of ideas and arguments, and I've made quite a convincing argument here that we wouldn't want to allow division by zero. It's not actually that we cannot divide by zero, it's more that if we allow that then maths wouldn't be the way we desire it to be. You might come back and say that how can it be such a matter of opinion, I thought maths was about ultimate truth?! Well I'm so confident in the argument I put forward, I think it is so convincing that basically anyone would agree that allowing division by zero is too problematic and that we should just skip it. It doesn't really matter if it is an opinion if your opinion is so strong and backed up with such good logic that no one would ever disagree with you.

And there is another quite bizarre thing about maths. It is all completely made up, deals with things that don't really exist, and yet somehow seems to get universal agreement despite age, race, language or up-bringing. Even some animals can understand that basic idea of counting, it seems to be a universal property of the universe that everyone can understand even though nobody can see it, we can only imagine and discuss it together. No other totally imaginary subject gets such broad agreement. For example take any fiction work, there will be lots of people arguing about lots of different aspects of it, interpretations of the ends of certain films etc. Yet when it comes to maths, as much a work of fiction as anything else, it is case open and shut in a few short lines of reasoning. Maths is so universal, we have even inscribed mathematical bits and bobs onto satellites we send out into space, assuming the only thing that would be easily understood by aliens is some maths.

And there are even bigger problems than just dividing by zero. Maths is actually full of strange contradictions and other unfortunate events, which I'll go over in my next post. Then we'll see that someone (Kurt Godel) decided to sit down and figure out where all the problems were coming from, and what he discovered was possibly more disturbing than any of the individual problems that prompted his research. So much so that the discoveries drove him mad. His paranoia was so great that he refused to eat anything his wife had not prepared, convinced he would be poisoned. When she was away for a period of time he actually starved to death. Read on next time if you dare!

(Published on 13 May 2016)


Crime doesn't pay, but it does make an awful lot of money

I teach Economics as well as Maths. Given my mathematical training, my background in insurance pricing, and personal interest and passion for the subject, it is something I have taught more and more over the years. Having recently done research on organised crime I was staggered to discover some of the statistics. Just in the UK alone organised crime groups make billions. If they were listed on the stock market, their share price would probably be in the FTSE 100.

And that's just the UK, globally the revenue of organised crime run into the hundreds of billions, with some estimates putting it at almost one trillion dollars. Holy Moly! That is an astonishing amount of money. Just to give some comparison, the top 100 corporations in the UK make around £514bn per year. Let's just stick with the UK though, so we can make a fair comparison. Taken from page 8 of this government Crime Report in 2012 the estimated revenue of organised crime in the UK was around £13.5 billion. To give some idea of comparison, Debenhams reported a mere £2.3 billion in revenue in 2015. So we're looking at a company roughly 5 times as large as Debenhams here.

Now these organised crime groups are not one unified force, in fact in some areas they are bitter rivals, but nevertheless we are looking at an economic force comparable to a large UK corporation. I was just quite surprised to see no mention of it in the Economics syllabus. Ok so it's not as if organised crime is taking over the economy, but I would have thought it was a large enough factor to merit if not an entire chapter, at least a paragraph. And if we're educating the next generation of potential policymakers wouldn't we want them to have an awareness of this issue? It's interesting to note that the government estimates the social costs of organised crime as quite a bit higher than their revenues, so we're looking at a significant player in the UK economy, and it's fair to assume that the problem is of a similar magnitude or even worse in other areas of the world.

What concerns me though is where is all this money going? And what are the profit margins, because I imagine they are quite high. The criminals are selling pretty inelastic goods and in most cases work hard (by murdering rivals) to get a monopoly on the market place, which creates very high profit margins. I'd also imagine they have much lower running costs, and also pay zero tax. So while their revenue may be lower than some legal organisations, their profits I would expect to be much larger relative to their size. All this wealth and money is just being piled up somewhere. Coupled with their brutal methods of influence, I can only imagine they are stockpiling large amounts of wealth and power. The main question is though, what is the trend. See if levels of organised crime have been static over the decades, then it's fair to assume that as the crime lords die or fight with each other, they will lose money and influence at roughly the same speed as they gain it, so over time no overall net gain will occur. They would under this scenario continue to have roughly the same level of power and influence in society over time.

However, if it is a growing trend, or they are somehow able to build on their gains over time and slowly accumulate more wealth and power, then what we have is a monster quietly growing in the background becoming harder and harder to deal with. As they buy more judges, intimidate more law enforcers, and tighten their grip on society they become more and more difficult to handle. I've seen a documentary about the top public lawyer in Columbia who is charged with tackling corruption in the country, and it is one of the most dangerous jobs in the world. She has had many assassination attempts and lives with a 24 hour armed guard. When you get to that level it is very hard to turn it around because anyone that tries to stand up to the criminals is instantly a target and so not enough people have the courage to stand up to them and the game is pretty much over by that point. I guess what I'm saying is, I don't know if it's all just business as usual when it comes to crime or is the world slowly sliding down the chute to a point where organised crime groups have comparable or more power than the elected governments? I don't know but if it's the latter or there is any risk of it being that then we need to do something about it now before it is too late.

(Published on 1 May 2016)


Why you suck at maths

I'm making a big assumption here, but as a mathematician I've calculated that most people either don't like maths or weren't any good at it at school; so I've got a very good chance that the person reading this is not great at maths, and a pretty decent chance they even consider themselves to be incompetent at it. I've recently developed a theory as to how that state of affairs has come to be. And to illustrate it, I'm going to introduce an entirely new subject I've developed, called complexology. It is a very important subject that everyone should learn, you'll need it in the world of work and in your career, and there'll be some pretty difficult exams on it that you'll need to study for and pass before you can do anything else in life. Basically, first of all you take something called a zag. These zags have different symbols, like so: '#+%&*(

The rules are as follows:
##% = *&

@ = #**

^%^ === @@#

and so on.

Ok time for your first test:

If I take the following zags, and imply them over a k-plane, what is the resulting fido-metric? The answer is of course &. I won't bore you with the proof of such a trivial example so let's move on to chapter two.

Ok I think you get the idea. See the great problem with maths, and also it's greatest asset, is that it is almost entirely symbolic. Let's look at the definition of a symbol: "a thing that represents or stands for something else" However, what I have found is often not made clear enough, is the link between the symbol and the something else. Without that crucial link, the symbols are about as meaningful as the mess that is complexology, and it's about as fun to study as power-sanding your own leg.

What should come first is a study of the something else, and later on we should introduce the symbols as a convenient way to refer to things already looked at and understood. What happens often though is the introduction of the symbols and the rules between them, aaand then a test about those things and move onto the next chapter. So in actual fact nothing has been studied there whatsoever except a bunch of meaningless relationships between meaningless symbols. What a big waste of time.

Let's look at an actual example of real maths. Take a look at the following grid of dots, I have put a certain number of dots there. At this point, I refuse to even name that quantity (although I'm sure you know what it is), to make the point that what is important is NOT the name or label which we give to the quantity, but the actual quantity itself. It's like constantly objectifying women and never getting to know the real person - only thinking about what they look like. So probably for the first time in a long time, maybe even ever, I want to introduce you to this guy:

He is a pretty interesting dude, not least because he can be rearranged in the following way:

See how he (I've just noticed I've defaulted to considering it to be a male here, but whatever) can be put nicely into two neat rows, each row the same size as the previous (and yes I did just use the label two, but I realised this will get very difficult if I refuse to use any names for numbers, which I think illustrates my point that the names are useful when trying to communicate or refer to different mathematical objects). Amazingly though, he can also be arranged like this:

How about that. Fits equally neatly into three rows. But what about 5 rows:

Oh dear. That didn't work did it. Well it's still pretty good going. But why is that, why is it possible to put him into 2, 3, or 4 rows, but not 5? Strange isn't it. What's even stranger, is that if I take just one dot away, and look at the guy that lives next to him, i.e. this guy:

Will only ever fit into one row. Try as you like, it won't go neatly into any equally sized rows other than one. That's pretty crazy isn't it, like how can we take just one dot away and rather than the nice jolly cooperative chap we got before, we're introduced to a stubborn, obnoxious, uncooperative bore? It still baffles me. Yet this rather intriguing problem, which has confounded the greatest minds to ever grace the face of the earth, for centuries, can be quite boringly and inconsequentially be stated in the language of standardly educated mathematics:

"Ahem, listen up everybody. A prime number is a number that can only be divided by one and itself. Ok class, make sure you remember that definition because it's going to be on the next class test. Also you need to know that the following numbers are prime: 2,3,5,7,11." And in my opinion that is why you suck at maths: you've probably never even come close to studying it in the first place. What you think you're not good at isn't even maths at all, it's a paper thin veneer with nothing underneath; masquerading as the bright, vibrant, solid block of wood that it should be.

(Published on 12 Apr 2016)


Why I don't speed (anymore)

Not too long ago I was caught speeding going from Hartfield to East Grinstead. At the time I was furious, and I was even more upset when I read my options in the letter that Sussex Police sent me:

  1. Accept points on my license
  2. Pay for a speed awareness course
I really really did not want points on my license, and so I was facing the £50 odd pounds (don't exactly recall what it cost) to pay for the course. I remember thinking, "What a con! They've turned the whole process into a money-making scheme. First I get fined for the ticket, now I have to pay for a course!" images The closest course to me was at the East Grinstead football club, and I had to take the afternoon off work to attend, which added insult to injury as far as I was concerned. I set off and rather grumpily sat myself down in the chair to tick the box and get it over and done with.

I must say my expectations about the course could not have been more wrong. After about 15 minutes I was riveted. The information presented, the people that ran the course, the exercises we did, it was a really great afternoon. By the time I left I really felt I'd got good value for money and I'd personally highly recommend the course. I jokingly said to my friends at the time that they ought to go and get caught speeding so that they can take the course! I won't give all the details, but there was one moment in particular that really stuck out for me. At the beginning, we'd been asked to write down all the reasons why we sped or felt it was ok to speed. Of course we all came up with things like "I was late", or "I wanted to get there faster", even things like "it's fun!" Right at the end of the course we were given an example of a little girl, her mum was about to take her to school but they'd forgotten something like her PE kit. So the mum went back inside the house for (get this), 30 seconds, when the little girl saw her friend across the road. The girl ran across the road to catch up with her and was hit by a car going 40mph. She died. This was actually made into an advert, you might remember it

Now this car was driven by a man who drove that way to work every day. He cut through a residential area to avoid traffic, and thought nothing of going just 10mph over the speed limit. The thing he did not take into account however, was the statistics involving the speed of the car and the mortality rate of people hit by cars. It's quite staggering, (for mathematical reasons I'll cover in another post), but the difference an extra 10mph makes is enormous.

Now think about this, every year there are around 1500 DEATHS from traffic accidents in the UK. That means, every day, 5 families get a call from a police person telling them that their father, husband, wife, daughter, etc., is not coming home that evening. Just think about the amount of pain and distress that causes, and the fact that it would be slashed in half or more if people simply stuck to the speed limit. We're not even including in that injuries and disablement, which run much higher than that.

One thing people often gripe about is speed cameras, but here's an interesting fact. In East Grinstead there is a speed camera on London Road heading out towards Felbridge. In the years before that was installed, the speed limit was 40mph and there were several road deaths in that period, one of them a school child. So the safety campaigners got it reduced to 30mph, and funnily enough the number of deaths on the road did not change too much, mainly because people were still speeding. Since that camera has been installed, the number of deaths on that road (they said in the course), has fallen to zero. That's right, none.

So my message to those of you that haven't had the fortune to do this course, next time you're trying to save 30 seconds by driving at 40mph or more in a 30-zone, is it really worth risking someone else's life for that? Have some sense of decency, and stick to the speed limit.

(Published on 4 Apr 2016)