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)