Aaaargh! He said the M-word! Run for your lives!
Wait up! If your reaction to the words mathematics is “Oh, it’s that really annoying boring thing”, or “Oh, it’s that really hard and complicated thing”, we’re not talking about the same thing. In fact, if you think like that, you probably never experienced REAL mathematics.
Sure, you did a thing called mathematics at school. But I could just easily make a subject at school called “sex” where you would memorize sexual positions and human anatomy and pass exams and get a degree. And yet it wouldn’t be anything like REAL sex.
So let me tell you what I love about real mathematics. Maybe, just maybe, I’ll make you curious enough to give it a try.
What real mathematics is like
If you only experienced mathematics at school, you might think it starts with some boring formal definitions, then some boring formal algebra, followed by a boring formal proof, from which you get a mildly interesting result.
In reality, mathematics starts from the other end. It always starts with a riddle.
Here’s a riddle you might have heard before:
A farmer has a fox, a goose, and some cabbage. He wants to cross a river on his boat, but the boat is small, so he can only take one item with him at a time. He can’t leave the fox with the goose, or the goose with the cabbage, because the former would eat the latter. How can he cross the river?
The answer is quite simple, so I won’t even tell you. And the answer is where it would end for any normal human. But not for a mathematician. As mathematicians, after we enjoy the glowing feeling of having solved the riddle, we wonder what would happen in similar cases. Let’s say the genetically modified cabbage suddenly becomes conscious and decides it loves eating foxes. Is there still a solution?
(The answer is yes. Screw the goose and the fox! You’ve got frickin fox-eating cabbage on your hands! Get on national TV, become famous, then sell the cabbage on e-bay and never have to work again. Or, alternately, build up an army of fox-eating cabbages and try to take over the world. Or kill the cabbage before it turns against you, because you’re obviously starring in a third rate horror flick.)
Then, if we feel like it, we might go on to make some generalizations. How many animals could we transport like this if we had two places on the boat? How about three places? What about the general case of having n places? What if the relationships between what animal eats what were more complicated than a simple top-down chain?
Here’s the fun thing. No one is forcing us to do this. If we get bored, we can just leave all the gooses and fox-eating cabbages behind, and go weigh balls on a balance scale, or square a circle with just a compass and an unmarked ruler (which is impossible btw).
But what are the applications?
I can hear some of you thinking… “But what are the applications of knowing how the farmer can transport his stuff across the river?” Let me give you a long and complicated answer:
Ok, now on to the short and simple answer…
What are the applications of paintings? Of playing music? Of playing chess?
Sure, painting skills can be used to make advertisements more effective, music can be used to add soundtracks to make movies sell better, chess can be used… well it can’t :). But anyway. The point is, we don’t paint, or play music, or play chess, because it can give some actual results. We just do it for fun. We enjoy it. To paraphrase Richard Feynman:
Mathematics is like sex. It can give practical results, but that’s not why we do it.
Get over the idea that mathematics is just something used by engineers and physicists to solve problems. Mathematics is an art in itself, just like music or drawing.
Some more examples of real mathematics
Here are a few more examples of mathematics. I’ll give you an easy example, a moderately hard example, and an evil example.
1. “The two kids” riddle
Imagine you’re chatting with a friend about one of your common acquaintances.
“I heard she has two kids,” you say.
“Yeah, that’s right. By the way, I met her yesterday at the supermarket, and she was with a small boy. We started chatting, and she told me it was her son. So at least one of her kids is a boy.”
What’s the probability that both of her kids are boys?
(Hint: Think hard. It’s not as straightforward as it seems.)
(Update 28/10/2008: Well, it turns out my original answer to this riddle was wrong. Check out the comments for more detail. Hey, you get to see me being proven wrong! )
2. The coin tossing riddle
Imagine you’re tossing a coin, and recording the results in a row like this: TTHTHHT…
I’d like you to consider two cases. In the first case, you keep tossing the coin until you get HTH, and then you stop. In the second case, you keep tossing the coin until you get HTT and then stop.
If you tried both of these cases a few thousand times, you would find that, on average, in one of the cases you will stop sooner than in the other. Which one and why?
3. The famous ball weighing riddle
This riddle is satisfyingly hard. (no, satisfyingly wasn’t the first word I thought of). It took me personally a couple of days to solve it.
Imagine you have twelve balls. One of them is either heavier or lighter than the rest, but you don’t know which ball, and whether it’s heavier or lighter. Can you find out which ball it is, and whether it’s heavier or lighter, by doing just three weighings with a standard balance scale? (one that simply tells you whether the balls you put on one half are heavier than the ones on the other half, but nothing else). You can number the balls for reference.
I won’t give you the answers to the above riddles. In fact, my favorite site for riddles is wu riddles. And you know why? Because it doesn’t tell you the answers (Yes, that’s a good thing. Look out for an article about “activation energy”, in which I will explain that in more detail.)
I’ll just let you enjoy thinking about the riddles.
Where are the numbers?
Wait a second. I’m writing an article about mathematics… and yet I haven’t written a single bit about numbers yet? (apart from the two kids riddle). Surely mathematics is all about numbers?
That’s because I don’t think numbers are the essential part of mathematics. It’s logic.
“When a problem has a correct solution, and the solution can be PROVEN to be correct, that’s mathematics.”
In other words, what I like about mathematics is the 100% certainty that a solution is correct. And if there is no solution… then there is a way to prove that with 100% certainty.
Sure, I still enjoy riddles that rely on real-world intuition. Where the answer makes a lot of sense, and any other answers are either unlikely, or too complicated. But it’s the 100% certainty that I really love about mathematics.
Is mathematics for you?
Did the riddles above make you think? Did you enjoy it? If yes, you would enjoy mathematics (the real mathematics, anyway). If not… then mathematics is not for you. And that’s fine too.
I hope I cleared up some misconceptions about what makes mathematics mathematics. (god, that was a pain to spell :p “mathematics mathematics mathe… argh! maths! why can’t I just call it maths!!!”)
If you think mathematics sounds like fun, and want to try some more, just google around a bit. I’m sure you will find plenty of interesting math questions out there. Or, for more fun riddles (that aren’t necessarily related to maths, or have a single solution), visit wu riddles. Cheers!
If you’re a math geek, you might be shouting at the computer screen by now.
“What the hell is this guy talking about? Mathematics is about numbers, and their sequences, and beautiful patterns, and wonderful geometrical theorems. Not some silly riddles! Aaargh!”
And you’d be right. Mathematics IS about all those things. But when they were first discovered, they started as a riddle. As a nagging question inside some mathematician’s head. Is the sum of angles inside a triangle constant? How long is the circumference of a circle, compared to its radius? Can the diagonal of a square be written as a fraction?
I believe mathematics isn’t about the knowledge. It’s about enjoying the thinking. And I just wanted to introduce the concept to non-math folks. Cheers!