Real Mathematics to Love Mathematics

Mostly people not like mathematics, they said,” Oh, it’s that really annoying boring thing”, or “Oh, it’s that really hard and complicated thing. I`m not talking about the same thing. In fact, if  people think like that, they probably never experienced REAL mathematics.
What real mathematics is like
If people only experienced mathematics at school, they might think it starts with some boring formal definitions, then some boring formal algebra, followed by a boring formal proof, from which they get a mildly interesting result.
In reality, mathematics starts from the other end.  It always starts with a riddle.
Here’s a riddle :
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 the answer. 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! We’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 us, because we’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 any 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).
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:
None.
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.
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 an easy example :
 “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.)
The answer is I’ve thought about it a bit, and here’s what I’ve come up. (And no, it’s definitely not what I expected I would come up with.) I see two possible trails of logic:
1. She was with a boy. That’s one kid. The other kid can be either a girl or a boy, which gives us a probability of 1/2 of her having two boys.
2. Imagine we did this as an experiment. Let’s assume she has two kids, who are each of random gender. Then, each day, she randomly picks one child to take to the store with her. (reasonable assumptions so far?)
The cases are, boyboy, boygirl, girlboy and girlgirl. Now clearly if she had two girls, she would never take a boy to the store, so that didn’t happen. When our friend encountered her, she had a boy with her, so she has to have boygirl, girlboy, or boyboy. So boyboy is only one of three cases. BUT, there are two boys in boyboy, so meeting her with a boy at the store would happen twice as often as with the other two possibilities. In other words, the chance is 2/4, or 1/2.
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.”
- Me 
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.
Afterword
“Mathematics is about numbers, and their sequences, and beautiful patterns, and wonderful geometrical theorems. Not some silly riddles! “
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.