Relationship between math and life

This will be a series of essays, pertaining to some relationship between math and life.

1. Life is a martingale

-- You go up and go up, but suddenly you lose everything.

Today when I was in the Martingale Theory class, the professor wrote a model on the board.

P(Y_i=2)=P(Y_i=0)=0.5
X_n=Y₁×Y₂×…×Y_n×

This is a submartingale. However, although we have E(X_n) monotonically increasing, X_n converges to zero as surely. Whenever Y takes a zero value, X_n will be zero after then.

After illustrating this, the old professor stared at the blackboard for a while and said:" Sometimes life is just like this, you go up and go up, but suddenly you lose everything."

When he said this, it seems as if all the memories of the past merged in his head.

Yes, it is. And when you lose everything, you will never get them back again.

2. Life is a game

-- Although no one did anything wrong, the result might be the worst.

Take a famous example in game theory: the centipede game, the payoff is as below:

We can solve the game from backwards induction. At each node, if the player chooses to stop, the payoff will be indicated at the below. At the last node, player 1 certainly wants to choose stop (101 vs. 100). Then seeing that the payoff will be 98 if continuing playing at the penultimate node, player 2 will also want to stop. So on and so forth, player 1 will choose to stop at the first node, which yields a payoff of (2, 0).

It is sad sometimes. Everyone can conceive the wonderful future, but at one time, one of them failed to trust the other anymore, and decide to be satisfied by the present. At that moment, all the promise is as pale as a paper. All the happiness collapses in a second.

But you cannot blame anyone, what he did was just playing her best strategy. And you are so naive to trust a promise.

3. You can't beat the system.

-- Things have their intrinsic properties, and you are not gonna change them.

This is also a principle associated with stochastic process. There are several interpretations of this.

One is a problem that always appears on bbs, saying that if every parents choose to give birth to children until they get their first boy, what the final ratio will be. However, one can show that in this setting, the expected ratio of boys and girls are still 1:1. It is just like making several break points in a long sequence.

Similar things also happen when you play a game or speculate in a stock market. You can set a rule for your speculation, for example, you buy a certain security only when the price of it is below "a" and sell it only when the price is above b>a. This plan sounds appealing, but the principle tells us that as long as the expected value of the stock is fixed, you can't make any money. This is what we call that" you cannot beat the system".

This means that sometimes, things have their intrinsic property, which will not be changed by your manipulation. What we do is just to change a path of life, but may not the result. Sometimes, you have to admit that there are some things that are destined, your job is just to make them realized. However, you still need to try hard, there might be miracles.

4. Life is an optimization problem

-- Life is to optimize

Having discussed about probability and stochastic process adequately, we are now turning to the field in which I am studying.

Yes, life is as an optimization problem. You can think every move in your life as a variable of your final prospective utility, which you are going to optimize. From those trivial things, to critical decisions, you are optimizing. You acquire information to increase your chance to succeed, you calculate the expected payoff, and you pick up the one that optimizes.

Conversely, the academic ramification of optimization also originated to aim to help people's lives, which also gives a justification of that idea. You can see many problems relating to our daily life. Look at your textbook; you can see how many of them come from real life.

Besides, life also resembles optimization problem in the way we solve it. As we know, there are many genius people devoting to make provable way to solve them. Some of the approaches are accurate, while more are just approximation. But in practical, maybe for many of the problems, heuristic algorithm prevails. Just like sometimes, what our first feeling is always right. When we consider and consider, we are clearer with every detail, however, we may not make as good decision as by just using our insight. Of course, we still have several good algorithms, i.e., when we want to do better, sometimes we need techniques. And if you want to be distinguished, you need to think more and to prove to other people.

5. Life is NP-Hard

-- You can't solve it even if NP=P.

As I promised, there will be some funny part. Here we go.

Proposition: Life is NP-Hard.

Proof: We can reduce and NP problem into a special case in our life. For example, we could face a TSP problem, we could face an integer programming problem, or even we could face the situation when our teachers want us to solve an NP-Hard Problem! Hence, life is NP-Hard. Q.E.D

Remark: Actually, by the above argument, any complexity class is a subset of what could happen in our lives. Therefore, generally you can't

1) Solve a problem arise in life efficiently
2) Approximate a problem arise in life by any factor
3) Judge any "decision" is right or wrong efficiently
...

So why bothering thinking about those annoying things, did you realize that when you think about them, you are just like trying to prove NP=P! However, is it possible to prove it? No! And you still may not solve the problem even if NP=P, since Life is strictly harder than that.

6．Life is life

Life is not mathematics. The delicacy of life lies in its resemblance to mathematics, but the beauty of life is the fact that it is not mathematics. As every great achievement human beings have attained, only a few people care about the delicacy parts, while most of us only enjoy its beauty.

Indeed, there are really few people who know mathematics (of course, most of us are not among them), but everyone has his understanding of life, probably in totally different versions. For most of us, life is not a martingale, because there are too many fluctuations to endure, and too few signs for us to predict; life is not a game, there can't be a possible definition of our payoff, and there are usually no winners nor losers; life is not an optimization problem, because too oftenly we are satisfied with what we have, and never bother to think about better, or too oftenly we perceive ourselves too high such that we forget about the potential unaffordable loss; also, we can sometimes beat the system, because when you have the will, there could be a way.

Life is temporary, it can't bear the delicacy which makes mathematics perpetual; while mathematics is rigid, and it can't own the dramatic scenes which make our lives beautiful and non-trivial. We marvel at those delicate equations, and sigh over the changes of lives.

As a pursuer of mathematics, we lead a life that is most close to what mathematics have interpreted for us, and consider our researches in a way that is most close to the way we lead our lives. We know there are certain things we can never deal with, we just make them our conjectures.