The Mathematical Principle of Traditional Ethics

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传统美德的数学原理

人类社会经过几万年的进化衍生出许多传统美德流传至今。例如，有福同享，有难同当，以及贞操观等等。在西方社会，不可奸淫这个概念在圣经中也有反复被强调。这些传统观念之所以会产生有着多方面的因素。本文就从数学角度解释一些传统的原理。

首先，这个世界上有两种快乐：当你关心、帮助、鼓舞、支持一个人取得成就时，你会感到快乐，这种快乐属于自我实现；而当你得到钱财等你想得到的物质财富时，你也会感到快乐，这种快乐属于物质上的满足。这两种快乐都能让你感到高兴，但是在数学特征上却存在一种本质区别，即前者为凸函数（斜率递增的函数），后者为凹函数（斜率递减）。

爱情观、贞操观也是如此。假设原先有两个完全一样的人，你关心、帮助或爱过其中的一个人，那么你自然会更喜欢那个人。所以，对一个人付出爱会提高你对他（她）的好感。而关心、帮助或爱一个你更喜欢的人时你所得到的快乐总是多于关心、帮助或爱一个你不太喜欢的人。所以，幸福针对爱情的函数是凸函数（斜率递增的函数，例如: y=x²），因为你越是对一个人付出爱付出关心，你就越喜欢那个人；而你越喜欢那个人你对他（她）付出爱时所得到的幸福就越大。也就是说，对一个人所付出的爱和对他的好感度会相互促进，所以，该函数斜率递增。也正是因此，把所有的爱集中在一个人身上，你才会得到最大的幸福，数学表达为 2²+0²>1²+1²，不等式左边为把两份爱付出在一个人身上，不等式右边为把两份爱分别付出在两个人身上。无论是多个人还是总共加起来更多份爱，结论都是一样的。强调一点，这里的爱指婚姻性爱，这也就解释的传统的贞操观念的原理。当然，在真实世界，不同人在性格、习性、爱好等方面会非常不同，所以，有些人合得来而有些人则合不来。因此，在相爱前应更多的彼此了解对方，只有把所有的性爱集中在彼此最适合的那一个人身上，你才会得到最大的幸福。

那么，有福同享、有难同当又该如何解释呢？先说有福同享。假如，你家被洪水淹了，你变得一无所有，饥饿之极，有个好心人送给你一个包子，你会觉得很快乐，因为他救了你一命。如果那个好心人又送给你100个包子，你会感到更快乐，但是不会感到100倍快乐。因为，你每得到一个包子，你对包子的需求量减少，你得到下一个包子时所得到的快乐也会相对减少。这也就是为什么同样捡到100块钱对于一个乞丐来说是莫大的欢喜，而对于一个亿万富翁来说，几乎不算什么。所以，快乐针对物质财富的函数是凹函数（斜率递减的函数，例如：y=x^0.5，如图二所示）。也正是因此，有福同享可以最大化一个群体的快乐，数学表达为 2^0.5+0^0.5<1^0.5+1^0.5，不等式左边为一个人独吞所有财富，不等式右边为两人同享财富。无论是多个人还是总共加起来更多份财富，结论都是一样的。不过在真实世界中，因为考虑到人有各种欲望、贪念等，这个函数的凸凹性会更为复杂，而且因人而异。所以，在此我们只考虑最基本的情况，那就是生存的角度。再说有难同当，假如你家境贫困，某天不小心丢了一千元，你会感到很伤心，然后你又掉了一千元。这后来的一千元损失对你所造成的伤害会更大，因为你掉了一千元之后，钱更少了，接下来的一千元对你来说更重要了。所以，一个人所感到的痛苦针对他所受到的灾难的函数是凸函数（斜率递增的函数，例如: y=x²），因为已经经受灾难的人对灾难的承受能力更差，相同的灾难对他来说打击更大。也正是因此，有难同当可以最小化一个群体在经历灾难时所受到的痛苦，数学表达为 1²+1²<2²+0²，不等式左边为两个人共同承担损失，不等式右边为一个人担当所有的损失。无论是多个人还是总共加起来更多的灾难，结论都是一样的。不过，在真实世界中，一个人的心理、意志和信念在历经灾难之后可能会变强或变弱，所以这个函数的凸凹性会更为复杂，而且因人而异。这里我们只考虑最基本的情况。

以上所用到的数学不等式叫做琴生不等式(Jensen’s Inequality)，概括的来说就是根据目标量针对资源这个函数的凸凹性，会得到两种相反的属性。对于凸函数（斜率递增的函数，例如 y=x²）来说，将所有资源集中在单一个体上最大化目标量，将所有资源平均分散在每个个体上最小化目标量（例如: 3²+0²+0²>...>2²+1²+0²>...>1²+1²+1²）；而对于凹函数（斜率递减的函数，例如 y=x^0.5）来说，将所有资源平均分散在每个个体上最大化目标量，将所有资源集中在单一个体上最小化目标量（例如 1^0.5+1^0.5+1^0.5>...>2^0.5+1^0.5+0^0.5>...>3^0.5+0^0.5+0^0.5）。这个原理除了本文用以解释传统美德的产生之外，在很多科学领域也有广泛的应用。

综上所述，一个人所得到的快乐和他所拥有的财富是不成正比的。如果我们从生存的基本角度出发，快乐针对物质财富的函数是凹函数，具有共享最大化快乐的属性，所以建立在金钱和物质基础上的爱情是经不住考验的。这也就自然而然的解释了为什么很多人穷困的时候和自己的爱人一起同甘共苦，度过了一段美好的时光，也成就了一番事业；而富有了之后，却时常容易闹分手离婚，错过那本应当更美好的时光。同时，这个数学原理也从另一方面说明了建立在自我实现基础上的爱情相较而言更经得住考验，更容易天长地久。最后，愿天下有情人终成眷属，也希望那些已经得到爱的人能学会如何珍惜手中的爱，真心呵护自己所爱的另一半。

The Mathematical Principle of Traditional Ethics

Through thousands of years of evolution, our mankind society has developed numerous ethics and social codes. Some of them have been passed down even until today. For example, the notion of sharing fortunes together, confronting hardships together, and the traditional chastity view. Also, in the western world, the notion of no adultery has been repeatedly emphasized in the bible. There are many reasons why these ethics evolve. This article will discuss some simple mathematical principles behind them.

Firstly, in our life there are two kinds of happiness, when you care for someone, help someone, encourage and support someone to achieve success, you will feel happy, this kind of happiness belongs to self-actualization; when you obtain money or material wealth, you will also feel good, this kind of happiness belongs to material satisfaction. Although both make you feel happy, their mathematical properties are essentially different: the former is characterized by a convex function (with an increasing gradient), and the latter is characterized by a concave function (with a decreasing gradient).

For the traditional chastity view, suppose there are two identical persons, you have cared, helped or made love with one of them, then you will naturally like him/her more. So, paying love to someone will enhance your relationship with him/her. Moreover, caring, helping and loving someone you preferred makes you feel happier than caring, helping and loving someone you don’t like much. Therefore, a function of happiness versus love is a convex function (with an increasing gradient, e.g., y=x²), because the more you pay love to someone, the more you like him/her; and the more you like someone, the happier you feel upon paying the same amount of love to him/her. In other words, the amount of love you devote to someone and your relationship with him/her mutually enhance each other. Therefore, the function has an increasing gradient. And because of this, you obtain maximum happiness by devoting all your love onto one person, as conveyed by the mathematical inequality 2²+0²>1²+1², the left hand side computes the happiness you will obtain by devoting two units of love onto one person, while the right hand side represents devoting them equally onto two persons. The inequality applies to the case involving multiple persons or more units of love as well. To emphasize, we are talking about marital love or sexual love here, so this explains the mathematical rationale behind the traditional chastity view. Of course, in our real life, different people can have very different behaviour, characteristics, preferences, etc. Some people get along with each other easily while some do not. So in general, people should try to get to know more about each other before falling into love; only if one can devote all his/her love on someone who is mutually the most suitable, one can obtain maximum happiness.

Then, for the notion of sharing fortunes or confronting hardships (sharing misfortunes) together, how can we explain them mathematically? First, let us talk about sharing fortunes. Suppose that your home was destroyed by flood, and you lost everything. While you were starving to death, a kind-hearted person gave you a bum, you would feel happiness because he/she saved you. If the same person gave you one hundred bums, you would feel much more happiness, but not 100 times more happiness. This is because after receiving each bum, your need for bums was reduced, thus, the amount of happiness you obtained upon receiving the next bum is slightly less. This is also the reason why picking up 100 dollars is considered a great fortune for a beggar, but almost nothing for a billionare. Thus, a function of happiness against material wealth is a concave function (a function with decreasing gradient, for example, y=x^0.5, as shown in Figure 2). As a result, sharing fortunes together maximizes the total happiness within a group of people. This is shown mathematically by 2^0.5+0^0.5<1^0.5+1^0.5, where the left-hand-side denotes the situation where all material wealth is own by one person, the right-hand-side denotes that the wealth is equally shared by two persons. The principle applies to multiple people as well. However, in the real world, the desire, lust, greediness, etc., of people will make the shape of the function more complex, and it will also differ from one person to another. Thus, here we only consider the simplest case, i.e., just from the aspect of survival. Then we talk about confronting hardships together. Suppose you were born in a poor family, on someday you accidentally lost $1000 by carelessness, you feel bad. And then you lost another $1000. The loss of the 2^nd $1000 is going to hurt you even more, because after you lost $1000, you have fewer money in total, thus, the subsequence same amount of $1000 became more important to you. Thus, the amount of suffering a person experiences forms a convex function (a function with increasing gradient, e.g., y=x²) against the number of disasters he/she encounters, because after one disaster, one becomes more vulnerable, the same amount of disaster will cause more suffering. As a result, sharing misfortunes minimizes the total suffering of a group of people. Mathematically, this is expressed as 1²+1²<2²+0², the left-hand-side represents the total suffering when the two persons share the misfortune, the right-hand-side represents when one person takes all the loss. The principle applies to multiple people as well. In the real world, a person’s heart, attitude and fortitude might strengthen or weaken after experiencing a course of hardship, so the shape of the function will become much more complicated and will differ from one person to another. Thus, here we only consider the simplest situation.

The mathematical inequality we mentioned above is called Jensen’s Inequality. Generally speaking, for a function of the objective yield against the resource, whether it is convex or concave will give rise to opposite characteristics. For a convex function (with an increasing gradient, e.g., y=x²), focusing all the resources on a single individual maximize the objective function, distributing all the resources equally on every individual minimizes the total objective function (e.g., 3²+0²+0²>...>2²+1²+0²>...>1²+1²+1²); for a concave function (with decreasing gradient, e.g., y=x^0.5), distributing all the resources equally on every individual maximizes the objective function, focusing all the resources onto one single individual minimizes the total objective function (e.g., 1^0.5+1^0.5+1^0.5>...>2^0.5+1^0.5+0^0.5>...>3^0.5+0^0.5+0^0.5). The principle is also widely applied in many scientific fields, apart from what is discussed here.

To summarize, the amount of happiness a person can obtain is not directly proportional to the amount of wealth he/she possesses. From the most basic starting point of view (i.e., survival), the function of happiness against material wealth is concave, it has the property that sharing maximizes total happiness, thus, love based on money and materialism is not durable. That also naturally explains why it is so common that some people, when not very rich, they had shared both fortunes and misfortunes with their loved ones, enjoyed a happy time, and achieved great things; but after they get rich, very often they get divorced and split, and missed the supposedly happier time in their lives. On the other hand, the mathematical principle also tells us that love based on self-actualization is relatively more durable and more stable. Finally, I wish all people with love can find the right other half, I also hope all married couples will know how to treasure love, and care for each other with full dedication of love.