The mathematics of love

爱的数学

by Hannah Fry

Today I want to talk to you about the mathematics of love. Now, I think that we can all agree that mathematicians are famously excellent at finding love. But it's not just because of our dashing personalities, superior conversational skills and excellent pencil cases. It's also because we've actually done an awful lot of work into the maths of how to find the perfect partner.

今天我想和你谈谈关于爱的数学。现在，我想我们都可以同意，数学家在寻找爱情方面是非常出色的。但这不只是因为我们的个性，优越的谈话技巧和优秀的铅笔盒。这也是因为我们在数学上做了大量的工作来寻找完美的伴侣。

Now, in my favorite paper on the subject, which is entitled, "Why I Don't Have a Girlfriend" -- (Laughter) -- Peter Backus tries to rate his chances of finding love. Now, Peter's not a very greedy man. Of all of the available women in the U.K., all Peter's looking for is somebody who lives near him, somebody in the right age range, somebody with a university degree, somebody he's likely to get on well with, somebody who's likely to be attractive, somebody who's likely to find him attractive. (Laughter) And comes up with an estimate of 26 women in the whole of the UK. It's not looking very good, is it Peter? Now, just to put that into perspective, that's about 400 times fewer than the best estimates of how many intelligent extraterrestrial life forms there are. And it also gives Peter a 1 in 285,000 chance of bumping into any one of these special ladies on a given night out. I'd like to think that's why mathematicians don't really bother going on nights out anymore.

现在，在我最喜欢的题为“为什么我没有女朋友”的文章中（笑声）彼得•巴克斯试图评价他找到爱情的机会。现在，彼得不是一个很贪婪的人。在英国所有的女性中，所有的彼得都在寻找一个生活在他身边的人，一个合适的年龄段的人，一个拥有大学学位的人，一个他很可能会相处融洽的人，有人可能会有吸引力，有人可能会发现他有吸引力（笑声），并提出了一个估计，26名女性在整个英国。它看起来不太好，是彼得吗？现在，只需要把这一点放在一个角度来看，这大约是对外星生命形式的最好估计的400倍。而且它也给了彼得一个285000的机会，在一个特定的晚上撞上任何一个特别的女士。我想这就是为什么数学家们再也不会在夜晚外出的原因了。

The thing is that I personally don't subscribe to such a pessimistic view. Because I know, just as well as all of you do, that love doesn't really work like that. Human emotion isn't neatly ordered and rational and easily predictable. But I also know that that doesn't mean that mathematics hasn't got something that it can offer us because, love, as with most of life, is full of patterns and mathematics is, ultimately, all about the study of patterns. Patterns from predicting the weather to the fluctuations in the stock market, to the movement of the planets or the growth of cities. And if we're being honest, none of those things are exactly neatly ordered and easily predictable, either. Because I believe that mathematics is so powerful that it has the potential to offer us a new way of looking at almost anything. Even something as mysterious as love. And so, to try to persuade you of how totally amazing, excellent and relevant mathematics is, I want to give you my top three mathematically verifiable tips for love.

问题是我个人并不赞同这种悲观看法。因为我知道，就像你们所有人一样，爱并不是真正的工作。人类的情感并不是整齐有序、理性和容易预测的。但我也知道，这并不意味着数学没有它能提供给我们的东西，因为，爱，和大多数生活一样，充满了模式和数学，最终，所有关于模式的研究。从天气预报模式到股市的波动，到行星的运动或城市的发展。如果我们诚实的话，这些东西没有一个是整齐有序的，也很容易被预测。因为我相信数学是如此强大，它有潜力为我们提供一种新的方式来看待几乎任何事物。甚至像爱一样神秘的东西。所以，我想试着说服你，你的数学是多么的惊人，优秀和相关，我想给你我的前三个Mathematica爱的可靠提示。

Okay, so Top Tip #1: How to win at online dating. So my favorite online dating website is OkCupid, not least because it was started by a group of mathematicians. Now, because they're mathematicians, they have been collecting data on everybody who uses their site for almost a decade. And they've been trying to search for patterns in the way that we talk about ourselves and the way that we interact with each other on an online dating website. And they've come up with some seriously interesting findings. But my particular favorite is that it turns out that on an online dating website, how attractive you are does not dictate how popular you are, and actually, having people think that you're ugly can work to your advantage. Let me show you how this works. In a thankfully voluntary section of OkCupid, you are allowed to rate how attractive you think people are on a scale between 1 and 5. Now, if we compare this score, the average score, to how many messages a selection of people receive, you can begin to get a sense of how attractiveness links to popularity on an online dating website.

好吧，那么TOP的第一个问题是：如何在网上约会中获胜。所以我最喜欢的网上交友网站是okcupid，特别是因为它是由一群数学家开始的。现在，因为他们是数学家，他们已经收集了每个人使用他们的网站近十年的数据。他们一直在寻找我们谈论自己的方式，以及我们在网上交友网站上相互交流的方式。他们提出了一些非常有趣的发现。但我特别喜欢的是，在一个网上交友网站上，你的魅力并没有决定你有多受欢迎，事实上，有人认为你的丑陋可以发挥你的优势。让我来告诉你这是怎么回事。在Okcupid的一个令人欣慰的自愿性部分，你可以评估你认为人们在1到5之间的比例有多大。现在，如果我们比较一下这个分数，平均分，一个选择的人收到多少信息，你可以开始了解在网上交友网站上，吸引力是如何与人气联系起来的。

This is the graph that the OkCupid guys have come up with. And the important thing to notice is that it's not totally true that the more attractive you are, the more messages you get. But the question arises then of what is it about people up here who are so much more popular than people down here, even though they have the same score of attractiveness? And the reason why is that it's not just straightforward looks that are important. So let me try to illustrate their findings with an example. So if you take someone like Portia de Rossi, for example, everybody agrees that Portia de Rossi is a very beautiful woman. Nobody thinks that she's ugly, but she's not a supermodel, either. If you compare Portia de Rossi to someone like Sarah Jessica Parker, now, a lot of people, myself included, I should say, think that Sarah Jessica Parker is seriously fabulous and possibly one of the most beautiful creatures to have ever have walked on the face of the Earth. But some other people, i.e., most of the Internet, seem to think that she looks a bit like a horse. (Laughter) Now, I think that if you ask people how attractive they thought Sarah Jessica Parker or Portia de Rossi were, and you ask them to give them a score between 1 and 5, I reckon that they'd average out to have roughly the same score. But the way that people would vote would be very different. So Portia's scores would all be clustered around the 4 because everybody agrees that she's very beautiful, whereas Sarah Jessica Parker completely divides opinion. There'd be a huge spread in her scores. And actually it's this spread that counts. It's this spread that makes you more popular on an online Internet dating website. So what that means then is that if some people think that you're attractive, you're actually better off having some other people think that you're a massive minger. That's much better than everybody just thinking that you're the cute girl next door.

这是Okcupid的家伙想出的图表。重要的是要注意的是，你越有吸引力，你得到的信息就越多。但问题是，这里的人比这里的人更受欢迎，尽管他们的吸引力是一样的。原因是，这不仅仅是简单的外表，也很重要。所以让我用一个例子来说明他们的发现。所以，如果你选择像波西亚·德罗西这样的人，每个人都认为波蒂亚·德罗西是一个非常漂亮的女人。没有人认为她丑，但她也不是超级名模。如果你把波西亚·德·罗西和莎拉·杰西卡·帕克这样的人比较一下，现在，很多人，包括我自己，我应该说，认为莎拉·杰西卡·帕克是一个非常出色的人，可能是世界上最美丽的生物之一。但有些人，也就是说，大多数人互联网，似乎认为她看起来有点像马。（笑声）现在，我认为，如果你问人们如何有吸引力，他们认为莎拉杰西卡帕克或波西亚德罗西，你要求他们给他们1分和5分的分数，我估计他们的平均分数大致相同。但是人们投票的方式会有很大的不同。因此，波西亚的分数都会聚集在4周围，因为大家都认为她很漂亮，而莎拉·杰西卡·帕克则完全区分了意见。她的分数会有很大的变化。事实上正是这种传播才有意义。正是这种传播使你在网上交友网站上更受欢迎。那么这意味着，如果有人认为你很有吸引力，那么你最好让其他人认为你是个大人物。比每个人都认为你是隔壁可爱的女孩要好得多。

Now, I think this begins makes a bit more sense when you think in terms of the people who are sending these messages. So let's say that you think somebody's attractive, but you suspect that other people won't necessarily be that interested. That means there's less competition for you and it's an extra incentive for you to get in touch. Whereas compare that to if you think somebody is attractive but you suspect that everybody is going to think they're attractive. Well, why would you bother humiliating yourself, let's be honest? Here's where the really interesting part comes. Because when people choose the pictures that they use on an online dating website, they often try to minimize the things that they think some people will find unattractive. The classic example is people who are, perhaps, a little bit overweight deliberately choosing a very cropped photo, or bald men, for example, deliberately choosing pictures where they're wearing hats. But actually this is the opposite of what you should do if you want to be successful. You should really, instead, play up to whatever it is that makes you different, even if you think that some people will find it unattractive. Because the people who fancy you are just going to fancy you anyway, and the unimportant losers who don't, well, they only play up to your advantage.

现在，我认为当你考虑发送这些信息的人时，这开始有点意义。所以，让我们说，你认为某人的吸引力，但你怀疑其他人不一定会有兴趣。这意味着你的竞争会减少，这是你获得联系的额外动力。相比之下，如果你认为某人是有吸引力的，但你怀疑每个人都会认为他们是有吸引力的。好吧，你为什么要去羞辱你自己，让我们诚实点？这里是真正有趣的部分。因为当人们选择他们在网上交友网站上使用的图片时，他们往往会尽量减少他们认为一些人会觉得没有吸引力的东西。典型的例子是人们，也许是有点超重，故意选择一张裁剪得很好的照片，或者秃头的男人，比如故意选择他们戴着帽子的照片。但事实上这与你的相反如果你想成功，就应该做。事实上，你应该去迎合那些让你与众不同的东西，即使你认为有些人会觉得它不吸引人。因为那些幻想你的人会喜欢你，而那些不重要的失败者，他们只会发挥你的优势。

Okay, Top Tip #2: How to pick the perfect partner. So let's imagine then that you're a roaring success on the dating scene. But the question arises of how do you then convert that success into longer-term happiness and in particular, how do you decide when is the right time to settle down? Now generally, it's not advisable to just cash in and marry the first person who comes along and shows you any interest at all. But, equally, you don't really want to leave it too long if you want to maximize your chance of long-term happiness. As my favorite author, Jane Austen, puts it, "An unmarried woman of seven and twenty can never hope to feel or inspire affection again." (Laughter) Thanks a lot, Jane. What do you know about love?

好的，顶级的：如何挑选完美的伴侣。所以，让我们想象一下，你在约会现场是一个巨大的成功。但问题是，你如何将成功转化为更长期的幸福，尤其是你如何决定何时才是安顿下来的合适时间？现在一般来说，只需现金和结婚的第一人谁来，并显示您的任何兴趣是不可取的。但是，同样地，如果你想最大化你长期幸福的机会，你也不想太久。正如我最喜欢的作家简·奥斯汀所说的那样，“一个七岁和二十岁的未婚女子再也不能指望能感受到或激发出爱了。”（笑声）谢谢，简。你知道什么是爱？

So the question is then, how do you know when is the right time to settle down given all the people that you can date in your lifetime? Thankfully, there's a rather delicious bit of mathematics that we can use to help us out here, called optimal stopping theory. So let's imagine then, that you start dating when you're 15 and ideally, you'd like to be married by the time that you're 35. And there's a number of people that you could potentially date across your lifetime, and they'll be at varying levels of goodness. Now the rules are that once you cash in and get married, you can't look ahead to see what you could have had, and equally, you can't go back and change your mind. In my experience at least, I find that typically people don't much like being recalled years after being passed up for somebody else, or that's just me.

因此，问题是，你怎么知道什么时候是适当的时间来解决所有的人，你可以约会在你的一生？值得庆幸的是，有一个相当美味的数学，我们可以用来帮助我们在这里，所谓的最佳停止理论。所以让我们想象一下，当你15岁的时候开始约会，理想的时候，你想在35岁的时候结婚。还有一些人，你可能会在你的一生中约会，他们会在不同程度的善良。现在的规则是，一旦你的现金和结婚，你不能向前看，看看你可以有什么，同样，你不能回去，改变你的想法。至少在我的经历中，我发现典型的人不太喜欢被别人追忆，或者仅仅是我。

So the math says then that what you should do in the first 37 percent of your dating window, you should just reject everybody as serious marriage potential. (Laughter) And then, you should pick the next person that comes along that is better than everybody that you've seen before. So here's the example. Now if you do this, it can be mathematically proven, in fact, that this is the best possible way of maximizing your chances of finding the perfect partner. Now unfortunately, I have to tell you that this method does come with some risks. For instance, imagine if your perfect partner appeared during your first 37 percent. Now, unfortunately, you'd have to reject them. (Laughter) Now, if you're following the maths, I'm afraid no one else comes along that's better than anyone you've seen before, so you have to go on rejecting everyone and die alone. (Laughter) Probably surrounded by cats nibbling at your remains.

因此，数学上说，在你的约会窗口的第一个37%，你应该做的是，你应该拒绝所有人的婚姻潜力。（笑声）然后，你应该挑选下一个比你以前见过的人更好的人。这里有一个例子。现在，如果你这样做，它可以在数学上证明，事实上，这是最好的方式最大化你的机会找到完美的合作伙伴。不幸的是，我不得不告诉你，这种方法确实带来了一些风险。例如，想象一下，如果你的完美伴侣出现在你的第一个37%。现在，不幸的是，你不得不拒绝他们。（笑声）现在，如果你在学数学，恐怕没有人会比你以前见过的任何人都好，所以你必须继续拒绝所有人，独自死去。（笑声）可能被猫在你的遗体上啃咬。

Okay, another risk is, let's imagine, instead, that the first people that you dated in your first 37 percent are just incredibly dull, boring, terrible people. Now, that's okay, because you're in your rejection phase, so thats fine, you can reject them. But then imagine, the next person to come along is just marginally less boring, dull and terrible than everybody that you've seen before. Now, if you are following the maths, I'm afraid you have to marry them and end up in a relationship which is, frankly, suboptimal. Sorry about that. But I do think that there's an opportunity here for Hallmark to cash in on and really cater for this market. A Valentine's Day card like this. (Laughter) "My darling husband, you are marginally less terrible than the first 37 percent of people I dated." It's actually more romantic than I normally manage.

好吧，另一个风险是，让我们想象一下，你第一次约会的人，你的第一个37%是令人难以置信的无聊，无聊，可怕的人。现在，没关系，因为你处在你的拒绝阶段，所以很好，你可以拒绝他们。但是想象一下，下一个跟你一起走的人比你以前见过的每个人都要少一些无聊、无聊和可怕。现在，如果你在学数学，恐怕你必须要和他们结婚，然后在一段关系中结束，坦率地说，这是不理想的。对不起。但我认为有一个机会在这里的Hallmark现金，并真正迎合这个市场。像这样的情人节贺卡（笑声）“我亲爱的丈夫，你比我约会过的37%的人稍微差一点。”实际上它比我通常管理的更浪漫。

Okay, so this method doesn't give you a 100 percent success rate, but there's no other possible strategy that can do any better. And actually, in the wild, there are certain types of fish which follow and employ this exact strategy. So they reject every possible suitor that turns up in the first 37 percent of the mating season, and then they pick the next fish that comes along after that window that's, I don't know, bigger and burlier than all of the fish that they've seen before. I also think that subconsciously, humans, we do sort of do this anyway. We give ourselves a little bit of time to play the field, get a feel for the marketplace or whatever when we're young. And then we only start looking seriously at potential marriage candidates once we hit our mid-to-late 20s. I think this is conclusive proof, if ever it were needed, that everybody's brains are prewired to be just a little bit mathematical.

好的，所以这个方法不会给你100%的成功率，但是没有其他可能的策略可以做得更好。事实上，在野外，有某些种类的鱼遵循和使用这种精确的策略。因此，他们拒绝所有可能的求婚者，在交配季节的前37%，然后他们挑选下一条在那扇窗户之后的鱼，我不知道，比他们以前见过的鱼更大更大。我也认为，在潜意识里，人类，我们无论如何都会这样做。我们给自己一点时间去玩这个领域，在我们年轻的时候有一个市场的感觉。然后，我们才开始认真对待潜在的结婚候选人，一旦我们打到我们的中到20岁左右。我认为这是一个确凿的证据，如果需要的话，每个人的大脑都是天生的，只是一点点数学。

Okay, so that was Top Tip #2. Now, Top Tip #3: How to avoid divorce. Okay, so let's imagine then that you picked your perfect partner and you're settling into a lifelong relationship with them. Now, I like to think that everybody would ideally like to avoid divorce, apart from, I don't know, Piers Morgan's wife, maybe? But it's a sad fact of modern life that 1 in 2 marriages in the States ends in divorce, with the rest of the world not being far behind. Now, you can be forgiven, perhaps for thinking that the arguments that precede a marital breakup are not an ideal candidate for mathematical investigation. For one thing, it's very hard to know what you should be measuring or what you should be quantifying. But this didn't stop a psychologist, John Gottman, who did exactly that. Gottman observed hundreds of couples having a conversation and recorded, well, everything you can think of. So he recorded what was said in the conversation, he recorded their skin conductivity, he recorded their facial expressions, their heart rates, their blood pressure, basically everything apart from whether or not the wife was actually always right, which incidentally she totally is. But what Gottman and his team found was that one of the most important predictors for whether or not a couple is going to get divorced was how positive or negative each partner was being in the conversation.

好吧，那是最高的小费了。现在，顶尖的“三好”：如何避免离婚。好吧，那么让我们想象一下，你选择了你的完美伴侣，你就和他们建立了一种终生的关系。现在，我喜欢认为每个人都希望避免离婚，除了，我不知道，皮尔斯摩根的妻子，也许？但现代生活中的一个可悲事实是，在美国的2次婚姻中，有1次婚姻以离婚告终，而世界上其他国家的婚姻却没有那么遥远。现在，你可以被原谅，也许是因为认为在婚姻破裂之前的争论并不是数学调查的理想人选。一方面，很难知道你应该测量什么，或者你应该量化什么。但这并没有阻止心理学家约翰·戈特曼，他确实做到了这一点。戈特曼观察到数百对夫妇进行了交谈并记录下了所有你能想到的事情。于是他录下了谈话中所说的话，记录了他们的皮肤导电性，他记录下了R面部表情，他们的心率，他们的血压，基本上一切除了妻子是否真的总是正确的，顺便说一句，她完全是。但是Gottman和他的团队发现，一对夫妇是否要离婚的最重要的预测因素之一是双方在谈话中的积极性或消极性。

Now, couples that were very low-risk scored a lot more positive points on Gottman's scale than negative. Whereas bad relationships, by which I mean, probably going to get divorced, they found themselves getting into a spiral of negativity. Now just by using these very simple ideas, Gottman and his group were able to predict whether a given couple was going to get divorced with a 90 percent accuracy. But it wasn't until he teamed up with a mathematician, James Murray, that they really started to understand what causes these negativity spirals and how they occur. And the results that they found I think are just incredibly impressively simple and interesting. So these equations, they predict how the wife or husband is going to respond in their next turn of the conversation, how positive or negative they're going to be. And these equations, they depend on the mood of the person when they're on their own, the mood of the person when they're with their partner, but most importantly, they depend on how much the husband and wife influence one another.

现在，低风险的夫妇在Gottman的比例上得分比负的多。而坏的关系，我的意思是，可能会离婚，他们发现自己陷入了一个螺旋的消极。现在，只要使用这些简单的想法，Gottman和他的小组就能够预测一对夫妇是否会以90%的准确率离婚。但直到他与一位数学家詹姆斯·默里合作，他们才真正开始理解是什么导致了这些负性螺旋，以及它们是如何发生的。他们发现我认为的结果是令人难以置信的简单和有趣。因此，这些方程，他们预测妻子或丈夫将如何回应他们的下一轮对话，他们将如何积极或消极。而这些方程，取决于他们自己的情绪，当他们和伴侣在一起时的心情，但最重要的是，他们丈夫和妻子之间的相互影响到底有多大。

Now, I think it's important to point out at this stage, that these exact equations have also been shown to be perfectly able at describing what happens between two countries in an arms race. (Laughter) So that -- an arguing couple spiraling into negativity and teetering on the brink of divorce -- is actually mathematically equivalent to the beginning of a nuclear war. (Laughter)

现在，我认为在这个阶段指出，这些精确的方程式也被证明能完美地描述两个国家在军备竞赛中所发生的事情。（笑声）这样，一对争吵的夫妻陷入了消极情绪，摇摇欲坠，濒临离婚。-实际上在数学上相当于一场核战争的开始。（笑声）

But the really important term in this equation is the influence that people have on one another, and in particular, something called the negativity threshold. Now, the negativity threshold, you can think of as how annoying the husband can be before the wife starts to get really pissed off, and vice versa. Now, I always thought that good marriages were about compromise and understanding and allowing the person to have the space to be themselves. So I would have thought that perhaps the most successful relationships were ones where there was a really high negativity threshold. Where couples let things go and only brought things up if they really were a big deal. But actually, the mathematics and subsequent findings by the team have shown the exact opposite is true. The best couples, or the most successful couples, are the ones with a really low negativity threshold. These are the couples that don't let anything go unnoticed and allow each other some room to complain. These are the couples that are continually trying to repair their own relationship, that have a much more positive outlook on their marriage. Couples that don't let things go and couples that don't let trivial things end up being a really big deal.

但是这个等式中真正重要的一个词是人们相互之间的影响，特别是所谓的负性阈值。现在，消极的门槛，你可以想到的是，在妻子开始真正生气之前，丈夫是多么的烦人，反之亦然。现在，我一直认为，良好的婚姻是关于妥协和理解，让人有空间自己。所以我想，也许最成功的关系是那些有一个真正的高负门槛的关系。如果情侣们让事情发生，如果他们真的是一个大问题，他们只会把事情提出来。但事实上，该小组的数学和随后的发现显示出了完全相反的事实。最好的夫妇，或最成功的夫妇，都是一个真正的低负门槛。这些夫妇不让任何事情不被注意，让对方有抱怨的余地。这对夫妇他们一直在努力修复自己的关系，对他们的婚姻有更积极的看法。不让事情发生的夫妇和那些不让琐碎事情发生的夫妻最终会成为一个大人物。

Now of course, it takes bit more than just a low negativity threshold and not compromising to have a successful relationship. But I think that it's quite interesting to know that there is really mathematical evidence to say that you should never let the sun go down on your anger.

当然，它需要的不仅仅是一个低的消极的门槛，而不是妥协有一个成功的关系。但我认为，知道有真正的数学证据表明你永远不应该让太阳落在你的愤怒上，这是很有趣的。

So those are my top three tips of how maths can help you with love and relationships. But I hope that aside from their use as tips, they also give you a little bit of insight into the power of mathematics. Because for me, equations and symbols aren't just a thing. They're a voice that speaks out about the incredible richness of nature and the startling simplicity in the patterns that twist and turn and warp and evolve all around us, from how the world works to how we behave. So I hope that perhaps, for just a couple of you, a little bit of insight into the mathematics of love can persuade you to have a little bit more love for mathematics. Thank you. (Applause)

因此，这些是我的三大秘诀：数学如何帮助你的爱情和人际关系。但我希望，除了他们的使用技巧，他们也给你一点洞察到数学的力量。因为对我来说，方程式和符号不只是一件事。他们是一个声音，讲述了大自然的惊人的丰富和令人惊讶的简单的模式，扭曲和扭曲，并在我们周围演变，从世界如何运作，我们的行为。所以我希望，也许，对于你们两个，一点点的了解爱的数学可以说服你对数学有一点点的爱。谢谢。（掌声）