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韦伯定律及其应用

Weber's Law - Numberphile

好 开始吧
Well, I thought we’d kick off
Brady 我们先在你身上做个小实验
by doing a little experiment with you, Brady.
嗯 我这里有一些秤砣
Er… I’ve got some weights here.
我想要测试一下 你的手有多敏感
And I want to see how sensitive your hands are.
我刚刚用了敏感这个词么
Do I mean sensitive hands?
我想让你拿一些秤砣
I want you to lift some weights.
先练习一下
This is a workout session.
通过这个实验 我可以知道你可以区分出
And I want to see if you can tell the difference
多大差别的重量
between how heavy different things are.
嗯 我可以拿着摄像机么
So I think, can I take the camera?
–你可以把这个当水平线–嗯 好的
– You’ll take it as horizon.– Yeah, ok.
Brady 你看起来不错
It’s fine, Brady. You’re fine, you’re fine.
虽然不是最完美的拍摄角度
It’s not the most flattering angle though.
这两个哪个更重一些
Which one of those is heavier?
这个
That one.
恭喜你 答对啦
Correct, correct answer.
好 这个是
All right, so that one is…
–我可以看一下么–当然
– Can I look?– Yeah.
120克
120 grams.
–好–你另一只手里 是100克
– Okay. – And in your other hand, you have 100 grams .
–好–好的 现在 我要加大难度了
– Okay.– Okay, now, I’m gonna make it slightly harder.
刚刚说明 你可以区分120克和100克
You could tell the difference between 120 and 100.
所以 你可以区分出20克的不同
So you can sense 20-gram difference then.
(汉娜去拍婚礼 处理公司的事儿了)
Hannah is available to film weddings and corporate events.
你现在拍摄这么模糊 可以用么
Is what you’re shooting vaguely usable.
不好意思
I’m sorry.

Okay.
好 那现在开始 现在开始
Okay, and now, okay now.
我感觉这个更重一些
I feel like that’s heavier.
啊奥 错啦
Incorrect.
这个是200克 这个是220克
So that is 200 grams and 220.
你没有分出这两个的不同
And you can’t tell the difference.
–这是情有可原的–好吧
– And there is a reason for that.– Okay.
好 那刚刚我们做的呢
Okay, so what was going on there
就是重演韦伯在1834年的发现
is something that Ernst Weber discovered in 1834.
我们想要做的就是这样一个实验
We want to do exactly this experiment.
在实验中他发现
He noticed that even though you can sense
即使当秤砣本身很轻的时候
a small change in between two weights,
你可以发现它们重量的差别
when the initial weight is very small.
但是当秤砣本身越来越重
As that initial weight gets bigger,
你就不能再识别这种差异了
you can no longer detect that change.
你的“最小可觉重量差”会根据
The just noticeable difference that you need changes,
秤砣本身的重量而变化
depending on the size of the weights.
这样 他从中得出一个公式
Now, this led him to come up with an equation
就是我们熟知的 韦伯定律
which is known as Weber’s law.
很小的不同 非常微小的变化
That small difference, that tiny change in the intensity,
是否可以被你察觉到 取决于(本身的重量)
if you like the feeling, depends…
这就是你感觉到的比率
It’s a ratio really that you’re sensing,
这个比率是个常数k
and that ratio is constant.
所以 虽然你可以识别出20克的不同
So although you can detect a 20-gram difference,
当你手里砝码是100克时
when you’ve got 100 grams in your hand,
你可以识别这个不同
you can detect it;
但是当砝码变成200克
when you’re at 200 grams,
你就更需要一个更大的区别
you need a much bigger difference
比刚刚那个最小可觉重量差要大
that’s more than its just noticeable difference
才能让你识别到
before you can detect it.
嗯 所以 虽然这个实验只是关于重量的
And the thing is… is that although in weights,
就像我们刚刚那个
you know often in that situation,
识别不同重量的实验一样
where you’re testing different weights,
事实上 我认为韦伯定律
actually, I think that this Weber’s law
适用于更广泛的情境
applies to a lot of different situations.
比如 假设你在一个小黑屋里
So, you know, if you’re in a really dark room
即使有人只是打开他的iphone手电筒
and someone maybe turns on their iPhone torch.
你会感觉 他照亮了整个房间
And you can see it kind of like lights up the entire room.
但是 你要是身处很亮的屋子
But when you’re in a really bright room,
有人打开他们的iphone手电
if someone flicked on their torch wouldn’t make…
你感觉不到丝毫不同
you wouldn’t notice the difference.
这也解释了为什么 随着年龄增大
And this explains why, as you get older,
你会觉得时光飞逝
years seem as though they’re going faster.
随着你长大 时间确实飞速流失 对么
Time kind of speeds up, as you get older, right?
即使一年的长度总是一样的
Even though a year is the same length always,
但是一年的长度/你的年龄 这个比值
the ratio of how long that year is to how long your life has been
却越来越小 越来越小
ends up getting smaller and smaller and smaller.
所以 随着年龄不同 感受也各异
So, how it feels changes over time.
这就是韦伯定律
So this here is Weber’s law,
但是 从中我们可以推论
but what it means is that the way
我们生活中对事物的体验 遵循对数定律
that we experience things in life actually follows a logarithm.
所以 当你想知道你对一些事情的体验或反应
So if you think about how something feels, your response to something,
你可以把它和那种你的感觉密度
and then you compare that to the sort of intensity
进行比较
of whatever it is your feelings.
这种感觉 可以是光 声音 或者砝码
So this could be light, could be sound, could be weights.
你的感受曲线是这个形状的
The way that this changes, it takes sort of this shape here.
即对数函数
It’s kind of a logarithmic shape.
这就意味着 如果你在这里开始
So what that means is if you start off down here,
你手里是很轻的砝码
and you’re at a very low weight, okay.
然后你改变一点点重量
And then you take a really tiny change in the weight,
比如增加20克
so this 20 gram change.
这就是 Ⅰ 这就是Ⅰ+Δ1
So this is gonna be I, this would be I plus delta I.
那你感觉到的不同就非常大了
The difference how that feels is gonna be quite big.
但如果你起点是在这儿
But then if you’re over here,
那这里是100克
so now this one will be sort of 100 grams down here.
假设你是在这儿
And let’s say when you’re over here,
砝码变成了200克
you’re up at 200 grams.
–哦 可能是y轴画的太短了–是的
–Ah, maybe the y-axis is too small.–That’s right.
即使你增加同样重量的砝码
Even if you take the same change in weight,
比如都是ΔⅠ
so, I plus delta I.
假设这是Ⅰ¹ 这是Ⅰº
Let’s make this I1, and I naught.
同样的刺激增量
That same change in stimulus
在你这里几乎感觉不到
is gonna feel like almost nothing in your response.
所以 这说明
So essentially what this means is
我们感知 理解生活中事物的方式
the way that we feel stuff, the way that we perceive stuff in life
并不是直线变化的
doesn’t follow a linear relationship.
实际上是对数型关系
It actually follows a logarithmic relationship
就像这条曲线一样
which is what this curve here is.
我觉得这个很有趣
I think that is quite interesting,
因为 学校里学对数函数时
because when you come across logarithms first in school
大概在我们16 17岁左右
which I think that, you know, when you’re sort of 16 or 17,
这些函数很难以理解
they feel really counterintuitive,
而且感觉也没什么用
they feel like they don’t really make much sense.
但是当我们理解 感知这个世界时
But actually this is exactly how you perceive the world.
这个曲线上的变量会跟着
The exact parameters on this curve change
我们讨论的东西而变化
depending on what you’re talking about.
会有光的对数函数曲线
So there’ll be one curve of light,
声音的对数函数曲线
there’ll be another curve of sound.
这些图像上升下降得好快啊
How quickly it goes up and how quickly it bends over.
但是这些我们在实验中
But the basic mathematical structure of this,
我们一遍又一遍展示的
which has been shown time and time again,
函数图像背后的数学结构
in all sorts of different experiments,
正说明 我们用对数的方式在感知世界
is that we perceive things logarithmically.
已知一个总量 一个你刚好可以感知的
So there is an amount, a change in stimulus
刺激增量
that you can just about notice.
我们就可以通过公式算出 这个增量是多少
And we know exactly, because of this equation, where that is,
结合我们所提到的不同刺激物的性质
depending on what you’re talking about.
销售人员就经常利用这个原理
People who do marketing use this completely to their advantage.
我打赌 Cadbury牛奶棒
So I swear the Cadbury Dairy Milk over time
随时间流逝 会越来越小
has gradually got smaller and smaller and smaller,
毫无疑问 它可能已经在变小了
and no doubt, it probably has.
那些决策者知道这个公式
But the people who does make these decisions know this equation,
知道我们是用对数方式在感知世界
know the way that we perceive things is logarithmic,
他们甚至知道
and know the exact amount
把巧克力棒缩小多少而不让你察觉
that they can shrink their chocolate bar by before you notice.
他们确实这么做了
That’s what they’ve done.
我能不能把这些砝码量一下
Couldn’t I put the mass, the weight on the…
没有人可以帮你检查这些东西
You have no one check those things.
或许 商人们也会 比如对于一个奢侈品
Or they also… a really expensive items,
他们知道 他们可以悄悄提高价格
they know that they can creep up the price by a much bigger sum
甚至你都没注意到价格变化
before you notice that the price has changed.
但是 那些超级便宜的东西 你懂的
Whereas things like… things that are really cheap, you know,
像牛奶 鸡蛋 或者什么的
buying sort of pints of milk, or eggs, whatever,
提价就没那么容易
you’ve got to be a lot more careful.
你只可以稍微提高几便士
You can only eke out by a couple of pennies
别人才会注意不到
before people start to notice.
嗯 我解释这么多 主要是因为
Well, so the reason why I came across this was
我正在做一个关于法官如何审案
because I was doing some research into how judges make sentences
如何判刑的实验
and how they decide on sentences.
事实上 这个研究非常重要
And actually this stuff becomes really important.
并不只是关于人们如何巧妙涨价
It’s not just sort of people trying to make a little bit more money
或者区分砝码重量的问题
or noticing different size weights.
三个月的监狱是实打实的刑罚
Three months in jail is three months in jail.
并不是关乎于 你已经在监狱
It doesn’t matter whether you have been in jail for,
待了 三个月
you know, three months already
或者待了三十年的问题
or if you’ve been in jail for thirty years already.
三个月监狱的刑罚 花费了纳税人同样的钱
Three months of jail cost the same amount of money through taxpayer,
还剥夺了一个人三个月的自由
still depriving someone of their freedom the same amount.
但是关键在于 六个月的刑期
But the thing is that a six-month jail term
比三个月的刑期感觉上长了不止一倍
feels a lot longer than a three-month jail term.
但是二十年零三个月的刑期
But a twenty-year and three-month jail term
并不比二十年听起来长很多
doesn’t really feel like that much more than twenty-year jail term.
于是 就有了我们这个
So as a result, there’s this study which looks at
研究人们受了多少刑罚
the sentences that people give out,
法官判了多重刑罚的实验
the judges around the world give out.
其实判刑的刑期可以
And there are these huge gaps in the timelines
有很多选择
that are available to them.
但因为对数定律
And it’s because of this, it’s because
法官在给一个犯人判刑时
it doesn’t… it just doesn’t feel like enough of a difference
判一些不同寻常的刑期
to give someone a sentence
并不会让人明显感到差异
that’s not one of these kind of preferred numbers.
所以我们才会经常看到刑期四舍五入
So you get lots and lots of fine granular sentences down low
变成了我们常常听到的20年或30年
and then you just get nice big round numbers like 20 years and 30 years.
因为人们的思考遵循对数规律
Because people think logarithmically.
谢谢你看这个视频
Thank you for watching now.
之前你可能听过我讨论过 Brilliant
In the past, you’ve probably heard me talk about Brilliant,
一个充满了谜语 难题和课程的节目
the site full of puzzles and quizzes and lessons,
都是数学 科学等等
all about mathematics and science.
那些成功人士都遵循八个学习原则
The people who make it are guided by eight principles of learning.
你可以读一读这些原则
You can read all about them.
其中我提到的一个就是激发兴趣
Cultivating curiosity is one I relate to.
例如 Brilliant里 有这样一个问题
For example, here’s a problem from Brilliant:
如果地球停止运转 灾难将接踵而至
If the earth suddenly stopped spinning, mayhem would ensue.
我认为这可能有点儿保守 我们继续
I think that is perhaps a slight understatement, but let’s continue.
地球表面的生物会怎样呢
What would happen to objects on the surface?
除了想这个问题的答案
Now quite aside from answering this question,
我希望大家点这个键 讨论解决方案
what I like is clicking here to discuss solutions.
好多人都在提出他们的解决方案
All sorts of people are explaining how they worked out,
其他人在评论
and others are commenting on it.
重要的不只是答案
It’s not just about the answer,
我对大家如何解释更感兴趣
I really like seeing how it’s explained.
这些解释激发了我的好奇心
It’s cultivating my curiosity.
现在 如果你也想浏览Brilliant 就去brilliant.org/numberphile
Now if you’d like to check out Brilliant, go to brilliant.org/numberphile,
他们就会知道你们从这儿过去的
so they know you came from here.
这里有大量免费资料
There’s loads of free stuff on the site,
但是下面的URL会给你打八折
but the URL below will give you 20% off
如果是会员 会有更多福利
a premium membership with access to even more stuff.
感谢Brilliant 给我们的支持
Our thanks to Brilliant for supporting this episode,
下期再见
and we’ll see you again soon.

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视频概述

见一枚好看的小姐姐;听一段纯正的英音;学一点数学;了解一个生活中的正宗心理学知识!

听录译者

jm

翻译译者

德克萨斯土豆丝

审核员

审核员YX

视频来源

https://www.youtube.com/watch?v=hHG8io5qIU8

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