Well, I thought we’d kick off
by doing a little experiment with you, Brady.
Er… I’ve got some weights here.
And I want to see how sensitive your hands are.
Do I mean sensitive hands?
I want you to lift some weights.
This is a workout session.
And I want to see if you can tell the difference
between how heavy different things are.
So I think, can I take the camera?
– You’ll take it as horizon.– Yeah, ok.
It’s fine, Brady. You’re fine, you’re fine.
It’s not the most flattering angle though.
Which one of those is heavier?
Correct, correct answer.
All right, so that one is…
– Can I look?– Yeah.
– 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.
You could tell the difference between 120 and 100.
So you can sense 20-gram difference then.
Hannah is available to film weddings and corporate events.
Is what you’re shooting vaguely usable.
好 那现在开始 现在开始
Okay, and now, okay now.
I feel like that’s heavier.
So that is 200 grams and 220.
And you can’t tell the difference.
– And there is a reason for that.– Okay.
Okay, so what was going on there
is something that Ernst Weber discovered in 1834.
We want to do exactly this experiment.
He noticed that even though you can sense
a small change in between two weights,
when the initial weight is very small.
As that initial weight gets bigger,
you can no longer detect that change.
The just noticeable difference that you need changes,
depending on the size of the weights.
Now, this led him to come up with an equation
which is known as Weber’s law.
That small difference, that tiny change in the intensity,
if you like the feeling, depends…
It’s a ratio really that you’re sensing,
and that ratio is constant.
So although you can detect a 20-gram difference,
when you’ve got 100 grams in your hand,
you can detect it;
when you’re at 200 grams,
you need a much bigger difference
that’s more than its just noticeable difference
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,
you know often in that situation,
where you’re testing different weights,
actually, I think that this Weber’s law
applies to a lot of different situations.
So, you know, if you’re in a really dark room
and someone maybe turns on their iPhone torch.
And you can see it kind of like lights up the entire room.
But when you’re in a really bright room,
if someone flicked on their torch wouldn’t make…
you wouldn’t notice the difference.
And this explains why, as you get older,
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?
Even though a year is the same length always,
the ratio of how long that year is to how long your life has been
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.
So this here is Weber’s law,
but what it means is that the way
that we experience things in life actually follows a logarithm.
So if you think about how something feels, your response to something,
and then you compare that to the sort of intensity
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.
The way that this changes, it takes sort of this shape here.
It’s kind of a logarithmic shape.
So what that means is if you start off down here,
and you’re at a very low weight, okay.
And then you take a really tiny change in the weight,
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.
The difference how that feels is gonna be quite big.
But then if you’re over here,
so now this one will be sort of 100 grams down here.
And let’s say when you’re over here,
you’re up at 200 grams.
–Ah, maybe the y-axis is too small.–That’s right.
Even if you take the same change in weight,
so, I plus delta I.
Let’s make this I1, and I naught.
That same change in stimulus
is gonna feel like almost nothing in your response.
So essentially what this means is
the way that we feel stuff, the way that we perceive stuff in life
doesn’t follow a linear relationship.
It actually follows a logarithmic relationship
which is what this curve here is.
I think that is quite interesting,
because when you come across logarithms first in school
which I think that, you know, when you’re sort of 16 or 17,
they feel really counterintuitive,
they feel like they don’t really make much sense.
But actually this is exactly how you perceive the world.
The exact parameters on this curve change
depending on what you’re talking about.
So there’ll be one curve of light,
there’ll be another curve of sound.
How quickly it goes up and how quickly it bends over.
But the basic mathematical structure of this,
which has been shown time and time again,
in all sorts of different experiments,
is that we perceive things logarithmically.
So there is an amount, a change in stimulus
that you can just about notice.
And we know exactly, because of this equation, where that is,
depending on what you’re talking about.
People who do marketing use this completely to their advantage.
So I swear the Cadbury Dairy Milk over time
has gradually got smaller and smaller and smaller,
and no doubt, it probably has.
But the people who does make these decisions know this equation,
know the way that we perceive things is logarithmic,
and know the exact amount
that they can shrink their chocolate bar by before you notice.
That’s what they’ve done.
Couldn’t I put the mass, the weight on the…
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,
they know that they can creep up the price by a much bigger sum
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,
you’ve got to be a lot more careful.
You can only eke out by a couple of pennies
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
because I was doing some research into how judges make sentences
and how they decide on sentences.
And actually this stuff becomes really important.
It’s not just sort of people trying to make a little bit more money
or noticing different size weights.
Three months in jail is three months in jail.
It doesn’t matter whether you have been in jail for,
you know, three months already
or if you’ve been in jail for thirty years already.
Three months of jail cost the same amount of money through taxpayer,
still depriving someone of their freedom the same amount.
But the thing is that a six-month jail term
feels a lot longer than a three-month jail term.
But a twenty-year and three-month jail term
doesn’t really feel like that much more than twenty-year jail term.
So as a result, there’s this study which looks at
the sentences that people give out,
the judges around the world give out.
And there are these huge gaps in the timelines
that are available to them.
And it’s because of this, it’s because
it doesn’t… it just doesn’t feel like enough of a difference
to give someone a sentence
that’s not one of these kind of preferred numbers.
So you get lots and lots of fine granular sentences down low
and then you just get nice big round numbers like 20 years and 30 years.
Because people think logarithmically.
Thank you for watching now.
In the past, you’ve probably heard me talk about Brilliant,
the site full of puzzles and quizzes and lessons,
all about mathematics and science.
The people who make it are guided by eight principles of learning.
You can read all about them.
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:
If the earth suddenly stopped spinning, mayhem would ensue.
I think that is perhaps a slight understatement, but let’s continue.
What would happen to objects on the surface?
Now quite aside from answering this question,
what I like is clicking here to discuss solutions.
All sorts of people are explaining how they worked out,
and others are commenting on it.
It’s not just about the answer,
I really like seeing how it’s explained.
It’s cultivating my curiosity.
现在 如果你也想浏览Brilliant 就去brilliant.org/numberphile
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so they know you came from here.
There’s loads of free stuff on the site,
but the URL below will give you 20% off
a premium membership with access to even more stuff.
Our thanks to Brilliant for supporting this episode,
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