我们理解为“真理”背后的不确定科学

大家好，我是艾拉·弗拉托，这里是《科学星期五》播客。今天我们要探讨的主题是：真理的不确定性科学。

Hey. It's Ira Flatow, and this is Science Friday. Today on the podcast, the uncertain science of what is truth?

新冠疫情正是这样一种情境，我们某种程度上需要从零开始构建大量知识体系。你无法奢望先观望几年再行动。

COVID was just this situation where a lot of this knowledge we, to some extent, had to build from scratch. You didn't get the benefit just to to wait and see for a few years.

人类最根本的特质之一就是追寻真理。我们可以追溯这段历史：从古希腊人对普遍真理的信仰，到牛顿等早期科学家揭示物理数学的核心原理，再到开国元勋们写下'我们认为这些真理不言而喻——人人生而平等'。如今我们与真理的关系似乎发生了变化。在这个充斥着虚假信息、个人主观事实和人工智能崛起的时代，真理该何去何从？我们又该如何界定真理？今天与我共同探讨的是《证明：确定性的艺术与科学》作者亚当·库查尔斯基。

One of the foundational qualities of the human race is a search for truth. We can trace this through history, from the ancient Greeks and their belief in a universal truth to early scientists like Isaac Newton uncovering core concepts in physics and math, to our founding fathers, writing, we hold these truths to be self evident that all men are created equal. Feels like these days, our relationship to the truth is a little different. In a world of disinformation, people who have their own different set of facts, the rising influence of artificial intelligence, where does the truth fit in, and how do we determine what it is? Joining me to discuss is my guest, Adam Kucharski, author of Proof, The Art and Science of Certainty.

他现居伦敦。欢迎来到《科学星期五》。

He's based in London. Welcome to Science Friday.

感谢邀请。

Thank you for having me.

不必客气。你个人如何看待不确定性？能坦然接受吗？

You're quite welcome. What's your personal relationship with uncertainty? Are you comfortable with it?

我认为自己越来越能适应了。我的数学背景本应属于那种永恒证明的确定性世界，但随着我深入现实数据与实际问题领域，不仅更清楚地认识到探索真相过程中的不确定性，也意识到人们在采取行动时对证据标准的主观判断。

I think it's something I've become increasingly comfortable with. So my my background was in mathematics, which is obviously this world of supposed certainty and and proofs that last forever. But, increasingly, I've moved into the world of real world data, real world problems. And I think that's really made me much more aware not only of the uncertainty we have in trying to work out what's happening, but also the subjectivity that people can have in the level of evidence you need to eventually act on something.

那么这条界限我们该划在哪里呢？

And where do we set that line?

这个标准其实变化很大。统计学中形成了一种可追溯至约百年的传统观念——既不能把门槛设得太高，否则可能忽略有趣或有用的发现；也不能设得太低，让大量错误结论蒙混过关。我们在统计领域看到的趋势是，医学文献普遍采用5%的显著性水平，即研究结果出现如此极端情况的概率应小于5%。

It it varies quite a lot. I mean, one of the things that has kind of become a bit of a tradition in statistics, and it can be traced back about a 100 years, is this idea that, you know, you don't wanna set the bar too high because then things that might be interesting or useful, you you'll ignore, but you don't wanna set it too low and let a lot of things that are false, through. And one of the things we've seen is convergence in statistics, a lot of the medical literature around this this 5% value of you want that, if you're doing a study, to be, a less than 5% chance that you'd you'd get a result as unusual or as extreme as that.

所以这是95%的确定性标准。

So it's it's the 95% certainty.

正是如此。这个标准常见于医学论文、临床试验和实验研究，但已成为硬性分界线。其设定其实相当随意，最初只是为了简化数学计算。

Exactly. Yeah. Yeah. So you see that, you know, in medical papers, in trials, you see that in experiments, but it's become this kind of hard cutoff, and it was it was fairly arbitrary how it's defined. It was actually just because it made the maths a bit easier.

当然，不同问题所需的确定性程度差异很大。二十世纪其他统计学家，比如吉尼斯公司的威廉·戈塞特，就采取了更务实的观点。他们认为本不该存在某个神奇阈值——高于它就确信不疑，低于它就全盘否定。实际应该考量：拟议改变可能带来的收益、潜在成本，以及获取更多证据的难度等因素。

But, of course, the the the type of certainty we need varies a lot with the problem. And other statisticians in in twentieth century, particularly William Gossett, who was working at Guinness, took a far more pragmatic outlook. And they said, well, actually, there there shouldn't be this magic value above which we were convinced and below which we're not because it will depend on, you know, well, what's the benefit of the change we're potentially gonna make? What's the cost it might have? You know, how hard is it to go out and get more evidence?

日常生活中也是如此。对于试错成本低的事情，人们往往愿意基于较弱的证据做出改变；而可能引发重大后果或带来巨大麻烦的决策，人们自然会要求更高的确定性。

I think we see that even in our daily lives, that something that isn't particularly costly for you to experiment with, you're probably happier to make that change with with lower levels of evidence. Whereas if something's gonna be enormously difficult or cause you a lot of headaches if you get it wrong, you're gonna want, a high level of certainty.

但人们总是很难应对不确定性，不是吗？

But people have a lot of difficulty with uncertainty, don't they?

我认为这是人类总爱刻意回避的事情。甚至有一项很好的研究，那是上世纪五十年代CIA内部进行的，当时有位分析师谢尔曼·肯特写了份报告，结论是苏联有可能试图入侵当时的南斯拉夫。但他在与人讨论时发现，每个人对'可能性'这个词的理解都不同，人们会谈论'很可能'和'有可能'。他感到非常沮丧，因为他说实际上人们会千方百计避免对某事做出明确表态。

I think it's something that humans love to go out of their way, to avoid. There's even this this nice study. It was internally at the sea in the CIA in the nineteen fifties, and someone had, Sherman Kent, one of their analysts had had written a report that concluded there was a serious possibility that the the then USSR might try and invade what was then Yugoslavia. And when he was talking to others, he realized that everybody had a different notion of what that term meant, and people would talk about probable and possible. And he he got quite frustrated because he said that actually people will kind of go out of their way to avoid being pinned down on something.

所以我认为其中一个原因是，当世界存在不确定性时，我们不喜欢面对这种情况。而且有些情况可能会非常违反直觉

So I think one of it is if there's uncertainty about the world, we don't like having to kind of deal with that. I think also you can get situations where things can be very counterintuitive

嗯。

Mhmm.

即使是在权衡不同类型错误的方式上，或者如何根据新信息更新信念时，这常常会造成很大压力，因为事情的发展可能不符合我们的预期。无论是试图说服他人，还是像某些医学检测的情况——比如存在少量假阳性而患病率很低时，你对检测结果的解读可能截然不同。你可能会想'检测呈阳性意味着我可能得病了'，但实际上数据组合方式可能会有些反直觉。

Even in terms of how you balance different types of errors or how you might want to have to, say, update your belief based on different information, that can often create a lot of tension because it it might not behave in the way we expect it to. Either way, whether we're trying to, for example, convince others or if in the case of some medical tests, you know, if you have things where there's a small force positive and not many people have a disease, your interpretation of that test result might be very different. You might think, oh, I've got positive test. That means I've probably got the disease, but, actually, the numbers can combine in slightly nonintuitive ways.

没错。你不仅是数学家，还是流行病学家。这让我们有很多话题可聊，特别是新冠疫情最初几年。请谈谈你当时对真相和确定性的理解。

Right. Not only, are you a mathematician, but you're also an epidemiologist. And that that opens up a whole bunch of stuff to talk about, especially the first few years of the COVID pandemic. Tell me how you understood what truth, certainty looked like in those early days.

是的。我认为新冠疫情是这样一个情境：我们某种程度上必须从零开始构建大量知识。虽然可以从其他病原体和其他情况中获取参考，但这次我们没有等待数年观察结果的余地。在这种情境下，不做决定本身就是一种决定。所以我们做了大量工作来建立早期证据，包括疾病严重程度、超级传播者范围、早期变种特性等各个方面。

Yeah. I think I mean, COVID was just this situation where a lot of this knowledge we, to some extent, had to build from scratch. We had examples of it from other pathogens, from other situations, but it was something where you didn't get the benefit just to to wait and see for a few years. I mean, in in that sort of situation, not making a decision is making a decision. So a lot of the work that we did was building up that early evidence around everything from the severity, the extent of super spreading, the characteristics of early variants.

通过大量以证据为基础的政策咨询工作，我深刻认识到：有时虽然存在很大不确定性，但仍能得出有用的结论。以阿尔法变种或德尔塔变种的出现为例，早期很难准确判断其传播性增强了多少——可能是20%，可能是40%，也可能是60%。

And I think one of the things working a lot with evidence to inform policy I became very aware of is that in some cases, might have quite a lot of uncertainty, but you can still say something useful as a conclusion. So to take the emergence of the alpha variant or the the delta variant, it's extremely hard early on to pin down exactly how much more transmissible it is. You know, it might be twenty percent. It might be forty percent. It might be sixty percent.

但实际上，所有这些都得出相同的结论——你将面临一场不断升级的流行病，并陷入困境。因此我认为，这再次体现了将看似充满不确定性的问题转化为可操作认知的过程：至少我们要明确自己站在哪一边，并以对政策制定者仍有价值的方式进行简化。不过，由于这是场影响广泛的公共危机，随着时间推移新认知不断涌现。我们实时目睹了一些因过度自信声明而适得其反的案例，可能是政府不愿承认未知因素所致。但另一方面，传达这种不确定性也很重要，因为形势会不断变化。

But, actually, all of those give you the same conclusion that you're gonna be seeing a rising epidemic, and you're gonna be getting into trouble. And so I think for that, again, it was that conversion between something that might feel like quite an uncertain problem, but actually saying, can we at least know which side of the fence we're on and simplifies in in way that's still useful for politicians. I think also, though, it because it was, you know, it was such a public crisis that affected so many people and, you know, knowledge did emerge over time. I think we saw some examples in real time where there was overconfident statements about certain features of of the pandemic that worked badly, perhaps because governments, you know, didn't want to acknowledge the unknowns. But on the other hand, sometimes communicating that uncertainty was important because the situation was gonna change.

但实际情况如此吗？公职人员是否成功传达了这种不确定性？还是说有更好的传达方式？

Was that, though, the case? Did public officials communicate this uncertainty successfully, or were was there a better way to do this?

我认为有些国家做得稍好，特别是丹麦和新加坡这样的地方表现突出——当政策必须调整时，比如面对新变种，虽然无法确知其具体风险，但需要立即决策并保留调整空间。他们更善于说明'这是现阶段措施，后续可能如此更新'。而其他地方更倾向于宣布'这就是最终政策'。

I think we saw some countries do it, a bit better, particularly I mean, it says place like Denmark and Singapore stood out where your policy was gonna have to change. And if you had, like, an emerging variant, for example, you might not know the exact risk of that. You have to decide what you're gonna do about it, and then you might wanna modify that. And I think they were much better in communicating that this is what we're gonna do at this point in time, and this is what we might update. I think other places were a bit more focused on saying, this is the policy.

这是我们必须采取的措施。而当他们突然转向时，民众就会感到困惑。

This is what we're gonna have to do. And then when they pivot, people get a little bit kind of confused about that.

你认为这是否最终导致部分人对科学失去信任？

Do you think this ultimately led some people to lose trust with science?

我们看到的情况比较复杂。某些指标显示，相比其他领域，科学信任度仍处于高位。但近年来各种因素相互交织：研究表明，人们对科学或公共卫生机构的看法，往往与对司法系统、政府等其他机构的信任相关联。我也从收到科学家愤怒留言的亲身经历中观察到，这与更广泛的社会共识有关。年初我做关于阴谋论的演讲时，那些充满细节——绝非随意评论——且坚信'我们看到了别人看不到的真相'的留言令人震惊，其中还透露出强烈的群体归属感。

We're seeing a bit of a mixed picture because, you know, under certain metrics, trust in science is still high, you know, relative to a lot of other industries. I think what we've seen, though, in recent years is a lot of things kind of intertwined together. I mean, there's there's been useful research in recent years that often these things are not just in isolation, that, you know, your relationship with science or your relationship with public health authorities is gonna be interlinked with your relationship with, you know, other institutions, judicial systems, governments, and so on. I think it also links a bit, and I've I've seen it kind of firsthand a lot from, you people who send scientists angry messages in in kind of your relationship with kind of wider consensus. So I gave a talk on on conspiracy theories early in the year, and it was really striking how many of the comments yeah.

首先这些言论极具细节性，绝非零星随意的评论；其次它们贯穿着'我们洞悉了他人未见的真相'这种强烈信念，几乎带有某种社群认同感。因此我认为，在影响个体与权力机构、权威体系关系方面，还存在许多其他动态因素。

First of all, there's there was a lot of detail. This wasn't just, you know, one off random comments, but it was also it it there was this very strong idea of we're seeing a truth that other people aren't seeing. It's very almost kind of a feeling of community underlying it. So I think there's there's a lot of these other dynamics in terms of how it Right. Influences your relationship with power, institutions, authority.

你们真的是这个掌握隐秘真相的群体吗？我认为除此之外还有很多其他维度，不仅仅是对科学事实的纯粹信任。

Are you actually this community that have this hidden truth? I think there's there's a lot of other dimensions beyond that, just pure trust in a scientific fact.

作为科学家，你对这里真相政治化的现状感到沮丧吗？

Well, you as a scientist, are you frustrated with the politics of truth here?

我认为最具挑战性的是科学证据与政治决策高度纠缠的情况。比如科学被限制或贬低，以追求某种政治或商业目标——这正是我们目睹的现象。回顾烟草行业的历史，就能看到大规模否定科学证据的努力。

I think one of the things that becomes, I think I think, very challenging is where scientific evidence and political choices get very intertwined. And I think, yeah, there there's there's kind of an obvious example of that is where science is either constrained or undermined to to kind of seek a a sort of political or commercial goal. I think that's what what we've seen. Yeah. If you look back at things like smoking, for example, a huge effort to to kind of undermine a lot of the scientific evidence.

这种情况在疫情期间尤为明显。所谓'遵循科学'的口号，很大程度上是政客用来逃避艰难决策的挡箭牌。科学只能提供有限指引，我们始终在思考如何呈现这些本质上都是糟糕选项的现实——声称疫情有简单解决方案的人显然错了，这始终是充满艰难权衡的过程。

I think it also becomes challenging. I think we saw, you know, during COVID, this this follow the science was this this sort of mantra, and I think a lot of it was essentially politicians using it as cover for not having to make very difficult decisions. I mean, the science could only take you so far. And I think one the things that we were always thinking a lot about how to present is, you know, you have essentially a series of bad options. And I think anyone who says that there was a simple solution to the pandemic is wrong, that there was a bunch of very difficult trade offs.

有趣的是，即便回顾1918年大流感，当时的报纸言论与2020年如出一辙。本质上是关于如何差异化评估社会各领域的争论，这本该是政治家的决策范畴，需要全社会共同权衡。但越来越多人误以为这是科学能做出的选择——实际上科学根本无法全面衡量这些社会维度。

And, actually, even if you look back to the nineteen eighteen pandemic, it's fascinating how many of those newspaper quotes you could have just pulled out of 2020. You know, it's just arguing about which bits of society should be valued in different ways. And then it's very much on politicians to to make those decisions and, yeah, as a society to to weigh those things up. And I think increasingly a lot of that has kind of got embedded more on that was a scientific choice when actually, you know, science can't weigh up all of those features of society in in those kind of ways. And it yeah.

在我看来，科学不应承担这种责任。流行病学只是为重大决策提供参考的一个维度。

In my view, I don't think it should. I think it's something like epidemiology is one thread contributing to that that wider, very difficult decision.

这种现象不仅存在于医学领域，在环境问题上同样明显——比如否认气候变化和气候危机的态度，就像在说'别往那边看'，试图假装问题不存在。

Well, we're seeing that not only, medically, but we're seeing that with the environmental situation with the denial of climate change and the climate crisis, like, oh, don't don't look over there. You know? Let's just ignore that and do away with it like it doesn't exist.

是的。而且分歧的程度也令人震惊。我认为，以气候变化为例，健康的辩论应基于我们所面临情况的记录证据。我们有证据表明现状，但对于应对措施存在更多不确定性，然后是关于应采取何种政策的决策。这正是令人震惊的一点。

Yeah. And it's also just really striking that the kind of the levels that the the disagreements have. And I think if you look at something like climate change, yeah, I think the healthy debate is where you you have the documented evidence about the situation we're facing. You have the evidence, which has a lot more uncertainty about what we might do about it, and then you have the policy decision about what we should do about it. And I think that was one of the things that was striking.

我为这本书采访了许多气候科学家，发现对于正在发生什么有很强的科学共识。但对于各种政策杠杆的具体效果及应对措施则共识较少。但正如你所说，人们现在正回归那些基本原理。由于可能不喜欢某些必要的改变，他们选择从根本上质疑。我们在其他干预措施中也看到类似现象，比如对于COVID，有些人反对疫苗强制令，认为这侵犯了自由。

I talked to quite a lot of climate scientists for the book that there is very good scientific consensus on the kind of the what is happening. There's less consensus on of all these policy levers exactly which one's gonna have which effect, what should we be doing about it. But I think what's what's been happening, as you say, is that people have kind of gone back to those those fundamentals. And rather than having debates and perhaps because they don't like some of those changes that can be required, they go down more fundamentally. And we see that with other interventions as well that I think, understandably, for COVID, there were some people who were not making up things like vaccine mandates because it was seen as infringement as freedom.

但与其讨论政策本身，争论退回到了更基础的层面——比如声称疫苗无效，或病原体本身根本不严重。我认为这本质上是对政策的不满导致人们转而攻击更底层的系统，尽管无论是COVID的严重性还是气候变化带来的危害程度，这些基础证据都非常坚实。

But rather than just talking about the policy, it went back to, okay. Well, vaccines don't work or, you know, claims that the pathogen itself isn't very severe at all. And I think, ultimately, it's it's dislike for the policy, which then you get people kind of trying to target the system further down instead, even though for something you know, whether we're talking about the severity of something like COVID or the extent of hazard that climate change is facing, that foundational evidence is very strong.

我们需要稍作休息，回来后我们将探讨技术如何模糊我们看待真相的视角。

We have to take a break and when we come back, how technology is fogging the lens through which we see truth.

在科学熏陶下成长的人总认为存在一个优雅的理论可以理解。即便作为数学家，我也想拿出纸笔推演——但那个时代已经过去了。

Growing up with science, you have this idea that there's a really elegant theory and you can kind of understand it. I think even the mathematician in me, you know, you want to get pen and paper out, and that that era has kind of moved on.

请继续收听。关于科学和医学，有许多我们不了解真相的简单事物，比如麻醉——我们知道它有效，但不知其原理。

Stay with us. There are a lot of things about science and medicine that we don't know the truth behind. Simple things like anesthesia. We know it works. We don't know how.

某些药物也是如此。在物理学领域，我们知道量子物理有效，但不知原因。著名诺贝尔奖得主理查德·费曼说过：如果有科学家声称知道原理，那是在说谎。既然无法验证'为什么'，我们是否还应该视其为真理？

Same goes for some medicines. And in physics, we know quantum physics works, but not why. Richard Feitman, a very well known and respected Nobel Prize scientist said, if any scientist tells you why, they know why it happens, they're lying. So we know it works, but we don't know why. I mean, should we still consider these things the truth even if we can't verify the why?

我认为这是一个非常好的观点，它实际上触及了我们与技术互动中许多紧张关系的核心，尤其是在现代，拥有对某事物运作的信心与理解其为何运作之间的重要性。这也真正触及了现代科学的本质。正如你所说，医学领域存在我们并不完全理解所有基础过程的情况，但我们有信心采取某种措施会得到相应效果。即使进行临床试验，它通常能告诉你某事物是否具有你测试的效果，但不一定能解释原因，你需要从其他来源获取。我认为，在其他领域，比如人工智能，人们对无法解释事故的自动驾驶汽车感到更加不安。

I think that's a really good point, and it's I think it gets to the heart actually a lot of tension with our interactions with with kind of technology and particularly in the modern era, the extent to which it's important to have confidence something works versus that understanding of why something works. It also really just gets the heart of what is science in the modern era. As you said, there's elements of medicine where we don't understand exactly all the underlying processes, but we have confidence if we do this, we'll get this effect. And even if you run a clinical trial, it will tell you with your often good confidence whether or not something has the effect you're testing, but it won't necessarily give you that why you need to get that from other sources. And I think, you know, in other fields and things like AI, for example, people are much more uncomfortable with self driving cars that that have accidents that we can't explain.

因此，即使自动驾驶汽车平均比人类安全得多——老实说，这并不难，因为人类在驾驶时会做出许多奇怪且无益的事情——我认为人们仍会对这种理解的缺失感到不适。我们也在与从事大量人工智能发现的科学家交流中看到这一点，比如某些蛋白质预测工作。我成长于科学环境，曾认为存在非常优雅的理论，可以理解它。即使作为数学家，我也想拿出纸笔解决它，或进行具体实验。例如，亚伯拉罕·林肯自学了所有希腊数学证明，因为他想更擅长论证。那个时代已经过去，我们现在进入了一个在数学和人工智能发现中，许多证明涉及计算机，难以手工验证，或科学中缺乏简单解释的时代，但我们可以拥有极其强大的预测，仍需接受这就是科学。

So even if a self driving car was on average much safer than a human, which isn't, let's be honest, massively difficult because humans, you know, do do a lot of very strange unhelpful things when they're driving, I think there would still be that discomfort about the that that that kind of lack of understanding. I think we've also seen that you're talking to scientists who've worked a lot of AI discoveries, so some of the kind of protein prediction work, where I think you know, and I've certainly had it myself that growing up with science, you have this idea that there's a really elegant theory, and you can kind of understand it. I think even the the mathematician in me, you know, you wanna get pen and paper out and and solve it, or you wanna, you know, be able to do the tangible experiment. Abraham Lincoln, for example, taught himself all the the Greek mathematical proofs he because wanted to get better at demonstrating things. And that that era has kinda moved on, and we're very much going to an era both in maths, a lot of the proofs that involve computers now and and can't really be easily verified by hand or in AI discovery and science where there isn't that simple explanation, but we can have predictions that can be enormously powerful and still having to come to terms with, well, this is science.

而且，正如我们拥有这些非常有价值的医疗工具一样，我们可以拥有我们有信心的科学发现，但或许我们失去了一些关于解释优雅性的浪漫主义。

And, you know, we have just as we have these medical tools that are very valuable, we can have scientific discoveries we can have confidence in, but we lose perhaps a little bit of that kind of romanticism around the elegance of our explanations.

对。那么，随着我们迈向这个技术未来，我们能从前辈那里吸取什么教训吗？

Right. Well, as we go forward into this technological future, are there any lessons we can take from our predecessors?

我认为对我来说最突出的一点是假设确定性的危险。近年来我们看到这种假设，即某些算法已达到超人类水平。例如，在理论上非常适合人工智能掌握的游戏领域，实际上人们发现并非如此。我认为这与大约一百五十年前关于物质已解决一切的假设有类比。

I think one of the things that that really stands out for me is the dangers of assuming certainty. I think we've seen it in recent years with this assumption that certain algorithms have reached superhuman status. For example, you know, in in games which are are very well defined should, in theory, be a perfect starting point for AI to reach that kind of mastery. But, actually, people who poked around have found that that's not true. And I think there's an analogy there even going back about a hundred and fifty years with that assumed idea that matter had solved it all.

我们曾拥有这些普遍真理，但人们开始发现不成立的例子。我认为还需要意识到我们必须平衡的那些方面。即使在关于错误信息的讨论中，我们非常关注不希望人们相信虚假信息。但处理信息时可能犯两种错误：一是相信虚假的事物，二是不相信真实的事物。

We had these universal truths, and people start to poke around and find examples where that doesn't work. I think also being really aware of those those balances we've got to strike. I mean, even in discussion of misinformation, we we focus a lot on not wanting people to believe falsehoods. But there's two errors you can make with information. One is believing things that are false, but another is not believing things that are true.

因此，越来越多的人意识到，仅针对虚假信息的干预可能有效，但这是通过减少对所有信息的信任实现的。我们需要关注的是，对于许多问题，实际上在许多情况下存在两种错误需要平衡，我们不希望无意中干预一种错误，同时削弱我们对其他信息系统的信任。

And so there's, I think, increasing awareness that interventions that focus just on false information might work, but they do it by just reducing belief in all information. So we need to focus on the fact that for many of these problems, there's actually two errors in many cases that we need to be balancing, and we don't inadvertently want to intervene on one but also undermine our trust in other in information systems along the way.

好吧，亚当，我们谈论了这么多话题，时间已经不够了。非常感谢你今天抽空参与我们的节目。

Well, Adam, we have run out of time talking about so many things. Want to thank you for taking time to be with us today.

是的，聊得很愉快。谢谢。

Yeah. Great to chat. Thank you.

亚当·库查尔斯基，《证明》一书的作者，这是一本非常非常优秀的书，适合夏日阅读。《证明》，关于确定性艺术与科学的著作。他现居伦敦。你可以阅读这本书的节选，请访问我们的网站sciencefriday.com/proof。

Adam Kucharski, author of Proof, a really, really good book, a good read for the summer. Proof, the art and science of certainty. He's based in London. And you can read an excerpt from the book. Head over to our website, sciencefriday.com/proof.

嘿，感谢收听。本期节目由凯瑟琳·戴维斯制作。下次见。我是艾拉·弗拉托。

Hey. Thanks for listening. This episode was produced by Kathleen Davis. See you next time. I'm Ira Flatow.