Chapter 1: What is the main motivation behind Adam Kucharski's book on certainty?
Hello, it's Eric Topol from Ground Truths, and I am really delighted to welcome Adam Kucharski, who is the author of a new book, Proof, a distinguished mathematician, by the way, the first mathematician we've had on Ground Truths, and a person who I had the real privilege of getting to know a bit through the COVID pandemic. So welcome, Adam. Thanks for having me.
Yeah, I mean, I think just to let everybody know, you're a professor at London School of Hygiene and Tropical Medicine. And also noteworthy, you won the Adams Prize, which is one of the most impressive recognitions in the field of mathematics. This is the book. It's a winner proof. And there's so much to talk about.
So Adam, maybe what I'd start off is the quote in the book that kind of captivates in the beginning. Life is full of situations that can reveal remarkably large gaps in our understanding of what is true and why it's true. This is a book about these gaps. So what was the motivation when you undertook this very big endeavor?
I mean, I think a lot of it comes to the sort of work I do in my day job where we have to deal with a lot of evidence under pressure, particularly if you work in outbreaks or emerging health concerns. And often it really pushes to the limits our methodology and how we converge on what's true subject to potential revision in the future.
I think particularly having a background in maths, I think you kind of grow up
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Chapter 2: How does the Monty Hall problem illustrate our understanding of probability?
with this idea that you can get to these concrete, almost immovable truths. And then even just looking through the history, realizing that often isn't the case, that there's these kind of very human dynamics that play out around them. And it's something I think that everyone in science can reflect on, that sometimes what convinces us doesn't convince other people.
And particularly when you have that kind of urgency of time pressure, working out how to navigate that.
Yeah, well, I mean, I think these times, of course, have really gotten us to appreciate, particularly during COVID, the importance of understanding uncertainty. And I think one of the ways that we can dispel what people assume they know is the famous Monty Hall, which you get into a bit in the book. So I think everybody here is familiar with the show Let's Make a Deal.
And maybe you could just take us through what happens with door numbers with one of the doors are unveiled and how that changes the mathematics.
Yeah, sure. So I think it's a problem that's been around for a while and it's based on this game show. So you've got three doors that are closed. Behind two of the doors are a goat and behind one of the doors is a luxury car. So obviously you want to win the car. The host asks you to pick a door. So you point to one, maybe door number two.
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Chapter 3: What are mathematical monsters and their significance in understanding proof?
Then the host who knows what's behind the doors opens another door to reveal a goat and then ask you, do you want to change your mind? Do you want to switch doors? And a lot of the, I think, intuition people have, and certainly when I first came across this problem many years ago, is, well, you've got two doors left, right? You know, you've picked one, there's another one, it's 50-50.
And even some quite well-respected mathematicians, people like Paul Erdosch, who has really published more papers than almost anyone else, That was their initial gut reaction.
But if you work through all of the combinations, if you pick this door and then the host does this and you switch or not switch and work through all of those options, you actually double your chances if you switch versus sticking with the door. So it's something that's counterintuitive. But I think one of the things that really struck me, and even over the years trying to explain it, is
convincing myself of the answer which was when i first came across it as a teenager i did quite quickly is very different to convincing someone else and even actually paul erdos one of his colleagues kind of showed him the what i'd call proof by exhaustion so go through every combination and that didn't really convince him so then he started to simulate and said let's do a computer simulation of the game a hundred thousand times and again you know switching was this optimal strategy but erdos wasn't um
wasn't really convinced because I accept that this is the case but I'm not really satisfied with it and I think that encapsulates for a lot of people their experience of proof and evidence it's kind of it's a fact and you kind of have to take it as given but there's actually quite a big bridge often to really understanding why it's true and feeling convinced by it.
Yeah, I mean, I think it's a fabulous example because I think everyone would naturally assume it's 50-50 and it isn't. And I think that gets us to the topic at hand. There's many things I love about this book. One is that you don't just get into science and medicine.
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Chapter 4: What lessons about proof and truth emerged from the COVID-19 pandemic?
but you cut across all the domains, law, mathematics, AI. So it's a very comprehensive sweep of everything about proof and truth. And it couldn't come at a better time, as we'll get into. Maybe just starting off with math, the term I love, mathematical monsters. Can you tell us a little bit more about that?
Yeah, this was a fascinating situation that emerged in the late 19th century where a lot of maths, certainly in Europe, had been derived from geometry because of a lot of the ancient Greek influence on how we shape things.
And then Newton and his work on rates of change and calculus, it was really the natural world that provided a lot of inspiration, these kind of tangible objects, tangible movements. And as mathematicians started to build out the theory around rates of change and and how we tackle these kinds of situations, they sometimes took that intuition a bit too seriously.
And there was some theorems that they said were intuitively obvious, some of these French mathematicians. And so one, for example, is this idea of kind of how things change smoothly over time and how you do those calculations. But what happened was some mathematicians came along and showed that when you have things that can be infinitely small,
that intuition didn't necessarily hold in the same way. They came up with these examples that broke a lot of these theorems. A lot of the establishment at the time called these things monsters.
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Chapter 5: How do different types of evidence rank in medical research?
They called them these aberrations against common sense and this idea that if Newton had known about them, he never would have done all of his discovery because they're just nuisances and we just need to get rid of them. There's this real tension at the core of mathematics in the late 1800s where Some people just wanted to disregard this and say, look, it works for most of the time.
That's kind of good enough. And then others really weren't happy with this quite vague logic. They wanted to put it on a much sturdier ground. And what was remarkable, actually, is if you trace this then into the 20th century, a lot of these monsters and these, particularly in some cases, functions which...
you know, could almost kind of move constantly, this kind of constant motion, rather than our intuitive concept of movement as something that's smooth. You know, if you drop an apple, it kind of accelerates a very smooth rate, would become foundational in our understanding of things like probability, Einstein's work on atomic theory.
A lot of these concepts where geometry breaks down would be really important in relativity. So actually, These things that we thought were monsters actually were all around us all the time and science couldn't advance without them.
So I think it's just this remarkable example of this tension within a field that's supposedly concrete and the things that were going to be shunned actually turned out to be quite important.
You know, it's great how you convey how nature isn't so neat and tidy and things like brownie emotion, understanding that. I mean, just so many things that I think fit into that general category.
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Chapter 6: What role do natural experiments play in evaluating health interventions?
In the legal, we won't get into too much because, you know, that's not the... so much the audience of Ground Truths, but the classic things about innocent until proven guilty and proof beyond reasonable doubt. I mean, these are obviously really important parts of that overall sense of proof and truth. We're going to get into one thing I'm fascinated about related to that subsequently.
And then, you know, in science, so before we get into the different types of proof, obviously the pandemic is still fresh in our minds and we're endemic of COVID now. And there are so many things we got wrong along the way of uncertainty and didn't convey that science isn't always evolving correctly.
search for what is the truth uh there's plenty no shortage of uncertainty at any moment so can you kind of recap some of the you did so much work during the pandemic and obviously some of it's in the book um what were some of the major things that you took out of proof and truth from the pandemic i think it was it was almost this the story of two hearts because on on the one hand
science was the thing that got us where we are today. You know, the reason that so much normality could resume and so much risk was reduced was development of vaccines and the understanding of treatments and the understanding of variants as they came to the next characteristic. So it was kind of this amazing opportunity to see this happen faster than it ever happened in history.
And I think ever in science, it certainly shifted a lot of my thinking about what's possible and even how we should think about these kinds of problems. But also on the other hand, I think where people might have been more familiar with seeing science kind of progress a bit more slowly and reach consensus around somebody's health issues, having that emerge very rapidly can present challenges.
And even we found with some of the work we did on alpha and then the delta variants, and it was the early quantification of these. So really the big question is, is this thing more transmissible? Because at the time, countries were thinking about control measures, thinking about relaxing things.
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Chapter 7: Why is understanding causation versus correlation crucial in research?
And you've got this just enormous social, economic, health decision-making based around, essentially, is it a lot more spreadable or is it not? And you only had these fragments of evidence. So I think for me, that was really an illustration of the sharp end. And I think what we ended up doing with some of those was rather than arguing over a precise number, something like delta,
Instead, we kind of looked at, well, what's the range that matters? So in the sense of arguing over whether it's 40% or 50% or 30% more transmissible is perhaps less important than being it's substantially more transmissible and it's going to start going up. Is it going to go up extremely fast or just very fast? That's still a very useful conclusion.
I think what often created some of the more challenges, I think the things that kind of on reflection people looking back,
pick up on are where there was probably you know overstated certainty we saw that around some of the airborne spread for example you know stated as a fact by in some cases some organizations I think in you know in some situations as well governments had a constraint and presented it as scientific you know so the UK for example would say testing isn't useful
And what was happening at the time was there wasn't enough tests. So it was more a case of they can't test at that volume. But I think blowing between what the science was saying and what the decision making. And I think also one thing we found in the UK was we made a lot of the epidemiological evidence available. I think that was really, I think, something that was important.
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Chapter 8: How can we better engage with differing views on scientific evidence?
I found it a lot easier to communicate if talking to the media, to be able to say, look, this is the paper that's out. This is what it means. This is the evidence. I always find it quite uncomfortable having to communicate things where there was, you know, you knew there were reports behind the scenes, but you couldn't actually articulate.
But I think what that did is it created this impression that particularly epidemiology was driving the decision making. lot more than it perhaps was in reality because so much of that was being made public and a lot more of the evidence around education or economics was being done behind the scenes. I think that created this kind of asymmetry in public perception about how that was feeding in.
It's really hard as well as a scientist when you've got journalists asking you how to run the country to work out those steps of, am I describing
evidence behind what we're seeing am i describing the evidence about different interventions or am i proposing you know to some extent my value system on what we do and i think all of that in in very kind of intense times um can be very easy to get blurred together in in public communication i think we saw a few examples of that where you know things were being the follow the science on policy type angle where actually once you get into what you're prioritizing within a society quite rightly you've got other things beyond just the epidemiology driving that
Yeah, I mean, that term that you just used, follow the science, is such an important term because it tells us about the dynamic aspect. It isn't just a snapshot. It's constantly being revised. But during the pandemic, we had things like the six-foot rule that was never supported by data, but yet still today. Like if I walk around my hospital and there's still...
the footprints of the six foot rule and, you know, not paying attention to the fact that this was airborne and took, you know, years before some of these things were accepted, the flatten the curve stuff with lockdowns, which, you know, I never was supportive of that, but the, You know, perhaps at the worst point, the idea that hospitals would get overrun was an issue.
But, you know, it got carried away with school shutdowns for prolonged periods. And, you know, in some parts of the world, especially, you know, very stringent lockdowns. But anyway, we learned a lot. But perhaps one of the greatest lessons is that people's expectations about science are Is that it's absolute and somehow you have this truth that's not there. I mean, it's getting revised.
It's kind of on the job training. It's, in this case, on the pandemic revision, but very interesting. And that gets us to, I think, the next topic, which I think is a fundamental part of the book distributed throughout the book, which is the different types of proof. in biomedicine, and of course, across all these domains.
And so you take us through things like randomized trials, p-values, 95% confidence intervals, counterfactuals, causation and correlation, peer review, the works, which is great, because a lot of people have misconceptions of these things. So for example, randomized trials, which is like the temple of the randomized trials, They're not as great as a lot of people think.
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