Latest news with #AdamKucharski
Fast Company
a day ago
- Health
- Fast Company
5 lessons on finding truth in an uncertain world
Adam Kucharski is a professor of epidemiology at the London School of Hygiene & Tropical Medicine and an award-winning science writer. His book, The Rules of Contagion, was a Book of the Year in The Times, Guardian, and Financial Times. A mathematician by training, his work on global outbreaks has included Ebola, Zika, and COVID. He has advised multiple governments and health agencies. His writing has appeared in Wired, Observer, and Financial Times, among other outlets, and he has contributed to several documentaries, including BBC's Horizon. What's the big idea? In all arenas of life, there is an endless hunt to find certainty and establish proof. We don't always have the luxury of 'being sure,' and many situations demand decisions be made even when there is insufficient evidence to choose confidently. Every field—from mathematics and tech to law and medicine—has its own methods for proving truth, and what to do when it is out of reach. Professionally and personally, it is important to understand what constitutes proof and how to proceed when facts falter. Below, Adam shares five key insights from his new book, Proof: The Art and Science of Certainty. Listen to the audio version—read by Adam himself—in the Next Big Idea App. 1. It is dangerous to assume something is self-evident. In the first draft of the U.S. Declaration of Independence, the Founding Fathers wrote that 'we hold these truths to be sacred and undeniable, that all men are created equal.' But shortly before it was finalized, Benjamin Franklin crossed out the words 'sacred and undeniable,' because they implied divine authority. Instead, he replaced them with the famous line, 'We hold these truths to be self-evident.' The term 'self-evident' was borrowed from mathematics—specifically from Greek geometry. The idea was that there could be a universal truth about equality on which a society could be built. This idea of self-evident, universal truths had shaped mathematics for millennia. But the assumption ended up causing a lot of problems, both in politics and mathematics. In the 19th century, mathematicians started to notice that certain theorems that had been declared 'intuitively obvious' didn't hold up when we considered things that were infinitely large or infinitely small. It seemed 'self-evident' didn't always mean well-evidenced. Meanwhile, in the U.S., supporters of slavery were denying what Abraham Lincoln called the national axioms of equality. In the 1850s, Lincoln (himself a keen amateur mathematician) increasingly came to think of equality as a proposition rather than a self-evident truth. It was something that would need to be proven together as a country. Similarly, mathematicians during this period would move away from assumptions that things were obvious and instead work to find sturdier ground. 2. In practice, proof means balancing too much belief and too much skepticism. If we want to get closer to the truth, there are two errors we must avoid: we don't want to believe things that are false, and we don't want to discount things that are true. It's a challenge that comes up throughout life. But where should we set the bar for evidence? If we're overly skeptical and set it too high, we'll ignore valid claims. But if we set the bar too low, we'll end up accepting many things that aren't true. In the 1760s, the English legal scholar William Blackstone argued that we should work particularly hard to avoid wrongful convictions. As he put it: 'It is better that ten guilty persons escape than that one innocent suffer.' Benjamin Franklin would later be even more cautious. He suggested that 'it is better 100 guilty persons should escape than that one innocent person should suffer.' 'We don't want to believe things that are false, and we don't want to discount things that are true.' But not all societies have agreed with this balance. Some communist regimes in the 20th century declared it better to kill a hundred innocent people than let one truly guilty person walk free. Science and medicine have also developed their own traditions around setting the bar for evidence. Clinical trials are typically designed in a way that penalizes a false positive four times more than a false negative. In other words, we don't want to say a treatment doesn't work when it does, but we really don't want to conclude it works when it doesn't. This ability to converge on a shared reality, even if occasionally flawed, is fundamental for science and medicine. It's also an essential component of democracy and justice. Rather than embracing or shunning everything we see, we must find ways to balance the risk that comes with trusting something to be true. 3. Life is full of 'weak evidence' problems. Science is dedicated to generating results that we can have high confidence in. But often in life, we must make choices without the luxury of extremely strong evidence. We can't, as some early statisticians did, simply remain on the fence if we're not confident either way. Whether we're sitting on a jury or in a boardroom, we face situations where a decision must be made regardless. This is known as the 'weak evidence' problem. For example, it might be very unlikely that a death is just a coincidence. But it also might be very unlikely that a certain person is a murderer. Legal cases are often decided on the basis that weak evidence in favor of the prosecution is more convincing than weak evidence for the defendant. Unfortunately, it can be easy to misinterpret weak evidence. A prominent example is the prosecutor's fallacy. This is a situation where people assume that if it's very unlikely a particular set of events occurred purely by coincidence, that must mean the defendant is very unlikely to be innocent. But to work out the probability of innocence, we can't just focus on the chances of a coincidence. What really matters is whether a guilty explanation is more likely than an innocent one. To navigate law—and life—we must often choose between unlikely explanations, rather than waiting for certainty. 4. Predictions are easier than taking action. If we spot a pattern in data, it can help us make predictions. If ice cream sales increase next month, it's reasonable to predict that heatstroke cases will too. These kinds of patterns can be useful if we want to make predictions, but they're less useful if we want to intervene in some way. The correlation in the data doesn't mean that ice cream causes heatstroke, and crucially, it doesn't tell us how to prevent further illness. 'Often in life, prediction isn't what we really care about.' In science, many problems are framed as prediction tasks because, fundamentally, it's easier than untangling cause and effect. In the field of social psychology, researchers use data to try to predict relationship outcomes. In the world of justice, courts use algorithms to predict whether someone will reoffend. But often in life, prediction isn't what we really care about. Whether we're talking about relationships or crimes, we don't just want to know what is likely to happen—we want to know why it happened and what we can do about it. In short, we need to get at the causes of what we're seeing, rather than settling for predictions. 5. Technology is changing our concept of proof. In 1976, two mathematicians announced the first-ever computer-aided proof. Their discovery meant that, for the first time in history, the mathematical community had to accept a major theorem that they could not verify by hand. However, not everyone initially believed the proof. Maybe the computer had made an error somewhere? Suddenly, mathematicians no longer had total intellectual control; they had to trust a machine. But then something curious happened. While older researchers had been skeptical, younger mathematicians took the opposite view. Why would they trust hundreds of pages of handwritten and hand-checked calculations? Surely a computer would be more accurate, right? Technology is challenging how we view science and proof. In 2024, we saw the AI algorithm AlphaFold make a Nobel Prize-winning discovery in biology. AlphaFold can predict protein structures and their interactions in a way that humans would never have been able to. But these predictions don't necessarily come with traditional biological understanding. Among many scientists, I've noticed a sense of loss when it comes to AI. For people trained in theory and explanation, crunching possibilities with a machine doesn't feel like familiar science. It may even feel like cheating or a placeholder for a better, neater solution that we've yet to find. And yet, there is also an acceptance that this is a valuable new route to knowledge, and the fresh ideas and discoveries it can bring.
The Guardian
29-03-2025
- Science
- The Guardian
Mathematician Adam Kucharski: ‘Our concepts of what we can prove are shifting'
Adam Kucharski is a professor at the London School of Hygiene & Tropical Medicine. As a mathematician and epidemiologist, he has advised multiple governments on outbreaks such as Ebola and Covid. In his new book Proof: The Uncertain Science of Certainty, he examines how we can appraise evidence in our search for the truth. What inspired you to investigate the concept of proof?Alice Stewart, an influential epidemiologist, used this nice phrase that 'truth is the daughter of time'. But in many situations, whether you're accused of a crime or thinking about a climate crisis, you don't want to wait; there's an urgency to accumulate evidence and set a bar for action. We're entering an era where questions around information – what we trust and how we act – are increasingly important, and our concepts of what we can prove are shifting as well. You examined the ways that mathematical proofs have changed throughout history. What did you learn?I was so fascinated by the cultural differences in what people have assumed to be obvious. In Europe, for example, negative numbers were shunned for a very long time. That's because a lot of our maths was built around ancient Greek geometry, where something like a 'negative triangle' doesn't make sense. In contrast, a lot of the early mathematical theories in Asia were driven by finance and the concepts of debt, where negative numbers make a lot more sense, so they were much more comfortable handling those concepts. I found that ideas like calculus were also based on physical intuitions about the world: an apple falls, and we can write down the equations that drive its motion. But when you started getting down into it, you could find exceptions where the intuitions weren't true. As a result, you had this tension with an entire community who wanted to ignore these nuisances – these 'monsters', as they called them – while other characters were pushing for an evolution of the field. Einstein's work on relativity, for example, relied on these controversial ideas. In the book, you draw some parallels between these mathematical debates and the politics of the time.A lot of democratic ideas in the foundation of countries like the US attempted a kind of mathematical precision. The statement 'we hold these truths to be self-evident' in the US constitution was originally 'we hold these truths to be sacred', for example. But Benjamin Franklin crossed it out because he wanted more mathematical certainty behind these ideas, as if they were axioms. And in parallel with mathematics crumbling in its foundations, you saw the US descending into civil war, because the idea that all people are created equal wasn't self-evident to some people. You had the same tension manifesting in two different ways in two very different fields. Tell me a bit about Abraham Lincoln. How was he influenced by mathematics in his rhetoric?Lincoln was renowned for his speeches and the precise nature of his arguments, and that didn't come about by coincidence. He made a conscious decision as a lawyer to get a better understanding of what it means to demonstrate something and construct an argument that is logically robust. So he went and taught himself the fundamentals of ancient Greek mathematics, Euclid's Elements, and he would deploy those principles in his debates. He took advantage of this notion that if you can find a flaw or contradiction, you can cause someone's entire argument to collapse. You describe how some core concepts arose from casual conversations. How did the perfect cup of tea inspire the design of clinical trials?There was a tearoom in Rothamsted agricultural station in Hertfordshire, and in the early 1920s, three statisticians were having a conversation when one of them, Muriel Bristol, observed that tea always tastes better if the milk is put in before the hot water. The other people around the table disagreed, but they thought it was a nice challenge to see how you could work out whether she could tell the difference or not. In the end, they calculated that you need to lay out eight cups, in which half have the milk added before the hot water, and half after, ordered randomly. This produces around 70 possible combinations, meaning there is only a 1.4% chance that Bristol would get that out of all the possibilities correct – which she did. One of the mathematicians was Ronald Fisher, who would end up writing a landmark book called The Design of Experiments that examined how you can separate an effect from chance or human biases – and this included the principle of randomisation and using probabilities to test the strength of a hypothesis. How is AI changing our understanding of proof?A few years ago, I was at a dinner with a lot of AI specialists, and there was a lot of talk about the fact that AI is often more efficient if you don't worry about it providing an explanation for what it's doing or the decisions it makes. I found it interesting that this was a concern, because in medicine, we have a lot of things that work without us understanding why. One example is anaesthesia: we know the combination of drugs that makes a patient unconscious, but it's not clear why they do. We've struggled with similar questions for a long time, right?In the 1970s, we had the first ever computer-aided proof of the four-colour theorem, which says that if you want to colour a map so that no two bordering countries have the same colour, you'll only need four colours to do this. Even if you simplify it down and you look for symmetries, there are still just too many combinations to go through by hand. But a pair of mathematicians used a computer to punch those combinations and get over the line. There was a lot of scepticism, because it was the first time that you had this major theorem that mathematicians couldn't verify by hand. But then, among the younger generation of mathematicians, there is scepticism about some of the old methods. Why would hundreds of pages of handwritten maths be more trustworthy than a computer? I think that's going to be an important debate in the future, as algorithms become increasingly useful for the predictions they can make, even if we don't have an easy understanding of the way they work. With the spread of misinformation, it may feel that many people have simply stopped caring about evidence. What do you think of this?There has been some interesting research looking at what people say they value online, versus what they actually do. If you ask people whether accuracy is important in what they share, they'll say they don't want to share things that are false. In the moment, however, they often get distracted: something triggers an emotional response, or aligns with political beliefs, and they share the information on those grounds, without checking the facts. But there are lots of layers to this. In some cases, there can be an incentive to go against what authority says, if you want to suggest that you're not constrained by the thinking of the crowd. Henri Poincaré, a mathematician who did a lot of work in the late 19th and early 20th centuries, described something like this: 'To doubt everything or to believe everything are two equally convenient solutions; both dispense with the necessity of reflection.' And I think that's the danger, at the moment – that there's this almost excessive doubt. People are just disengaging with truth altogether. Proof: The Uncertain Science of Certainty by Adam Kucharski is published by Profile (£22). To support the Guardian and Observer order your copy at Delivery charges may apply
The Guardian
17-03-2025
- Science
- The Guardian
Did you solve it? Here's (not) looking at Euclid
Earlier today I set you these mind-mangling puzzles about non-Euclidean geometry, in which the internal angles of triangles do not add up to 180 degrees. 1. Right, Right, Right. Assume the Earth is a perfect sphere. Imagine drawing a straight line from the North Pole to a point on the Equator. Can you draw two more identical lines to make a triangle where all the internal angles are right angles (i.e. they add up to 270 degrees overall)? Solution Add one line going a quarter of the way around the Equator, and another going back to the North Pole.2. Full circle Next, let's go bigger, angle-wise. Can you find a way to cover the Earth with equilateral triangles that have internal angles of 120 degrees (i.e. they add up to 360 degrees overall)? These triangles must all be the same size and there must be no overlaps or gaps between them. (Hint: think about drawing triangles side by side.) Solution Imagine starting with one triangle that meets the criteria on the sphere's surface. Now, add three more identical triangles, one along each edge of the original triangle, so that their edges align. Because each of these triangles has internal angles of 120 degrees, then at each corner where three triangles meet, the angles will add up to 360 degrees. As a result, the triangles completely cover the surface of the sphere! We can show this more intuitively by imagining a triangular-based pyramid made of 4 'flat' equilateral triangles. If we 'inflate' this pyramid until it becomes a sphere, then its surface will still be covered by 4 triangles, but they'll now each have internal angles of 120°. Each triangle will cover a quarter of the sphere. 3. Tasty triangles Now imagine a donut instead of a sphere. Can you draw two identical right-angled triangles on the donut so they perfectly cover its surface? And what will the sum of the six internal angles of these two triangles add up to? (A donut is a 'torus', a cylinder that curves and joins itself in a loop, as in the image above.) Solution The trick is turn the donut back into a cylinder by 'cutting' along one of its loops and re-straightening. Then, cut the cylinder open along its side to 'unwrap' it into a rectangular strip. Now, if we divide this rectangle diagonally, we create two identical triangles. Since this rectangle originally came from the donut, we can wrap it back up to recreate the donut, now covered with these two triangles. The six angles of the two triangles will meet at a single point: marked by where the 'cut' and 'wrap' lines form a cross on the surface of the donut. The angles of the four segments of the cross (with two segments split in two, to give the six internal angles of the triangles) will add up to 360 degrees. Thanks to Adam Kucharski for today's puzzles. Adam is the author of a fascinating and beautifully-written new book, Proof: The Uncertain Science of Certainty. One of the many stories it tells is of how non-Euclidean geometry made mathematicians reassess what they had assumed were fundamental truths. The book is out on Thursday in the UK, and you can buy it at the Guardian Bookshop. I've been setting a puzzle here on alternate Mondays since 2015. I'm always on the look-out for great puzzles. If you would like to suggest one, email me.
The Guardian
17-03-2025
- Science
- The Guardian
Can you solve it? Here's (not) looking at Euclid
The ancient Greek geometer Euclid presented a list of five axioms he held to be self-evidently true. They are (or are equivalent to): You can draw a line between any two points. You can extend lines indefinitely. You can draw a circle at any point with any radius. All right angles are equal. All triangles have internal angles that add up to 180 degrees. Euclidean geometry is what we learn at school, and only applies to flat surfaces. The internal angles of a triangle on a curved surface do not add up to 180 degrees – the topic of today's puzzles.1. Right, Right, Right. Assume the Earth is a perfect sphere. Imagine drawing a straight line from the North Pole to a point on the Equator. Can you draw two more identical lines to make a triangle where all the internal angles are right angles (i.e. they add up to 270 degrees overall)?2. Full circle Next, let's go bigger, angle-wise. Can you find a way to cover the Earth with equilateral triangles that have internal angles of to 120 degrees (i.e. they add up to 360 degrees overall)? These triangles must all be the same size and there must be no overlaps or gaps between them. (Hint: think about drawing triangles side by side.) 3. Tasty triangles Now imagine a donut instead of a sphere. Can you draw two identical right-angled triangles on the donut so they perfectly cover its surface? And what will the sum of the six internal angles of these two triangles add up to? (A donut is a 'torus', a cylinder that curves and joins itself in a loop, as in the image above.) I'll be back at 5pm UK. PLEASE NO SPOILERS. Instead discuss your favourite axioms. Today's puzzles were set by Adam Kucharski, who is a maths professor at the London School of Tropical Medicine and a popular science author. In his brilliant new book Proof: The Uncertain Science of Certainty Adam tells the story of how nineteenth century thinkers began to challenge Euclid's self evident truths – and how this shaped the history of mathematics. It's a great read that covers many fields, including history, politics, statistics, computer science and epidemiology, which is Adam's area of professional expertise. Proof by Adam Kucharski is out in the UK on Thursday and available at the Guardian Bookshop. I've been setting a puzzle here on alternate Mondays since 2015. I'm always on the look-out for great puzzles. If you would like to suggest one, email me.
