Whether it’s football stickers in the UK or baseball stickers in the US, kids have been buying, coveting, trading, swapping and sharing collectibles of their favourite sports icons for decades. With the arrival of the World Cup comes the excitement of a new sticker book to complete and the expectation that every new packet of stickers purchased will fill in gaps in the album, taking you one step closer to completion.

But as the tournament progresses and your collection grows, it seems harder and harder to fill those elusive gaps – to find the missing shiny or the obscure defender. Buying a new packet that yields only swaps with no new gap-filling stickers can feel like throwing money down the drain. The despondency can leave many, initially avid collectors, to question how much it will cost them to complete the album and many more to abandon their collection before reaching the end.

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So how much should it cost to complete an album? Well that depends on your strategy. If you are planning to simply buy packet after packet to complete your set then you’re going to have to have deep pockets. The 2026 Panini World Cup sticker album comes complete with 980 different blank spaces to fill. With each pack of seven stickers costing £1.25 you’re looking at a minimum possible outlay of £175. But this would require each sticker you find to be a new one, with no swaps, and while this is a statistical possibility, it’s so astronomically unlikely as to make it practically impossible.

In reality, for each gap we want to fill, we are going to have to buy multiple stickers. The first sticker we get is guaranteed to be completely new to us, so the expected number of stickers we need to buy to fill the first empty spot in the book is just one. To fill another spot in the album our next sticker can be any of the 979 stickers we haven’t already got, which happens with probability 979/980. This means that on average we will need to buy 980/979 (just over one) stickers to fill this next slot. And so the trend continues with the next spot needing 980/978 stickers and the one after that requiring 980/977 stickers on average.

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Because these numbers are all quite close to one, it’s likely that we won’t get any repeats for a while. However, as we draw closer to completing the album, the expected number of stickers we need to buy to fill the next gap balloons massively. We find the penultimate sticker, which is only one of two that we don’t yet have, with probability 2/980. This means we need to buy 980/2=490 stickers on average to fill the penultimate gap. The last sticker is even worse. On average we need to buy 980 stickers to find the elusive one we are looking for.

Adding all these up, in total we would expect to have to buy

stickers at a cost of £1307.50 (remembering you have to buy stickers in packs of 7).

In general, if your sticker album contains spaces for N stickers, then you will need to buy

stickers, on average, to complete your set.

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The diminishing set of fractions inside the brackets are a well-known mathematical object called the ‘Harmonic Series’. The name comes from the concept of ‘overtones’ or ‘harmonics’ in music, which give each instrument its distinctive sound. The fundamental frequency of vibration of a string determines its basic pitch. The string’s harmonics have frequencies that are whole number multiples of the string’s fundamental frequency – that is to say they oscillate 2, 3, 4 etc times more often than the fundamental wave does. Correspondingly, the wavelengths of the harmonics are 1/2, 1/3, 1/4, etc of the string’s fundamental wavelength (as illustrated in the image below).

The formula we found for completing the sticker album crops up in a huge variety of different areas because it relates to the cost or effort required to fully explore an unknown set. In market research if you want to sample the full range of different consumer preferences then the harmonic series tells you how many people you will expect to have to survey. For DNA sequencing, where DNA is broken into fragments, sequenced and then reassembled, the harmonic series helps you approximate how many fragments you should expect to sequence before you have coverage of the whole genome.

In ecology, scientists use the harmonic series to estimate the number of species in a particular habitat. The underlying theory is not dissimilar to the sticker album filling problem. Imagine there are a number, N, of species in the ecosystem the scientists are studying. Each time they observe the system they note how many different species they find. If it were a collector’s album of species then the question would be ‘how many observations would the scientists expect to make before they observed all N species?’ Instead, since the number of species in an ecosystem is not known in advance but something we would like to know, the problem is often the other way around: given that I’ve seen a certain number of species from my observations of this ecosystem so far, how many are still out there that I’ve not seen. The harmonic series is key to estimating the distribution of this unseen biodiversity.

Heading back to our sticker collection problem, the harmonic series formula not only tells us how much it will cost to fill a sticker album, but also explains why it is so much more satisfying (not to mention quicker) to fill the first half of the album than the second. To fill the first 490 gaps of a 980-sticker album we would expect to have to buy just 679 stickers at a cost of £121.25. But to fill the second half we would need to buy another 6637 stickers, at a cost of £1186.25.

If we thought it were going to cost us well over £1000 to fill our sticker albums then we’d probably never even buy them in the first place. In reality, once you start to get duplicates then you can begin to swap them with your friends, filling in the gaps in both your albums. There is a mathematical formula relating to the harmonic series which can tell you how many stickers you and your friends would need to buy to complete all your albums, but it’s complicated. Imagine you’ve got 10 friends who are all collecting and you want to build up 11 complete albums (one for each of you). With our 980-sticker album example, you would need to buy roughly 26,230 stickers, which works out at roughly 2385 stickers each at a cost of about £426 – a substantial saving compared to the £1300 you’d spend collecting alone. Even with just four other friends, the cost would come down to £533 each – still a substantial saving.

Even without the help of your friends, once you get close to the end of your collection, you can usually head to the Panini website and order any missing stickers. They typically cost you twice as much as a normal sticker and there’s a strict limit on the number you can purchase. If you’re interested, the most cost effective way to complete the album would be to collect the first half (490) of the stickers by buying packets at a cost of about £121 and then to order the rest from the website for about £175. This would set you back £296, which isn’t too far from the minimum possible outlay of £175. Whether Panini let you order the 490 missing stickers directly is a different matter.

Of course, in this modern age, you can collect your ‘stickers’ in an online album. There’s no need to go to the shops to buy your sticker packets, no call for spending time swapping with your friends and no requirement to painstakingly stick the stickers into the album in the right place. But where’s the fun in that?

A version of this article was originally published on BBC Futures.

The last ten days have been a bit of a busy/surreal for me, so I thought I’d write a summary of everything I’ve been up to.

The main thing that’s been going on is that my new book, You don’t Know What You’re Missing, came out last Thursday (04/06). On Wednesday I went down to Old Broadcasting House in London to record an episode of BBC Radio 4’s All in the Mind (with the excellent Claudia Hammond), about the more psychological aspects of the book. That episode comes out in a couple of weeks, so if you’d like to hear it, then watch this space.

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On Thursday morning – publication day – I woke up early, headed over to the ITN studios on Gray’s Inn Road to appear on Jeremy Vine’s Channel 5 TV show. Jeremy has always been super kind to me, ever since he read my second book - How to Expect the Unexpected. It was lovely of him to have me on the show to promote this new book.

I also got a little taste for what his regular guests are letting themselves in for, presenting and commenting on a few of the day’s news stories that had been handed to me just a few minutes earlier. I have to say, I don’t envy them the job of having to talk coherently about a topic they’ve had less than five minutes (on average) to prepare for. Stressful.

If you’d like to see some of the more visual examples from the book, then I’ve included the full clip of the segment here.

I spent a quiet hour in the British Library before heading over to New Broadcasting House to appear live on Jeremy Vine’s Radio 2 show. Where I also bumped into Michael Berkeley from Radio 3’s Private Passions.

On Jeremy’s show we talked about the different sorts of missingness, from the gaps in our perception to the power of expectation, the ways in which information can be biased before it even reaches us and even the ways in which missingness can be useful.

You can catch up here from 1:06:30 (just so you know what you’re missing!).

The publicity drive seems to have had an impact, as You Don’t Know What You’re Missing made its way up into the low 100s across all books (it’s currently still the number one best seller in popular maths!).

After a quick pit stop at home on the Thursday evening, I headed over to Cheltenham on Friday to host a session in the Town Hall called “How to Win the World Cup”. On the panel were Liverpool’s former head of Data, Dr Ian Graham, and sports psychologist at Liverpool John Moores University, Dr Gillian Cook. On the night, their insights into everything from forming a cohesive squad to how to manage penalty shootouts were absolutely fascinating.

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Ian also has an excellent book called How to Win the Premier League, which is well worth a read if you’re interested in football (particularly the evolution of English football over the last 15 years). I had a lovely chat with him over dinner afterwards gleaning lots of insight, but also chatting nonsense about football for an hour, which is always high up my list of priorities for a Friday night.

I took the chance later in the evening to catch up with a few science communication colleagues, including Andrea Sella, Mark Miodownik and, a bit later, (too late really – I need early nights) Adam Rutherford, Kevin Fong and Alice Roberts. This is the really nice thing about Cheltenham Science festival: the opportunity to meet up with old friends and to make new connections.

And then came the most surreal part of the evening. As I was sitting down at one table, someone else was just leaving. I introduced myself as “Kit”, and he replied, “Kit Yates? I know you”. It turns out that this was the comedia John Harvey (aka Count Binface – not in costume, otherwise it might have been quite obvious!). He works with another maths communication friend of mine, the comedian Matt Parker, hence the connection. So let me finish by wishing John (Count John?) the best of luck in next week’s Makerfield by-election: you can read his manifesto here.

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So that was it for my surreal week. You can tell how hectic it was because it’s taken me a whole week to recover form it and write this post.

If you think you’d like to join the other people who are reading the book, then you can pick up your own copy:

Amazon: https://www.amazon.co.uk/Dont-Know-What-Youre-Missing/dp/1529438039

Bookshop.org (supports independent bookshops): https://uk.bookshop.org/p/books/you-don-t-know-what-you-re-missing-the-science-of-what-s-lost-and-how-to-find-it-kit-yates/497ab9dcf971763a

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Ahead of the men’s Football World Cup in a few days, former England captain Kevin Keegan shared the news that he had been diagnosed with cancer. While much of the focus on the reporting of the news has rightly been on how the footballing world has rallied around the former Liverpool and Newcastle great, there is an interesting lesson to be learned from the way in which Keegan’s cancer was discovered.

Keegan was in a car crash just a few weeks before his diagnosis. Multiple studies have shown that people who have the sorts of injuries caused by car crashes are also found to have a higher incidence of cancer. That is to say, there would be a larger number of cancer diagnoses in a group of people who’ve had car crashes than there would be in the same size group of people with similar backgrounds who haven’t had car crashes.

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Most experts believe that there is little evidence to support the hypothesis that traumatic injuries sustained in car crashes can cause cancer. So if it isn’t the effect of traumatic injury sustained in the accident, what could explain this link? Maybe there is an underlying common cause? Perhaps people who drive more are likely to live more sedentary lifestyles which comes with increased cancer rates? Maybe frequent drivers are exposed to more cancer risk factors (such as obesity or UV exposure), while the increased driving time means they are also more likely to be involved in accidents? Maybe sleep deprivation leads both to more accidents and a higher cancer risk?

To correctly resolve the conundrum, we have to think about how we find cancer. Typically, we diagnose cancer in people who undergo some type of medical examination – either because their cancer has caused them to feel unwell or for some other reason. Often, people who are involved in car crashes end up in hospital. If we run a CT scan or an MRI on them to understand if there has been any internal damage as a result of the accident, then we may incidentally find a tumour. It’s not that being in a car crash has increased their chance of having cancer, but rather the chance that it is detected.

Indeed this is exactly how Keegan recently described the detection of his cancer. “I was in a car accident and, through that, I had to have an operation. Whilst having the scan for the operation, they found out I had cancer.” The underlying rate of cancer isn’t any higher in a group of people who’ve been in car crashes than in a similar group of people who haven’t. It’s just that we have a better chance of diagnosing it in car crash victims because of the increased medical scrutiny they undergo.

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That’s not to say car-crash victims are the only group of people with higher rates of cancer diagnoses. People who attend accident and emergency for almost any reason are subject to increased medical attention and, as a result, will also have higher rates of incidental diagnosis.

As I outline in my new book, You Don’t Know What You’re Missing, this is a classic example of a detection bias - increased monitoring of particular scenarios in comparison to another can induce the perception that one phenomenon has a higher probability than another. For example, despite shark attacks being incredibly rare (around eighty unprovoked attacks and only a handful of fatalities worldwide each year on average), the public are disproportionately worried about sharks.

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In large part, this is probably due to an availability bias – shark attacks are so graphic and feature prominently in popular culture (think Jaws, the Reef, the Shallows, etc.) that they occupy a disproportionate amount of space in our imaginations. However, in part this may also be due to the misconception that sharks are attracted to popular beaches, where many people like to go to swim.

While there is some evidence that some species of sharks might be attracted by splashing, as it sounds like struggling prey, other species are put off by it. There is no strong evidence to suggest that more humans in the water will lead to a higher probability of attracting a shark.

It is true, however, that there are more shark attacks in places where lots of people swim, but this isn’t primarily because sharks are disproportionately attracted to these popular areas. Gavin Naylor is the director of the Florida Program for Shark Research. His data show that the number of shark attacks in the US is strongly correlated with the numbers of people attending the beach. Naylor argues that ‘The more sharks and people there are in one place, the greater the chance of them bumping into each other’; so popular beaches may see more attacks simply because there are more people in the water, not because sharks have a preference for popular beaches.

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It’s also true that busy beaches are disproportionately more likely to report shark sightings. But again, this is purely a function of the fact that more people are around to spot their tell-tale fins poking out of the water – another detection bias. Indeed, well-frequented beaches are more likely to have lifeguards who may be on the lookout for sharks or even to employ drones to help reassure beach users that it’s safe to go in the water.

Other detection biases can mean that interventions can seem to have the opposite impact to the one we might naively have expected. When policing is increased in an area, you might expect that key crime measures would go down, but instead they often go up. Why? Because more police usually means that more crime is detected and recorded even if the underlying crime rates remain the same or even, in fact, go down. Similarly, workplaces with better safety protocols can counterintuitively appear to have higher rates of workplace safety breaches. This can simply be a result of the fact that more breaches are detected and reported, rather than a result of the safety protocols making things less safe.

Multiple studies suggest that, similarly to Kevin Keegan’s case, many cancers are first spotted incidentally in accident and emergency in people who have attended for completely unrelated reasons. It’s important to remember that just because we see a higher incidence of cancer in people who attend accident and emergency, it doesn’t mean that being acutely unwell for other reasons is linked to having cancer. In fact, there is evidence to suggest that being unwell to such an extent that you require increased medical attention can actually prolong your life expectancy through the incidental finding of other diseases like cancer.

It’s publication day, and this is the final post in this short series summarising the chapters of my new book, You Don’t Know What You’re M ss ng. The aim has been to give you a sense of the shape of the argument, the kinds of stories I use in the book, and the ideas I hope will stay with you after you’ve finished it.

In this post I’m going to summarise Chapter 7, The silence that speaks.

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“Shape clay into a vessel. It is the space within that makes it useful.” - Lao Tzu

This final chapter’s opening story is about the Soviet physicist Georgy Flyorov. During the Second World War, he noticed something strange. Papers on nuclear fission, which had previously appeared in major Western journals, had suddenly stopped - a piece of negative data. Flyorov inferred that the work had not dried up, but was instead being hidden as a result of wartime censorship because the Allies were pursuing an atomic bomb. He wrote to Stalin, urging the Soviet Union to act. His reasoning from negative data helped trigger the Soviet nuclear programme.

This chapter is all about that kind of reasoning - what I call constructive missingness. Earlier in the book, missingness often involved our brains filtering things out or smoothing things over (intrinsic missingness), or the world presenting us with incomplete and biased information (extrinsic missingness). In this chapter we consider a third type of missingness. This time the key is not just that something is absent, but that its absence is itself informative: sometimes not finding what you expected to find is a vital piece of information.

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Detectives (especially fictional ones) understand this well. Sherlock Holmes’s famous “curious incident of the dog in the night-time” revolves around exactly this sort of reasoning. A guard dog at a stables doesn’t bark during the night a prize thoroughbred is stolen. That absence of action, Holmes realises, is a key piece of information. If the dog did not bark, the intruder who is linked to the crime must have been known to it. The absence of an expected reaction becomes evidence. The chapter traces that same logic into modern court cases and forensic puzzles, where missing fingerprints, absent noise, or the failure of some expected trace to appear can be almost as telling (if not more) than the clues that are physically present. The challenge, of course, is that negative data are often harder to notice precisely because they are not there to stare us in the face.

But this kind of reasoning comes with an important warning label. The absence of evidence is not always evidence of absence. If you open the fridge and cannot find the butter after a proper rummage, that is pretty good evidence that the butter is not in the fridge. But when the thing you are looking for is more elusive, or when the search has been only partial, or the evidence would be hard to detect anyway, that logic becomes much shakier. Not finding evidence for something does not automatically prove that it is not there. That is why the chapter spends a bit of time deconstructing an oft-used truism: “absence of evidence is not evidence of absence” – sometimes it is and sometimes it isn’t. What the absence of evidence can never be, however, is proof of absence.

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The case of the coelacanth is an important example that reinforces this idea. For decades, all the available evidence (no-one had ever found a live one) suggested this fish had gone extinct tens of millions of years ago. Then in 1938, a living specimen turned up. One fish was enough to overturn a century of increasingly confident scientific assumption. Negative evidence can be strong without being definitive. The more extensively we have looked, the stronger the evidence from absence becomes - but proof is another matter altogether.

That tension leads into one of the chapter’s bigger themes: the difference between science and mathematics. In mathematics, if the axioms hold and the logic is sound, a valid proof gives certainty. In science, things work differently. We gather evidence, we develop theories, we test them, and we gain confidence - but we never prove anything in the same absolute sense. Newton’s theory of gravitation worked remarkably well until it didn’t. Einstein’s general relativity repaired some of the gaps, but it too is incomplete. Scientific theories are robust not because they are ever proven beyond all doubt, but because they survive repeated attempts to break them.

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The chapter becomes somewhat philosophical towards the end as we consider the importance of falsifiability. The real power of a scientific theory is not that it can be endlessly confirmed by examples that fit it (although that may give us more confidence), but that it makes risky predictions that could, in principle, prove it wrong. If your theory is “all mammals give birth to live young”, then observing yet another dog having puppies doesn’t really give you much more evidence than you already had. Finding a platypus that lays eggs, however, can completely debunk the theory. A single contradictory example can do what a million confirming examples cannot.

That sounds straightforward, but psychologically it is harder than it should be. We are naturally drawn towards confirmation. We look for examples that fit our current ideas rather than cases that might undermine them. The chapter revisits Peter Wason’s classic selection task, where most people choose options that might confirm a rule rather than the options that could show it to be false. It is a succinct little demonstration that the habits of good scientific reasoning do not come naturally. We prefer examples that reassure us. Contradictions are more informative, but they are also less comfortable.

If we want to know anything new about the world, we have to give up the dream of perfect certainty. Mathematics can build towers of logic, but it cannot tell us by itself what the world is like. Science can tell us about the world, but only by living with uncertainty, by testing ideas against evidence, and by accepting that what survives scrutiny has not been proved true, only not yet been proved false. Constructive missingness is the way in which we use the absence of evidence to reason constructively about the world. Sometimes the thing that matters is what fails to appear. Sometimes the most important clue is the one that isn’t there. But to use that clue well, you have to think carefully about whether the silence really means something, or whether you simply haven’t listened hard enough.

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A favour: order the book

If you’ve enjoyed this summary and the other summaries in the series, you’ll find much more detail in the book itself, with the stories and the science explained in more detail.

If the book sounds like your sort of thing, please consider ordering it this week. First-week orders matter far more than most readers realise. They’re one of the strongest early signals that a book has an audience, which influences everything from how many copies are stocked to how widely it’s recommended.

Amazon: https://www.amazon.co.uk/Dont-Know-What-Youre-Missing/dp/1529438039

Bookshop.org (supports independent bookshops): https://uk.bookshop.org/p/books/you-don-t-know-what-you-re-missing-the-science-of-what-s-lost-and-how-to-find-it-kit-yates/497ab9dcf971763a

Thanks,
Kit

Today is the penultimate of the short series of posts I’m writing for this Substack summarising the chapters of my new book, You Don’t Know What You’re M ss ng, ahead of publication later this week (4^th June). If you’ve been enjoying them, then take heart because the book is almost here. If not then take heart because this series of posts is almost over! Either way the aim is to give you a feel for the shape of the argument, the kinds of stories I use in the book to get the message across, and the ideas I hope will stick with you long after you’ve finished reading it.

In this post I’m going to summarise Chapter 6, Where the light is.

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“Where the light is brightest, the shadows are deepest” - Johann Wolfgang von Goethe

One of the easiest mistakes to make when thinking about the past is to assume that what survives is somehow representative of what that era or epoch in history was like. We speak casually of “cavemen” as if our prehistoric ancestors mostly lived in caves, when in reality caves are simply the places where the evidence of their existence is most likely to have survived. Tools, bones, art and ornaments are preserved there because caves protect them from the elements. Most of the homes, objects and other traces of the people who lived out in the open have vanished. Caves are where the evidence lasted long enough for us to discover it.

That is the story of Chapter 6: Where the light is. Again and again, we focus on what survives, what is visible, what is memorable, what is measured, what is easiest to observe. And because those things are available to us, we instinctively treat them as representative. But very often they are not. They are simply where the light happens to fall.

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One form this takes is survival bias. Survivors are the ones who remain visible. They are the successful people whose stories get told, the organisations that endured, the species that made it, the products that still exist. The casualties, both literal and metaphorical, are often missing from view. That is why stories of success can be so misleading. We hear from the author who persisted through rejection and became famous, but not from the many equally talented people who gave up on the twelfth rejection letter and never made their mark on the cultural tapestry. The danger is that we miss all these representative failures and build our theories of success from the winners alone.

A close cousin is the spotlight fallacy: the assumption that something highly visible is representative of the whole situation. A high-profile error can make a person seem more generally incompetent. A few widely publicised cases can make an entire group seem dangerous, unruly or untrustworthy. Media coverage can exacerbate the effect. The chapter explores the spotlight fallacy through the treatment of Liverpool and Liverpool supporters in the years before and after the Hillsborough disaster. Because football fans - and Liverpool fans in particular - had already been cast in the media spotlight as hooligans, the disaster was initially understood through that lens. Drunkenness and disorder amongst fans were blamed, even though the underlying causes were failures of crowd control, policing and infrastructure. The spotlight did not merely distort public understanding after the event through biased media coverage, it almost certainly shaped the decisions that made the disaster worse while it was unfolding.

The spotlight fallacy is one of a broader family of biases known as availability biases: what comes to mind comes to matter. If an example is vivid, shocking or easy to recall, we tend to give it disproportionate weight when making a judgement. That is why people often worry more about shark attacks than about far more likely dangers in the sea like rip currents. Shark attacks are graphic, memorable and heavily represented in popular culture. Rip currents are deadlier, but much less cinematic. Availability biases, in this context, can result in a skewed sense of risk.

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And because memorable things are often the strange or unusual ones, the bias feeds on itself. The most bizarre and unusual items in a list stand out while the ordinary ones dissolve into background. In the chapter we encounter the story of a bank robber whose brightly coloured orange trainers end up giving him away because they were so memorable to witnesses. Distinctive details are easier to notice, easier to remember and easier to over-interpret. The flip side, also covered in the chapter, is hiding things in plain sight: if you want to make something go missing, one strategy is to hide it among more of the same. The best place to hide a leaf is in a forest.

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Another major theme of the chapter is detection biases. Sometimes a phenomenon appears more common in one place than another not because it really is more common, but because we are looking harder there. Busy beaches report more shark sightings partly because more people are there to spot sharks. Better-policed areas can seem to have more crime because more crime is detected and recorded.

This idea becomes especially important in medicine. The chapter uses the example of incidental cancer diagnosis to show how increased scrutiny can create the appearance of a relationship where none exists. People who have had major accidents or acute illnesses can seem to have higher cancer rates simply because they are subjected to more imaging and investigation. It is not that the accident caused the cancer. It is that the accident caused the cancer to be detected.

In some cases, as with my brother-in-law, Simon, that increased scrutiny is life-saving. But detection bias also complicates how we evaluate screening programmes and treatment success. Catching a disease earlier can make survival “after diagnosis” look longer even if no extra life has actually been added. That is why we should be wary of simplistic uses of five-year survival rates when trying to sell the virtues of screening.

Then there are observer effects: the fact that measurement itself can alter the thing being measured. Sometimes this is psychological, as in the Hawthorne effect or white coat syndrome. Sometimes it is physical, as in quantum measurement, tyre-pressure gauges, code profilers, or even (in very rare circumstances) tumour biopsies that can in some circumstances alter what happens next. The act of looking is not always passive. Sometimes the very process of trying to gather information perturbs the system we want to understand.

One of my favourite ideas in the chapter, though, is the stranger subcategory of observer selection effects. These are cases where what you observe is constrained by the fact that you are only in a position to observe it under particular conditions. That is (almost tautologously) why the thing you lose is always in the last place you look: once you find it, you stop looking. It is why your friends, on average, have more friends than you do: the highly connected people are overrepresented in your experience of the network. Observer selection effects even explain why we should be cautious about marvelling at how finely tuned the universe seems for life. We can only observe a universe in which observers like us are possible.

All of this leads to the chapter’s titular metaphor. Scientists (asking the questions they can easily find the answers to and not necessarily the most relevant questions) and really, more generally, all of us, are often like the drunk searching for his keys under the streetlight. Asked whether he is sure he lost them there, he replies: “No, I lost them in the park, but this is where the light is.” We tend to study the things we can measure, count or recall, and we mistake that convenience for importance. We ask the questions to which we know we can get an answer, rather than the questions that would get us the answer.

That doesn’t mean measurement is futile, or that studying what is available is foolish. It means we have to remember that there are areas on which the light does not shine. The danger is not in looking where the light is, but in forgetting that there is darkness beyond it.

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A favour: pre-order the book

If you’ve enjoyed this summary, you’ll find much more detail in the book itself, with the stories, the science, and the slightly uncomfortable implications.

If the book sounds like your sort of thing, please consider pre-ordering it. Pre-orders matter far more than most readers realise. They’re one of the strongest early signals that a book has an audience, which influences everything from how many copies are stocked to how widely it’s recommended.

Amazon: https://www.amazon.co.uk/Dont-Know-What-Youre-Missing/dp/1529438039

Bookshop.org (supports independent bookshops): https://uk.bookshop.org/p/books/you-don-t-know-what-you-re-missing-the-science-of-what-s-lost-and-how-to-find-it-kit-yates/497ab9dcf971763a

In May 1845, two ships, the HMS Erebus and the HMS Terror, set sail from England under the command of Sir John Franklin, carrying 129 officers and crew on a mission to chart the last unknown stretch of the Northwest Passage. Instead, it became one of the great disasters in the history of polar travel. Within little more than a year, the ships were locked fast in Arctic ice in the vicinity of King William Island, where they remained trapped through successive winters. Franklin himself died in 1847, and by April 1848 the remaining survivors (105 men in all) abandoned the vessels and attempted a desperate trek south across the frozen landscape, hauling boats and supplies by hand. None would make it to safety.

For over a hundred years, despite search parties being sent out, little was discovered about the ill-fated expedition. Search parties in the nineteenth century recovered scattered bones, fragments of equipment, and a few written records, while contemporary Inuit testimony and modern forensic analysis suggested the hapless sailors were subject to starvation, and even cannibalism. But although the names of every man on the expedition were known, the identities of the discovered remains largely were not. It was announced yesterday that, modern DNA analysis has begun to identify who some of the recovered human remains belong to. By matching genetic material from skeletal remains with living descendants, researchers have been able to identify several of the sailors, including four announced recently, finally putting names to bones that lay anonymous for over 170 years and shedding new light on the expedition’s final, fatal journey.

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Long before the more recent and successful expeditions to discover the fate of Franklin’s expedition, in 1852, Sir Edmund Belcher set out to find and even potentially rescue Franklin and his colleagues. His mission was ultimately fruitless, abandoning four of the five ships he set out with, but ultimately returning safely to England. His mission is notable today, however, for spawning the most expensive typo in history.

Before Belcher set out on his expedition, the brewery Samuel Allsopp and Sons provisioned him and his crew with a bespoke “Arctic Ale” to take with them on the long journey. Not all the ale was drunk on the mission and some of the unopened bottles made their way back with the crew on the one remaining ship to England.

Fast forward to 2007 and eBay user petere92346 finds himself the recipient of one of these bottles – a little piece of Arctic history passed down through his family for over half a century. In June of that year, he tried to sell the full sealed bottle on the auction website with a reserve price of $299. Sadly, when he had typed in the name of the collectors’ item in his listing, Peter missed off the second ‘p’ in Allsopp’s. It meant that potential collectors simply couldn’t find the item when they searched. When the allotted time for bidding was up, the winning bid (of two that were placed) from ‘CollectorDan’ stood at just $304. When Dan relisted the bottle just a couple of months later, with the correct spelling, it attracted 157 bids with 74,000 users watching the item at one point. When the auction ended, the winning bid stood at over $500,000.

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This is just one of the stories about missingness from my forthcoming book: You Don’t Know What You’re Missing. Peter missed his typo when proofreading – not out of carelessness, but simply because our brains have been trained to read what we expect to see. And needless to say, what we expect is not always what is actually there. We have fine-tuned our brains to smooth over typos and efficiently extract the information we need. When proofreading, we often ‘read’ what we meant to write, rather than what we have actually written. Peter’s listing goes down as one of the most expensive missing letters in history. What, in hindsight, would he have given for an insight into his own proofreading blind spots?

Peter’s expensive spelling mistake illustrates a crucial lesson from The Missing – that we see what we expect to see. Our expectations influence our perceptions, making it hard to spot mistakes. We know what we want to see and this is often what we think we have seen, irrespective of what is actually there. Once we are conscious of this, however, we can start to implement fixes and use techniques to override these subconscious shortcuts, so that we take in the finer points of the scene that confronts us or we really do read every word in situations where attention to detail matters.

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As I mentioned, the latter part of this article (about the typo) is a short, adapted excerpt from You Don’t Know What You’re Missing.

If the book sounds like your sort of thing, please consider pre-ordering it. Pre-orders matter far more than most readers realise. They’re one of the strongest early signals that a book has an audience, which influences everything from how many copies are stocked to how widely it’s recommended.

Amazon: https://www.amazon.co.uk/Dont-Know-What-Youre-Missing/dp/1529438039

Bookshop.org (supports independent bookshops):

https://uk.bookshop.org/p/books/you-don-t-know-what-you-re-missing-the-science-of-what-s-lost-and-how-to-find-it-kit-yates/497ab9dcf971763

Thanks,
Kit

We’re now into the second half of the short series of posts I’m writing for this Substack, summarising the chapters of my new book, You Don’t Know What You’re M ss ng. Publication is under two weeks away on the 4th June. As before with these summaries, the aim is to give you a feel for the shape of the argument, the kinds of stories I use in the book to get the message across, and the ideas I hope will stick with you long after you’ve finished reading it.

In this post I’m going to summarise Chapter 5, Select Company.

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“The greatest obstacle to discovery is not ignorance - it is the illusion of knowledge.” Daniel J. Boorstin, American historian.

“A QUARTER of students catch an STI during their first year at university - and half are too drunk to remember who gave it to them.”

So read a headline in the Daily Mail in October 2013. Timed to coincide with the first couple of weeks of a new university year, the article brought a number of questions immediately to my mind; prosaically and perhaps indicative of my mathematically preoccupied mindset, the most pressing one was ‘How had the headline statistics been gathered?’

The answer, it turned out, was not a university-led study, or a medical survey, or anything you would reasonably call research. The figures came from a self-selected poll of the subscribers to a dating website - ShagAtUni.com - whose “sole purpose”, in their own words, “is to help students meet up for sex”.

You don’t need to be a statistician to see the problem. If you only ask the people most likely to have lots of sexual partners, and you exclude people in stable relationships, you are going to get a very skewed picture of student sexual health. Whole groups are missing before you even begin.

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That is the central idea of this chapter: we can end up making mistakes, not because we have missed something in our own reasoning, but because the information we’re reasoning from has already been filtered. Something is missing by the time it reaches us. In the book I call this extrinsic missingness: missingness in which important data is removed from view by some external process before it is even set in front of us, so our view is partially blocked and we are not seeing the full picture.

Surveys are common places where extrinsic missingness is perhaps most readily found, because surveys come with a built-in illusion of objectivity: there is a number; it looks precise; it feels scientific. But surveys go wrong in predictable ways, and this chapter is, in large part, a guide to those predictable failures.

One obvious issue is representativeness. Even if every respondent answered perfectly honestly, you can still be wrong if the group you asked isn’t representative of the population you’re trying to talk about. And you can also be wrong simply because you didn’t ask enough people. If the true preference between two options is 52:48, small samples will often tell you the wrong answer, purely through randomness. Even at sample sizes that feel large, close-run things (like the Brexit referendum) are hard to call.

But the bigger, and arguably more interesting failures of surveying are the ones that don’t go away with bigger samples. Because if the method is biased, taking more data doesn’t pull you towards the truth. It pulls you more confidently towards the wrong answer.

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A lot of the chapter is then about different flavours of bias that cause the wrong people to appear in your results, or cause some of the right people to disappear.

One set of problems comes from how people end up in the survey at all. There is a distinction between how you choose the sample and who ends up in the results. Even if you start with a decent sampling plan, you can still end up with skewed outcomes through non-response, drop-out, missing data, and all the other ways people can vanish from the final dataset.

A prominent example of overrepresentation was recently illustrated in a BBC headline about chronic traumatic encephalopathy (CTE) in NFL players: “Brain disease affects 99% of NFL players in study”. The underlying study examined brains in a brain bank, many donated by families who suspected their loved one had neurological problems. That is not a random sample of players. It is a sample heavily enriched for the very thing you’re trying to measure. To their credit, the scientists who wrote the study warned against drawing population-level conclusions from this sample. But that warning is exactly the kind of nuance that is easily stripped away as results travel from paper to press release to headline.

Underrepresentation can be just as damaging. The classic cautionary tale is the 1948 US presidential election, when the Chicago Tribune printed “DEWEY DEFEATS TRUMAN” before the results were in. The call was supported by a Gallup poll that looked reliable, but had a fatal flaw: quota sampling meant interviewers had targets by demographic category (in an attempt to ensure representativeness), yet they were free to fill those quotas with whoever was easiest to find and willing to talk, which tended to miss working-class Truman voters on long shift patterns and over-represent more affluent respondents. Truman won comfortably, and the paper was immortalised in embarrassment.

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One almost surefire way to create a biased sample is to let it pick itself. Open-access surveys are cheap, quick, and often completely unrepresentative. People who respond tend to be those with the strongest views, the most time, or the biggest grievance. This creates a familiar distortion: the moderate majority goes missing and the extremes dominate.

HOPE not hate’s “National Conversation on Immigration” report illustrates the point neatly. The same question (“On a scale of 1–10, with 1 very negative and 10 very positive, do you feel that immigration has had a positive or negative impact on the UK, including your local area?”) was asked in two ways: an open online survey and a nationally representative poll. In the open survey, responses clustered at the extremes. In the representative poll, far fewer people held the most extreme views. Changing the sampling method made people with differing views become more or less visible.

And even when you try to correct for missingness, you can make things worse. Pollsters weight survey results to match demographic proportions, but if turnout varies strongly by demographic group, then “making your sample look like the population” can still misrepresent who will actually vote. The 2015 UK election polling miss is a cautionary tale: pollsters corrected for underrepresented younger voters (who tend to be more left-leaning), but failed to account properly for the fact these same younger demographics tend to have lower turnout. Consequently, many polls ended up with an overly optimistic picture of Labour support.

Quite apart from the question of whether your sample is representative, there is a different, more human set of issues: even if you ask the right people, they might not answer truthfully.

Sometimes people simply misreport. In a closed population over a fixed time period, heterosexual partnerships contribute one partner to the male count and one to the female count, so average reported numbers should not diverge dramatically. Yet in survey after survey, men report far more opposite-sex partners than women do. This is a big red flag that responses are not purely measuring behaviour, but are also affected by the truthfulness (or otherwise) of the respondents.

And beyond misremembering, there are predictable psychological pressures. People say what they think is expected of them. They say what makes them look good. They say what avoids embarrassment. They say what they think people want to hear. Surveys can suffer from acquiescence bias, courtesy bias, compliance or coercion, demand effects, and social desirability bias. In authoritarian settings, “approval ratings” can be meaningless because dissent is dangerous. In less extreme settings, people still tweak their answers towards what feels socially acceptable.

So what can you do when you want to measure something people don’t want to admit to?

You get clever about how you ask. Chapter 5 ends with an illustration of an elegant survey technique known as a randomised response trial. When trying to find out what proportion of the population practice an embarrassing or taboo habit, the idea is to introduce randomness so individual answers carry plausible deniability, while still allowing us to estimate the true prevalence of a behaviour across a whole population. The simplest version gets everyone to flip a coin: heads answer truthfully, tails answer “yes” to the embarrassing behaviour regardless. A “yes” answer becomes deniable - “the coin made me do it”. The cost is that you’ve deliberately contaminated your data with false answers. The benefit is that people can answer honestly without feeling exposed, and because we know how many default “yes” answers we expect due to people flipping a tail, we can recover an estimate of the true prevalence by working back from the overall proportion of “yes” responses.

Extrinsic missingness doesn’t just mislead us. It gives us the confidence of a number while hiding the fact that the number may have been born biased.

So the take-home message from this fifth chapter is about questioning where the statistics that are put in front of us come from. When you see an alarming statistic or a suspiciously neat survey result, your first question should be “how was this measured?” To help you answer, ask: “Who was missing?” “Who was overrepresented?” “What incentives shaped the answers?” “What might have been filtered out before the number reached you?”

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A favour: pre-order the book

If you’ve enjoyed this summary, you’ll find much more detail in the book itself, with the stories, the science, and the slightly uncomfortable implications.

If the book sounds like your sort of thing, please consider pre-ordering it. Pre-orders matter far more than most readers realise. They’re one of the strongest early signals that a book has an audience, which influences everything from how many copies are stocked to how widely it’s recommended.

Amazon: https://www.amazon.co.uk/Dont-Know-What-Youre-Missing/dp/1529438039

Bookshop.org (supports independent bookshops):

https://uk.bookshop.org/p/books/you-don-t-know-what-you-re-missing-the-science-of-what-s-lost-and-how-to-find-it-kit-yates/497ab9dcf971763

Thanks,
Kit

We’re about halfway through the short series of posts I’m writing for this Substack in order to summarise the chapters of my new book, You Don’t Know What You’re M ss ng, ahead of publication on 4th June, and things are starting to feel real. I managed to get my hands on a copy of the the new book at a festival I was speaking at on the weekend before I’ve even been sent my own copies. So nice to feel a physical copy of it in my hands. Some lucky people even walked away from the festival owning a copy of the book before I did. You can get your own hands on a copy by clicking the button below.

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As a reminder, the aim of these summaries is to give you a feel for the shape of the argument, the kinds of stories I use in the book to get the message across, and the ideas I hope will stick with you long after you’ve finished reading it.

In this post I’m going to summarise Chapter 4, Inside information.

“Information is a measure of one’s freedom of choice” - Warren Weaver

If you can read a sentence with jumbled letters (see last week’s post about Chapter 3 - Reading Between the Lines), or skim past the odd word without losing the meaning, it suggests something slightly strange, perhaps counterintuitive: not every part of a sentence is doing the same amount of work. Some words are carrying the load. Some are almost redundant. The context surrounding a particular word in a sentence is doing a lot of work when we read.

So it’s natural to ask a question that sounds philosophical but turns out to have a huge amount of practical relevance: how much information is actually contained in a sentence, a paragraph or even a whole language? And if we can measure it, can we strip a message down to its essence - a distilled informational concentrate?

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This fourth chapter (Inside Information) is about the art - and the risk - of doing exactly that.

When we compress data, we’re trying to remove what is predictable so we can store or transmit what remains more efficiently. That’s the heart of the information revolution that has transformed daily life in the last few decades: squeezing photos, calls, videos, documents and messages down so they travel quickly and reduce the amount of storage needed.

But there’s a catch. The moment you boil down message down to its bare informational bones, you strip away the helpful redundancy that makes communication robust. Perfect compression gives you no safety net. If something goes wrong, you may have no way to spot the error, let alone correct it. The challenge is finding the balance between economic efficiency and reliable resilience.

Information theory is not concerned with whether a message is true, wise, or worthwhile. In that sense, the information content of a sentence is quite different from its meaning. Instead, information theory is concerned with the mechanics: how messages can be encoded, stored, transmitted, and recovered.

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The person who, more than any other, showed us how to think about information properly was Claude Shannon. Shannon’s breakthrough was to quantify information using a concept called entropy.

Thinking about entropy as a measure of surprise is a useful analogy. If something is very predictable, it contains little information. If it’s genuinely uncertain, you gain more information when you discover the outcome. In Scrabble, picking a Z gives you more information (because you don’t expect to see a Z because it’s rare) than drawing an E would (because you expect to see the common letter E). The surprise you get is different because the probabilities of the letters are different.

The maximum information you can gain from a yes-no question is one bit (a portmanteau of ‘binary’ and ‘digit’) - and you only get close to that maximum if “yes” and “no” are equally likely. This is why the best strategy in Guess Who is not to ask something unlikely like “Are they bald?” (high payoff if yes, but usually no), but to ask questions that roughly split the remaining options in half (“Are they female?” is usually a good starter question). The same logic underpins the most efficient way to play Twenty Questions: keep dividing the space of possibilities as evenly as you can. It’s not the most fun way to play, but it’s a good illustration of how you can gain as much information as possible.

Another of Shannon’s brilliant insights takes us back to language. Letters don’t appear independently of each other. They come in patterns - words. They come with context - sentences. That context makes them predictable, which means each letter carries less information than you might naïvely think.

Shannon tried to estimate how much information there really is per character in English, once you include the predictability created by surrounding letters and words. The startling conclusion is that context can reduce the effective information content of English to around a bit per character (whereas 5 binary digits would be needed to encode the 26 letters of the alphabet – 00000 encoding A, 00001 encoding B, 00010 encoding C, 00011 encoding D etc). In other words, a lot of written language is redundancy - usually helpful redundancy.

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This is where compression comes in. Standard text encodings like ASCII use seven bits per character, which is robust but not very efficient. If English can be encoded, in principle, far more economically, then a lot of what ASCII stores is redundancy - space we could make go missing. Compression algorithms are built to exploit exactly this sort of structure: they remove predictable parts while keeping enough to reconstruct the message.

But once you start stripping away redundancy, you start living dangerously. A beautifully compressed message can become brittle. If a single error creeps in, the receiver may not be able to tell.

Morse code is a lovely historical stepping stone here because the choice of encodings for the letters was an early form of compression in action. Morse assigns short codes to common letters and longer codes to rare ones. But Morse comes with a vulnerability: its symbols are not naturally self-delimiting. If spacing is sloppy, messages can become ambiguous. A pause too short can turn clarity into confusion.

The most haunting illustration is the story of the Star Dust, a plane that vanished over the Andes in 1947. Its final transmission ended with the enigmatic word “STENDEC”, repeated three times, and then silence. No one knew what STENDEC meant. What makes the tragedy instructive rather than just intriguing is not the mystery, but the mechanism: in Morse code, missing or misplaced gaps can destroy meaning. When context is stripped away, a small formatting error can convert a message into a riddle with no solution.

It’s similar to the risk we run when we compress information too aggressively. If all the context is removed and a mistake occurs, the message may become ambiguous or unintelligible.

So what do we do about that fragility? We add redundancy back in - but deliberately and in a way that helps us catch and correct errors.

This is where error-detection and error-correction codes come in. The point is not to avoid redundancy entirely, but to choose the right kind. Add a single parity bit to a string of binary digits and you can detect that something has gone wrong. Add more structured redundancy and you can often pinpoint the error and correct it. These techniques sit behind the everyday miracle of modern computing: the fact that we can store and transmit huge volumes of data through noisy, unreliable physical systems and still have it arrive intact.

If you want an example of how small an error can be and how dramatic the consequence, you could do worse than look to the Belgian federal election in 2003, where an electronic voting anomaly turned out to be consistent with a single flipped bit - a tiny physical event in memory leading to a large jump in the number of votes and almost to an incorrect winner being announced.

The lesson is not “computers can’t be trusted”. The lesson is that without redundancy and checking, even a naturally occurring one-bit glitch can change the result.

This information theory chapter ends by returning to the broader theme: the key to efficient, robust communication is not maximal compression, nor is it no compression. It’s a finely struck balance between the two. We need enough context to give messages clarity and the possibility o correcting errors, but not so much that we bury the signal in superfluity. It sounds oxymoronic, but the main take-home from the chapter is that we really need this essential redundancy.

A favour: pre-order the book

If you’ve enjoyed this summary, you’ll find much more detail in the book itself, with the stories, the science and the slightly uncomfortable implications.

If the book sounds like your sort of thing, please consider pre-ordering it. Pre-orders matter far more than most readers realise. They’re one of the strongest early signals that a book has an audience, which influences everything from how many copies are stocked to how widely it’s recommended.

Amazon: https://www.amazon.co.uk/Dont-Know-What-Youre-Missing/dp/1529438039

Bookshop.org (supports independent bookshops): https://uk.bookshop.org/p/books/you-don-t-know-what-you-re-missing-the-science-of-what-s-lost-and-how-to-find-it-kit-yates/497ab9dcf971763a

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Thanks,
Kit

Heard melodies are sweet, but those unheard are sweeter - John Keats, ‘Ode on a Grecian Urn’.

First off, I wanted to say a massive thanks to all of you who’ve already gone and pre-ordered a copy of the book. Your support means so much to me. As you’ll probably know by now, I’m writing a series of posts summarising the chapters of my new book, You Don’t Know What You’re M ss ng, ahead of publication on 4th June. The aim is to give you a feel for the shape of the argument, the kinds of stories I use in the book to get the message across, and the ideas I hope will stick with you long after you’ve finished reading it.

In this post I’m going to summarise Chapter 3, Reading between the lines.

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It mhigt spusiere you to konw taht dispete the fcat I hvae jbumled up all the ltteres in the mddile of the wdros in tihs scetnene you can sltil raed it woutiht mcuh erffot: as lnog as the fsirt and lsat ltteres are in the rghit pacle you can wrok it out. 51M1L4RLY, 3V3N 1F 50M3 L3773R5 4R3 5UB5717U73D F0R NUMB3R5 Y0U C4N R34D 7H15 53N73NC3 W17H0U7 3V3N BR34K1NG 57R1D3. P#rh#ps #v#n m#r# s#rpr#s#ng #s th#t #f #ll th# v#w#ls #n th# s#nt#nc# g# m#ss#ng #nd #r# r#pl#c#d w#th #n#ther ch#r#ct#r y## c#n st#ll j#st ab##t r##d #t. It was harder, but you could still manage, right?

We are incredibly adept at decoding garbled messages, in which seemingly important linguistic structures are missing. One version of this is disemvoweling: a technique internet moderators sometimes use to tread the fine line between overzealous censorship and an unregulated free-for-all. It renders potentially offensive posts less objectionable, while ensuring readers c#n st#ll r##d th#m #f th#y r##lly try. What I’m really concerned about in this chapter, though, is what this says about how our brains operate when confronted with missing information.

A big part of the answer is context. The words in a message prime us to expect particular words to come next. Even if a word is ambiguous because we haven’t looked at it carefully or it has become corrupted, context combined with out prior experiences can allow us to interpret what it should be. At a finer scale, the letters of a word prime us to expect other letters. If you see a “q” you’re already leaning towards “u” coming next before you even get to it.

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But it isn’t only the context that comes before the missing piece that’s important. Often it’s the context that comes after that helps us out. With heavily jumbled text, you might not truly parse each word as you encounter it. You can feel, in hindsight, as if you understood everything smoothly, when in reality later words have allowed your brain to go back and fill in gaps you didn’t even notice were there. Reading isn’t just a predictive process; it’s also a repair job.

I’ve had first-hand insight into this while helping my son learn to read. A couple of years ago, he reached the stage where he no longer needed to sound out every letter in order to parse a word. He lets context help him out. If he gets stuck, he knows that reading on often resolves the ambiguity. That is a tremendous advantage. It speeds up reading dramatically. But it comes with a familiar trade-off: the more your brain relies on expectation, the more it risks seeing what it expects rather than what is actually on the page. Which is why proof-reading our own work is so difficult.

When re-reading, we tend to automatically interpret many of our own mistakes as correct. We know what we meant to write, so we expect to see the correct form on the page when proofing, and this is often what we end up “reading”. I sometimes type “form” when I mean “from”. Word doesn’t flag it because it isn’t a spelling error. And I often don’t spot it because my brain knows what it is expecting to see and that is what it sees. I’ve resorted to searching for both words and checking each instance manually.

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This general phenomenon also explains why certain “spot the mistake” puzzles are so effective. Try this one one.

70% per cent of the public can’t spot the mistake in this text.

Apart from it probably being poor English to start a sentence with a number in digit form, did you manage to spot the mistake? I certainly didn’t the first time I read it. The answer is the repetition of % and per cent. Our brains tend to simply filter this out. Even when you are primed to catch mistakes it isn’t always easy to spot them.

We miss repeated little words. We glide straight past a duplicated “the” or an extra “a”. We can stare at a sentence and confidently feel we have inspected it, when our brains have actually filtered out exactly what we were trying to catch. Even when you are primed to detect errors, your mind is still optimised for extracting meaning, not for conducting forensic audits.

The same is true for sound. In this chapter we spend some time looking at the ways we mishear and misinterpret language, not because we are careless, but because we are trying to make sense of noisy signals in real time. Mondegreens and eggcorns are the harmless, funny end of this spectrum: plausible substitutions that fit the context and make a surprising kind of sense. They are reminders that, while context can help us interpret missing parts of language, it can also lead us astray.

Sometimes the consequences are not funny at all, but deadly serious. Under the wrong conditions, an ambiguous phrase or a misheard command can become fatal. When people are anxious, excited, frightened, or simply primed to expect one thing over another, they can fill in the gaps with the wrong word and not even realise they have done so. This is one of the uncomfortable themes of the chapter, that misunderstanding can feel, from the inside, exactly like understanding. Sometimes we don’t even know we are making a mistake.

All of this pushes us towards a natural question: if the brain is constantly filling gaps, how does it decide what to fill them with?

One possible explanation for how our brains reason under uncertainty is called the Bayesian Brain Hypothesis. Many neuroscientists think it’s useful to treat the brain as a statistical organ: one that combines prior knowledge with new evidence in order to reach the best current guess about what is happening. This is sometimes called Bayesian integration. The basic logic is simple. Your prior experiences give you a set of expectations about what is likely. Incoming information nudges those expectations. The brain updates.

The Bayesian Brain Hypothesis helps explain why we can read meaning into degraded text, why we can often reconstruct what someone said in a noisy room, and why we can predict the next word in a sentence. It also helps explain why we make systematic errors. Sometimes our prior expectations are too strong. Sometimes they are applied in the wrong domain. Sometimes we mistake familiarity for predictive power and end up with the cognitive equivalent of the gambler’s fallacy: feeling that the world “ought” to balance out in the short term, even when it doesn’t work that way.

The brain (Bayesian or otherwise) is a fantastic tool. It often allows us to draw correct inferences from incomplete information. But it is so fluent at filling in gaps that we are sometimes not even aware that anything is missing. And if you don’t realise something is missing, you don’t realise you might have filled it in incorrectly.

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A favour: pre-order the book

If you’ve enjoyed this summary, you’ll find much more detail in the book itself, with the stories, the science and the slightly uncomfortable implications.

If the book sounds like your sort of thing, please consider pre-ordering it. Pre-orders matter far more than most readers realise. They’re one of the strongest early signals that a book has an audience, which influences everything from how many copies are stocked to how widely it’s recommended.

Amazon: https://www.amazon.co.uk/Dont-Know-What-Youre-Missing/dp/1529438039

Bookshop.org (supports independent bookshops): https://uk.bookshop.org/p/books/you-don-t-know-what-you-re-missing-the-science-of-what-s-lost-and-how-to-find-it-kit-yates/497ab9dcf971763a

Thanks,
Kit

If you’ve been reading this substack over the last few weeks you’ll know that I’m writing a series of posts summarising the chapters of my new book, You Don’t Know What You’re M ss ng, ahead of publication on 4th June. The aim is to give you a feel for the shape of the argument, the kinds of stories I use in the book to get the message across, and the ideas I hope will stick with you long after you’ve finished reading it.

In this post I’m going to summarise Chapter 2, Putting things in context.

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On the face of it, the story that opens this chapter feels like just a fun anecdote. Eddie Vedder, frontman of Pearl Jam, steps out on the street after a Chicago Cubs game, hears a band busking, and ends up playing some of his most famous songs with them. A world-famous musician, playing some of his best-known songs, right outside a stadium that had hosted Pearl Jam a year earlier. And hardly anyone notices.

Paul McCartney once did something similar outside Leicester Square tube station, playing Yesterday for spare change as part of a film shoot. A globally recognisable artist singing an iconic song in a busy public place. And again: almost no-one stops to listen.

It’s tempting to think this is about distraction, or maybe the death of our attention spans, or just the sheer pace of modern life. And those things might be part of the story, but the more revealing explanation is simpler and more general. It’s about context. If you’re leaving a stadium after a game, or rushing through a crowded station, you’re not in “spot a global superstar” mode. You’re in “get home” mode. The context of a situation shapes your expectations, and your expectations shape what you experience. Out of context, even something extraordinary can go missing.

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That idea - context as a guiding determinant of perception - runs throughout the rest of the chapter.

A good way to understand the impact of the surrounding context is to look at what happens when something genuinely is missing from your sensory input. You’re probably aware that you have a physiological blind spot: a small region of the retina where there are no light-detecting photoreceptors because that’s where the optic nerve exits the eye. In daily life you don’t notice it, partly because you have two eyes and your brain blends their images together. But when you test it with one eye (see below), the striking thing is not that you get a hole in your vision. You don’t. The object you’re looking at disappears and the brain simply fills in the gap with whatever pattern it thinks should be there. White background produces white. Black background produces black. The missingness is plastered over.

Position the screen about 35cm from your face (or between three and four times the distance between the cross and the dot if that is markedly different on your device. You might have to experiment with this positioning. Cover your left eye so you can’t see out of it at all. Line up your right eye with the cross on the left hand side of the page and focus on it. You may need to adjust the distance at which you hold the book slightly, but you will find a relatively large range of distances for which the black spot on the right hand side disappears. What remains, however, is not a hole in our field of vision. Instead the brain has filled in the gap with white, the colour of the background surrounding the black dot.

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The same basic thing happens in time as well as space. Your eyes make rapid saccadic jumps as they scan the world, but the in-between bits are not delivered to your conscious experience as a blur or a flicker. Instead, your brain down-regulates your visual input for the movement and fills the gap with the stable image after the movement has finished. This is known as saccadic masking. The result is that you feel like you saw the scene continuously, even though you didn’t. You can experience this happening in practice as part of the stopped-clock illusion: moving your eyes to focus on a clock face, the first “second” can feel strangely long, as if time has briefly frozen and then restarted – consistent with the visual system filling in the gap during which your eyes moved with the image input at the end of the movement.

Once you are aware of these effects, you start to understand why the phrase “Sorry Mate, I Didn’t See You”, often heard after traffic accidents, is not always a confession of carelessness. It can be a description of a real perceptual mechanism. When we pull out of junctions we often feel as if we’ve checked the whole panorama, but what we may actually have is a small set of snapshots stitched together, with the missing bits smoothed over. The stitching is normally a blessing. In the wrong place, at the wrong time, though, it can be disastrous.

That brings us to one of the most unsettling ideas in the chapter: the illusion of absence. This is not just failing to see something because it was behind an obstruction. It’s the stronger, more dangerous belief that the obstructed region is empty. Not “I couldn’t see what was there”, but “I effectively experienced it as not being there at all”.

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Driving is full of opportunities for this sort of mistake, because cars come with built-in occluders: pillars, headrests, passengers, windscreen stickers, mirrors that don’t quite show everything. The A-pillars of modern cars are particularly pernicious. They don’t need to be large to hide something important, and under certain conditions another road user can remain hidden behind a pillar for long enough that by the time they emerge into view there’s no time left to do anything useful.

We all know that blind spots exist. But perhaps what is harder to understand is that a driver can genuinely feel they have checked, genuinely feel the road is clear, and still have been wrong - because their brain has treated the occluded region as “seen and empty”, rather than “unseen and unknown”.

For a more general psychological handle on why this happens, we explore a phenomenon called amodal completion: our propensity to fill in gaps when objects are partially occluded. We do it all the time. In many cases it’s incredibly useful. A scene full of partial information becomes coherent because the brain completes it. The problem is that the same mechanism can also suppress alternative possibilities. Once the brain has completed a scene in one plausible way, it can be surprisingly hard to hold in mind what else might be behind the occluder.

The chapter then widens from occlusion to a broader class of context effects: situations where the same physical stimulus can be experienced in dramatically different ways depending on what your brain assumes about the surrounding conditions.

The classic visual example is light constancy. In the real world, your brain tries to keep the perceived lightness of surfaces relatively stable even as illumination changes. Normally we would regard that as a triumph of perception. But it also means that ambiguous lighting can produce wildly different interpretations of the same image. The Dress is the famous case: people genuinely saw The Dress in different colours because their brains made different assumptions about the illumination, and then corrected for it. The assumed context of the image was vital to how The Dress was perceived.

And although many of the examples in the book are visual, our other senses employ context to help them out as well. For example, it turns out that, labels matter for smell and names matter for taste. If you tell someone a pine-like odour is “Christmas tree” they don’t just enjoy it more than if you call it “disinfectant” - many people seem to experience it as a different smell altogether. Likewise, taste can be influenced by expectation: what you think you’re about to taste can change the taste you actually experience.

Hearing perhaps even more vulnerable to these context-dependent overwritings. The internet has produced some wonderfully compelling illustrations, like the Brainstorm/green needle clip, where the same ambiguous sound can flip depending on what you’re primed to hear. If you hear the stimulus thinking about one phrase, then this is probably what you’ll hear. But if you prime yourself by thinking about the other, then that’s what you’ll hear. Your expectations don’t merely bias a judgement after the fact. They shape the perception itself.

Although many of the stories in the chapter emphasises the places where context can mislead us to fill in the gaps incorrectly, there’s also a more useful side to context-aided perception. Cochlear implants can initially sound fuzzy and distorted to new users, and yet with time many people report that their perception adjusts. The brain reroutes in order to fill in the gaps using the context it previously learned. The same top-down machinery that can mislead us can also help us adapt, sometimes in ways that feel close to miraculous.

All of these phenomena sit under a broader umbrella of top-down processing: the brain starting with higher-level frameworks - memories, expectations, assumptions - and applying them to make sense of lower-level sensory inputs. It’s a key driver of what I call intrinsic missingness. It helps us cope with ambiguity and incomplete information.

Once you accept that expectations can change perception at this level, it becomes easier to believe that disagreements about the world are not always about one person being irrational or dishonest. Two people can be looking at the same thing and genuinely experiencing it differently, because they are bringing different contexts and different life histories to the scene. That doesn’t mean “everyone’s right”. But it does mean we should sometimes be a little less certain that our own interpretation is the only reasonable one, and a little more willing to ask: is there something missing here?

Buy the Book

A favour: pre-order the book

If you’ve enjoyed this summary, you’ll find much more detail in the book itself, with the stories, the science and the slightly uncomfortable implications.

If the book sounds like your sort of thing, please consider pre-ordering it. Pre-orders matter far more than most readers realise. They’re one of the strongest early signals that a book has an audience, which influences everything from how many copies are stocked to how widely it’s recommended.

Amazon: https://www.amazon.co.uk/Dont-Know-What-Youre-Missing/dp/1529438039

Bookshop.org (supports independent bookshops): https://uk.bookshop.org/p/books/you-don-t-know-what-you-re-missing-the-science-of-what-s-lost-and-how-to-find-it-kit-yates/497ab9dcf971763a

Thanks,
Kit

Over the next few weeks, I’m running a series of posts summarising the chapters of my new book, You Don’t Know What You’re M ss ng, ahead of publication on 4th June. The aim is to give you a feel for the shape of the book, the kinds of stories I use to illustrate the concepts, and the ideas I hope will stick with you long after you’ve finished reading it.

In this post I’m going to summarise Chapter 1, Making sense of it all.

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Late in the evening of 22nd December 1978, routine Alitalia flight 4128 took off from Rome, ferrying passengers home to Sicily for Christmas. Past midnight, on final approach to Palermo, the runway lights appeared as a thin strip against otherwise total darkness. The pilots descended a little more, held steady, waiting until they were above the runway - and then hit the sea. It is suspected they fell victim to a trick of the mind known as the “black hole illusion”. 108 of the 129 people on board lost their lives.

That story is extreme, but it illustrates the fundamental idea of the chapter: that the mind can be tricked and the consequences can be huge. More importantly, it’s not just pilots who suffer misconceptions and it’s not just extreme events that trigger them. We are all subject to intrinsic missingness - the way our own minds and bodies filter, compress, and sometimes simply drop information. Our senses can seem to make things disappear.

When you look around you, it’s easy to believe you are experiencing an objective reality. You can see the texture of the ground beneath you, you can feel the rough pages of a book, you can hear birds or traffic. It feels unmediated, as if the world is simply arriving in your mind fully labelled and correctly sorted.

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But the compressions and rarefactions of the air that hit your eardrums do not arrive with tags saying “blackbird” or “car engine”. The electromagnetic waves entering your eye don’t say “green” or “blue”, much less “grass” or “sky”. It is your brain’s job to interpret the signals it is constantly bombarded with and to figure out what they mean. This is the perception problem.

Perception is not simply the narration of a pre-written story. It is an inventive and creative act: the brain attempts to weave the threads of the signals it receives into a coherent narrative. Most of the time the working rules it uses are good enough for everyday life, which is precisely why we so readily assume they must be accurate. But “good enough to get by” and “true” are not the same thing.

Natural selection is driven by fitness - survival and reproduction - not by a commitment to metaphysical accuracy. Our sensory systems have evolved towards mechanisms that allow us to complete tasks efficiently. They spare us the world’s excruciating detail and present us with something closer to a usable interface than a perfect representation.

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That interface comes with a cost. In taking shortcuts that subjectify reality, in conjunction with our senses, our brains select which information we see and which we don’t. Information is filtered without us necessarily being able to see the mesh through which it is being strained. And because the system is designed to be seamless, the absences often don’t feel like absences. They feel like the world.

Much of the chapter is focussed then on the ways in which this filtering happens.

Sometimes it’s because our brains rely on implicit rules that usually work. These rules are so deeply embedded that we barely notice them - until they misfire. We can learn to compensate in familiar situations, but that doesn’t mean we literally “see” correctly. It means we’ve learned that our perception can be misleading, and we’ve learned how to behave accordingly.

Sometimes it’s because our senses conflate different properties of the world, so the signal doesn’t uniquely determine what’s out there. A simple example is touch. Metal cutlery can feel colder than a wooden spoon sitting in the same kitchen, even when both objects are at the same ambient temperature. What we “feel” as cold depends on heat transfer and material properties, not just temperature. So even something as basic as “how cold is this?” isn’t as straightforward to answer with our fallible sense as we might hope.

For me, the most practically relevant part of intrinsic missingness is the set of mechanisms that make us miss things even when they’re present.

Adaptation is one mechanism which causes things to go missing. It’s the reason smells fade. It’s why background noise can melt away. It’s why constant stimuli stop clamouring for attention. Adaptation is often beneficial because it stops us being overwhelmed, leaving us sensitive to change. And yet it can be disastrous when it makes us tune out the very thing we most needed to notice.

Here’s a pertinent example. Natural gas is odourless, so an unpleasant smell is added to make leaks detectable. But if you’re exposed to that smell over time you can become acclimatised, and then the warning signal stops feeling like a warning signal. In 2009 the Conagra food production plant was having new boiler fitted. Relying on smell rather than detectors, the workers fitting the boiler unknowingly allowed the room they were working in to fill gas which eventually ignited, resulting in multiple deaths and many injuries. The brain’s capacity to “get used to” a stimulus did exactly what it evolved to do - and in that context it was catastrophic.

Habituation is another filtering mechnism. Unlike adaptation, habituation isn’t that you stop perceiving the stimulus. It’s that you stop responding to it because experience has taught you it isn’t important. That can be sensible, but it can also be deadly. Alarm fatigue is the most obvious example: if an alarm goes off repeatedly and nothing happens, people learn that the alarm can be ignored. Expected alarms don’t provoke the same automatic responses as unexpected ones, which means that when the real emergency finally arrives, the system is already biased towards complacency.

Then there’s attention. We like to think we take everything in, but attention behaves more like a spotlight than a floodlight. When attention is engaged, we miss stimuli we are not expecting to perceive that fall outside of the narrow cone of the spotlight. This is the territory of inattentional blindness. It’s also why phone conversations can be dangerous in cars even when your eyes are on the road and your hands are on the wheel: the issue isn’t physical control, it’s the allocation of cognitive capacity.

Finally, there’s the extra twist that makes all of this so difficult to manage: metacognitive errors – errors in what we think about the way we think. We are not just prone to missing things. We are prone to being overconfident about our ability to notice. Even when people are told how common and substantial these misses can be, they still massively overestimate their own detection abilities. That overconfidence is part of the problem. It makes intrinsic missingness feel like something that happens to other people.

The aim here isn’t to encourage paranoia, or to suggest we should doubt every piece of information that comes through our senses. The key is not to doubt or question everything, but to be more conscious of our blind spots, and more alert to the situations in which we might be missing information without even realising it. Once you start seeing intrinsic missingness for what it is - a feature of the system, not an occasional glitch - you begin to understand why illusions work, why errors can feel impossible in retrospect, and why “I would definitely have noticed” is such a dangerous sentence.

A favour: pre-order the book

If you’ve enjoyed this summary, you’ll find much more detail in the book itself, with the stories, the science and the slightly uncomfortable implications.

If the book sounds like your sort of thing, please consider pre-ordering it. Pre-orders matter far more than most readers realise. They’re one of the strongest early signals that a book has an audience, which influences everything from how many copies are stocked to how widely it’s recommended.

Amazon: https://www.amazon.co.uk/Dont-Know-What-Youre-Missing/dp/1529438039

Bookshop.org (supports independent bookshops): https://uk.bookshop.org/p/books/you-don-t-know-what-you-re-missing-the-science-of-what-s-lost-and-how-to-find-it-kit-yates/497ab9dcf971763a

Thanks, Kit

Yesterday I wrote a short post about US Health Secretary Robert F. Kennedy Jr. (RFK Jr.) attempts on Wednesday to redefine percentages in his Senate testimony in order to justify the dodgy math(s) claims of his boss, President Donald Trump.

Yesterday we had an update, which shed a little more light on the thinking behind the dodgy math(s). Standing beside a seated Trump, RFK Jr. gives an account of his maths testimony in the Senate: “If the drug was $100 and you raise the price to $600 that’s a 600% rise.”

In fact, on Wednesday in the Senate, RFK had attempted to justify the same 600% figure, claiming a fall from $600 to $10 was a 600% fall.

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So yesterday’s claim started off differently but was already mistaken. An increase of $500 (from $100 to $600) as a percentage of the original $100 price is a 500% rise. We could argue that RFK was getting close to the correct math(s), but then comes the nonsense bit.

“Well if it [the price] drops from 600 to 100, that’s a 600% saving.”

Trump is seen to nod along with this and even interject with “Right” in the middle of RFK’s last sentence.

Whoa. Hold up. This doesn’t make sense at all.

If the price drops from $600 to $100 that’s a $500 drop. As a percentage of the $600 dollar price that’s about an 83.3% drop.

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RFK’s error is reminiscent of a fundamental but surprisingly common mathematical mistake and something that I’ve written about extensively in How to Expect the Unexpected: the nonlinearity of reciprocal relationship

Perhaps surprisingly, a seemingly big percentage rise can be wiped out by a smaller percentage fall. Similarly, even a small percentage fall always needs to be offset by a larger percentage rise. At first it seems counter-intuitive that, despite the Nasdaq growing over 400% during the dot-com bubble’s ascendancy, a fall of just 70% was enough to erase almost all of the gains that were made over the preceding five years, but percentage rises and falls do not combine additively as our linear expectations would lead us to believe.

If we invest £100 in a company and their share price falls by 10% to £90, a 10% rise from that position only takes us back up to £99. The rise required to get us back to £100 is just over 11%. For bigger percentage losses, the commensurate rise required to level-up is even larger. For a fall in share prices of one quarter, the corresponding rise to recoup the loss is one third. For a fall of 50% the share price must double – a rise of 100% - to leave us even. When the Nasdaq rose 400% an 80% fall would have been enough to set it back to square one. This is an example of a nonlinear relationship at play.

The relationship (illustrated for different percentage losses in the figure above) is clearly nonlinear (the tops of the bars don’t form a straight line). In this case it is known as a reciprocal relationship. The sense in which the descriptor ‘reciprocal’ should be understood here is in its interpretation as an ‘inverse’ or ‘inverted’ relationship. For example, the reciprocal of two is a half, 1/2. When two reciprocal numbers are multiplied together, they return us to one - the whole. To recoup a loss of half (1/2) in the share price (a 50% decrease) the share price must double (x2 - a 100% increase).

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Exploiting our propensity to think linearly in this context is one way in which companies offering financial products can cook the books. By showing a percentage return averaged over a number of years they can make their performance look better than it actually is. For example, a fund which gains 50% in one year and loses 50% the next year doesn’t break even at the end of two years. The 50% loss and the 50% gain don’t simply add together and cancel each other out. Instead, the relative gain and loss must be multiplied together: 50% of 150% isn’t 100%, it’s 75%, corresponding to a 25% loss over the two years.

In How to Expect the Unexpected I go on to write about how these sorts of tricks are also exploited by organisations or individuals wishing to push a particular agenda when the facts don’t support their viewpoint. And it feels like this is exactly what RFK/Trump are trying to do here – to win people over with some impressive sounding numbers and hope that they don’t look too closely.

Perhaps the most frustrating (note not surprising) thing about the whole affair is RFK’s unwillingness to admit that he made a miscalculation or simply misunderstood percentages. Seemingly, this regime would prefer to try to rewrite the fundamental rules of mathematics rather than climb down over a simple mistake.

While the American public fail to hold their politicians accountable at the ballot box for these transgressions, they will continue to be saddled with the post-truth politics of bluster and bullshit that characterise populism, where facts take a back seat to spin and disinformation. Rather than voting for politicians that insist they are making America great again, will a majority of Americans be able to choose politicians who improve Americans’ resilience to fake news by (to borrow Andrew Yang’s campaign slogan) Making America Think Harder (MATH)?

*Header image by Karim MANJRA on Unsplash

Yesterday, US Health Secretary Robert F. Kennedy Jr. (RFK Jr.) appeared before the Senate Finance Committee.

Democrat Senator Elizabeth Warren took him to task about the prices on the federal government’s new prescription drug website TrumpRx. She argued that some drugs on the website were massively overpriced compared to generic drugs available on the open market.

Perhaps, by now, the American public should not be surprised. What really caught my attention during Senator Warren’s question, however, was Trump’s claim that drug prices on TrumpRx are lower than those of comparable countries by as much as 600%.

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Let’s just stop for a second to try to interpret what that means.

If you reduce the price by 50% you have halved the price.

If you reduce the price by 100% then you are effectively giving it away for free.

Plausibly, you can only reduce the price by between 0 and 100%. Arguably you could reduce the price by a negative percentage, but that would actually be an increase in the price, so let’s not go there.

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What does it mean then to reduce the price by 600%. Well, theoretically it means you are paying people to take your product away – paying them five times as much as it was originally worth.

So what does it mean when President Trump says he has reduced the price by 600%? In reality it means nothing. It is nonsense. But to people not paying attention, perhaps it sounds impressive.

And this is not Trump’s first dodgy percentage rodeo. Remember when he claimed the price of eggs had come down by 400%? Or when he claimed to have reduced the price of drugs by up to 1500%? Clearly, Trump has his own ‘special’ way of calculating percentages, one in which maths goes out of the window.

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Perhaps the most surprising thing in RFK Jr.’s response to Warren’s questioning was that he tried to toe the line and justify Trump’s calculations:

“President Trump has a different way of calculating, there’s two ways of calculating percentages,” he argued. “If you have a $600 drug and you reduce it to $10, that’s a 600% reduction.”

What? Feasibly you might say that is a 60-fold reduction, but not a 600% reduction. That doesn’t make sense. The actual percentage reduction is about 98.3% (again it has to be between 0 and 100%), not 600%.**

This sort of redefinition of a basic mathematical concept feels, to me, almost more Orwellian than any other piece of Newspeak the US regime has yet come up with. And that is saying something.

Can you just go about redefining how you calculate percentages? Only in a world where Ignorance is Strength.

** In the 18th and 19th Century, in parts of the English-speaking world, it was not uncommon to refer to changes in stock price relative to a baseline (or par value) of £100. A fall of £800 - from £1000 to £200 say - may have been referred to as a fall of 800 percentage points (£800 being 800% of the baseline £100). However, it is clear that this is not what Kennedy had in mind. Even under this esoteric system, the change that Kennedy proposed from $600 to $10 is a fall of $590, which would have been referred to as a fall of 590 percentage points relative to a baseline of $100, not a 600% fall. The fact that Kennedy was not attempting to revive this old method became even more clear with his follow-up statements, when he claimed that, in fact, a fall from $600 to $100 was a 600% fall. Even under the arcane system, this fall of $500 relative to a baseline of $100 would have been referred to as a fall of 500 percentage points, not a 600% fall.

*Header photo by Myriam Zilles on Unsplash

I’m delighted to share some news with you, my wonderful substack audience: my new book is coming out in the UK on 4^th June.

It’s called You Don’t Know What You’re M ss ng, and it’s about something that lies hidden in the background of almost everything we do, yet rarely gets named. Missingness.

Not missingness as in “I can’t find my keys” (though that too). By missingness here, I mean the hidden gaps that shape how we think, decide, argue, and even how we determine what counts as true.

If you’ve ever confidently sung the wrong lyric, trusted a statistic that later fell apart, or felt sure you understood a situation only to realise you’d been working with part of the story, then you already know the subject. You just might not have known you knew it!

Buy the book here

Different kinds of Missingness

In the book I explore three complementary types of missingness that show up everywhere, from everyday life to science, to politics, and even the stories we tell to ourselves.

1) Intrinsic missingness

Sometimes missingness is produced by our own minds/bodies. Our brains are brilliant editors. They filter out “noise”, fill in blind spots, and create a smooth, coherent experience from messy inputs. That’s usually a feature, not a bug - but it can also cause us problems. It can make us confidently wrong, and it can make it hard to notice mistakes (especially the ones we we’re “sure” we’d spot).

2) Extrinsic missingness

Sometimes the missingness is imposed from outside. Data can be biased before it even reaches us. Samples can be skewed. Evidence can be selectively reported. Information can be suppressed. In these cases, the problem is not that our brains are inventing a story - it’s that the story we’re handed already has parts removed, often without any of the tell tale signs that we might not be seeing the whole picture.

3) Constructive missingness

And then there is the kind that I find most philosophically interesting: when absence itself becomes informative. Sometimes not finding something, after a proper search, tells you something important. “Negative data” can be as revealing, if not more so, than traditional positive date. Silence can speak. Reasoning from what’s missing isn’t just gimmick of detective stories, but an important part of what underpins the scientific method, and the way we build confidence in our beliefs.

Know what you’re missing

My hope is that the stories in this book do two things.

Firstly, that they make you notice the occasions when The Missing is likely to be distorting your picture of reality - when your senses are editing too aggressively, when context is doing too much work, when your data may be biased, when your “certainty” is dependent on an unseen gap.

Secondly, that they give you practical tools; ways to slow down the brain’s overconfident shortcuts, warning signs that something important may have been filtered out, and questions to ask when you suspect the absence itself is trying to tell you something.

Over the next seven weeks leading up to publication (4th June) I’m going to share summaries of each of the seven chapters so you can get a better feel for the book and the interesting stories that are contained within it.

A favour: pre-order the book

You’ve been so great in reading the writing I’ve been sharing on this substack for the last few years, but here’s the slightly awkward bit. If the book sounds like your sort of thing, please consider pre-ordering it.

Pre-orders matter far more than most readers realise. They are one of the strongest early signals that a book has an audience, which influences everything from how many copies are stocked to how widely it’s recommended. For authors, it’s one of the most direct ways you can help a book find its readers.

And if you’ve been enjoying my writing here, buying the book is also a rather lovely way of supporting the work - and helping to make the next one possible.

You can pre-order You Don’t Know What You’re M ss ng from Amazon here:

https://www.amazon.co.uk/Dont-Know-What-Youre-Missing/dp/1529438039

or to support independent bookshops around the country from Bookshop.org here:

https://uk.bookshop.org/p/books/you-don-t-know-what-you-re-missing-the-science-of-what-s-lost-and-how-to-find-it-kit-yates/497ab9dcf971763a

And if you’re feeling especially generous: please do share this post with a friend who likes science, data, psychology, or the occasional wobble in reality would. It would mean a lot.)

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Thanks for your continued (and often unseen support) that makes my writing possible.

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Last week, a colleague of mine sent me a picture of a leaflet from his local Reform Party that had been posted through his door. The prominently positioned bar chart (have a look at the image below) made the popularity of Lib Dems (yellow) and Reform (turquoise) appear almost neck‑and‑neck, while Conservatives (dark blue) seemed to be lagging a long way behind. The headline of that section of the leaflet was “Only Reform can challenge the Lib Dems here”.

Despite the fact that the leaflet went out across numerous different wards in Bournemouth, the results that were purportedly graphed were actually from a by-election in a single ward in the Bournemouth area: The Talbot & Branksome Woods by-election last September. The results from this ward have, at best, tenuous relevance to the other wards. Presenting a local snapshot as if it were a universal trend is a classic election material fiddle. But this isn’t the leaflet’s worst statistical crime.

If you follow the link to the results you can see that the Liberal Democrats won with 910 votes (32.4%). Reform UK came second on 791 (28.2%), with the Conservatives just 21 votes behind on 770 (27.4%). That’s a 4.2 percentage point gap between the Lib Dems and Reform, and a mere 0.8 percentage points between Reform and the Conservatives.

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So why do the bars on the chart appear to show Reform so close to the Liberal Democrats while the Conservatives are so far behind. An innocent mistake? Unlikely. We’ve seen numerous examples of party political election material featuring a number of graph crimes, from truncated y-axes to tilted bar charts and even vibes-based bar charts that are more or less made up.

These are textbook misuses of a bar chart. Bar charts typically encode magnitude through length, so when you compress the scale or subtly change the baseline, you can exaggerate differences you like and shrink those you don’t. Tilting the bars on the chart can accentuate or diminish the perceived differences between parties’ popularity. And if you can’t fiddle the charts using the actual numbers then just guessing at what the figures should be is a popular option.

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When challenged, a local Reform representative who was involved in creating the leaflet, didn’t apologise or plead mathematical incompetence; he explained that the misleading bar chart was a deliberate tactic. “…it’s done for exactly this reason - to get people (mainly conservatives) sharing and complaining about it - as that way the conversation is about ‘how much Reform beat them by’ rather than who is polling higher. Having them share that very same image on their social media is doing our work for us.”*

For me there is something qualitatively different about this. I found the frank admission that the graphic was engineered to provoke outrage‑amplification - a deliberate ploy for attention relying on opponents spreading their misleading message for them – quite shocking. Rather than correct, retract or apologise, the response to being caught out was to celebrate the deception as a strategy.

If I were a voter in the area I would certainly think hard about voting for a party that is so gratuitous in its celebration of the seemingly deliberate deceit of the electorate. I would also question its opinion of my intelligence as a voter if it thought that such an obvious manipulation would be enough to win my vote.

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Local statistician and recipient of the leaflet Barney Maunder-Taylor said “This has crossed a line for me, taking the graph beyond the disingenuous sort of misrepresentation that I’m used to seeing from political leaflets and placing it firmly in the territory of outright lying. The numbers were deliberately changed, because the real data didn’t provide enough evidence for the message that the politician wanted to convey.”

All this matters because trust is cumulative in politics - won slowly and squandered quickly. If we normalise the idea that it’s OK to play fast and loose with data visualisation - that graphics are there to generate heat rather than light - we train voters to distrust all numbers on election paraphernalia, not just the cooked ones.

But maybe this is what’s needed: for the electorate to look at the numbers with due scepticism. We need people to common-sense check the differences between the bars to see if they stack up. If something doesn’t seem right then compare the picture to the official source - in this case, the council’s own tally, which gives the lie to the fanciful claim “only Reform can challenge” when the third place party were within 21 votes of them.

Above all, we need voters to remember that graphics can be corrupted. There are myriad ways to lie with statistics. The next time a leaflet insists your vote “can only” do one thing, check to see whether their story survives contact with reality. If it doesn’t, use your vote to demonstrate that you won’t be taken for a fool.

*I’ve seen no evidence that this tactic is being successful. Through a reverse image search I’ve failed to find any online versions of the image on the internet at all and none of the local Conservative social media accounts seem to have posted anything about it.

I am a big fan of Sherlock Holmes. I’ve read the whole canon multiple times over. I love how Holmes uses analytical reasoning to unravel problems that look mysterious, but ultimately prove to have simple explanations. Yes of course some of the things he spots and the leaps of reasoning he makes to devise his solutions are occasionally far-fetched, but each case has a degree of plausibility about it.

***Warning this article contains spoilers about the new Amazon Prime series Young Sherlock. If you don’t want to ‘ruin the surprise’ then go and watch the series now – or don’t bother.***

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I have just finished the first series of Guy Ritchie’s Young Sherlock. Sadly the “mysteries” that are set up and solved in this series don’t come close to offering the intrigue and cerebral pleasure of the original stories. In the books, once a mystery is unravelled we can usually follow Holmes’ train of thought recognising why his solution makes sense.

In the TV series the problems feel contrived; set up as an afterthought to fit the overarching plot line. The solutions seem to spring miraculously to Holmes with no convincing explanation, relying heavily on coincidence or chance, failing to offer the same satisfaction as the books.

The young Holmes himself seems inconsistent. Sometimes he shows flashes of genius, while at others he is stumped by seemingly simple problems. He is not the consistent indefatigable reasoner we have come to know and admire through the books. Many of his insights in the series rely on him being coincidentally at the right place at the right time.

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Maybe we can put those character inconsistencies down to the younger version of the famous detective, not having honed his craft sufficiently. This is, perhaps, where he cut his teeth and learned his trade.

I think that the most damning thing you can say about the series is that, if it weren’t called “Young Sherlock” and the characters names were blanked out, you probably wouldn’t guess it was supposed to be a Sherlock Holmes story. There are some funny parts of the series. It is light-hearted and warm in parts, but the errors and incongruities are too distracting for me to really enjoy it.

Mathematical crimes

A key part of the plot relies on mathematics. Holmes first meets Moriarty in a maths lecture. All the male students are sat around in mortar boards (naff, but whatever!) while the Professor points at stuff he’s already written on the board (also unrealistic, but whatever!).

As the professor waffles on, Sherlock can’t help jumping in to correct the professor on something he’s said (which is seemingly largely unrelated to what’s written on the board he is pointing at – lets let that go as well). It turns out that what the professor is saying actually is wrong, but that the point Sherlock corrects him on doesn’t need correcting. Indeed Sherlock makes the same mistake in his correction as the Professor makes in his original speech.

The maths on the board is interesting enough. It’s demonstrating the solutions to a quintic equation x⁵ + x⁴ + x³ + x² + x + 1 = 0. As shown nicely in the video below, the equation has five solutions. Some of these solutions involve imaginary numbers (this will turn out to be an important plot point).

Imaginary numbers result from taking the square roots of negative numbers. In the maths many of us will have learned at school we are taught that ‘a positive times a positive makes a positive’ and that ‘a negative times a negative also makes a positive’. So square numbers (when you multiply a number by itself) should seemingly always be positive. But if we want to take the square root of -1, say, then we need to venture into the realm of imaginary numbers to find a solution. To solve this problem we define a new number:

and then we can find the square root of any negative number as a multiple of i.

It’s plausible that the sort of equation written on the board might appear in an early first year undergraduate tutorial. Something approaching a passable solution is written on the board in excruciating detail (the sort of detail you wouldn’t use at school, let alone in a maths degree at Oxford), but ultimately there are mistakes in the maths, which is annoying.

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Towards the end of the lecture the professor sets the students homework to find all the solutions to the equation, even though they are already written on the board. Despite this, the end of the scene sees Sherlock spending some time trying to think of the solutions before Moriarty comes up and shows him two of the five solutions (as if they were the only ones), which he too writes down incorrectly, but in a different way to the incorrectness already on the board.

As he writes down the complex solution (a solution which contains both real and imaginary numbers) he says “These solutions, they’re not real. They’re imaginary.” (although technically he means complex, but fine!). He goes on to say, “That means even if you can’t see the target, you can still shoot for it.” Which is nonsense, even as a metaphor.

OK, so lots of annoying mistakes, which undermine the credibility of the maths professor, Holmes and Moriarty (who in Doyle’s canon will eventually become a maths professor himself) as mathematicians, but this isn’t the worst of the maths blunders.

Creeping Death and Corpley roots

In the last episode, Holmes and his team are battling to halt the distribution of a deadly chemical weapon known as the “Creeping Death”. They find a scrap of paper in a secret room which they say is the “equation for turning it into the creeping death.” “It” here is a mineral “Thanatite” that the baddies have been extracting in large quantities to make stockpiles of the nerve agent.

I was expecting to see some complex chemical reaction formulae sketched on the page showing the different reactions needed to turn this mineral into the deadly gas. But when it’s held up to the camera, I was surprised to see the mathematical equation:

What does this have to do with the chemical process for creating the deadly nerve agent?

Nothing, it turns out. Or at least nothing I can imagine. In fact it’s a device to allow Holmes and Moriarty to hark back to that moment in the lecture theatre when they first met “We create new numbers, but the numbers, they aren’t real. They’re imaginary. Imaginary numbers created to solve unsolvable equations. Two solutions, opposite or complementary. There’s always another solution to every equation. Positive to mirror… the negative.”

This is all passable. Of course not every equation has two solutions. z – 1 = 0, has only one solution: z = 1. But it’s true for common class of equations (polynomials with real coefficients) that, if the equation has complex solutions, those solutions come in what are known as complex conjugate pairs and sometimes they are called complementary to each other. So this isn’t completely wild. But what follows is just nonsense.

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“If we have the positive equation… then we can come up with the negative. And thus create a compound to neutralise the threat of Creeping Death.”

Perhaps they meant “positive solution”, because equations aren’t positive or negative, but either way it doesn’t really make sense. Yet there’s more. Moriarty goes on to point out that they have a problem, “This equation is not finished.” By this I think he means that the three solutions to the equation are not written out explicitly.

One solution, z = -6 is given. Indeed if you plug z = -6 into (z³ + 4z² - 10z + 12 = 0) you find it does indeed satisfy the equation. What is on the remainder of the page is a reformulation of the equation (a factorisation) which shows that the remaining solutions can be found by solving z² – 2z + 2 = 0 a quadratic equation.

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A quadratic equation is just an equation built around a squared term (in this case z²), which has two solutions. The formula for the solutions may be familiar to GCSE-age students (normally aged 15 to 16 years old) across the world.

For a general quadratic equation az² + bz + c = 0, the two solutions are

Yet, we are supposed to believe that, despite having supposedly solved a far more complicated equation than this in the first episode (in which he probably used the quadratic formula), Moriarty seemingly can’t find the solution to this much simpler equation. So stumped is Moriarty – the future maths professor and current Oxford maths student – that he spends precious time, as a bomb is about to detonate, searching for this missing solution. He almost loses his life because this mathematical prodigy apparently can’t solve a quadratic equation that many GCSE students would be able to rattle off.

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The piece of paper he eventually finds at the last minute (see the image below), with the last few lines of the solution, contains an incorrect statement of the quadratic formula, although the solutions are at least correct. But there is a more damning indictment of the whole charade to come. In summing up the solutions, the precious paper reads:

“THERE ARE 2 CORPLEY ROOTS ONE POSITIVE, ONE NEGATIVE”.

This struck me as being very weird. I had never heard of CORPLEY ROOTS before. And then I realised, the word was supposed to be COMPLEX ROOTS (meaning a mix of real and imaginary parts and roots just meaning solutions to the equation). It was as if the piece of paper had been generated by AI (notoriously not good at generating coherent text) and no one had thought to check it. Arguably the most important piece of paper in the whole show, providing the motivation for the characters to do what they did for most of the series, was complete bullshit. It was as if the plot hinged on a crucial piece of physics and they had written Einstein’s theory General Melativity or Newton’s Laws of Botion.

I appreciate my dissection of the maths is high-grade nerdery. Most people will have watched the series without pausing like I did (as in literally freeze-framing!) to look at the maths and probably won’t have noticed. That’s fine.

This is also not a crack at people who aren’t expert in maths. Not by any means. I don’t have a problem if people don’t know or even don’t care deeply about maths (although I try to encourage everyone to appreciate the importance of our subject). There are many subjects of which I myself have only scant knowledge and little passion for. That’s fine. But, and I suppose this is the overriding message of this piece, if maths is going to be a pivotal plot point in your blockbuster series then you’d sure as shit better get yourself a maths consultant who knows what they’re doing.

As well as writing this substack, I also have an actual job as a Professor of Mathematical Biology and Public Engagement at the University of Bath. My group has a new open access paper out this week in the journal Advanced Science. It’s all about consensus formation and consensus change. Together with the press team at the University of Bath we’ve put together an explainer/press release for the paper. Have a read below.

Trying to persuade people to abandon deeply held views often backfires, leaving groups entrenched and unable to move forward. A new study by researchers at the University of Bath proposes a strategy that is both surprising and more effective: encourage neutrality.

The researchers, led by Professor Kit Yates (me!) from the Department of Mathematical Sciences, found that when individuals are encouraged to step back and adopt a neutral position – for example by abstaining in a vote – groups become more responsive, decisions become easier to reach, and shifts in consensus happen more smoothly.

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Neutrality does not stall progress – it creates valuable breathing space in which people can reassess their stance, making it easier for a consensus to form or for a group to change its mind when circumstances evolve.

Professor Yates said: “Allowing people to take a neutral stance creates breathing space for reassessment, making it easier for a consensus to form or for a group to change its mind.

“By recognising neutrality as a feature – not a bug – of group decisions, our study resolves a long standing trade-off: you don’t need elaborate, many person dynamics or sophisticated social structures for consensus and flexibility to emerge.

“Instead, once neutrality is allowed as an option, very simple interactions between pairs of individuals, where A influences B or B influences A, are enough to produce the observed group level behaviour.”

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In the new study – published today in Advanced Science – the researchers developed a simple mathematical model to explore how groups make decisions. Their model shows that groups can reach agreement in two ways.

One is the familiar route of persuading undecided individuals to join one side. The other, which has received much less attention, is a ‘de-escalation’ route, in which disagreement pushes people into a neutral state before they later choose a side independently.

The team found this de-escalation route to be particularly effective, allowing groups to change direction more quickly. This occurs because the number of active decision-makers becomes smaller when more individuals become neutral, giving chance a greater influence and allowing a new consensus to form faster.

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Neutrality is an option for both animals and humans

The researchers tested their finding in two real systems: locusts and humans.

In marching locusts, they found clear evidence that whenever a swarm switches direction, it first enters a brief phase where many locusts stop moving, effectively becoming neutral.

With most of the group paused, only a small minority remains on the move. Those few individuals have a much stronger influence on what happens next. This temporary shrinking of the active group magnifies small fluctuations, allowing a new collective direction to take hold quickly.

Next, the team ran voting games with human participants and found that when abstention was allowed, groups shifted their overall decision more quickly and cleanly than when the option of abstention was removed.

The researchers envisage their behavioural insight scaling from animal groups and voting games to boardrooms and online communities. The findings suggest practical strategies for adaptive decision making: if you want to overturn an entrenched consensus, it can be more effective to cool down strong opponents so they adopt a neutral stance, rather than targeting only the stereotypical ‘floating voter’.

“It might be annoying when someone is on the fence about an important topic that you feel passionately about, but in fact this can be a useful strategy to help groups make better decisions in the long run,” said co-author Professor Tim Rogers.

“Our model and experiments suggest a de-escalation tactic speeds up responsive consensus change.”

The team carrying out the study were University of Bath academics Professors Kit Yates and Tim Rogers from the Department of Mathematics, Dr Janina Hoffmann from the Department of Psychology and Dr Andrei Sontag, (now at University College London).

I’ve always loved Google’s Doodles honouring famous scientists or nice pieces of science and maths. I’m also a big fan of Pi Day and I’ve written celebrating the mathematical constant before. So when I was contacted a few weeks ago and asked if I would act as the math(s) consultant for Google on a Doodle they were planning for π Day (3.14 in US date format) it was a no-brainer for me.

The designers of the Doodle told me they wanted to explain Archimedes’ method for approximating pi. Although there were approximations of π before Archimedes (from Babylon and Egypt – both within 1% of the true value) it wasn’t until Archimedes that we had an algorithm for rigorously calculating pi. Around 250 BCE Archimedes used a method known as the method of exhaustion to bound π above and below.

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His idea was that he could draw two regular polygons (closed two-dimensional shapes made of straight lines of the same length) – one on the outside (circumscribing) of a circle and one on the inside (inscribing). It would be possible to calculate the perimeters of these polygons. Knowing that the perimeter of the inscribing polygon would be smaller than that of the circle (the circumference), while the circumscribing polygon’s perimeter would be bigger, he would be able to bound the circumference of the circle between two values. Finally, recognising that the circumference of the circle is given by 2πr, where r is the radius, he would be able to bound π between two values. Better still, if the radius, r, were 1 then this would make the calculation even easier.

Archimedes started out using hexagons as the bounding polygons, likely because they could be broken down into equilateral triangles making the calculation of the perimeter easier. But hexagons don’t look very much like circles (see the first still from the Google Doodle below). This initial attempt would have bounded π between 3 and approximately 3.4641. Not very good, given the Egyptians and Babylonians had both done much better centuries earlier.

But Archimedes’ real trick was to develop an algorithm which allowed him to calculate the perimeters of polygons with twice the number of sides from polygons whose perimeters he already knew. So once he had the perimeters of the 6-sided hexagons, he could go on and calculate the perimeters of the 12-sided dodecagons (see figure below).

This is the formula which is displayed on each side of the Google Doodle. a_n is the perimeter of the circumscribing polygon with n sides and b_n that of the inscribing polygon with n sides, so that b_n < 2π < a_n. The first step in Archimedes’ formula is to calculate the perimeter of the circumscribing polygon with 2n sides, a_2n, using a_n and b_n, and then to calculate the perimeter of the inscribing polygon with 2n sides, b_2n, using a_2n and b_n.

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Once he had the perimeters of the dodecagons he could iterate to find the perimeters of the 24-gons (see below) and thence the 48-gons. Each time his bounds on π improved.

Archimedes only had the power to take the calculations to 96-sided polygons (don’t get me wrong, this is still incredibly impressive). If he could have done the calculations precisely then he would have found bounds for π between 3.1410 and 3.1427. As it was, he made approximations during his calculations in order to express his final answers as fractions. He ended up bounding π between 223/71 and the famous approximation⁠ ⁠22/7⁠ (that is, in decimal, 3.1408 < π < 3.1429). This upper bound is accurate to 0.04% and gives rise to Pi Approximation Day 22/7 (the 22^nd of July) in the UK date format. It’s worth noting that Pi Approximation Day is actually a slightly more accurate approximation to π than the 3.14 of π Day itself.

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Archimedes’ approximation of π was the most accurate known for around 400 years until Ptolemy gave a value of 3.1416. Around 480 AD, Zu Chongzhi bounded π between 3.1415926 and 3.1415927 using Archimedes’ method of bounding a circle by polygons with 12,288 sides. Indeed Archimedes’ method would remain the dominant method for approximating π for centuries after his death until it eventually gave way to infinite series approximations. So associated did the constant become with Archimedes, that π is still occasionally referred to as Archimedes’ constant.

As for me, I had a great time working with the creative team in order to bring Archimedes’ method to life in the 21^st century. If you’d like to have a look at the Doodle in action then you can find it here.

At a secret meeting in 2025, some of the world’s leading mathematicians gathered to test OpenAI’s newest large language model, o4-mini.

Experts at the meeting were amazed by how much the model’s responses sounded like a real mathematician when delivering a complex proof.

“I’ve never seen that kind of reasoning before in models,” Ken Ono, a professor of number theory at the University of Virginia said at the time. “That’s what a scientist does.”

But was the artificial intelligence (AI) model being given more credit than it deserved? And do we run the risk of accepting AI-derived proofs without fully understanding them?

Ono acknowledged that the model might be giving convincing — but potentially incorrect — answers.

“If you say something with enough authority, people just get scared,” Ono said. “I think o4-mini has mastered proof by intimidation; it says everything with so much confidence.”

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In the past, confidence and the appearance of a good argument were good signs because only the best mathematicians could make convincing arguments, and their reasoning was usually sound. That has changed.

“If you were a terrible mathematician, you would also be a terrible mathematical writer, and you would emphasise the wrong things,” Terry Tao, a mathematician at UCLA and the 2006 winner of the prestigious Fields Medal, told Live Science. “But AI has broken that signal.”

Naturally, mathematicians are beginning to worry that AI will spam them with convincing-looking proofs that actually contain flaws that are difficult for humans to detect.

Tao warned that AI-generated arguments might be incorrectly accepted because they look rigorous.

“Unfortunately, the AI is much better at sounding like they have the right answer than actually getting it … right or wrong; they will always look convincing,” Tao said.

He urged caution on the acceptance of AI ‘proofs’. “One thing we’ve learned from using AIs is that if you give them a goal, they will cheat like crazy to achieve the goal,” Tao said.

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While it may seem largely abstract to ask whether we can truly ‘prove’ highly technical mathematical conjectures if we can’t understand the proofs, the answers can have significant implications. After all, if we can’t trust a proof, we can’t develop further mathematical tools or techniques from that foundation.

For instance, one of the major outstanding problems in computational maths, dubbed P vs. NP, asks, in essence, whether problems whose solutions are easy to check are also easy to find in the first place. If we can prove that, we could transform scheduling and routing, streamline supply chains, accelerate chip design, and even speed up drug discovery. The flip side is that a verifiable proof might also compromise the security of most current cryptographic systems. Far from being arcane, there is real jeopardy in the answers to these questions.

Proof is a social construct

It might shock non-mathematicians to learn that, to some extent, human-derived mathematical proofs have always been social constructs — about convincing other people in the field that the arguments are right. After all, a mathematical proof is often accepted as true when other mathematicians analyse it and deem it correct. That means a widely accepted proof doesn’t guarantee a statement is irrefutably true. Andrew Granville, a mathematician at the University of Montreal, suspects there are issues even with some of the better-known and more scrutinised human-made mathematical proofs.

There’s some evidence for that claim. “There have been some famous papers that are wrong because of little linguistic issues,” Granville told Live Science.

Perhaps the best-known example is Andrew Wiles‘ proof of Fermat’s last theorem. The theorem states that although there are whole numbers where one square plus another square equals a third square (like 3²+4²=5²), there are no whole numbers that make the same true for cubes, fourth powers, or any other higher powers.

Wiles famously spent seven years working in almost complete isolation and, in 1993, presented his proof as a lecture series in Cambridge, to great fanfare. When Wiles finished his last lecture with the immortal line “I think I’ll stop there,” the audience broke into thunderous applause and Champagne was uncorked to celebrate the achievement. Newspapers around the world proclaimed the mathematician’s victory over the 350-year-old problem.

During the peer-review process, however, a reviewer spotted a significant flaw in Wiles’ proof. He spent another year working on the problem and eventually fixed the issue.

But for a short time, the world believed the proof was solved, when, in fact, it hadn’t been.

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Mathematical verification systems

To prevent this sort of problem—where a proof is accepted without actually being correct—there’s a move to shore up proofs with what mathematicians call formal verification languages.

These computer programs, the best known example of which is called Lean, require mathematicians to translate their proofs into a very precise format. The computer then goes through every step, applying rigorous mathematical logic to confirm the argument is 100% correct. If the computer comes across a step in the proof it doesn’t like, it flags it and doesn’t let go. This encoded formalisation leaves no room for the linguistic misunderstandings that Granville worries have plagued previous proofs.

Kevin Buzzard, a mathematician at Imperial College London, is one of the leading proponents of the formal verification. “I started in this business because I was worried that human proofs were incomplete and incorrect and that we humans were doing a poor job documenting our arguments,” Buzzard told Live Science.

In addition to verifying existing human proofs, AI, working in conjunction with programs like Lean, could be game-changing, mathematicians said.

“If we force AI output to produce things in a formally verified language, then this, in principle, solves most of the problem,” of AI coming up with convincing-looking, but ultimately incorrect proofs, Tao said.

Buzzard agreed. “You would like to think that maybe we can get the system to not just write the model output, but translate it into Lean, run it through Lean,” he said. He imagined a back-and-forth interaction between Lean and the AI in which Lean would point out errors and the AI would attempt to correct them.

If AI models can be made to work with formal verification languages, AI could then tackle some of the most difficult problems in mathematics by finding connections beyond the scope of human creativity, experts told Live Science.

“AI is very good at finding links between areas of mathematics that we wouldn’t necessarily think to connect,” Marc Lackenby, a mathematician at the University of Oxford, told Live Science.

A proof that no one understands?

Taking the idea of formally verified AI proofs to its logical extreme, there is a realistic future in which AI will develop “objectively correct” proofs that are so complicated that no human can understand them.

This is troubling for mathematicians in an altogether different way. It poses fundamental questions about the purpose of undertaking mathematics as a discipline. What is ultimately the point of proving something that no one understands? And if we do, can we be said to have added to the state of human knowledge?

Of course, the notion of a proof so long and complicated that no one on Earth understands it is not new to mathematics, Buzzard said.

“There are papers in mathematics where nobody understands the whole paper. You know, there’s a paper with 20 authors and each author understands their bit,” Buzzard told Live Science. “Nobody understands the whole thing. And that’s fine. That’s just how it works.”

Buzzard also pointed out that proofs that rely on computers to fill in gaps are nothing new. “We’ve had computer-assisted proofs for decades,” Buzzard said. For instance, the four-colour theorem states that if you have a map divided into countries or regions, you’ll never need more than four distinct colours to shade the map such that neighbouring regions are never the same colours.

Almost 50 years ago, in 1976, mathematicians broke the problem into thousands of small, checkable cases and wrote computer programs to verify each one. As long as the mathematicians were convinced there weren’t any problems with the code they’d written, they were reassured the proof was correct. The first computer-assisted proof of the four-colour theorem was published in 1977. Confidence in the proof built gradually over the years and was reinforced to the point of almost universal acceptance when a simpler, but still computer-aided, proof was produced in 1997 and a formally verified machine-checked proof was published in 2005.

“The four-colour theorem was proved with a computer,” Buzzard noted. “People were very upset about that. But now it’s just accepted. It’s in textbooks.”

Uncharted territory

But these examples of computer-assisted proofs and mathematical teamwork feel fundamentally different from AI proposing, adapting and verifying a proof all on its own — a proof, perhaps, that no human or team of humans could ever hope to understand.

Regardless of whether mathematicians welcome it, AI is already reshaping the very nature of proofs. For centuries, the act of proof generation and verification have been human endeavours — arguments crafted to persuade other human mathematicians. We’re approaching a situation in which machines may produce airtight logic, verified by formal systems, that even the best mathematicians will fail to follow.

In that future scenario — if it comes to pass — the AI will do every step, from proposing, to testing, to verifying proofs, “and then you’ve won,” Lackenby said. “You’ve proved something.”

However, this approach raises a profound philosophical question: If a proof becomes something only a computer can comprehend, does mathematics remain a human endeavour, or does it evolve into something else entirely? And that makes one wonder what the point is, Lackenby noted.

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This piece is adapted from my original piece which appeared in Live Science.

A couple of weeks ago I had the BBC Crowd Science team - producer Emily Knight and presenter Alex Lathbridge - in my office to talk probability. In particular, they wanted to try to answer a question, posed by listener Griffith in Ghana, about another of the BBC’s World Service Science show – Unexpected Elements. Observant Griffith had noticed that in the show’s weekly multiple choice quiz, the answer was often B or C, but very rarely A and wanted to know why.

While I can’t speak for the producers who set the questions on the show (they do give an explanation themselves in the programme), it seemed to me that the problem was probably a result of our species’ difficulty with dealing with randomness. One of the major themes of my second book, How to Expect the Unexpected, is that we are bad both at spotting randomness, but also (and perhaps more importantly in this context) at being random.

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Ideally in order for the answer to be hard to guess, it should be labelled A, B, or C at random with equal probability. Any systematic divergence from that allows astute listeners like Griffith to pick up on a pattern and to be able to guess at the answer without knowing it and have a better-than-random probability (one third in this quiz with three possible answers) of getting it right.

One of the most important biases that prevents us from being random in the context of multiple choice questions is known as middle bias - the tendency to exclude extreme options in favour of more central ones. Behavioural scientists have shown that when choosing between two pricing-plan options, most people tend to choose the basic rather than the premium option, but that when a third ultra-premium option is introduced, the now ‘mid-range’ premium option becomes the most popular. It’s worth questioning, when hedging your bets against the future with an insurance policy, whether the ‘platinum option’ really offers any tangible benefits or is just there to make the ‘gold option’ more palatable.

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Similarly, educational psychologists have found that students who genuinely don’t know the answer to multiple-choice questions tend to favour the middle two options of four, or the centre-most option of five for their guesses. The same effect occurs when playing Battleship (where the tendency is to guess coordinates away from the edges disproportionately often when trying to sink an opponent’s ship), when choosing items on a shelf or options from a computer drop-down menu, and even when visiting the toilet (where a middle cubicle is up to 50 per cent more likely to be chosen than an outer one).

Whether middle bias was actually at play when the producers were picking their answers is unclear. Middle bias is just one of the many reasons why we are poor at being random and multiple choice quizzes are perhaps just one of the many places where randomness can surprise us. I had a great time exploring some of the others – like the classic Birthday problem and the Monty Hall problem – with Alex and Emily on the show.

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If you want to hear more about these and other counterintuitive probability problems then check out the episode here or pick up a copy of How to Expect the Unexpected here.

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