Let me start by telling you three stories. The first is one you very likely know. A fraudulent tailor with no money to buy cloth nevertheless persuades a wealthy emperor that he will make him the finest suit of clothes imaginable. The emperor, deceived by the flattery of the tailor, believes that he is wearing a beautiful new outfit (made of very lightweight material). In fact, he goes out to greet his subjects stark naked. The rest of the crowd falls prey to the same mass delusion until a little boy points out the truth and then the spell is broken and the emperor becomes a laughing stock.

Now let’s also consider the story of the fraudulent Hitler diaries. Unlike the fable of the emperor’s new clothes, this is a true story (although I don’t swear that every detail here has been forensically researched). A fairly incompetent forger named Konrad Kujau wrote them himself during the early 1980s, staining the pages with tea to make them look old, before he sold them to the German news magazine Stern for several million pounds. Stern relied on the analysis of English historian Hugh Trevor-Roper who pronounced them genuine. Only after the fraud was uncovered did Trevor-Roper reflect on his thought process which included factoring in that Stern was paying a vast sum for the publication rights. Since fraudulent Hitler diaries had no value, he reasoned, these must therefore be authentic.

Finally, let us consider Edmund Landau and Fermat’s Last Theorem. As many people know, the brilliant 17th century mathematician Pierre de Fermat once scribbled in the margin of a book the equation xn + yn = zn and asserted that he had discovered a truly marvellous proof that there were no whole number solutions where n > 2. (If n = 2, this is Pythogoras’s theorem.) He did not alas provide the proof, claiming that the margin was too small to contain it. The theorem was finally proved in 1995 by English mathematician Andrew Wiles, whose proof ran to many dozens of pages and build on the work of Gerhard Frey and Ken Ribet and required a deep understanding of semistable elliptical curves. Evidently, this was not the proof which Fermat had (or thought he had).

Fermat’s Last Theorem was for three hundred years the outstanding unsolved problem in mathematics. Not because it was particularly important (although Wiles’ work did open up new avenues of exploration) but because it had remained unsolved for so long and yet it could be stated in terms which anyone with a passing knowledge of algebra could understand. This meant that it attracted the attentions of a great number of enthusiastic amateurs, all of whom imagined that they would be the ones to solve the unsolvable.

The problem was compounded in 1908 when industrialist Paul Wolfskehl offered a large cash prize for a valid proof. Edmund Landau was the mathematician who was, for a while, responsible for assessing entries. Overwhelmed by the sheer number of submissions, many of which were little more than gibberish, he eventually had a stack of cards printed which read: “Dear [BLANK]. Thank you for your manuscript on the proof of Fermat’s Last Theorem. The first mistake is on line [BLANK]. This invalidates the proof.” He gave the job of filling in the blanks and returning the cards to his students.

What does all of this tell us about vaccines, conspiracy theories and the nature of evidence?

I had high hopes that the natural experiment currently being run in which tens of millions of healthy adults are being given a vaccine to help prevent them from contracting COVID-19 would silence the anti-vax brigade. If it were true that vaccines were toxic in any way at all, then we would see a huge wave of whateveritis sweeping the globe. No such wave has in fact materialised. The best (ugh) the anti-vaxxers could muster were a few cases of blood-clotting, which were at such a low incidence it was barely possible to connect them to the vaccine at all.

Although something of a side issue here, the case of blood clotting is instructive when it comes to vaccines in particular. Even though the incidence was so low that it was almost impossible to distinguish from the base level (some people will get blood clots anyway), it was determined that one brand of vaccine could increase the likelihood of developing a blood clot in some people. Okay then. We have identified a risk. Somebody fearful about developing a blood clot would be sensible to refuse the vaccine. Even if the risk is small – blood clots are nasty and can be fatal.

Except… that by not getting the vaccine, you are increasing your risk of contracting COVID-19. And one of the possible complications of COVID-19 is – blood clots. In fact, you are many, many times more likely to develop a blood clot as the result of contracting COVID-19 than you are from getting the vaccine.

Life is a game of risk vs benefit. The problem is that we tend to see risks where we take positive action (getting a shot) and ignore them where we are passive.

But that alone can’t explain the demented anti-vax brigade rapidly pouring on to social media to tell me that vaccines are toxic, that they aren’t properly tested, that diseases don’t exist and that I should do my own research. There’s an enormous appeal for some people in the conspiracy theory. It enables the conspiracy theorist to be the brave little boy in the story of the emperor’s new clothes, whose plain common sense and straightforward approach cuts through all of the sophisticated bullshit to expose the obvious truth. Of all the characters in that story, the one we want to be is the little boy. We don’t want to be the villainous tailor whose nefarious plans are ruined and we certainly don’t want to be the foolish emperor, taken in by such a simple trick. The appeal is obvious.

The trouble is that fate is manifestly unlikely to present us with such a situation. Randall Munroe in one of his XKCD comics, imagines a riposte to a parent who asks “If all your friends decided to jump off a bridge – would you?” as follows: “Which scenario is more likely: every single person I know, many of them level-headed and afraid of heights, abruptly went crazy at exactly the same time… or the bridge is on fire?”

Thus, Edmund Landau’s pre-printed cards. Landau didn’t need to read these crazy would-be proofs, and nor did he fear missing the mathematical discovery of the century. Anyone who had done the necessary work to even begin to attack this prodigiously difficult problem would of necessity be someone in the mathematical community who was publishing on a regular basis, and that is where the proof would turn up (if it were ever found). Landau didn’t live to see Wiles’ proof which was published nearly a century after his death, but its form and the manner of its revelation would not have surprised him.

That brings us to the key question. How can I, with a 2:2 in mathematics, know that Wiles’ proof is correct? I have read an outline of the proof intended for the curious lay reader, and I failed to keep the concepts clearly in my head. I do not have a mental model I can use for considering modular groups or elliptic curves or how they relate to each other. In fact, when Wiles’ proof was first published, an error was found and it took Wiles and a colleague several further months of work to patch the problem. Probably there are less than 100 people living who can read and appreciate every line of the final version – such is the obscurity of modern mathematics, all of it resting on pre-existing understanding of already fairly abstruse and difficult material.

In practice, I cannot “do my own research”. It is too late for me now to begin a decades-long career in number theory and I do not believe that the investment of time and energy would have a satisfactory payoff. So, how can I be sure that I am not poor foolish Hugh Trevor-Roper, who takes the fact that Stern believes in the Hitler Diaries as a reason to vouchsafe to that same organ that they are worth the sum proposed for their publication rights?

This is the charge that the conspiracy theorists level at those of us who subscribe to scientific scepticism. Some of the more obviously loopy ones will happily (and loudly) present themselves as the little boy pointing at the naked emperor. The idea that a minority opinion is likely a false one has never entered their head. They so enjoy the heady rush of being in the select group of the clear-thinkers and the far-seers that they never stop to question why no-one else can think those same clear thoughts and see those same far-off things.

But the charge of blind obedience or fatuous credulity is a harder one to dismiss. You’re just believing what you’re bring told. You’re Hugh Trevor-Roper falling for the Hitler diaries because some authority figure tells you they’re genuine. And look! They proclaim. Here is a paper which proves what I am saying is correct. Here is a doctor who doesn’t believe in germs. Here is an engineer who doesn’t believe the Twin Towers could have fallen without demolition charges. Here is an Air Force pilot who has seen flying saucers. Here is an astronaut who believes the Earth is flat. Here is a photo of Bigfoot. Here is a NASA scientist who can read minds. Et cetera and so forth.

Particularly with medicine, online debate-via-link-to-scholarly-journal is rife. Don’t take my word for it, look at the conclusion reached by this research team. This exchange of technical papers which neither party can properly understand is the height of pointlessness. I was (unwisely) drawn into a debate about the events of 9/11 on Facebook not long ago. A “new” paper had been published (several years ago), paid for by a group whose only purpose is to find any contradiction at all in the “official story”. This piece of mathematical modelling purported to show that the collapse of Building 7 was inexplicable without some additional force acting (they coyly stop short of saying the word “explosives”). Online it was easy to find people with far more expertise than me taking this paper apart, but the technical details were beyond me and so I saw no reason to comment on its contents.

What I found fascinating was the existence of another paper which had been published many years earlier which had modelled the collapse and found that the weakening of the structure due to the immense heat of the fires which burned for nearly an hour was entirely sufficient to explain what was observed on that horrific day. I was angrily told by defenders of the later paper that they had been researching the subject for years and were expert in this field (and by those same defenders that the online criticism which I provided links to was too technical). But the existence of this earlier paper seemed not to have registered.

Surely, if a second analysis has been performed which comes up with a different result, then the first question to ask would be: what was the difference in approach and what were the faulty assumptions made by the first team? Otherwise, what sense does it make to accept the conclusions of the second paper wholesale and completely dismiss all of the conclusions of the first? Without an understanding of what led to the difference, we’re simply being asked which black box we prefer. How can we possibly learn any objective truths about the world by doing this?

So, are we doomed to just saying “nobody knows” or committing to decades of re-education to get to the truth? I don’t believe so. In practice, we don’t need to even ask these basic questions. We don’t need to take the paper(s) apart. We don’t need to begin a lengthy career of detailed study to be able to critique the work of both teams. We can just ask instead: who else believes this?

Science proceeds in general not because of individual mavericks whose ideas seem crazy at first. Instead, evidence gradually piles up in favour of one conclusion more than the others. It was well-known and agreed by almost all medical practitioners that stomach ulcers were caused by stress, until it was shown that they were caused by bacterial infection. Now, virtually every doctor prescribes antibiotics. Palaeontologists used to believe that dinosaurs were likely scaly and reptilian. Now the preponderance of evidence suggests instead that many of them were feathered. The evidence moved the consensus of opinion over time.

With COVID-19, we have seen this play out as we’ve watched. Early guidance emphasised handwashing, because (for a variety of reasons) the importance of aerosolised viral particles was underappreciated. Now, both mask-wearing and crucially ventilation for indoor events is understood to be key. We didn’t know that when the pandemic started. But the consensus of medical opinion moved as the evidence accumulated.

And this brings me to my last bugbear regarding Internet vaccine warriors. We have seen that “do your own research” is ridiculous. We are simply not equipped to do anything of the sort. Instead, we need to appreciate that we are not likely to be the little boy pointing out that the emperor has no clothes – if everyone else sees clothes and we don’t, it’s far more likely that we are having a stroke. But the world is full of plausible sounding people who write articles and share videos (and post blogs!) and it’s easy to get seduced by their rhetoric.

“Watch this video, then you’ll understand.” “This doctor gets it, read what she writes.” “This book really changed my mind about this topic.”

So now we have an expert who is explaining complicated ideas in ways that we lay people can understand. We don’t have to worry about whether we can master the technical details. If this expert can explain this in a way which makes sense to us – then we will understand it. Right?

Wrong.

This is in fact the very error made by Hugh Trevor-Roper, who took the opinion of Stern as the basis for his conclusion. You can’t learn about scientific consensus by listening to one expert, no matter how slick, folksy, friendly or straightforward they seem.

Watching well-produced explainer videos on YouTube is a great way to learn about stuff and there are some great creators out there. But you don’t discover objective truths about the world by opinion shopping. You can’t sift through a few different takes on a complex subject and then decide “I’ve found my guy.” That only works if your guy is just saying what everyone else is saying. But if your guy is an outlier, you need to ask why. If your guy’s evidence – which might be highly convincing to you, a non-expert – has failed to convince the other experts, you need to ask why not.

So, if 99 doctors tell you that the vaccine is safe and it’s in your best interests and society’s best interests for you to take it as soon as you are offered it, the fact that you can find a 100th doctor who tells you something else is not relevant. At all. Because if that one doctor had real evidence that the other 99 were wrong, the other 99 would change their minds. How do we know? We have seen it happen in real time as the pandemic has played out, and we have seen it happen throughout the history of science. Your one paper “proving” that masks don’t slow the spread of this, or any disease, does no such thing. It’s an outlier. The consensus is that they do help. Your one paper is wrong. (Or it isn’t widely applicable. Or you’ve misunderstood it. Or it doesn’t say what you claim it does.)

Salinsky’s Second Law: You can prove anything with one study.

This has become a very long post, for which apologies. My point really is a simple one.

We believe our dentist when we are told we need a filling. We believe the mechanic when we are told that our car needs a new fan belt even though we couldn’t pick a fan belt out of a line-up. We trust experts all of the time and don’t feel the need to “do our own research” – unless and until a campaign grows around a particular topic and a small but insistent band desperately needs to believe that the emperor has no clothes on.

So, I don’t really mind where you fall on the scale from “I’m not anti-vax, I just don’t feel this has been tested enough,” to “You’ve been lied to! Disease isn’t real!” Both those positions are wrong because both of them misunderstand what science is and how it works. You can’t possibly know what “tested enough” means. You don’t (and I don’t) have even a basic working knowledge of how vaccines are routinely developed or tested and what happened in this case. The first error is in word eight. This invalidates the argument.

So, by all means “do your own research”. Research is fun. You can teach yourself a lot and you can end the day understanding more about virology, immunology, biomedical research and the spread of infectious diseases than you did at the beginning. But if you fail to take into account where the conclusion you are satisfied with sits in relation to the scientific consensus, you are not avoiding the error made by Hugh Trevor-Roper, you are committing it.

Don’t be Hugh Trevor-Roper. The bridge may very well be on fire.

So… what do I think of the news that Chris Chibnall and Jodie Whittaker are leaving Doctor Who?