Adam Kucharski’s Proof was published in the UK with the subtitle The Uncertain Science of Certainty. In the United States it went out as The Art and Science of Certainty. I always wonder how the powers that be decide on the titles that the same work (be it a book, film, short story) carry in different countries, even when the same nominal language is spoke. Anyway, I ready the book originally expecting a statistical epistemology tour. What I got was rather more interesting, somewhat uncomfortable, and considerably more useful.
The monsters turned out to be the furniture
Kucharski is a mathematician by training (yay) and an epidemiologist by trade, so he begins where you would expect: Euclid, axioms, and the fantasy that you can start from self-evident foundations and walk in a straight line to truth. Spinoza thought you could prove God this way. Various constitutional drafters thought you could do the same for a nation.
Then the late nineteenth century arrives and mathematics starts producing objects it does not want. Functions that are continuous everywhere and differentiable nowhere. Curves that refuse to behave. The establishment called them monsters: aberrations against common sense, nuisances that Newton would have swerved to avoid. The tidy response was to declare them pathological and move on. He covers how Karl Weirestrass came up with his trying to find an example of a real-valued function that is continuous everywhere but differentiable nowhere.
Within a few decades the monsters were underwriting Brownian motion, probability theory and Einstein’s work on atoms. The things geometry could not accommodate turned out to be how the world actually behaves. It is a lovely reminder that “this breaks my framework” and “this is wrong” are not the same sentence, though we treat them as synonyms with impressive regularity.
The Monty Hall problem gets the same treatment, actually I try to exaplain this in my own book “Statistics and Data Visualisation with Python”. Anyway, I diverge, it is here that Kucharski quietly states his thesis. Paul Erdős, a man with more published papers than almost anyone alive, got it wrong, and stayed unconvinced when shown proof by exhaustion. A hundred thousand simulations eventually forced his agreement, but he was still not satisfied. Being shown a thing is true and being brought to understand why are separate operations, and the second one is much harder to industrialise.
That gap between what convinces me and what convinces you is the actual subject of the book. Everything else is worked examples. And that is not a bad thing!
Courts, constitutions, and Gödel’s very bad afternoon
The legal sections could have been a detour. They are not, because the law solved a problem science largely pretends it does not have: what do you do when you cannot wait for certainty but must still act? “Beyond reasonable doubt” is not a proof. It is an engineered threshold, arrived at through centuries of institutional argument about acceptable error rates in both directions.
The Gödel anecdote is the one everyone knows (?) and nobody takes seriously. At his 1947 US citizenship interview, with Einstein and Morgenstern in tow as witnesses, Gödel announced he had found a logical route by which the Constitution could permit a dictatorship. It is normally told as a charming story about an unworldly genius. Kucharski, to his credit, went and asked what a mind of that calibre might genuinely have spotted, and the leading theory is not a clause enabling a strongman, but a recursive one: mechanisms within the amendment process that make further amendment progressively easier. A slow loosening rather than a single breach. Gödel had spent the 1930s watching exactly that happen in Europe.
Kucharski admits he wondered whether the passage was too deep in the weeds of American history to include. The book was published in 2025. It reads rather differently now.
The pandemic chapters, which are the ones that stay with you
Kucharski sat on SPI-M, the modelling subgroup feeding SAGE. These are the sections where the book stops being about the history of ideas and becomes an account of doing this work with the clock running and a government waiting.
The methodological insight is genuinely good: when Alpha and then Delta emerged, the arguments about whether the variant was 30% or 40% or 50% more transmissible amy seem to be largely beside the point. That is the power of hindsight. The decision-relevant question was whether growth would be very fast or extremely fast. Precision that does not change the action is precision you cannot afford to chase. Anyone who has watched an organisation stall for a fortnight over the third decimal place of a metric will recognise the pathology.
The admissions are sharper. Certainty was overstated in places: airborne transmission stated as settled before it was. And the UK line that testing was not useful was, at the time, a capacity constraint wearing scientific clothing. There were not enough tests. That is a supply problem, and dressing it as a finding is an error with consequences that outlive the shortage.
The author makes a subtler point that I had not previously considered. The epidemiological evidence was published; the economic and educational modelling largely was not. The asymmetry created a public impression that epidemiology was driving policy far more than it actually was. Transparency applied unevenly does not produce partial trust. It produces a distorted picture, and epidemiologists then absorbed the blame for trade-offs they had not made. “Follow the science” became, in practice, a mechanism for laundering value judgements through people who had not been asked to make them.
Against this, the practice he describes from one advisory group is the best thing in the book: presentations that closed by inviting colleagues to explain why the presenter was wrong. Simple, structural, nearly impossible to sustain once a public persona is attached to a position. It is the difference between seeking evidence and seeking vindication, and it is exactly what most conspiracy theorising lacks, despite the folders of peer-reviewed papers such theorists frequently have to hand. Kucharski is careful here: these are rarely lazy people. They are often assiduous. They have simply removed the one step that would let them lose.
The most human moment is a single aside. His wife was pregnant during the vaccine rollout, when the pregnancy data was thin. You will have to read the book to find out what they decided to do.
Modelling for a living
Angus Deaton appears with the line “Gold standard thinking is magical thinking”, and the book largely earns it. Rigid evidence hierarchies fail in exactly the situations where evidence matters most, because they bias you towards inaction. William Gosset, working out experimental statistics at Guinness a century ago, framed it better than most modern practitioners manage: the relevant question is how much we lose by deciding now.
Two threads should worry anyone in our field. In any field.
First, prediction as an evasion. Kucharski notes how many disciplines drift towards forecasting because it sidesteps causality altogether, and then papers avoid causal language for twenty pages before closing with a policy recommendation. We do this constantly. A model that predicts which vehicles will fail, or which patients will deteriorate, is not a model that tells you what to change. I have written about this tension in the context of AI in the NHS; the procurement question is almost always about accuracy, and almost never about whether the decision actually moves.
Second, the loss of scientific satisfaction. The AlphaFold team, Kucharski reports, have broadly made their peace with excellent predictions they cannot mechanistically explain. Biology is complicated; the tool works; take the win. This is the Deep Thought problem made real: the answer arrives, and nobody can reconstruct the question well enough to know whether to trust it. We are all going to have to decide what standard of evidence gets us into the self-driving car, and “the benchmark scores are strong” is not obviously it.
Proof is a good book, occasionally the connective tissue between domains is thinner than the individual chapters deserve. having said that, that does not matter. This is the most useful thing I have read this year on how evidence actually behaves when it meets deadlines, institutions and people who do not already agree with you.
If you produce evidence for a living, e.g., models, trials, dashboards, business cases, anything that ends with someone else deciding something, read it! Then go and find the one belief you would least like to see overturned, and write down what would overturn it.
I will wait. Let me know!