Kucharski, a professor of mathematics at the London School of Hygiene and Tropical Medicine, explores how proof is not just a mathematical concept but a vital tool in decision-making, justice, and survival. From the medieval Islamic world to the recent pandemic, scientific progress has relied on different methods of establishing fact from fiction. Today, in the face of ever-increasing disinformation, how we prove things – to ourselves and others- has never felt more urgent. There is far more to proof than axioms, theories and scientific §       of someone’s guilt, or deciding whether to trust a new type of financial transaction, weighing up evidence is rarely simple. 

Kucharski received a threatening email from a meeting of Britain’s Science Advisory Group for Emergencies (of which he was a member during the pandemic). “ That month had felt like a tug of war between the two theories about Delta  ( a new Covid variant), each piece a competitor pulling in one direction or the other,” Kucharski recalls in Proof.

Looking at the US, the several conspiracy theories and non-factual beliefs that spread along with the coronavirus in the early 2020s seem rather low key by 2025 standards. Proof starts with the well-known Monty Hall problem, whose correct answer almost nobody can understand. Game show host Monty offers a contestant the choice of three doors, one of which hides a new car, and the other two less desirable goats. The correct answer is to switch, as there is a higher probability the car is behind door 1. Yet one of the disbelievers, the book tells us, was the genius mathematician Paul Erdös. He could not understand is the whole programme of basing decisions on evidence doomed to flounder on humans’ inability to understand probabilities and logic?

In 1951, Kurt Gödel demonstrated that axiomatic mathematical systems are either inconsistent or incomplete. Early 20^th -century mathematics reveals “why software engineers sometimes struggle to develop successful decision-making algorithms”, Kucharski writes. Algorithms cannot cover all possible contexts or give internally consistent outcomes.

A pro-Publica investigation in 2016 suggested Correctional Offender Management profiling for Alternative Sanctions, a system widely used in the US to help judges make decisions about bail and parole through automated risk assessment, which feeds variables such as age, education and history of violence into an algorithm to produce a risk score. Although race was not explicitly used in the algorithm, it assigned Black defendants to a higher risk category, which was clearly unfair. Such examples hold warnings for policymakers cantering towards artificial intelligence to cut costs or improve efficiency. According to Kucharski the correct answer on the basis of evidence and quantitative methods in the law, in medicine and indeed in all areas where strong claims are made for “evidence-based” conclusions. 

In one chapter which might surprise some readers, casting serious doubt on the claim that randomised control trials (RCTs) are actually the “gold standard” for evidence on medical efficacy. There are many situations where other types of proof- including experience- are more relevant.

Proof: The Uncertain Science of Certainty by Adam Kucharski, Profile £22/ Basic Books $32, 368 pages.