Dr. Adrianna Teriakidis

Lincoln and the Founding Fathers were influenced by mathematical proofs when constructing civic arguments. Inspired by Euclid, who used a set of axioms and logic to prove a wide number of statements about geometry, Lincoln treats the U.S. constitution as a set of axioms to argue that slavery is not justified using a proof by contradiction. What it boiled down to is that if A could construct an argument to enslave B, for example on the basis of skin colour, then B could also make the same argument to enslave A, therefore it was not legitimate for any person to enslave any other. Although being able to know political truths through axioms and proofs is a satisfying idea, it sadly did not settle political debate for the rest of time. Even mathematics started to get weird (i.e. counter-intuitive or hard to conceptualise) when proving statements by logic rather than relying on intuition, eventually leading to Kurt Gödel's famous result that some true statements could not be proved. Funnily enough, shortly after publishing his incompleteness theorems, Gödel was applying to get US citizenship and while researching the constitution "he had found some inner contradictions and that he could show how in a perfectly legal manner it would be possible for somebody to become a dictator and set up a Fascist regime." When it came time for his interview he mentioned this to the judge who quickly moved on. "As a result, there is no record of what loopholes Gödel had identified in the constitution." Even Lincoln softened his stance over time, and by the time is gave the Gettysburg address he called them "propositions" instead of "axioms".

Covid exposed scientific inference and uncertainty to the public at a very consequential time, and Kucharski, as one of the epidemiologists involved in the UK government’s response to COVID, was right in the middle of it. He goes into some detail about the different methods they used and how they tried and sometimes failed to communicate the results to the public, noting that scientists from different fields also didn’t always agree on the evidence, and sometimes seemed to prefer their own methods. “For certain biologists, it seemed Alpha wouldn’t be more transmissible unless the evidence came from methods they had expertise in.”

The two chapters that cover ways of scientifically synthesising evidence and how that can go wrong (5&6) cover a very broad set of philosophical ideas interspersed with interesting anecdotes, but it takes a lot of work from the reader to make the connections and I doubt someone unfamiliar with the area would be able to really understand all the different threads.

The book ends on an interesting discussion about proof without human understanding. Mathematicians had to start contending with this from the 1970s when the first “proofs by exhaustion” were performed. The classic example is the proof that any map could be filled in with just four colours so that any countries that bordered each other had different colours. This was demonstrated in 1976 when a computer program found a solution for the set of maps layouts that any map could be reduced to. There is something unsatisfying about proving something by brute force rather than through logic, but that has become more common as computers have become more powerful. In fact, this type of proof may actually be easier to verify by more people, than a complex derivation that spans thousands of pages and requires specialised mathematical knowledge.

“When Wolfgang Haken’s son, a PhD student at Berkley, gave a lecture on his father’s proof of the four-colour theorem, he recalled the audience reaction: “The older listeners asked, “How can you believe a proof that makes such heavy use of a computer?” The younger listeners asked, “How can you believe a proof that depends on the accuracy of 400 pages of hand verification of detail?”“ One era’s definition of certainty had become the next generation’s source of fallibility.“

This leads on to a discussion of AI and the general discomfort with black-box models. Kucharski makes the point that in other domains we regularly use methods that we don’t fully understand, like defibrillators, general anaesthesia and flying planes so why not black box algorithms in self-driving cars or large language models.

Overall, fascinating topic, with good stories and anecdotes peppered throughout and a pretty good structure, but so many concepts packed in and the relationship with the stories not always explicit making it sometimes hard to follow.

📚 Proof by Adam Kucharski (2025)