Earlier this month, I wrote a story about how OpenAI had inadvertently produced two versions of its proof of the Navier-Stokes problem. You can read the details here, but one thing that has stuck with me is that some non-mathematically minded people I spoke to about the story struggled with the idea that two different proofs of the same problem could both be correct. It got me thinking about how and why mathematicians prove things multiple times – and what happens when it goes wrong.
One thing it’s important to grasp is that proofs are not homework problems. A couple of mathematicians have made this point to me recently: because most people’s experience of maths begins and ends at school, there is a perception that mathematicians are simply working down a big list of problems and writing answers to be ticked off by some sort of cosmic maths teacher. This is not the case – proofs are a way of organising and developing mathematical thinking. You can think of a proof as a journey, from statement to conclusion – and as with many journeys, there are many routes you can take to get from A to B. This is where we get multiple proofs of the same problem.
This act of reproving, of taking the mathematical road not yet travelled, dates back to antiquity. One of the earliest examples is the statement that an isosceles triangle always has two equal sides and two equal angles at the base. Writing around 350 BC, Aristotle describes a proof of this that involves drawing lines from the centre of a circle to its circumference – the original author of that proof is lost. Around 50 years later, Euclid described a proof that only requires triangles, not circles. This neatly avoided a mathematical disagreement about how to define angles formed between a straight line and a curved line – a question that wasn’t settled for thousands of years, as it happened.
Euclid also crops up in another example that shows how useful it is to keep attacking “solved” problems. He recorded a proof that there are infinitely many primes, and you may have learnt it at school. Suppose you have a finite list of primes. If you multiply them all together, you get a new number, . Now, define = + 1. If is prime, it can’t have been on the list, because it is bigger than . If isn’t, it must instead be divisible by a prime, as all non-primes are. If that prime we divide by had been on the list, it must also divide , but that must also mean it divides – = 1, which is impossible. Either way, we have identified a prime not on the original list.
According to a 2023 survey, they are now 200 different proofs of Euclid’s theorem, as it is known. One that particularly added to mathematical knowledge at the time was published by Leonhard Euler in 1737, who found a formula that related the sum of all numbers to the product of all primes. He also used this to hint at the structure of the primes – not only that there are infinitely many, but that they become sparser as you go up the number line. Euler’s formula inspired Bernard Riemann over a century later to propose the Riemann hypothesis, a deep question about the structure of prime numbers that is considered one of the greatest unsolved mysteries in maths today – so it was well worth Euler having another crack at Euclid.
Perhaps surprisingly, there are even times when mathematicians have produced multiple proofs of the same theorem before revealing a flaw in the reasoning. In 1933, mathematician Wilhelm Grunwald published a proof showing that a number, , with a certain property in relation to all prime numbers, must always be able to be written as some other number, , raised to the th power, so = . A second mathematician, George Whaples, gave a new proof of this using slightly different maths in 1942. But then in 1948, Shianghao Wang found that actually, this isn’t true for 16, because 16 isn’t equal to 8 for any . The correct result is now known as the Grunwald-Wang theorem.
Doing maths on the maths
Pleasingly, researchers have even studied the idea of taking proof journeys mathematically. In a 2022 paper, Scott Viteri at Stanford University and Simon DeDeo at Carnegie Mellon University in Pennsylvania point out that if a mathematical proof is a sequence of logical steps, each with a tiny probability of a mistake, the odds are that a long enough proof is more likely to be wrong than not. Roll enough times, and the dice will be extremely likely to land on a 1. How then can mathematicians prove anything?
The pair show that a mathematical proof is structured like a network, with “truth” being transmitted along paths between each step, exactly like a journey. With the potential for error at each step, the strength of this truth diminishes along longer paths. But proofs aren’t random walks around a conceptual arena – they have structure, and claims are often reinforced by multiple routes. This structure allows truth to be transmitted along the strongest paths, even if it fails along other routes. Because of this, the network undergoes what the pair call an “epistemic phase transition”, a scenario under which you can be extremely certain the proof is correct.
This is just a model, and not necessarily something accepted by all mathematicians, but I find it appealing. It also fits with what you would naturally expect: the more proofs mathematicians have of a result, the more paths that are carved from A to B, the more certain you can be that it is true. In an age when AI is beginning to produce maths that humans don’t necessarily understand, we might find that having multiple routes to the truth, whether human or AI derived, can bring confidence where we might otherwise have uncertainty.