A few years ago, on my way to Rio de Janeiro, I made a stopover at Heathrow, London’s famous airport. I wore a T-shirt emblazoned with a formula tied to the standard model of particle physics, followed by the phrase “Our universe so far.” A boy, wearing a Flamengo shirt and a pronounced British accent, looked at my shirt and asked: “Is that so? – is that right?”. I remember answering “hopefully!”, hopefully!, delighted with the boy’s curiosity.
“Is that so?” This almost childish question is one of the premises of modern research in mathematics. Given a fact, or a conjecture, or a simple possibility, part of the job is trying to figure out if “this is it.” And here we are talking about proving theorems, like that of Pythagoras. When (daily) we use this theorem, we are convinced that its content is true, no one doubts. So, when I went from Gávea to Jardim Botânico, I took a shortcut through the garden, since the hypotenuse is the shortest path.
But there is a more fundamental question: “Why is that so?”. Why do we want to prove theorems? When a plane flies, we witness turbulence; when a boat sails, we perceive the formation of waves in its surroundings; when water is injected into an oil well, oil is extracted — and we know so much that it is even conjectured that extraction is more efficient if we inject fresh water.
In short: we know how fluids behave, whether air, water or oil. Why do we need a mathematical object such as the Navier-Stokes equation to model fluid mechanics? Furthermore, why does the Clay Mathematics Institute offer a $1 million prize to anyone who can demonstrate that solutions to this equation really exist? This is a very difficult question. After all, it deals with the reasons to prove that the world we know is mathematically as we know it.
Another class of examples may sound different. Facts that seem obvious, but we still want to demonstrate. Let’s do an exercise, which we’ll call 3N+1. We choose any number. If it’s even, we divide it by two. If it’s odd, we multiply it by 3 and add it to 1. And we continue. So if we start with 5, we multiply by 3 and add 1, which gives us 16. Because 16 is even, we divide by 2 and we get 8. Since 8 is still even, we divide by 2, which gives 4. The next step returns 2 and the next returns 1. Since 1 is odd, we multiply by 3 and add 1, just to get 4! And we are forever in a 4-2-1 cycle!
More than the tactical scheme of a team that had three players sent off and bet on the back, 4-2-1 is a very special cycle. The reason is simple: no one has found a number that, when starting 3N+1, reaches another cycle. All attempts end the same way: 4-2-1. The reader can do an exercise with their favorite number.
There is no theorem that guarantees this rule. It has never been possible to prove this fact. It is not even known if there is any number that, submitted to this algorithm, generates a different cycle. Now, if all known attempts ended the same way, could we convince ourselves that it will always be this way?
In 1993, Arthur Jaffe and Frank Quinn published an article in the Bulletin of the American Mathematical Society discussing the role and importance of proofs. The article is interesting in several dimensions, but it slips by suggesting that excessive use of rigorous proofs makes math slower and, above all, more difficult. A series of letters, written by prestigious mathematicians and mathematicians, were published in the same newsletter confronting certain aspects of the article.
The debate generated by these publications is very rich. In particular, the idea that the function of mathematicians is to advance people’s mathematical understanding. In this sense, a demonstration is like a goal: a fundamental aspect of sport, seen as a collective effort.
In the end, any real mathematical breakthrough—necessary to complete a proof—can lead us to new ways of thinking and apprehending the world. At a more elementary and less abstract level, this seems to happen with the teaching of basic mathematics: there is evidence that the understanding of a result is much better when it is presented with its proof.
In hindsight, perhaps the role of rigorous proofs in mathematics—and our mathematicians as well—has to do with the response to the Flemish boy at Heathrow. In fact, I should have replied, “What (and how) do you think? – what (and how) do you think?”
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Edgard Pimentel is a researcher at the Center for Mathematics at the University of Coimbra and a professor at PUC-Rio.
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