There is no Nobel Prize for mathematics. Whether Alfred Nobel merely overlooked this or he had some other reason we may never know, but whatever the case, we were left with no Nobel for mathematics. But there are other prestigious awards – the Fields Medal, for instance, which is awarded to mathematicians under 40 years every four years, or the Abel Prize, which is awarded every year.
Much like the Lindau Nobel Laureate Meetings create a forum between Nobel Laureates and young researchers, the Heidelberg Laureate Forum (HLF) creates similar events involving Laureates and young researchers from the fields of mathematics and computer science. The HLF was modeled after the Lindau meetings, and to symbolize the bond between them, every year, there’s a Heidelberg Lecture in Lindau and vice versa.
This year, Fields Medal Laureate Efim I. Zelmanov held the Heidelberg lecture, speaking about the beauty in mathematics, and whether mathematics is a science or an art.
Her Majesty, Proof
Mathematicians don’t really try to calculate or experiment – they try to understand. So if you want to talk about mathematics, you first need to talk about proof.
“In experiment, the criteria of proof is the repetition of an experiment. Whatever you claim has to be independently confirmed. In mathematics, it is proof,” says Zelmanov. “But what is proof? It’s a very delicate question.”
Proof is ever-changing, and yet it is very constant. Famous mathematician and physicist John von Neumann once famously said “don’t even mention proof to me, it’s changed so much during my lifetime”. But somehow, proofs are also stable. Proofs from Ancient Greece are still valid. So what exactly is proof?
There are many formal definitions and most seem straightforward enough. For instance, you could define a mathematical proof as an inferential argument for a mathematical statement that demonstrates that the assumptions guarantee the conclusion. But when you try to put that into practice, especially in more abstract mathematics, it becomes more elusive. So Zelmanov suggests that proofs have a human component to them as well, which also makes them a bit more akin to art.
“In reality, a proof is what is conceded to be a proof by all mathematicians. It’s a bit informal, but since I teach students, I know that it is virtually impossible to define what a proof is. Students need to read many proofs and then they will develop this ability to see what is a proof and what is not.”
A proof can also be ‘beautiful’ or ‘ugly’, says Zelmanov, drawing another parallel between maths and art, but beauty also lies in the eye of the beholder. Still, much like someone recognises a musical masterpiece when they hear one, a mathematician should recognise what makes a proof beautiful – “How do mathematicians know what is beautiful and what is not? In the same way as musicians know. In musical schools, they are taught that the music of Mozart, Bach, etc, is beautiful. Students in art schools copy paintings of great masters just to develop taste. In the same way, mathematicians learn the work of great predecessors and they develop taste (though some do not),” Zelmanov quips.
A Young Genius
While music has the likes of Mozart or Bach, a golden standard for beauty in mathematics is the Galois Theory. The work of Évariste Galois, a mathematician who was killed in a duel at just 21 years old, is nothing short of monumental. Galois never obtained any formal degree after high school, quite possibly because he understood mathematics better than his university examiners – and wasn’t shy about telling them – but that didn’t prevent him from solving century-old problems and pioneering complex fields of mathematics.
When Galois was still a teenager, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, answering a problem that had been open for 350 years. Galois also invented something called finite fields. Finite fields, and Galois’ work in general, ended up being foundational for the field that was to become abstract mathematics.
“To tell you the truth, I understand why Galois worked on solutions on equations, many mathematicians were working on it … but why did he invent finite fields, I don’t know. He was just a genius,” says Zelmanov.
Finite fields are so abstract and hard to truly grasp that more than a century later, when Zelmanov was an undergrad at Novosibirsk University, he was taught that “only specialists can be interested in such useless things.” But as it turns out, they’re not useless at all.
The Unreasonable Effectiveness of Mathematics
“Never in history has mathematics interfered with what people are doing as much as today,” Zelmanov mentions, and it’s not hard to see why he says it. When you turn your computer and send an email, you’re using mathematics to keep the message private. When you’re buying something online, guess what makes it happen? Again, mathematics.
Much of today’s cryptography relies on something called the Diffie-Hellman-Merkle key exchange. The key exchange ensures that a shared secret between two parties can be used for secret communication even over a public network.
Think of it this way, if two parties (say, Alice and Bob) want to exchange a secret over a public network, they can start from their own secrets (a and b) and use a generator of a field g. So Alice shares ga, Bob shares gb, and now both know gab. Now, if a malicious hacker (let’s call her Catherine) hears the entire thing, she knows ga, she knows gb, but she still doesn’t know gab – if these were real numbers, she could calculate, but keep in mind, these are not real elements, they are elements of a Galois field. This clever little trick is a key part of what enables us to use the internet safely.
Recent studies suggest that very well-funded attackers, such as countries’ security services, could still crack this, but it’s a great example of how abstract, “useless” mathematics is actually very important in daily life. We also owe X-rays to abstract mathematics and several advancements in the field of geology and astronomy communications. There’s also a distinct irony here: the more mathematics affects our life, the fewer people are able to understand it.
“Mathematics is a very elitist art. How many billions of people can enjoy music and the Gioconda painting? How many mathematicians can enjoy a very deep paper in number theory? Maybe 10.”
But this shouldn’t take away from the fact that, in addition to sometimes being beautiful, mathematics can also be very useful and practical – though not always in the most straightforward way. Mathematics is, as physicist Eugene Wigner put it in a 1960 article, unreasonably effective.