Exploring the Beauty of Mathematics in the Natural Sciences

When researchers describe an equation or set of equations as beautiful, it is not necessarily instantly clear exactly what they mean. An elusive indefinable quality, they could be saying that the equations are pleasurable to perceive at a primal level, either in their simplicity, symmetry or some other aspect or combination of aspects of their form. They could mean they are beautiful in their mathematical content, succinctly and satisfyingly describing new or complex concepts. Or they might mean the equations are beautiful in a different way, perhaps in how they describe a natural process in an unerring and precise way.

Regardless, scientists are attracted to beauty in mathematics, just as we are all attracted to beauty in the physical form, art, music and pretty much every area of life. Which is why it was little surprise that Gerardus ‘t Hooft’s Agora Talk ‘The Beauty of Mathematics in Science’ was packed to the rafters on the first day of #LINO24.

‘t Hooft’s experience of the beauty of mathematics comes from theoretical particle physics. Indeed, his 1999 Nobel Prize in Physics was awarded jointly alongside his supervisor Martinus Veltman (who passed away in 2021) for placing particle physics theory on a firmer mathematical foundation, more precisely “for elucidating the quantum structure of electroweak interactions in physics”.

When the Dutch pair began working together in 1969 on fundamental aspects of quantum mechanics that would go on to form the Standard Model of particle physics (a set of equations that brings together all the fundamental forces, except gravity, with all the fundamental particles), significant progress had already been made. The wave functions of quantum mechanics and the fields of electromagnetism had been united into a quantum field theory, and there was another more complicated quantum field theory that united electromagnetism and the weak force (a mechanism responsible for the radioactive decay of atoms) into an electroweak interaction for which Sheldon Glashow, Abdus Salam and Steven Weinberg received the 1979 Nobel Prize in Physics.

A Beautiful Particle

However, when used for calculating the properties of particles, the latter would sometimes give unreasonable results. “The mathematical equations didn’t come together, and many of our colleagues said ‘Well, who cares? You just want to get the best possible approximation that makes this weak force work’,” recalled ‘t Hooft. “But Veltman insisted that this had to be mathematically precise.”

Using Veltman’s previous successes and insights, and his computer program for testing different ideas, ‘t Hooft managed to find the missing piece of the puzzle. “We tried a spin zero particle and suddenly everything fell into position,” he said. “We could now get the equations to meet, and the discrepancies disappeared.”

This spin zero particle (which had already been predicted by 2013 Nobel Prize in Physics recipients François Englert and Peter Higgs) would later become famous through its discovery at CERN’s Large Hadron Collider. But for ‘t Hooft it was what it did to the underlying quantum field theory that made the Higgs special. “We needed this extra particle that we didn’t understand what it was good for, but it worked. And this made a beautiful theory … mixing charged particles and neutral particles together in a completely new way, so the symmetries of old-fashioned electromagnetism were mixing with the symmetries of Yang-Mills theory.”

‘t Hooft also feels that the Standard Model, for which his work forms a fundamental pillar, is beautiful, though incomplete. “There are all these nasty questions that are still left over. And I believe they’re still here because the people who are addressing them have something in their minds about what the theory should look like,” he opined. “I trust nature to find mathematical solutions to these problems, which we have not yet discovered.”

Galois’ Gold Standard of Beauty

‘t Hooft’s regard for mathematical beauty is multifaceted, including how it successfully draws together disparate mathematical and physical concepts, and how it precisely describes nature at a fundamental level. Efim Zelmanov, presenting #LINO24’s Heidelberg Lecture entitled What Do Mathematicians Think About? delved even deeper into this topic on day 3.

Zelmanov was one of four recipients of the 1994 Fields Medal, widely considered the world’s most prestigious prize for mathematics. His work included a proof of the restricted Burnside problem, where he showed that certain mathematical constructs known as periodic groups are finite. Thankfully for audience members who were not mathematically minded, he did not dwell on the highly advanced group theoretic concepts underpinning this proof in his talk. Instead, he took the audience back to the founding of group theory. Group theory is the study of groups, and groups are sets equipped with an operation (like multiplication, etc) that satisfies certain basic properties.

Early 19^th Century French mathematician Évariste Galois pioneered fundamental mathematical concepts that would later form the basis for group theory. Galois was a teen prodigy who died at the age of 20: “He never had any formal mathematical education, he failed entrance exams for university twice,” said Zelmanov. “People who knew him later said it was because he knew mathematics much better than his examiner’s and never tried to hide it.”

Many of the concepts Galois pioneered were created to solve a problem that had been leaving mathematicians scratching their heads for 350 years. Polynomial equations of degree 2 (quadratic equations), 3 and 4 were known to be solvable by standard means. “But people tried to find a similar formula for equations of degree 5 unsuccessfully,” explained Zelmanov. The problem can be posed as follows: does there exist a formula for the roots of a fifth or higher degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc)?

“People started to suspect that maybe it doesn’t exist,” said Zelmanov. “Galois explained what was going on, he asked the right question: what is the group of symmetries of the equation?” Following this approach, he explained why it is possible to solve some equations, including all those of degree four or lower, and why it is not possible for most equations of degree five or higher. 

As became clear much later when mathematicians started to appreciate Galois’ insights, Galois’ theory acts as a fundamental bridge between the important mathematical disciplines of field theory and group theory, making such problems – including the quantum field theories ‘t Hooft spoke about earlier – easier to understand.

As well as seamlessly connecting seemingly disparate areas of mathematics, another important part of what makes Galois theory beautiful in Zelmanov’s opinion is its enduring applicability to a great diversity of different topics. For instance, Galois theory introduced the abstract algebraic concept of finite fields. As it turned out, finite fields have become central to everything from defining an algorithm, to public cryptography, tomography and building good computer networks. “The golden standard of beauty in mathematics is Galois theory,” said Zelmanov.

Beauty, Mathematics, Art and Nature

Zelmanov has thought deeply about the nature of mathematics more generally: “Mathematics is not only a technological engine, it is much more than that – it is art,” he told the enraptured audience. “If you view mathematics as art, it is very elitist. Millions of people can enjoy Mozart but how many people can appreciate a very deep and beautiful paper on number theory – maybe 10.”

He also shared his thoughts on exactly what is meant by mathematical beauty: “What is the purpose of a proof? The purpose is understanding. For a mathematician, it is not enough to know if something is correct or not correct. They need to know why it is correct or not correct.” He continued: “A proof can be beautiful or ugly… [But] how do mathematicians know what is beautiful, what is not? I should say that that is like how do you know the music of Mozart is beautiful … it is impossible to explain what is beautiful. But there are some [common] features like simple statements, complicated proofs, unexpected ideas sometimes coming from a different area, generality.”

Zelmanov ended by echoing ‘t Hooft’s thoughts and those of physicist Eugene Wigner (1963 Nobel Prize in Physics) in the title of his 1960 article ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’, hinting that perhaps beauty in mathematics does indeed stem from its uncanny ability to describe nature.