A stylized image of a rose passing through several planes of interpretation by physical theories..

Illustration by Ana Kova

The Universe Speaks in Numbers: How Modern Maths Reveals Nature’s Deepest Secrets Graham Farmelo Faber & Faber (2019)

Mathematics is an immensely powerful tool for understanding the laws of the Universe. That was demonstrated dramatically, for instance, by the 2012 discovery of the Higgs boson, predicted in the 1960s. Yet an ongoing, often fervid debate over the direction of theoretical physics pivots on the relationship between physics and maths — specifically, whether maths has become too dominant.

The worry — expressed by a number of theorists and writers over several decades — is that theoretical physics has become a monoculture too focused on a small clutch of concepts and approaches. Those include string theory, overstated predictions of new discoveries, over-reliance on mathematical elegance as a guide and a general drift into what physicist and writer Jim Baggott, in Farewell to Reality (2013), called “fairytale physics”, divorced from its empirical base. Notable critiques have come from theoretical physicists including Peter Woit, Lee Smolin and, more recently, Sabine Hossenfelder (see A. Ananthaswamy Nature 558, 186–187; 2018). Science writer Graham Farmelo clearly intends The Universe Speaks in Numbers as a riposte.

Farmelo takes us on a tour through the history of the field. His main protagonists are James Clerk Maxwell, Albert Einstein and Paul Dirac (subject of Farmelo’s outstanding 2009 biography, The Strangest Man). The unification of electricity, magnetism and light in Maxwell’s equations is a highlight of any good physics degree. I suspect most physicists can remember the moment when, after a few algebraic tricks with currents and voltages, the speed of light appeared, as if by magic. The Universe isn’t just speaking in numbers: it’s singing and dancing.

That constant value of the speed of light led to Einstein’s special theory of relativity in 1905. From this, in an amazing conceptual (and mathematically abetted) leap, Einstein conjured up general relativity in 1915, then the curvature of space-time, and eventually the gravitational waves discovered by the Laser Interferometer Gravitational-Wave Observatory (LIGO) 100 years later. And in 1928, Dirac, demanding mathematical consistency between quantum mechanics and special relativity, gave us both an understanding of the spin of the electron — without which the periodic table of the elements makes no sense — and predicted the existence of antimatter, discovered experimentally a few years later.

These are brilliant successes of the mathematical approach, and Farmelo leads us through them adeptly, with a mixture of contemporary accounts and scientific insight. He also casts a sceptical eye on the stories the players tell about themselves — and here the tensions start to be felt. Take Einstein’s warning to those who want to learn about theoretical physicists’ methods: “Don’t listen to their words, fix your attention on their deeds.” As Farmelo recounts, this is given interesting context by studies of Einstein’s notebooks, showing how he later overstated the role of mathematics, and underplayed that of physical insight, in his own breakthrough.

A productive union

Farmelo’s argument is that mathematics and physics work effectively together, to the benefit of both. Dirac and Einstein were evangelists for mathematically led physics, but their pleas were more or less ignored by their younger colleagues, such as Richard Feynman and Steven Weinberg, who were developing the standard model of particle physics. During what Farmelo calls “the long divorce” between mathematics and theoretical physics from the 1930s to the 1970s, our current understanding of fundamental physics was assembled. Dirac and Einstein were hardly involved in those developments.

That the most fruitful period in the development of particle physics coincided with its estrangement from pure mathematics could be seen as undermining Farmelo’s case. However, the pace of progress probably had more to do with the rapid experimental advances of the time than with any intrinsic issue in the relationship between the two subjects.

This was a fertile patch for experimentation, and theorists were continually buffeted by new and startling results, from the appearance of the muon to the observation of structure inside the proton; these demanded explanation. Although the few mathematical physicists engaged in the field, notably Freeman Dyson, made important contributions, most physicists didn’t need to go beyond well-established mathematical techniques to progress. Dyson himself (quoted by Farmelo) says that “we needed no help from mathematicians. We thought we were very smart and could do better on our own.” And, as Farmelo puts it, the feeling was mutual: physicists “rarely generated ideas that were of the slightest interest to mathematical researchers”. Many on both sides of the divorce were content with the situation.

There has been a re-engagement since the 1980s. In the mainstream of particle physics, theorists and experimentalists were calculating and confirming multiple results that established the standard model as, at the very least, a remarkably precise ‘effective theory’. But others, led by luminaries such as Michael Atiyah, Edward Witten and pioneers of string theory including Michael Green and John Schwarz, were probing its mathematical boundaries.

Whether the mathematical approach eventually became too dominant, taking over in terms of academic recognition and funding, is the crux of much of today’s debate. Farmelo gives a lively description of the back-and-forth of contributions typical of any thriving interdisciplinary area, with physical problems stimulating mathematical breakthroughs and mathematics throwing up new insights and techniques in physics. He steers clear of discussing the infeasibly large ‘string landscape’ of possible physical theories to which the mathematical approach seems to have led — contrary to hopes of a unique ‘theory of everything’. Instead, he concentrates on developments more directly useful and testable in physics, where some of this mathematical sophistication begins to feed back into an understanding of the standard model.

The standard model is a complex, subtle and immensely successful theoretical structure that leaves significant questions unanswered. Farmelo makes a convincing case that, in attempting to answer those questions, mathematics has a crucial role. Yet whether theoretical physics has become too enamoured of beautiful mathematics will, I suspect, remain a topic of hot debate.

The long experimental search for the Higgs was motivated by the fact that, before we accepted the existence of a quantum energy field that fills the whole Universe — part of the theory that predicted the particle — we demanded more evidence than ‘it makes the maths come out right’. The need for evidence is even stronger if the argument is ‘it makes the maths look beautiful’. The Universe might speak in numbers, but it uses empirical data to do so.