Reviews Physics World  December 2019
(David Parker / Science Photo Library)

A mathematical mindset

Matin Durrani reviews The Universe Speaks in Numbers: How Modern Maths Reveals Nature’s Deepest Secrets by Graham Farmelo

For the last 40 years, a dedicated group of theoretical physicists has been beavering away on an ambitious project to create a quantum theory of gravity and unify the four forces of nature. Using extended 1D objects called strings, their theoretical framework could help answer questions that no other theory can tackle. Guided largely by mathematics, string theory – to its protagonists – underlines the supreme power of mathematical reasoning when it comes to understanding the natural world.

But not everyone sees things (or strings) that way. Some physicists feel that string theory is simply too far divorced from reality and makes no simple predictions that can be tested, or at least not with the current generation of experiments. String theory is a kind of exotic mathematical fantasy, they complain, and isn’t how physics should be done. Where is the close interplay with experimental findings that ought to be the hallmark of “proper” physics?

In an attempt to explain just why physicists should pay heed to mathematical deliberations of theorists – and especially those in the string-theory community – the veteran science writer Graham Farmelo has now published The Universe Speaks in Numbers: How Modern Maths Reveals Nature’s Deepest Secrets. In crisp, refined and polished prose, Farmelo offers a bracing defence of mathematically inclined theoretical physics. String theorists and other mathematically minded physicists will lap this book up.

In revealing how mathematics has led the way to a deeper understanding of nature, Farmelo begins with Newton’s work on mechanics and gravity, then moves on to Maxwell’s realization that electricity and magnetism are two sides of the same coin, before examining Einstein’s special and general theories of relativity. His chapter on quantum mechanics is particularly strong, which is not surprising given that Farmelo wrote the definitive biography of Paul Dirac, who excelled at using mathematics to understand the natural world.

For some, Dirac will always be the culprit for taking physics in an unnecessarily mathematical direction by insisting on the principle of mathematical “beauty” as the driving force for science. As Farmelo reminds us, Dirac’s view was that researchers should “strive to maximize the amount of beauty of the mathematical structures that underpin their theories of the natural world”. Einstein – Dirac’s intellectual brother-in-arms – also felt the pull of mathematics. He wanted theoretical physicists to focus on developing the mathematics of their best theories and to keep an eye on new maths that might one day be relevant to them.

Farmelo’s attention turns next to the period after the Second World War, which saw physics and mathematics go through what he calls “a long divorce”. With oodles of data emerging from particle accelerators, telescopes and solid-state physics labs, physicists simply got by with the maths they’d been taught at college. It was only in the 1970s that mathematics and physics began to come together again. The discovery of the J/Ψ particle in 1974 indicated that all the forces that shape atoms are described by gauge theory – a descendant of Maxwell’s account of electricity and magnetism – and led to the development of the Standard Model of particle physics.

This is the launching-off point for the second half of the book, in which Farmelo looks at how theoretical physicists have, since the late 1970s, “imagined their way to new concepts and fundamental theories almost entirely without the stimulus of experimental discoveries”. It’s tough going for those involved. As Farmelo points out, any new idea that stands a chance of being correct has to be consistent with both quantum theory and special relativity. That, coupled with a paucity of data from particle experiments, has led theorists into territory previously occupied only by pure mathematicians – or what’s loosely called “physical mathematics”.

Given the complexity of the subject matter, you’ll need your wits about you in reading later chapters of the book, which touch on esoterica such as the “no-ghost theorem”, Calabi–Yau spaces, the Malcadena duality, twistors and amplituhedra – a word I can hardly pronounce let alone understand. Thankfully, Farmelo is an authoritative, reliable and trusted guide, as he is throughout the book. With a firm hand at the tiller, he cuts through the theoretical murky waters with panache.

I doubt anyone could do a finer or more stylish job of describing the field’s development

Indeed, I warmed to this book the further I got into it. Farmelo knows his subject well and he writes engagingly, choosing his words carefully. The book’s also backed up by many interviews with top researchers, including Steven Weinberg, the late Michael Atiyah and mathematician-turned-physicist Freeman Dyson. Wisely, Farmelo steers clear of explaining every nuance, which means that this isn’t a book to read if you truly want to understand string theory. But I doubt anyone could do a finer or more stylish job of describing the field’s development, even if he is a little too in thrall of the protagonists.

Farmelo is also too polite to name in the main text those “influential commentators” who are disenchanted with the state of modern theoretical physics. But a quick look at the book’s footnotes reveals he’s taking aim at the likes of fellow science writer Jim Baggott, who has used the phrase “fairy-tale physics”, as well as theorists Peter Woit, who writes a blog called Not Even Wrong, and Sabine Hossenfelder, whose 2018 book accused a generation of theorists of being “lost in math” (September 2018).

In a withering put-down, Farmelo declares himself “troubled by the dismissiveness of some of the critical commentators, especially those who write with a confidence that belies the evident slightness of their understanding of the subject they are attacking”. He feels that their attacks are part of an “especially regrettable” trend, whereby “anyone can have a valid opinion on any subject, regardless of their technical knowledge and appreciation of it”. Ouch!

Farmelo’s underlying message is that we should not knock the endeavours of today’s leading theoretical physicists, or at least those involved in strings. Just look at theorists like Einstein and Dirac, he says; they weren’t always understood at the time but were proved right in the end. Farmelo admits, however, that the pace of progress is likely to slow, with future developments occurring over decades and centuries, rather than years. Remember the Higgs boson? It was predicted in 1964 but discovered only in 2012.

The question is whether theorists of the future will be prepared for such glacially slow progress – “sedate” as Farmelo puts it – and instead be tempted into fields where the action is faster. Do theoretical physicists really want to be like mathematicians, making only incremental progress on a few, hard problems during the course of their career? Farmelo thinks “they would do well to cultivate the virtue of patience”, but the danger – or, depending on your point of view, the hope – is that those theorists might switch away from strings altogether.