The Marriage between Maths and Reality: An Interview with Graham Farmelo

Dr Graham Farmelo is an award-winning biographer and science writer. Based in London, he is a Fellow at Churchill College, Cambridge and a regular visitor at the Institute for Advanced Study, Princeton. His latest book, The Universe Speaks in Numbers, is a historical account of the relationship between mathematics and physics. What follows is my interview with Farmelo about his new book.

Logan Chipkin: In your book, you suggest that we will never know “whether the laws of nature are invented or discovered.” Throughout history, scientists have provided satisfying explanations and answers to questions that people had assumed to be forever mysterious. What makes this problem different?

Graham Farmelo: Mathematicians have discussed for centuries whether they’re looking at an invented structure or a discovered structure, whether it preexists “out there,” ready for them to find, or whether it’s an imaginative construction. You can find brilliant mathematicians on both sides of that question. It’s similar in physics. Most physicists would say that fundamental laws of nature are out there to discover. They’re out there, perhaps in some mystical sense. Other people say, particularly critics of some aspects of physics, that they’re a human construction.

I don’t see how it can ever be demonstrated one way or the other, in either mathematics or physics. But there are strong views on both sides.

Chipkin: It seems that philosophy is always intertwined with whatever progress is being made in physics. So I don’t know if that would cease, just because it seems like maybe now it’s more common to work on math rather than philosophy.

Farmelo: It’s a good point, and I’m not anti-philosophy. Philosophy looks at questions about the nature of physics that could well become increasingly important.

Chipkin: You write about the different philosophical approaches that Einstein and Dirac thought would best aid our quest for deeper understanding of the physical world. With hindsight, which of their philosophies do you think has been more fruitful?

Farmelo: Einstein came to believe that the road to discovering the fundamental laws of nature was through mathematics. He got that view from his development of the theory of gravity, which was a monumental achievement. And in the final stages of completing that theory, this is in the late fall of 1915, he really did believe that mathematics had guided him to those final equations, which was arguably the greatest achievement of his life.

Now, Dirac was way ahead of his time. He thought that physicists should attend to mathematics, and mathematicians should attend to physics. In his view, mathematics is a game in which mathematicians invent the rules, whereas physics is a game where nature supplies you with the rules, and it’s our job to try and guess them. So he’s arguing that there is a common cause in physicists and mathematicians working together. That has proved a very prescient observation, I think.

Chipkin: So they were both ahead of their time.

Farmelo: Very much so. Some people think that this mathematics has gotten out of hand, that physicists have gotten too interested in mathematics, it’s misled them, and what have you. Now, I respectfully disagree with that. I think that we’re in an unusual time, because in the last few decades, there have been relatively few really juicy clues from nature about the way that theoreticians should develop their theories. Instead, what theoreticians have done is that they’ve developed the best theories that we have and found themselves in the territory of mathematics. There’s one very important point to make about these speculations—that the great work that’s being done now in theoretical physics is largely based on the two foundational theories of twentieth-century physics, which are the basic theory of relativity and quantum mechanics.

Chipkin: The relationship between physicists and mathematicians over time reads like a romantic drama—you even call the post-Second World War period a “long divorce” between the cultures. Why did they separate in the 1940s, and what ultimately brought them back together?

Farmelo: One factor is that there was a movement called Bourbaki in France which was very influential, which sought to take the whole of mathematics and rebuild it bottom-up from completely rigorous statements and paid no attention to applications at all. It was a useful initiative, but it did move mathematicians away from close collaboration with physicists.

Meanwhile, physicists were having a ball. They were making very good discoveries about the subatomic world. There were discoveries in low-temperature physics, in nuclear physics, all over the shop, but without the need for advanced mathematics.

What brought them back together was that physicists found that quantum field theory was the right way to describe the forces that shape atoms. The great mathematician Michael Atiyah saw that this was rich with mathematical potential. The theory contained a lot of mathematical questions that he and his colleagues could benefit from. When Michael Atiyah met Edward Witten in an office in the Massachusetts Institute of Technology in the fall of 1976, that was a coming together of a great mathematician and a brilliant theoretical physicist.

Michael Atiyah said that the quantum field theorists were building their tunnels in physics and making some success. The geometers, meanwhile, were digging a completely separate tunnel. What happened at Oxford when Atiyah went back and worked with Witten and other great mathematicians and physicists, is that those tunnels intersected. And yet, what happened was, to the surprise of everyone, it looked like the tunnels had been perfectly engineered to cross.

Chipkin: Einstein’s general relativity was thought of as “one of the least promising fields” not long after its establishment in the early twenty-first century. This sentiment seems to have faded in the past few decades, replaced by much more excitement and optimism around the subject. What changed?

Farmelo: When Einstein came up with his theory of general relativity, quite a few people saw this as a brilliant piece of work, something you admire from afar, so to speak. Einstein explained some anomalies, he set up a beautiful theoretical construction, but it didn’t seem terribly relevant.

What happened later was that, owing to developments in technology and some brilliant work in particular by John Wheeler, Roger Penrose and Stephen Hawking, there was a renaissance in our appreciation that relativity was not just about tiny effects, but was about real things, like black holes, for example. Earlier this year, we saw that amazing image that was the first ocular proof, so to speak, that they really exist.

Now, we’re in a wonderful time in cosmology and astronomy, where, owing to these fantastic developments in telescopes and numerical simulations and what have you, cosmologists and astronomers can look back at the very early stages of the universe. So general relativity now, as you rightly say, is hot, hot.

Chipkin: Many have heard about the oft-repeated “unreasonable effectiveness of math in physics,” but you describe the converse—the “unreasonable effectiveness of physics in math.” What do you mean by that?

Farmelo: Mathematics is a wondrous tool. Even in high school, you see how you use basic maths in order to account for experiments. And that works all the way up to the superstar physicists who use mathematics. What has been so striking since the early 1970s is that the intuitions you get from the geometry that arise out of physics are amazing insights into mathematics. The great mathematician Pierre Deligne constantly finds stimulation from string theory, and he’s by no means alone. I’ve never known pure mathematics and physics to be closer in my lifetime.

Chipkin: That was a very thought-provoking idea.

Farmelo: It is! Dirac wrote in 1939, perhaps farsightedly and a bit fancifully, that one day the subjects [mathematics and physics] might unify. That would be an astonishing thing. And we’d want the explanation of why they unify. That’s way beyond anything I might speculate about, but it’s incredibly exciting.

Chipkin: You mention in your book the largest discrepancy between observation and theory currently plaguing physics—the prediction of the energy of empty space versus its measured value differ by 120 orders of magnitude. Do you think that any recent theoretical developments may eventually solve this problem?

Farmelo: Probably the biggest discovery in physics in the last twenty years is the discovery by astronomers that the rate of the expansion of the universe isn’t slowing down, but it’s actually accelerating! Now, this is evidence for something called the energy of the vacuum, of empty space. When you do a simple calculation of the energy of empty space using basic ideas of atomic physics, you get an answer that is 120 orders of magnitude out. The problem of understanding dark energy, the energy of empty space, is my candidate for the most pressing problem.

Chipkin: Do you think that any of the theoretical developments you write about in the book could pave the way to finding a solution to this problem?

Farmelo: I hope so. Prognostications about these areas always humiliate the person who makes them. What we’re all waiting for is for experimental guidance. We’re not going to go back to natural philosophy and believe, along with Descartes, that you really could have a kind of narrative philosophy of nature. We want it anchored in comparisons between mathematical theories and the best numerical experiments we can do.

Chipkin: You write that “the slow rate of progress of the string theory framework may presage a more sedate pace in fundamental physics that may persist for centuries to come.” Could it be that there’s something fundamentally wrong with physicists’ current strategies?

Farmelo: My hunch is that we’re definitely missing something, maybe something fundamental. You’d have to be a fool to rule that kind of thing out. My belief is that the thing that is tough for many people to come to terms with is that for decades, in the physics of the atom and of gravity, physicists made incredibly strong progress. I remember when I was a student, discoveries were coming seemingly every few months. Now, we’re in a different world. You see theoreticians who never make an experimental prediction, and you even see experimenters who get PhDs on simulations, without doing any actual experiments. We’re looking at a different pace, where it takes a lot longer to set up these huge projects. I think we might have to get used to progress that’s been a good deal slower than the feast that we saw forty, fifty years ago.

Chipkin: What is the inspiration behind The Universe Speaks in Numbers?

Farmelo: Einstein wrote in 1952 that he believed it to be a miracle that our universe—which seems so complicated and even chaotic—has an underlying order, which we human beings can discover. Now, those patterns are what physicists are trying to explain. Mathematicians, meanwhile, also study patterns—patterns of ideas. What we find is that the patterns studied by the mathematicians are incredibly useful for understanding those patterns that we find in nature. That’s essentially why physicists and mathematicians work together—they’re both fundamentally interested in patterns. They really do need each other’s input, and I wanted to understand how we came to that situation from a historical perspective, beginning back with Isaac Newton, through electromagnetism in the nineteenth century, right up to the present day, and try to tell one coherent story about that astonishingly symbiotic relationship between physicists and mathematicians.

You can learn more about Dr Farmelo’s work at https://grahamfarmelo.com/. His new book, The Universe Speaks in Numbers, is out now in Kindle, eBook and hardcover format. Follow him on Twitter @grahamfarmelo.