Graham Farmelo’s new book The Universe Speaks in Numbers has recently been published in the UK, US publication is next week. The topic of the book is one close to my heart, the relationship of mathematics and physics. I’m very much in agreement with the main argument of the book, which is that our most fundamental theories about physics have turned out to naturally be expressed simply and beautifully in terms of deep ideas about mathematics. This surprising congruence between the deepest ideas in two seemingly different fields strongly indicates that they share an unexpected unity. As fundamental physics reaches technological limits on the experimental side, investigating the underlying mathematical structures may be the best and only route open to further progress.
Much of the book is historical (and often Anglo-centric), beginning with Newton, a figure who made huge tightly intertwined advances in both mathematics and physics. Looking at Newton’s career, it makes no sense to characterize him as a mathematician or a physicist, he’s both in equal measure and at the same time. Farmelo then moves on to Maxwell, who also revolutionized physics while at the same time introducing important new mathematics into the subject. He tells the story of Maxwell’s 1870 talk “On the relations of mathematics to physics”, then a couple chapters later it’s Dirac’s 1938 talk “The Relation between Mathematics and Physics”.
In Farmelo’s account:
Dirac quickly arrived at what was, in effect, a manifesto for research into theoretical physics. He proposed a new principle – the principle of mathematical beauty – which says that researchers should always strive to maximize the beauty of the mathematical structures that underpin their theories of the natural world…
He concluded that “big domains of pure mathematics will have to be brought in to deal with the advances in fundamental physics”…
Eventually the two subjects might possibly become unified, Dirac suggested, with “every branch of pure mathematics then having its physical applications, its importance in physics being proportional to its interest in mathematics.”
A major reason for Dirac taking this sort of view was the example of Einstein’s work on general relativity, which to reach fruition had required Einstein to become expert in the newly developed and rather challenging abstract mathematical machinery of Riemannian geometry. The two great pillars of modern physics, relativity and quantum theory, are deeply related to central modern ideas about mathematics, in particular, respectively, geometry and representation theory. While in the case of relativity the mathematics came first, for the case of quantum theory, the mathematics underlying the subject was mostly developed later.
Farmelo goes on to explain that relations between math and physics entered a fallow period during the 1950s and 1960s, but the advent of gauge theory and the Standard model led to a productive renewal of healthy relations during the 1970s. He does a good job of explaining how this came about, discussing in particular the central role played by Witten. Farmelo has benefited from getting to talk to Witten himself in some depth about this, and gives a nuanced portrayal of Witten’s rather complex and evolving feelings about the relations of the two subjects and the role that he and his immense talents have played in this story.
There’s a great deal of the usual sort about the history of string theory, emphasizing its points of contact with new mathematics. The next to last chapter is about Nima Arkani-Hamed and the amplituhedron story, portrayed as the latest exciting development on the math-physics front. Farmelo is clearly enthralled by Arkani-Hamed and his intense enthusiasms. The evolution of Arkani-Hamed from phenomenologist to mathematical physicist is definitely a fascinating thing to observe, and I’ve often written about it here (you might want to for instance read this posting). Farmelo also points to an excellent lecture by Greg Moore on Physical Mathematics and the Future (discussed here), which I think is based on a much deeper understanding of the current state of the math/physics relationship, and gives a much broader perspective than the narrow one of the amplituhedron. By the way, I see on Moore’s website that he’s writing up for the 2019 TASI school what appears to be an excellent set of notes about Chern-Simons theory and related topics.
A major problem though with this book is that it pretty much completely avoids the big problem raised by the program of pursuing progress in fundamental physics through beautiful mathematics: how do you know whether people doing this are on the right track or headed down a blind alley? Farmelo starts the book off with a very odd preliminary chapter comparing Einstein’s work at the IAS in his later years to that of his modern day successors:
[Einstein] was seeking a new theory, not in response to puzzling experimental discoveries, but as an intellectual exercise – using only his imagination, underpinned by mathematics. Although this approach was unpopular among his peers, he was pioneering a method similar to what some of his most distinguished successors are now using successfully at the frontiers of research.
The fact of the matter is that, in retrospect, Einstein’s work of this era was a huge failure, as he got stuck deep down a blind alley. He was seduced by a specific speculative idea about how to get unification out of mathematics, by using simple extensions of the differential geometry that he had such success with in the case of GR.
How does Farmelo know that string theory enthusiasts following Einstein haven’t run into the same problem he did? In essence, Farmelo just assures us that he has talked to them and they tell him that, like Einstein before them, they think they’re on the right track. The existence of skeptics is mentioned, but their writings are carefully excluded from the 200+ item bibliography. Jim Baggott, Sabine Hossenfelder and I (and our writings) appear only in a short footnote on page 6, with bloggers described as complaining that “modern physics should get back on the straight and narrow path of real science”. But the three of us are complaining about, not “modern physics”, but one small subset of it, and at least in my case, the path I argue for is almost exactly what Farmelo is arguing for: absent help from experiment, pursue the path advocated by Dirac.
Here’s part of Farmelo’s summation of the current situation in the book’s last chapter:
The great majority of today’s leading theoretical physicists are, however, confident that they are motoring steadily in the right vehicle, despite the problems they are having in trying to drive it.
In the public domain, the debate about the merits of the string framework has been raging for years, especially in print and online. Some of these onslaughts are useful correctives to the hype lavished on this programme and to the superciliousness of pronouncements made by some string (although rarely by the best ones in my experience). Experts on the string framework have every reason to be proud of the progress they have made, but until such time as experiments confirm its validity, there is no room for smugness. Yet I am often 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. Opposing the view taken by leading theoreticians might be interpreted as a healthy disrespect for orthodoxy. However it may be part of the worrisomely common view that anyone can have a valid opinion on any subject, regardless of their technical knowledge and appreciation of it. In scientific matter this trend is especially regrettable.
The first part of this I think is simply not true, with most “leading theoretical physicists” these days unsure what the right direction is for how to get beyond the Standard Model. As for the last bit, I’ll just say that I think it can accurately be described as “sleazy” (by the way, Farmelo at one point came to see me in New York when he was doing research for the book, and we had a quite pleasant conversation, he’s rather charming). Besides the ad hominem attack on unidentified critics, there’s nothing anywhere in the book about the actual problems of the vehicle some people are motoring in. For instance, the problem of the landscape and the multiverse is dealt with by just ignoring it, it’s not mentioned at all. If it had been mentioned, Farmelo might have had to deal with the fact that it’s mathematically hideous, so a direction which should be abandoned by his own arguments.
In the end, my feelings about this book are much the same as in the case of Farmelo’s biography of Dirac (see here): a wonderful book in many ways, but marred by a bizarre degree of string theory fanboyism. While there’s a lot to like about this book, and much of it makes a good case for a controversial point of view that I strongly agree with, unfortunately the problems with it are even more serious than in the case of the Dirac biography.
The IAS is having a public event next week, convening a panel to discuss the math-physics issues brought up by the book. I may add something here about this after it happens. Perhaps someone in attendance can get a show of hands from the assembled leading theorists to see if they really feel that they’re steadily motoring in the right vehicle or not.
Update: I’m listening to a live-streaming version of the IAS event here. After a talk by Farmelo, mainly about history, and a discussion with Karen Uhlenbeck and Freeman Dyson (moderated by Natalie Wolchover), Greg Moore is now giving a talk (available here) on TQFT and gauge theory which is quite good. I’m very much in sympathy with his take on “physical mathematics”.
Update: Besides an advertisement by Nima Arkani-Hamed and Thomas Lam for their work on amplitudes, the only discussion of the current state of the math/fundamental physics interface was the final short conversation between Dijkgraaf and Witten. Looking back at expectations from 30 years ago, Witten said he had expected progress on understanding the laws and principles behind string theory, but that has not occurred. Dijkgraaf tried to end with a defense of string theory as “well, by AdS/CFT it’s connected to QFT, so it somehow is connected to the real world”, to which Witten said something like “we shouldn’t give up hope for string theory as a unified theory, we might yet find the right string theory vacuum”. In the end Witten said that he found it hard to accept the possibility that the unexpected things discovered through string theory didn’t mean that it had to do with a unified theory, but admitted he didn’t have a scientific justification for this.
Update: There’s a review of the book by Tony Mann at Times Higher Education. It includes:
Farmelo’s book is a response to these contrarians [Hossenfelder and myself]. He is confident that the beauty of the mathematics is significant in indicating that we are on the right track, and that eventually, even if we have to wait for many years, we will be able to test string theory against new experimental evidence. As a spirited defence of the idea that beautiful mathematics should be a guide for physicists, Farmelo’s book is a timely response to critics such as Woit and Hossenfelder, defending what science writer Jim Baggott has called “fairy-tale physics”. Ultimately I am not sure, however, that he makes his case anything more than a matter of faith.
I think Mann gets it right that the Farmelo book is intended as a defense of string theory against the Baggott/Hossenfelder/Smolin/Woit critique (with the odd feature that he refuses to allow mention of our books in the bibliography, or to engage in any way with the arguments we make), and that his argument comes down to little more than “trust certain famous theorists”, especially those at the IAS. As usual, I’m inaccurately portrayed as opposed to the idea that progress can be made by following beautiful mathematics. The problem with string theory unification is that it’s a failed physical idea, with the failure indicated not just by lack of predictivity, but also by the fact that string theory models compatible with observations are horrendously ugly.
Update: The Dijkgraaf-Witten conversation is available here. Some extracts from near the end.
Witten: I’m actually personally reasonably confident that what we’re doing is a lasting contribution. But I’m less confident that we’ll really be able to put it on a completely solid footing. It depends on being lucky with experiment, I would say…
Dijkgraaf: Often I hear questions about whether is string theory wrong or right, but I often answer there’s no way in which we’ll ever get rid of string theory, because in some sense it’s an integral part of the theories that we already are using.
Witten: That’s true but not completely satisfying. Let’s not give up on the dream of finding the vacuum that describes the real world…
The honest answer is that personally as I told you before I have confidence that the general enterprise is on the right track, but I don’t claim that the argument I’ve given is scientifically convincing.
Witten makes clear that at this point he has no likely forseeable experimental results relevant to string theory he can point to, no “scientifically convincing” argument that it’s on the right track, just a feeling that since string theory research has turned up various points of contact with important math and physics, there must be something right about the idea.
Update: Siobhan Roberts at the New York Times has a piece about the centenary of the eclipse that made Einstein famous that also discusses Wednesday’s IAS event.