Fundamental Limits
Benjamín Labatut's "When We Cease to Understand the World" and technical subjects in literary art
Towards the end of Benjamín Labatut’s acclaimed 2020 novel Un verdor terrible (English: When We Cease to Understand the World), one character seems to sum up the book’s main thesis about modernity, as he describes his horror at
[…] the sudden realization that it was mathematics—not nuclear weapons, computers, biological warfare or our climate Armageddon—which was changing our world to the point where, in a couple of decades at most, we would simply not be able to grasp what being human really meant. Not that we ever did, he said, but things are getting worse. We can pull atoms apart, peer back at the first light and predict the end of the universe with just a handful of equations, squiggly lines and arcane symbols that normal people cannot fathom, even though they hold sway over their lives.
This may seem like a relatively abstruse point to most people (especially as compared to nuclear weapons and their ilk), but for the literary novelist that problem, of the inscrutability to a layman of important parts of the working of the modern world—especially those which concern mathematics—is an absolutely central one.
The problem is not new, of course. Fiction authors have struggled to understand and present mathematical ideas ever since the birth of the discipline. The anxiety provoked by these difficulties is nicely encapsulated in a telling (but false) literary anecdote which nonetheless continues to circulate about the English mathematician Charles Dodgson, a.k.a. Lewis Carroll. The story goes that Queen Victoria was so delighted by his Alice’s Adventures in Wonderland that she wrote to its author personally, asking to be sent a first edition of the next book he published. Prof. Dodgson, the story continues, obediently sent her the following year a copy of his last publication: An Elementary Treatise on Determinants. The Queen, naturally, could make neither head nor tail of it. The real Dodgson would never have been so cheeky with the sovereign, but this story remains a favorite for a reason. It is what many people fear about mathematicians: that, being delighted by the surfaces of their work, we will find ourselves forever unable to penetrate into the essence; their real selves, and the true nature of the things they care about, are forever out of our intellectual grasp.
Now, in point of fact, an understanding of the sorts of mathematical concepts that most excite a literary imagination (chaos theory, incompleteness, entropy) while not a cakewalk, would not be out of reach for the kind of clever person who is intellectually engaged with the world of art and literature. They are just not easy, and unlike the discoveries of biology or lunar astronomy, they cannot be “translated back” into concrete reality once they have been learned, or paraphrased verbally with almost any freedom — at risk of turning false and laughable, or simply incoherent.
The number of mediocre literary treatises that play fast and loose with quantum mechanics or Gödel’s incompleteness theorem are, of course, numberless. Even the greats (and those seemingly well-qualified) are not immune from mathematical error: David Foster Wallace delivered a senior thesis on modal logic at Amherst, but Infinite Jest is marred by pointless, seemingly unintentional mathematical mistakes, and his non-fiction book on the nature of infinity is notoriously useless as an actual guide to the subject, as it is riddled with basic misstatements of fact. Tom Stoppard’s award-winning play Arcadia offers a laughable account1 of the mathematical subjects (chaos theory and fractals) that it purports to address, not just in details but in basic essences. An example of an honorable exception, to prove the rule, is Robert Musil, who in his 1906 novel Die Verwirrungen des Zöglings Törleß (English: The Confusions of Young Törless) presents a credible and correct account of how imaginary values are necessary for intermediate steps in finding even the real roots of certain polynomials — and this point is brilliantly woven into the central theme and ideas of the novel with no painful misstatements along the way. It is to be noted that Musil had a technical background himself (he matriculated at the Technical College in Brno, Austria, where he studied mechanical engineering), he confined himself to an elementary mathematical point of which his understanding was fully secure, and he wrote from the perspective of a young student struggling with the subject. There’s no way to know how many of these constraints Musil could have shed without producing a mathematical embarrassment. It is evident that the trick is harder than it looks: trying to speak clearly and poetically about mathematics — trying to attach any meaning to it at all, to make it sing in any language that is not its own — is like moving water in the palms of your hands. The farther you go, the harder it is to keep it with you.
As comprehended in the quote that opens this piece, Bejamín Labatut is aware of what a minefield he has entered. When We Cease to Understand the World comes across as an attempt to confront the issue head-on, to try to re-enchant the 20th-century metaphysics of math and quantum theory, without falling into either the scylla of mere tedium or the charybdis of clownish misstatement.
A skilled reader will be attuned to certain “red flags” in any work of non-fiction. One of the most serious of these is when an author seems convincing and forceful on subjects you (the reader) are not familiar with, yet seems to do a very slipshod job of covering those subjects you actually know something about.2 This same heuristic can be applied, with some caution, even to fiction. If a globe-spanning novel seems horribly unconvincing and caricatured when it portrays your own neighborhood, it’s worth questioning whether the foreigners in its cast have been just as ill-served. This is necessarily a rough heuristic, which is why I am going to immediately qualify my statement after telling you that the section of this book that concerned people and ideas I am even passingly familiar with (the chapter entitled “The Heart of the Heart”, about Mochizuki and Grothendieck) seemed to me by far the worst, and the least faithful to its subject matter.
The first qualification I will add is that the five main sections of this book are all extremely dissimilar, to the point that it seems actually to be a handful of short works on vaguely similar themes that have been bound together in one volume, rather than a “novel” per se (although the final piece does make some effort to tie them together). The second qualification is that while the ostensible subject matter of “The Heart of the Heart” (Grothendieck’s work on mathematical foundations and Shinichi Mochizuki’s claimed proof the abc conjecture) is more familiar to me than the others,3 it is also the most abstruse and non-concrete. Undegraduates in physics and engineering are expected to be familiar with the Schrödinger wave equation (another topic covered in the novel), but there are dizzyingly few human beings alive who have even the remotest grasp of Inter-Universal Teichmüller Theory.4
Since the first four sections of this “novel” are so distinct from one another in style and subject matter, I will discuss them separately, and in order.
The best of all the pieces is the first, which is titled “Prussian Blue”. The original Spanish title of When We Cease to Understand the World is Un verdor terrible — an allusion to this first piece (the English title is taken instead from section 3). “Prussian Blue” is not about math or fundamental physics at all, but rather about chemistry, and it differs from the rest of the book as well in being almost entirely true.5 This increases its power, as history is a better inventor than Labatut. Its thesis is compact, and is forcefully dramatized by the facts he has arranged — the beauty and the misery of the twentieth century are depicted here, all helplessly unfolding from the tainted seed of human curiosity, the amoral splendor of science. It’s a genuinely moving and fascinating piece, which really should be read on its own. It sets the tone and (very broadly) the thematic currents of this novel, but is disconnected from the technical subject matter of the rest, and its style is very different. If I am going to recommend any part of When We Cease to Understand the World, it is likely to be “Prussian Blue” first and foremost.
The book’s second section, “Schwarzschild’s Singularity”, is somewhat more fictionalized than the first, although still sticking mostly to the framework of actual events. It focuses on Karl Schwarzschild, and imagines him agonizing over the singularity implied by his solution to Einstein’s equations. Labatut’s discussion of singularities produced only the faintest cringe from me as a mathematically literate reader — his touch is light, although I might have more to complain about if I was versed in astrophysics. It’s an interesting idea, depicting Schwarzschild as haunted by the black hole in much the same way the e.g. Cantor is often depicted with his infinities.6 The extension of this metaphor (the lurking black hole) to Germany’s approaching catastrophe is layered on perhaps a little thick, but the piece is overall effective. This is another short piece, and in a way is really a dress rehearsal for part four. Before we get there, however, we will need to pass through…
The third section of the book, which is entitled “The Heart of the Heart”. The piece presented in this section is, essentially, a work of fanfiction about Shinichi Mochizuki and Alexander Grothendieck in a Lovecraftian key, bearing only the most passing resemblance to any historical persons or events. Although Labatut cultivates the impression in this book that his is a sort of “God of the gaps” fictionalizing style, in this section the central facts around which the story is structured are blatantly ahistorical. “The Heart of the Heart” reaches its climax with Shinchi Mochizuki standing up an international conference on his ideas in 2014 and then abruptly withdrawing all his papers on Inter-Universal Teichmüller Theory — an event that most assuredly did not take place; Mochizuki has continued to defend his ideas in public through the present day, and had given no indication of stopping while this book was being prepared (in 2020). “The Heart of the Heart” should really have been about two fictional mathematicians, drawing inspiration from Grothendieck and Mochizuki but not explicitly about them (Labatut also seems to want Grothendieck to have dramatically ceased mathematical research in the 1970s, which isn’t quite right either).
(The title “The Heart of the Heart”, for what it’s worth, is apparently a reference to certain objects in algebraic geometry known as “motives”. As far as I know these objects do not possess any special power to drive people mad who learn about them, nor do they contain the terrible secret of any future superweapon. I am not familiar enough with algebraic geometry to comment on how arbitrary this choice was — as far as I can tell Grothendieck did consider motives an important and useful construct.)
Now, this is where the matter of the heuristic about familiarity comes in. Does this third section, “The Heart of the Heart” only seem so laughable to me because I happened to have followed the ITT saga in the mathematics blogosphere while it was happening? Would the other sections too seem laughably ahistorical if I was as familiar with the biographies of 20th-century theoretical physicists? Am I deceiving myself, in short, that the rest of When We Cease to Understand the World is making a serious attempt to understand anything at all?
For one thing, as far as I can tell, no absolutely central facts from the other chapters are as baldly fabricated — Schwarzschild really did die of an immune disease after being exposed to gas on the Eastern front; Schrödinger and Heisenberg really did develop their theories with the illnesses and in the remote locations depicted in the book. And no other section is structured as a lazy potboiler, Lovecraftian or otherwise,7 like “The Heart of the Heart”. So I think it really is different — and worse.
In the English translation, Labatut’s title comes from the fourth section — “When We Cease to Understand the World”. This is by far the book’s longest section, and is really the meat of the novel (along with “Prussian Blue”). The section is structured around two independent tales of troubled discovery: Werner Heisenberg developing the matrix formulation of quantum mechanics on the German island of Heligoland while sick with hay fever, and Erwin Schrödinger developing his wave equation while convalescing with tuberculosis in a sanatorium in the alps. Both are depicted as troubled by bizarre psychosexual and apocalyptic visions which culminate in their respective breakthroughs. Here Labatut makes full use of the freedom granted by writing about a man six decades dead by imputing ephebophilic agonies to Schrödinger, but nothing in this section is gratuitous — the stories of Heisenberg and Schrödinger, obviously fictionalized, are skillfully deployed to create an atmosphere of dread and reflection on the theme of the terrors produced by scientific knowledge. Its climactic scene, involving Heisenberg’s uncertainty principle, is especially deft. There are obvious directions to go with this theme in terms of the history of the 20th century, and Labatut to his great credit deploys a very light touch, quietly reminding us of terrible facts which he can be certain we already know.
Labatut, in this fourth section, also wisely allows the actual physics that is at issue fade into the background — whereas in “Schwarzschild’s Radius” the question of black holes provides the story’s central metaphor (and hence Labatut exposes himself to possible trouble, as he has to describe a mathematical concept over and over again in different ways), here the pangs of discovery are center stage, with the scientific controversies they would resolve a relatively muted backdrop, albeit one which places these scenes in a rich, foreboding historical context. Does this save the technical matters? I cannot personally say, although it’s certainly not as obviously laughable as a lot of literary depictions of quantum mechanics.
The book’s final section, “The Night Gardener”, is the most fictional and novelistic of all, attempting to collate and summarize the events in the disparate sections preceding. It is in this section that the titular night gardener offers the observation which began this review — that mathematics is the most terribly occult of all scientific disciplines, and that it is impossible to escape its influence. In my view, this final section was not Labatut’s most powerful, and it comes across as almost an anticlimax, or a failed bundling together of unlike strands — but others may feel differently. The fact that it leans a little on the weak material about Grothendieck may be another reason it failed to land for me.
Overall, I think Labatut has recognized a serious and interesting problem in contemporary art, and with this novel he has made a concerted stab at trying to resolve it — how to write about technical subjects, and most of all deep mathematics, as a philosopher of the world who simply cannot come to grasp these technical subjects in the way the experts do. It is a noble effort, even if it is one at which no one can ever expect to perfectly succeed. When We Cease to Understand the World is vital and thrilling in its presentation, and only when it reached the knottiest matters of all—those of mathematical foundations—did it begin to embarrass itself, at least to me (a decided non-specialist). I can still recommend it — despite, or perhaps even because of, the fact that its central dilemma remains stubbornly unresolved.
The author of the linked review, a mathematically educated novelist himself, also introduced me first to Wallace’s mathematical problems.
Some people use the term “Gell-Mann amnesia” in connection with this phenomenon — a phrase coined originally in reference to newspaper coverage, which virtually always has this sad property due to the tight schedules and very broad readership which constrain even the rare conscientious newspaper reporter.
Not in the sense of any kind of expertise, but I can often tell when somebody is bullshitting about higher mathematics, and I followed the news about Mochizuki’s proof at the time that it was happening, which makes some of the events depicted in “The Heart of the Heart” more than a little eyebrow-raising.
To the extent that there is anything to grasp, of course — a controversy I will not enter into any further than this footnote, on grounds of abject ignorance.
The Acknowledgements at the back of the book admit to one false paragraph in the first chapter, although I have been unable to discover from any source which paragraph — the most memorable and striking facts have all held up to my casual, google-based scrutiny.
The truth about Cantor is another matter, but what kind of a re-enchanted world would it be if no one is driven mad by their discoveries?
No knock on Lovecraft is intended — unlike his many imitators, Lovecraft didn’t simply leave a blank at the center of his better stories, but filled in the horrifying climax with spectacularly purple prose. Not everybody’s taste, but he didn’t typically resort to “tastefully” omitting any description of the horrible truth.