Somehow, without really intending to, I absorbed many years ago a line from The Brothers Karamazov that still comes to mind, verbatim, with surprising frequency: “The impotent and infinitely small Euclidean mind of man.” It burbles up whenever a book I’m reading or a podcast I’m listening to prompts me to envision something impossible—the vast infinity stretching out before the big bang, or the notion that linear time is not real. It echoes reliably in the back of my mind when I read about AI systems that are so complex even their makers cannot fathom how they produce their results. I nearly said it aloud the other day after realizing that I’d forgotten, for the second time in a row, to turn off the burner after making a cup of tea.

The line is spoken by Ivan Karamazov, who is talking to his brother Alyosha about the impossibility of grasping the divine will, which, if God were to actually exist (Ivan is not so sure), must be far beyond human understanding. He’s also talking about the paradoxical nature of the universe itself, which pioneering mathematicians of the day proposed might operate in four dimensions, according to geometrical axioms that were totally incomprehensible. Dostoevsky, I think, was using Ivan to make a point about how modern science requires as much faith—and runs up against the same cognitive frontiers—as religion. Just as the believer will never understand why an omnipotent and merciful God allows evil, so atheist intellectuals like Ivan cannot force their brains to imagine an infinity in which two parallel lines meet. This is owing not to a dearth of knowledge but to the limits of the operating system. “All such questions,” as Ivan puts it, “are utterly inappropriate for a mind created with an idea of only three dimensions.”

The irony, of course, is that Ivan cannot stop himself from asking those inappropriate questions. By the end of the novel, he goes mad, pushing reason to its absurd outer limits. It’s a persistent human error; we cannot resist trying to understand what we are hardwired not to. If anything, the death of God has left a conspicuously empty seat in the rafters that we keep trying to inhabit—that purely transcendent, objective vantage outside the totality of things. Spinoza called it sub specie aeternitatis. Hannah Arendt named it “the Archimedean point.” Thomas Nagel termed it the “View from Nowhere.”

William Egginton, a literary scholar at Johns Hopkins, calls this tendency “metaphysical overreach.” His new book, The Rigor of Angels: Borges, Heisenberg, Kant, and the Ultimate Nature of Reality, is a joint biography of three figures who called attention, much as Dostoevsky did, to the problems and paradoxes that emerge when we try to extend our ordinary way of seeing the world beyond the human scale. These contradictions—or what Kant called antinomies—arise when we try to eject ourselves beyond space and time, imagining earthly life from an eternal perspective. (Egginton calls this illusory vantage “the god of very large things.”) They also arise when we zero in too closely on things at a minute scale. (Egginton calls this, borrowing a phrase from Arundhati Roy, “the god of very small things.”)

Whether large or small, the point of view is presumed to be objective, unmoored, and impartial: hence the false divinity. This is not a new problem, but Egginton argues that it slyly persists in many thought experiments and theories, including the multiverse hypothesis, interpretations of quantum mechanics, and debates about free will. Borges, Kant, and Heisenberg each cautioned against such confusions and, as Egginton puts it, “shared an uncommon immunity to the temptation to think they knew God’s secret plan.”

Borges’s 1942 short story “Funes the Memorious” is about a man, Ireneo Funes, who has a perfect memory. Funes can remember every tree in every forest he has ever walked through, as well as every leaf that appeared on those trees—and every leaf on every tree he has ever dreamed or imagined. He can perfectly recall the position of the clouds on any given morning from his childhood and can, if he chooses, replay every detail of any day of his life—though doing so, of course, takes an entire day.

But in classic Borges fashion, the story is pushed to its logical extreme to consider the hidden liabilities of a godlike memory. Because Funes is incapable of forgetting any detail, he has problems understanding language. He cannot fathom why the word “dog” is used for so many distinct dogs, all of which vary in size, color, and form. He cannot even understand why the same dog is called by a single name—why, for instance, “the dog at three fourteen (seen from the side) should have the same name as the dog at three fifteen (seen from the front).”


It’s Borges’s narrator who points out that despite his perfect recall, Funes “was not very capable of thought.” “To think is to forget differences, generalize, make abstractions,” the narrator observes. “In the teeming world of Funes, there were only details, almost immediate in their presence.” What Funes lacks is the conceptual stability that allows for continuity across time and space. A memory that is granular enough to recall every vein of every leaf cannot (quite literally) see the forest for the trees. Funes’s mind is so focused on recording every detail, in fact, that he begins to lose his sense of self: “His own face in the mirror, his own hands, surprised him every time he saw them.”

Borges first became interested in the paradoxes of time and the self in the late 1920s, when the woman he loved, the poet Norah Lange, abandoned him abruptly for another man in the middle of a party. In the throes of heartbreak and despair, Egginton notes, he became obsessed with Zeno’s paradox, which demonstrates that any distance or span of time can be infinitely divided into smaller and smaller segments. The more one tries to isolate the elusive moment of change (in the ancient puzzle, the instant when Achilles overtakes the tortoise in a race), the more the very notion of change recedes from one’s grasp.

Borges’s story of Funes dramatizes essentially the same problem. Our ability to understand the world—our sense of continuity, time, cause and effect—requires keeping some distance from what we observe. When we look too closely, we lose the continuous self who can span discrete moments in time and relate them to one another. In fact, Egginton points out that if Borges were to push the logic of his story just one step further, Funes would be unable to perceive, period:

A being who was truly, exclusively saturated in a present moment wouldn’t be able to observe anything at all. Observation, any observation, installs a minimal distance from what it observes, for the simple reason that for any observation to take place, one here and now must be related to another here and now, and that relation needs to be registered by some trace or connector between the two.

This is the problem encountered by “the god of very small things,” a vantage so immersed in particulars that the world becomes incomprehensible—and one that has a habit, Egginton shows, of creeping into science and philosophy under different guises. David Hume fell prey to it when he argued, in the conclusion of A Treatise of Human Nature, that there is no such thing as a continuous self. Hume believed that the self was a meaningless abstraction we impose on a collection of unconnected impressions—heat or cold, light or shade. Because of this, we cannot know anything for certain about the world that comes to us through our senses; no perception in isolation can be an example of a general rule.

Kant claimed that this argument, which he first encountered in a German translation in 1771, woke him from his “dogmatic slumber,” and his attempts to wrestle with it led to the insight that gave birth to Critique of Pure Reason. What he realized, basically, was that Hume’s proposition was incoherent. The very notion that we can discern the difference between heat and cold, light and shade, necessarily entails a self embedded in time and able to bridge two disparate moments. Without a self, Kant argued—without a sense that someone is looking and can register the beginnings and endings of sensations—experience itself would not exist.

Quantum physics, too, has suggested that reality cannot be described without reference to an observer. And it’s telling, given Borges’s and Kant’s conclusions about the relationship between paradox and granularity, that the most baffling scientific observations arise when we bore down into the building blocks of reality and try to witness the moment of change at extremely small scales. Heisenberg realized this when he outlined his uncertainty principle, which demonstrates that the closer you zoom in on the location of a particle, the less you know about its momentum.

This is not, Egginton insists, because there is something “spooky” going on with reality; it’s because the ability to simultaneously observe an electron’s position and momentum would require the impossible: perfect presence in a single moment in time. The very nature of observation requires the observer to generalize. “Without this slight blur,” Egginton writes, “this ever so subtle distancing, this lifting up and holding steady of a standard so as to register some infinitesimal alteration, all there would be is an eternal present.”

The larger truth that Borges, Kant, and Heisenberg uncovered was that we often confuse our knowledge of the world with the world itself. Science and philosophy fall into error when they treat time, space, or the self as objects of observation, as opposed to the necessary means by which we can observe anything at all. Heisenberg was well aware that his principle did not just identify an anomaly in the physical world; it suggested that there is no coherent reality independent of our observations of it. As he put it: “We have to remember that what we observe is not nature in itself but nature exposed to our method of questioning.”


Egginton’s title is drawn from a Borges line: “Enchanted by its rigor, humanity has forgotten, and continues to forget, that it is the rigor of chess masters, not of angels.” What we forget, in other words, is that the world ceases to make sense when we imagine we are transcendent beings who can stand outside it, not the players of a board game whose rules are formulated by our own limitations. “Our reason propels us to incredible heights—understanding the fundamental components of matter and laws of the universe, seeing almost to the edges of the cosmos and the beginning of time,” Egginton writes. “But it also leads us woefully astray. For the very ability we have to map our world and hence see our way through the dark also treats that map as though it were the world.”

For Kant, that mental map included space, time, and causality, which he took to be a priori structures of the mind, not objects of experience. Kant referred to this insight as his “Copernican revolution” because it displaced the world itself from the center of philosophy, just as Copernicus removed the Earth from the center of the cosmos—though in hindsight, Kant’s revolution did not go far enough. He still believed that some mathematical propositions and geometric laws were logical necessities, which is to say inviolable features of reality itself—a belief that modern physics upended.

The possibility that so troubles Ivan Karamazov, a non-Euclidean universe that the human mind cannot grasp, turns out to be true (hyperbolic geometry laid the foundations for general relativity). It was Einstein who pointed out that modern physics had more or less demolished Kant’s belief that human reason might still bear some shadowy formal relation to the world itself. “As far as the propositions of mathematics refer to reality, they are not certain,” Einstein wrote in 1921, “and as far as they are certain, they do not refer to reality.”

Despite this admission, Einstein famously resisted the implications of quantum physics, particularly the idea that the laws of nature were somehow dependent on the observer. The Rigor of Angels traces the long walk Einstein and Heisenberg took through Berlin in the spring of 1926 and their debates about whether physics could reveal objective truths about reality. Heisenberg, for his part, came to believe that Kant’s epistemology remained relevant and had actually anticipated some of the insights of modern physics. While the a priori could no longer be presumed to have a metaphysical dimension, it remained useful so long as we thought of it as the hardwired structure of the human mind—our distinctive map of the world that had emerged through biological evolution and, because it was rooted in classical physics, had only limited use in the quantum realm. For Heisenberg, physics had simply widened the chasm between the knower and the object of knowledge, proving that “it will never be possible by pure reason to arrive at some absolute truth.”

While we tend to think of these revelations as modern, Egginton points out that they have ancient roots. When the Christian philosopher Boethius wrote in the sixth century that “things are known not according to their natures but according to the nature of the one who is comprehending them,” he was expressing an idea that has a deep lineage in religious thought: the fundamental disjunction between humanity’s temporal nature and God’s eternal truth. For the theologians like Boethius and Augustine, eternity was not an endless extension of time but a reality that exists outside of time, an idea that entered Christian thought through Plotinus. (There is a conceptual ancestry here that Egginton traces in a rather unsystematic way—which is fitting, given how chaotically ideas cross-pollinate across centuries: Heisenberg and Borges read Kant, and all three men were conversant with the Greeks. Borges read Dante, who’d read Boethius. Kant was aware of Augustine’s theology, which he came to via Martin Luther.)

It was obvious, even to these ancient thinkers, that weird things happen when humans try to cast off the shackles of linear time and assume the vantage of eternity. In his Consolation of Philosophy, Boethius asks how divine foreknowledge can be compatible with free will. If God already knows what will happen at each moment, aren’t our actions predetermined? In the end, though, he concludes that we are free because any knowledge that exists outside of time is irrelevant to the very nature of choice, which must necessarily take place in time.

“The god of very large things,” the second of Egginton’s major fallacies, is the delusion that humans can inhabit the perspective of Boethius’s God—beyond time and space, hovering somewhere outside the totality of things. This pitfall emerges in thought experiments like Nietzsche’s idea of eternal recurrence, the notion that, given infinite time and a finite number of atoms, the universe must inevitably run through all possible permutations of reality and start repeating itself. Borges encountered this idea in Thus Spoke Zarathustra, and as he tried to entertain the possibility that his life had been repeated many times before, he discovered an internal contradiction: the very notion of repetition is meaningless without an observer who stands outside the repetitions and registers them. “Without a special archangel to keep track,” Borges wrote in a 1936 essay, “what does it mean that we are going through the thirteen thousand five hundred and fourteenth cycle and not the first in the series or number three hundred twenty-two to the two thousandth power?”

Borges’s short stories play with the paradoxes that arise when we try to comprehend infinity. “The Library of Babel” conjures a universe-size collection of books that contain every possible permutation of language. “The Garden of Forking Paths” envisions a novel that functions like an infinite choose-your-own-adventure book. As opposed to most fiction, in which characters who face alternative possibilities must choose one and forgo the others, in this novel the characters simultaneously choose all of them, creating “an infinite series of times, a growing, dizzying web of divergent, convergent, and parallel times” that “contains all possibilities.”

The story eerily anticipates the “many worlds” interpretation of quantum mechanics, which Hugh Everett proposed in the late 1950s, more than fifteen years after Borges had published his forking paths story. What if, Everett proposed, measuring a particle did not change its location but instead created two separate universes that realized both possibilities? The theory was an attempt to eradicate the radical implications of Heisenberg’s uncertainty principle and return to the naive objectivity of classical physics. But as Egginton points out, the many worlds thesis “requires a presumption of godlike knowledge.” Just as the infinite possibilities of a choose-your-own-adventure story are only apparent to someone who exists outside the novel—a reader—so the many worlds hypothesis posits an outside observer, a reader of the universe who can see all forking time-paths simultaneously. That godlike perspective is, he argues, our own imaginative projection: “The invisible pullulations of those infinite other worlds can remain only that, shadows, impossible to discover, alternate realities conjured to assuage the discomfit caused by the inconsistencies of our own.”

“The god of very large things” is a subtle and insidious delusion, Egginton maintains, one that creeps into discussions about cosmological fine-tuning or attempts to imagine what happened “before” the origins of the universe. Some fourteen centuries after Boethius, he insists, arguments about free will still defer to a godlike knowledge. As an example, he considers a thought experiment proposed by Sam Harris, an outspoken determinist. Harris asks the reader to imagine that all the atoms in his body have been replaced with the atoms of a murderer’s; in that case, he argues, there would be no remaining part of himself that could resist committing the same crime. For Egginton, Harris’s thought experiment smuggles metaphysics back into the picture and relies, implicitly, on some transcendent awareness that exists outside the conditions of the thought experiment. (Harris’s example, Egginton writes, is “in fact doing precisely what he disallows: positing a secret extra part of himself able to clock those differences and report back.”)

Egginton is relying here on the familiar compatibilist argument that freedom is nothing else but the ability to imagine roads not taken. The simple fact that we cannot know the outcome of any situation means that “choice” is a meaningful idea and condemns us “to that very freedom that the godlike knowledge of a mechanistic universe seeks to absolve me of.” (Harris would surely object that this understanding of freedom is a clever bit of redefinition and not what ordinary people mean by “free will.”)

In the end, Egginton is not fatalistic about the scientific endeavor but is offering a cautionary tale about the errors that result when we try to see beyond the human apparatus. Many efforts at transcendence inevitably fold in on themselves, returning us, paradoxically, to ourselves and the structures of our own minds. “No matter where we train our gaze on the starry skies above, we look inward toward the very origin of space and time,” he writes. “Stretching out from ourselves to resolve that incompatibility, we ultimately realize what we are striving for lies inside us; we find ourselves in the world and the world in ourselves.”

I realize that the book I’ve been describing does not sound much like a biography. I should probably mention that Egginton’s discussion of these ideas comes between long narrative passages about Borges’s heartbreak, Kant’s home renovation headaches, and Heisenberg’s impressive skill as a downhill skier. Almost every new chapter begins with a scene from one of the men’s lives, narrated relentlessly in medias res. (“When Heisenberg entered the room for the Bohr Festival in Göttingen on a balmy spring day in June 1922, he was still only twenty years old.”)

I’ve resisted recounting all this here because these passages, although lucid and well written, arrived as unwelcome distractions and felt largely irrelevant to the book’s drama of ideas—though not from any lack of effort on Egginton’s part. He is, if anything, too eager to synthesize the life of each man with his thought, supplying psychological conjectures about how personal tragedies and professional setbacks may have informed their work. Kant’s annoyance at the “theatrically loud prayers” issuing from the prison next to his house is linked to the aversion to religious fanaticism in his moral philosophy. Borges’s “Garden of Forking Paths,” Egginton proposes, might be read as a Frostian meditation on roads not taken after the writer accepted an invitation from Augusto Pinochet, which may have cost him the Nobel Prize.*

I did not find many of these connections persuasive, and they tended to come at the end of each section, just as the wild strands of ideas promised to coalesce. One particularly engaging thread about eternity ended, much to my disappointment, with some back-of-the-envelope psychoanalysis. (“Time is loss. Time is heartbreak. Time is desire for timelessness,” Egginton writes of Borges’s unrequited love.) It was the same letdown I’d felt years ago while watching Christopher Nolan’s Interstellar, a film that plumbs the depths of quantum weirdness only to arrive at the conclusion that love is the one thing in this crazy universe that transcends space and time.

Maybe it’s stupid to seek cosmic insights in the byways of popular culture. Maybe I, like Ivan Karamazov, am guilty of wanting answers to “inappropriate” questions. I got the sense that Egginton, too, was haunted by inappropriate questions—not metaphysical ones but questions more expansive than those typically asked in the twenty-first century—and was driven by a curiosity that extends well beyond the lives of his subjects. The Rigor of Angels reminded me of another recent biography, John Higgs’s William Blake Versus the World (2022), which uses Blake as a kind of launchpad for far-ranging explorations of neuroscience, Jungian theory, and psychedelics and was criticized in The New York Times for revealing “more about the mind of John Higgs than that of William Blake.”

But it’s arguably the subjective nature of such explorations that makes them so meaningful. At its best, reading The Rigor of Angels feels like watching someone chase down a wild and possibly ill-advised thought experiment, one that might metastasize at any moment into a paranoiac string map. If I was annoyed by the occasional reminders that it was, in fact, a biography, it’s because they felt increasingly like a smoke screen, a literary scholar’s attempt to smuggle a hypothesis that spans physics, philosophy, and theology into the acceptable, and acceptably modest, conventions of that genre.

Egginton wrote several scholarly books on literature and philosophy before turning to popular nonfiction. (The Rigor of Angels most closely resembles his 2016 biography of Miguel de Cervantes, The Man Who Invented Fiction, which explores how Don Quixote sent ripples through literature, art, and politics and ultimately helped form what we think of as the modern world.) Last fall, when the theoretical physicist Sean Carroll interviewed Egginton on his Mindscape podcast and asked whether he considered The Rigor of Angels a “contribution” or merely an attempt to explain things to the person on the street, Egginton replied that it was “equally both.” Carroll acknowledged that it was an impressive work of scholarship and noted that the depth of Egginton’s research (which included reading source material from the pioneers of quantum mechanics) was rare, even among physicists. But later in the interview, Carroll bristled at the loose connections Egginton made between quantum mechanics and Kantian epistemology. “Immanuel Kant did not know about quantum mechanics,” Carroll said. “He was talking about a different thing than the fundamental laws of physics…so I think it’d be wrong to give people the impression that quantum mechanics is a version of the same problems we had in the classical world.”

Carroll was expressing an all-too-common skepticism among scientists and academics toward grand philosophical conclusions and sweeping connections across centuries and domains. The human mind, so this logic goes, is prone to flights of fancy, and disciplinary guardrails are crucial to keeping it rooted in concrete particulars. If such reminders to “stay in your lane” are more often aimed at humanities scholars who dare to venture into the cathedral of hard science (I don’t think many readers of Gödel, Escher, Bach questioned Douglas Hofstadter’s authority, as a computer scientist, to speak about the fugue), it’s owing to a widespread suspicion of metaphor, that vehicle of sloppy thinking. What’s interesting, however, is that Egginton’s book suggests that “metaphysical overreach” is actually more common in disciplines that maintain a narrow focus and become captive to the language of specialization.

Among the most memorable passages in The Rigor of Angels is a discussion of Heisenberg’s 1942 essay about the use of language in physics. Humans need language in order to think, and natural language is entirely built on metaphor. This was why, Heisenberg argued, the emphasis on linguistic precision in his field so often prevented physicists from genuine understanding. There was also truth in the way that poets use language, calling upon associations, multiple meanings, and intuitive connections. Just as the physical world ceases to make sense when we zoom in too far, so language (as Borges’s story of Funes demonstrates) becomes meaningless when it’s too static and precise. “What is sacrificed in ‘static’ description,” Heisenberg writes, “is that infinitely complex association among words and concepts without which we would lack any sense at all that we have understood anything of the infinite abundance of reality.”

To read Heisenberg, Kant, or Borges—or, for that matter, Dostoevsky—is to be dumbfounded by the breadth of knowledge that was once expected of intellectuals (Heisenberg once claimed “that one could hardly make progress in modern atomic physics without a knowledge of Greek natural philosophy”), and it’s clear that the ability to think and express themselves in broad, analogical strokes deepened their work. On the Mindscape podcast, Egginton noted that after spending half his career writing scholarly books that were read only by colleagues and graduate students, he found popular writing to be a revelation—not because it was easier but because it demanded more rigorous thought. “And then it began to seem to me that some of my past writing was relying on, say, jargon,” he said, “or skipping steps in thinking through a problem by using a kind of shorthand that I felt that my colleagues and students would totally understand but that we hadn’t necessarily really thought through.”

Another way to put it: perhaps true understanding requires an observer who maintains enough critical distance from a given field to perceive the continuities that extend beyond it. The advance of knowledge might even benefit occasionally (who knows?) from an ambitious mind that attempts to span centuries of thought, drawing connections between the ideas that keep resurfacing and that feel increasingly destined to vanish in a deluge of data and the granular fixations of isolated expertise. The disciplinary boundary policing is itself evidence that we keep forgetting the most important insights these very fields have produced. If it’s true that our knowledge of the world is filtered through language and cognitive structures—if it’s true that what we’re studying is, at least in part, our own Euclidean minds—then commonalities, and common errors, are bound to recur across different contexts. The through line is us.