Do the Numbers Lie?

Here are some things scholars have recently tried to count. The estimated cost of the Covid-19 pandemic: $16 trillion in the United States alone. Premature deaths from air pollution: 10 million a year worldwide, most caused by burning fossil fuels. The number of speaking characters in Jane Eyre: 67.

Conservative pundits and left-wing scholars are besotted with the idea of the professor as subversive visionary — the transgressive theorist-guru who punctures dogma and ushers in social upheaval. A seductive vision, useful for well-funded propagandists waging culture war, and psychologically gratifying for glazed-eyed adjuncts mustering the will to write letters of recommendation while eating breakfast cereal in a bathrobe.

The truth is that most scholars are more like accountants. Behind the scenes, they weigh and measure, itemize and tabulate, sort and categorize. They check the facts. They correct errors, deceptions, and distortions, and make sure everything “adds up.” This ethos is discernible in scholars working with a great diversity of objects, from labor practices in Mexico to injury patterns among ballet dancers to bioluminescence in the deep sea. These quiet efforts of taking account are rarely glamorous. But this labor helps make the world more transparent and knowable.

Because counting is one of our basic intellectual procedures, it is easy to forget that questions about when we should count, and what can be counted, have long attracted controversy. Is a human life worth $10 million? Is pleasure unitary, varying only in intensity and duration (therefore quantitatively measurable), or are there different kinds of pleasure, each of which is qualitatively distinct? Can life satisfaction be scored on a 1-to-10 scale? Our intellectual regime, ruled with cool impersonality by Number, is populated on the margins with heretics and dissenters. These dissenters say that not everything in life can be counted, and that rationality is incomplete without qualitative judgment. The demure scholar-accountant totting up figures at her desk faces a chorus of objectors who insist that quantification can obscure as well as reveal.

In Uncountable: A Philosophical History of Number and Humanity From Antiquity to the Present (Chicago, 2021), David Nirenberg and Ricardo L. Nirenberg join this circle of dissidents. They are skeptical about what they call the “imperial claims of number.” They express the familiar worry that as quantification extends its reach, the insights of art, poetry, and dream are in danger of being lost.

The challenge they present against quantification is idiosyncratic, highly abstract, and grounded in a lifetime of reading in philosophy, literature, and mathematics. (David Nirenberg, a medieval historian, is the new director of the Institute for Advanced Study, in Princeton, N.J.; his father, Ricardo L. Nirenberg, did research in mathematics before turning to literature and philosophy, a pivot that suggests some disillusionment with mathematical tools.) They think proponents of number are making a philosophical mistake. When we count, we are treating the objects we are counting like “pebbles,” as commensurable units that can be totted up and aggregated. But few objects, outside of highly artificial mathematical constructs, are really like this. We live in a world of flux and change. To treat objects as countable, the Nirenbergs argue, is a choice, one that we could always make differently.

The book is a failure, but an interesting one. In an attempt to defend humanistic knowledge, it retreats into irrationality and mysticism. It shows, moreover, that dazzling erudition cannot save us — and may even be fatal.

To count or not to count? In the tempestuous early decades of Europe’s 20th century — the years of Einstein’s theories of relativity, artistic modernism, and global war — several prominent thinkers saw the centrality of mathematics as a world-historical calamity. The German historian Oswald Spengler thundered that the tyranny of the exact sciences had drained Western civilization of its vitality. Max Weber thought that man’s ability to “master all things by calculation” had led to the disenchantment of the world. After the physicist Friedrich Adler shot and killed the prime minister of Austria, in 1916, his lawyers tried for an insanity plea, arguing that the assassin suffered from “an excess of the mathematical.” In these episodes of derangement, mathematics entered as villain, or scapegoat — the source of Europe’s triumphs, now the root of its malaise.

These attacks on mathematics were part of a pained response to modernity. But thinkers have been debating the proper role of numbers for centuries. The poles of argument are visible as early as the Greeks. The Nirenbergs contrast the pre-Socratic philosophers Parmenides and Heraclitus. Parmenides, as they read him, saw true reality as unchanging, eternal. He rejected the senses as false. Although our senses tell us we are living in a dynamic, ever-changing world, logic tells us of immutable laws. Heraclitus, by contrast — associated with the dictum “everything flows” — saw the world as fluid, variable. “The way up and the way down are one and the same,” he wrote in a fragment. Oppositions of up and down, right and left, are relative, dependent upon what perspective we are taking.

Heraclitus is the most radical dissenter from what the Nirenbergs call “Greek fantasies of sameness” — the idea that our knowledge can be built on stable foundations, such as Plato’s eternal Forms. And the relativistic position they take in this book is to a large degree Heraclitean. Even Aristotle, who is alert to the limits of mathematics and the indispensability of qualitative reasoning — he opens his Ethics by advising that each of the arts and sciences has its proper method and appropriate degree of precision — does not go far enough for them. They fault Aristotle for endorsing the principle of noncontradiction, according to which something cannot be A and not-A at the same time. Such a belief, the authors suggest, reveals a certain poverty of imagination.

In numbers, human beings have sought a realm of eternal sameness, a refuge from the mutability of the world and the body. This quest for permanence through number has often had a religious element. This is the idea that mathematics brings us closer to God. The book traces this belief from medieval Christian and Islamic theologians through the 20th-century writer Simone Weil. In our universe of flux and change, there exists “a grounding and eternal sameness that we can glimpse in number.” Mathematics answers to the eternal within us. These religious arguments, expertly retrieved by the Nirenbergs, remind us that questions about counting are not merely technical disputes, but involve agonizing metaphysical uncertainty: human frailty confronting the shadow of everlasting permanence.

The authors pursue their theme — humankind’s search for “kernels of eternal sameness” at the foundations of knowledge — into a range of disciplinary areas, including philosophy, physics, and economics. Throughout, they emphasize the presence of instability and contradiction. In math, they stress irrational numbers. In physics, they focus on wave-particle duality, as well as more general problems in measuring and observing quantum objects like electrons. They take note of contradictory desires in human psychology — stunningly observed in literature, but allegedly denied by economists. Instability, they find, is at the foundations of everything.

The hero of their story is poetry. Philosophers, mathematicians, scientists, and scholar-accountants of all kinds impose a “fantasy of sameness” on their objects of study. Poets do not make this mistake. They sing of mutability and contradiction. Their business is with the irrational and ephemeral. By giving up on countable “pebbles,” poets unlock insights more profound and primordial.

The authors argue that “there is no necessary sameness or difference” in any set of objects, apart from purely mathematical objects. How we divide or aggregate things is a matter of perspective. Whether we emphasize sameness or difference in our objects of study is a choice, “not forced on us by nature or necessity. We are always free to make that choice differently.” They end on a note of mysticism. They urge the reader: “Strive to cultivate within yourself the simultaneous mysteries of sameness and difference.” And so this ambitious but flawed study concludes not by opening up new lines of research, or proposing alliances between the humanities and the sciences, but with a therapeutic invocation, an image of a reader sitting alone in a room pondering the ineffability of the world.

The Nirenbergs deserve praise for the polymathic breadth of their study, and the courage with which they confront deep epistemological questions. But their book falls short of its aims in three respects. (I hope they will forgive me for counting them.)

First is the idea of counting, or not counting, as a “personal” or spiritual choice. The authors repeatedly state that we are free to choose, in any context, “between emphasizing abiding sameness or the inescapable potential for difference.” It is not clear what this means. If it means that scholarly communities could form new consensuses about how to count or measure certain objects or systems, fine. But this “liberty of choice,” if taken up ad hoc, would destroy the uniformity of knowledge and plunge many academic fields into chaos. The idea that quantification is a “choice,” a private matter, is dubious, because the whole point of using numbers is to create knowledge that can be standardized and shared. Quantification, argues the historian of science Theodore Porter in his classic study Trust in Numbers (Princeton, 1995), is not merely an internal aid to thought. It is a “social technology,” a strategy for enhancing the trust of outsiders and legitimizing the decisions of experts.

Intervening at the level of solitary contemplation, in other words, will do little to halt quantification. The authors lament the “imperial claims of number,” but they do not seem interested in how quantification established its empire. They mention only in passing the rise of the modern bureaucratic state. They give us sketches of dozens of philosophers, scientists, and poets; but people relying on numbers for practical tasks — merchants, engineers, accountants, bureaucrats — are missing entirely from this history of “number and humanity.” Unless we understand how numbers wield power in the real world, we will not be able to challenge the sovereignty of quantification.

The book’s second fault is its alignment of numbers with rationality, on the one hand, and poetry with irrationality and dream, on the other. The authors warn that “to attend only to the countable, or to live only by laws of reason,” does violence to our humanity. They say: “We want to learn from our dreams and our poems and also from our science.”

These are fine sentiments. But this framework sets up a false binary. It places dreams, poems, and the irrational on one side, and science, number, and rationality on the other. What falls out of the picture here is qualitative reasoning. This way of handling the problem plays into the hands of people who want to quantify everything, because it suggests that the alternative to quantification is irrationality. Our authors end the book by asking: “Where does the greater danger lie … in the tyranny of axioms, or of unreason?” Luckily, these are not our only options. The rational is not identical with the numerical. On the contrary: Without qualitative judgment, rationality is incomplete — crude about matters of value and blind to relevant concepts and distinctions.

The authors’ schematic division between poetry and number neglects how poetry, too, is reliant on number. The connections between poetic creation and acts of counting (words, syllables, stresses, intervals, meter) are many. Similarly, the fact that music, our most dreamlike and purely formal art, is also the art most closely tied to mathematical pattern suggests that there are mysteries worth exploring here. Mathematics is no enemy of the imagination.

The third problem is that the book’s central analytic terms of “sameness and difference” fail to provide a clear organizing principle. The book repeatedly mentions “choices between sameness and difference,” “ideas of sameness and difference,” “mysteries of sameness and difference.” The phrase “sameness and difference” runs through the study like a refrain. The authors use “sameness” to refer to the quantitative and the numerical: Numbers are “kernels of sameness.” But I must confess I found this usage unclear. Why, for example, would the anti-mathematical Heraclitus, with his refusal to make any distinctions at all (“up and down are one and the same”), not be on the side of “sameness”? If your central terms are so vague as to be reversible, then your argument may need further shaping.

David Nirenberg’s admirable Anti-Judaism (Norton, 2013) took on a similarly vast historical sweep. But it did so with a pointed argument: the claim that Western intellectuals have for centuries deployed a projective anti-Judaism to criticize elements of their civilization (as in Marx’s identification of Judaism and capitalism). The present book, lacking a clear line of argument, offers variations on a broad theme. This creates problems for the selection and treatment of sources. The authors offer hurried exegesis of hundreds of illustrious thinkers under the muddled rubric of “sameness and difference.” They examine a 19th-century science-fiction writer for several pages only to dismiss his work as uninteresting for their purposes and move on to Proust; they are fond of noting which Arabic philosophers or ancient treatises they wish they had more time to discuss.

The sense one gets while reading this book is of being locked in a private study with a bespectacled Bouvard and Pécuchet who show you one book, then another — mathematical proofs, economic treatises, lectures on physics, modernist poetry. The books, eagerly taken up and hastily discarded, pile up on the floor. “You see?” the authors say, reaching for another shelf. “Sameness and difference … sameness and difference again!”

I enumerate my disappointments because this book troubled me. It troubled me because we need good arguments to resist the quantification of everything. But it also troubled me for a more personal reason. I have long thought that erudition was a purely positive good. The more you read, the wiser you are. But these authors seem paralyzed by their immense learning, so full of negative capability that they cannot make an argument. This celebration of humanistic knowledge becomes, in the end, a warning against it. Even the most stunning accumulations of learning may leave you stranded in the wilds of irrationality, or locked among bookshelves in an empty room.