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A Philosophical History of Number and Humanity from Antiquity to the Present


A Philosophical History of Number and Humanity from Antiquity to the Present

Ranging from math to literature to philosophy, Uncountable explains how numbers triumphed as the basis of knowledge—and compromise our sense of humanity.

Our knowledge of mathematics has structured much of what we think we know about ourselves as individuals and communities, shaping our psychologies, sociologies, and economies. In pursuit of a more predictable and more controllable cosmos, we have extended mathematical insights and methods to more and more aspects of the world. Today those powers are greater than ever, as computation is applied to virtually every aspect of human activity. Yet, in the process, are we losing sight of the human? When we apply mathematics so broadly, what do we gain and what do we lose, and at what risk to humanity?

These are the questions that David and Ricardo L. Nirenberg ask in Uncountable, a provocative account of how numerical relations became the cornerstone of human claims to knowledge, truth, and certainty. There is a limit to these number-based claims, they argue, which they set out to explore. The Nirenbergs, father and son, bring together their backgrounds in math, history, literature, religion, and philosophy, interweaving scientific experiments with readings of poems, setting crises in mathematics alongside world wars, and putting medieval Muslim and Buddhist philosophers in conversation with Einstein, Schrödinger, and other giants of modern physics. The result is a powerful lesson in what counts as knowledge and its deepest implications for how we live our lives.

432 pages | 1 halftone | 6 x 9 | © 2021

History: General History, History of Ideas

Mathematics and Statistics

Philosophy: Ethics


“Erudite  . . . it draws on science, philosophy, and history, with frequent detours into literature. Balzac, Nabokov, Kafka, and other literary giants are enlisted to endorse the message that certain things ‘can’t be grasped by counting, statistics, or any logical or scientific analysis.’ Moreover, by ‘opting to attend only to the countable . . . we do violence not only to the objects of our thought but also to our own being.’ This claim lacks a clear definition of violence, but it is the first step towards overthrowing ‘the tyranny of axioms.’”

Times Literary Supplement

"This book offers a fascinating series of sophisticated and insightful reflections on the many ways the appraisal of numbers has shaped culture and civilization. . . . [It] will yet be an intellectual treat for anyone who may delight in original philosophical musings without having to cross Greco-Abrahamic borders."


“Ricardo and David Nirenberg, father and son scholars of mathematics and history, have teamed up in a breathtaking voyage examining the foundations and limits of knowledge in western thought. Not content with secondary sources, they have translated from the literature in their original languages: Arabic, French, German, Greek, Hebrew, Italian, Latin, and Spanish.  In particular, they target mathematics and the natural sciences, and the way the concepts of sameness and differences affect our understanding of the natural world. But in the process, the authors touch upon many other facets of human endeavor, all named after their Greek roots: poetry, philosophy, psychology, economy.  Along this wildly entertaining journey, we meet dozens of erudite thinkers, scientists, and writers such as Anaximander, Al-Fārābī, Fyodor Dostoevsky, Ludwig Wittgenstein, Werner Heisenberg, and Rainer Maria Rilke. The book arrives just in time to give us ammunition as attempts are being made to put truth itself into the supercollider. It is a source of inspiration and comfort to learn how the far-flung ideas about numbers, our existence, and the world we live in have been debated in the past.”

Joachim Frank, Columbia University, Nobel Prize in Chemistry

“This is an erudite and insightful exploration of the collision between two distinct ideas: sameness and difference. The authors brilliantly unify wide-ranging bodies of thought across millennia around this theme. They forcefully and eloquently demonstrate why this dichotomy matters today and how deeply it shapes our vision of the world and the actions we take.”

James J. Heckman, the Henry Schultz Distinguished Service Professor of Economics at the University of Chicago, Nobel Memorial Prize in Economics

"Drawing on a broad range and telling selection of examples from economics, quantum physics, literary studies, and more, the Nirenbergs combine their strength as a professional medievalist and a mathematician, respectively, to take the reader on a guided, remarkably enthralling tour through the universe and universals of formalization. Beautifully and engagingly written, this book will greatly appeal to a wide, interested reading public and change the critical terms of debate regarding mathematical versus other forms of reasoning in the natural, social, and human sciences as well as literature and the arts, even ordinary life, for years to come."

Hent de Vries, the Paulette Goddard Professor of the Humanities, New York University

“Uncountable is erudite, imaginative, and articulate as well as existentially relevant, even revelatory. David and Ricardo L. Nirenberg trace the epistemic habits, ideals, and practices that over millennia have presumed the knowledge of numerical relations to be the only true knowledge and numbers the only true form of being. The result is a philosophical and poetic exhortation to take responsibility for knowledge.”

Chad Wellmon, author of Permanent Crisis

Table of Contents

Introduction: Playing with Pebbles
1 World War Crisis
2 The Greeks: A Protohistory of Theory
3 Plato, Aristotle, and the Future of Western Thought
4 Monotheism’s Math Problem
5 From Descartes to Kant: An Outrageously Succinct History of Philosophy
6 What Numbers Need: Or, When Does 2 + 2 = 4?
7 Physics (and Poetry): Willing Sameness and Difference
8 Axioms of Desire: Economics and the Social Sciences
9 Killing Time
10 Ethical Conclusions
Index of Names


 A remarkable attribute of the species biologists call Homo sapiens is that its members have so often asked themselves about the nature of their own sapientia: the knowledge or wisdom for which they are named. Equally remarkable is the fact that in answering these questions, humans have been so willing to tear themselves apart. Over and over they have divided their cognitive capacities into “good” ones and “bad.” They have even imagined that some ways of thinking make humans eternal and godlike, while others lead to mortality, deceit, damnation, even the destruction of the world.

Often enough over these past three thousand years, we humans have pursued these divisions to the death, clashing over differences of opinion about what we should know and how we should know it. We are not talking here only of the many clashes between different religions and cultures of knowledge in the distant past. Even the two world wars of the twentieth century were understood by many who lived through them as the consequence of bad choices about what kinds of knowledge to pursue. World War I, for example, was explained by leading European and American intellectuals as the result of mathematics gone bad, an inhuman fusion of arithmetic and geometry that destroyed Western civilization... Plenty of ideologues found it easy enough to cast the Cold War as a struggle between two different theories of knowledge, “Marxism” and “liberalism,” “determinism” and “freedom.” Perhaps future generations will come to see the current arguments about the human impact on climate change as yet another chapter in this long history of humanity’s division over the nature of knowledge.

Today, mathematical forms of knowledge—computation, artificial intelligence, and machine learning, for example—touch many more aspects of the world than they did in the first half of the twentieth century, or, indeed, in any previous period of this planet’s history. Divisions between types of knowledge, such as those between the humanities and the sciences—“the two cultures,” as C. P. Snow dubbed them in the year 1959—are if anything deeper than they have ever been. Yet unlike our predecessors from a century ago, few people today—except perhaps panicked humanities professors who feel their habitat melting away beneath their feet—would consider these divisions deeply threatening. Even fewer would claim that understanding them is in some way essential to humanity.

We are not writing an Apocalypse. Ours is an attempt to understand these millennial divisions so that we might better live with them. How has humanity pitted its various powers of thought so fiercely against itself? And why have the truth claims of numerical relations emerged so powerful from this conflict? Achieving this understanding is a historical task, and the first half of this book set out to provide that history. In chapters ranging from ancient Greek philosophy and the rise of monotheistic religions to the emergence of modern physics and economics, we trace how ideals, practices, and habits of thought formed over millennia have turned number into the foundation stone of human claims to knowledge and certainty...

Learning to live humanely with these divisions is the goal of the second half of this book. These divisions and conflicts of our faculties and our knowledge are not necessary ones. The fragments of our humanity can be brought together in different ways, even in ways that might be truer to basic aspects of the questions we want to ask and the objects we want to know, truer even to our own human being.

This book is therefore not only a history. It is also a philosophical and poetic exhortation for humanity to take responsibility for that history, for the knowledge it has produced, and for the many aspects of the world and of humanity that it ignores or endangers. We seek to convince you that how we humans think about our knowledge has deep consequences for how we live our lives and that we need to become more conscious of the first if we wish to change the second…

The discovery and mathematization of repetition and periodicity in the movements of the sun, moon, planets, and even of the stars that so densely packed the preindustrial skies gave many ancient societies a sense that they could project the past into the future: a comforting predictive power in a vast and variable cosmos. Long before written record we find the constellation we call Orion painted by Paleolithic hands on the cave walls of Lascaux. And on the cuneiform tablets of the first Mesopotamian scribes, the verb to count was applied to the skies in the production of astral omens...

Today oneirology (the science of dreams) is scarcely a word. The study of magic is confined to anthropologists, historians, or “the ignorant.” Astronomy, however, is beneficiary of billions in annual investment from science foundations and is a monument to the ability of the human mind to make some sense of the structure of the universe.

The point here is not that knowledge has progressed. (When it comes to dreams, it may even be that a certain kind of self- knowledge has been lost by not attending to them.) Our point is rather that the form of attention we today call scientific has been oriented toward certain kinds of sameness—in this case formalizable, axiomatic, mathematical—and not toward others. There are many reasons why this is the case. But one, noted already by the Roman natural philosopher Pliny, writing some two thousand years ago, is that mathematical astronomy provided some seemingly stable powers of prediction about an otherwise overwhelming universe...

Pliny thought mathematical astronomers praiseworthy because they derived procedures through which they could predict the movements of the planetary deities, thereby binding “gods.” And to the degree that the planets were thought to determine the fate of a person, a science (today we call this astrology, not astronomy) capable of predicting the future position of the planets also offers knowledge about human fate, thereby binding “men.”

The Aztecs provide a different astronomical example, instructive because it reminds us of the contingency of certainty and the tenacity of fear. They were sophisticated astronomers, but they believed (as did the ancient Egyptians) that the system that kept the sun appearing regularly in the sky was unstable and that the sun might someday fail to rise if humans did not do their part by making offerings to the gods...

How could anyone think that human action (let alone sacrifice) is necessary to ensure the dawn? What could be more certain from our experience than the sun’s rising? And yet the Aztecs were not wrong in worrying about their knowledge of the sunrise. Their refusal to deduce future certainty from previous dawns is defensible in the most sophisticated probabilistic terms. As solar system dynamicists today would tell us, the system is unstable. Yet we moderns expend very little of our still considerable religious energy in keeping the solar system going. Again, the point is not that the Aztecs were bad scientists or that we moderns should be more worried about the sunrise. Our point is simply that the desire for certainty can lead us to extend—often inappropriately, sometimes disastrously—lessons, methods, and sciences of sameness from one domain of knowledge into another where perhaps they do not apply.

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