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Uncountable: 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.
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, Hardcover
Published October 20, 2021
About the author
David Nirenberg
20 books40 followersI have spent most of my intellectual life shuttling between the micro and the macro, trying to understand how life and ideas shape and are shaped by each other. One stream of my work has approached these questions through religion, focusing on the ways in which Jewish, Christian, and Islamic cultures constitute themselves by interrelating with or thinking about each other. My first book, Communities of Violence: Persecution of Minorities in the Middle Ages, studied social interaction between the three groups within the context of Spain and France in order to understand the role of violence in shaping the possibilities for coexistence. In later projects I explored the work that “Judaism,” “Christianity,” and “Islam” do as figures in each other’s thought. One product of that approach, focused on art history, was (jointly with Herb Kessler) Judaism and Christian Art: Aesthetic Anxieties from the Catacombs to Colonialism (2011). In Anti-Judaism: The Western Tradition (2013), I attempted to apply the methodology to a very longue durée, studying the work done by pagan, Christian, Muslim, and secular thinking about Jews and Judaism in the history of ideas. More or less simultaneously in Neighboring Faiths: Christianity, Islam, and Judaism Medieval and Modern (2014), I tried to bring the social into conversation with the hermeneutic, in order to show how, in multireligious societies, interactions between lived experiences and conceptual categories shape how adherents of all three religions perceive themselves and each other. Then in Aesthetic Theology and Its Enemies: Judaism in Christian Painting, Poetry, and Politics (2015), I focused on how thinking about Judaism shaped the ways in which Christian cultures could imagine the possibilities and limits of community and communication.
Beginning with my book Anti-Judaism, which stretched from ancient Egypt to the twentieth century in order to try to understand the work done by a family of concepts across history, I have tried to cultivate a new approach to the “long history” of ideas. My most recent book, Uncountable: A Philosophical History of Number and Humanity from Antiquity to the Present, written in collaboration with Ricardo Nirenberg (a mathematician who happens also to be my father), follows this path as well. It explores the long history of the various types of sameness that underpin the claims of different forms of knowledge (from poetry and dreams, to monotheism, math, and physics), using these to think critically about the powers and the limits of the sciences and the humanities. I am now at work on the long history of yet another family of concepts, namely the inter-connected history of race and religion from the Neolithic to the present.
Beginning with my book Anti-Judaism, which stretched from ancient Egypt to the twentieth century in order to try to understand the work done by a family of concepts across history, I have tried to cultivate a new approach to the “long history” of ideas. My most recent book, Uncountable: A Philosophical History of Number and Humanity from Antiquity to the Present, written in collaboration with Ricardo Nirenberg (a mathematician who happens also to be my father), follows this path as well. It explores the long history of the various types of sameness that underpin the claims of different forms of knowledge (from poetry and dreams, to monotheism, math, and physics), using these to think critically about the powers and the limits of the sciences and the humanities. I am now at work on the long history of yet another family of concepts, namely the inter-connected history of race and religion from the Neolithic to the present.
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Displaying 1 - 5 of 5 reviews
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July 26, 2026DNF warning, but this is a good book. Read why below:
Uncountable: A Philosophical History of Number and Humanity from Antiquity to the Present in a nutshell is the tale of “sameness” and “difference” throughout the ages.
Are numbers dangerous? And how do they tie in with the history of the modern world, with philosophy, and with math—oh yes, with math if that wasn’t obvious enough. Uncountable is an anomaly unlike any other in that somehow it combines so many disparate fields into one narrative and out of all the ways to link it, did so by focusing on something that is both very much countable and as the title alludes, anything but. Books like this don’t come along often and perhaps the only reason this one was able to be what it became was the father-son mathematician/historian duo working together to prove that sometimes 1 + 1 = 2 is both accurate and a contradiction and why that actually matters not just in arithmetic, but for the sake of humanity.
Don’t let the cover fool you like it did me: this isn’t a “pop” science book about the history of math (oh, if only it was!). Its subtitle (see above) nails what’s in store: a tour de force from Ancient Greece through the modern era perhaps serving both as an educational trip through time and a warning of where we went wrong and how to now right ourselves:
No righter words spoken and while the claim alludes to a rough road—and this is not an easy read by any means—the authors push hard. Stacked footnotes aplenty in concert with a relentless pummel driving down the almost weird history of numbers—especially one of them—and how ‘sameness’ has been one can say the cornerstone of Western thought since the days of Anaximander.
With full disclosure, Uncountable defeated me. About halfway through what began as interesting quickly transitioned to (surprisingly) hard math and economic theory and stayed that way seemingly to the very end (I started to skip around). Thus, this is a personal DNF, but this is far from a bad book! If you come from a STEM or economics background, this may be a fascinating read. Otherwise, extreme caution is advised. It’s kind of ironic writing this because the book’s introduction actually notes: “Readers who have a low tolerance for ancient history, philosophy, or religion may want to skip chapters 2–4.” (page 2) Ironic because those chapters plus the first one were the ones I found most intriguing. But the authors write after that are clear in their intention that it’s the second half of the book that is their main forte and the introductory chapters more or less exist to build up to them.
Uncountable: A Philosophical History of Number and Humanity from Antiquity to the Present in a nutshell is the tale of “sameness” and “difference” throughout the ages.
Are numbers dangerous? And how do they tie in with the history of the modern world, with philosophy, and with math—oh yes, with math if that wasn’t obvious enough. Uncountable is an anomaly unlike any other in that somehow it combines so many disparate fields into one narrative and out of all the ways to link it, did so by focusing on something that is both very much countable and as the title alludes, anything but. Books like this don’t come along often and perhaps the only reason this one was able to be what it became was the father-son mathematician/historian duo working together to prove that sometimes 1 + 1 = 2 is both accurate and a contradiction and why that actually matters not just in arithmetic, but for the sake of humanity.
Don’t let the cover fool you like it did me: this isn’t a “pop” science book about the history of math (oh, if only it was!). Its subtitle (see above) nails what’s in store: a tour de force from Ancient Greece through the modern era perhaps serving both as an educational trip through time and a warning of where we went wrong and how to now right ourselves:
“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.”
Nirenberg, David; Nirenberg, Ricardo L.. Uncountable: A Philosophical History of Number and Humanity from Antiquity to the Present (pp. 2-3). The University of Chicago Press. Kindle Edition.
No righter words spoken and while the claim alludes to a rough road—and this is not an easy read by any means—the authors push hard. Stacked footnotes aplenty in concert with a relentless pummel driving down the almost weird history of numbers—especially one of them—and how ‘sameness’ has been one can say the cornerstone of Western thought since the days of Anaximander.
With full disclosure, Uncountable defeated me. About halfway through what began as interesting quickly transitioned to (surprisingly) hard math and economic theory and stayed that way seemingly to the very end (I started to skip around). Thus, this is a personal DNF, but this is far from a bad book! If you come from a STEM or economics background, this may be a fascinating read. Otherwise, extreme caution is advised. It’s kind of ironic writing this because the book’s introduction actually notes: “Readers who have a low tolerance for ancient history, philosophy, or religion may want to skip chapters 2–4.” (page 2) Ironic because those chapters plus the first one were the ones I found most intriguing. But the authors write after that are clear in their intention that it’s the second half of the book that is their main forte and the introductory chapters more or less exist to build up to them.
August 27, 2025
I found this a frustrating read that I had every reason to enjoy more than I did. The Nirenbergs, père et fils, start with the foundations of mathematics but quickly expand that into a more generalized critique of the social-psychological construction of number and an exhortation to grasp the nettle of our existential challenge of living within that uncertainty. They contrast the rational with intuition under many names over the history of thought, beginning with ancient Greece up to the debate between Einstein and Bergson at Davos. They also end with some ruminations on Kierkegaard, Kafka, and Borges in understanding the fundamental incalculability at the heart of the human.
I found salutary the Nirenbergs' explication of the concern with sameness and difference that undergirds mathematics, and how the ghost of that concern haunts Western philosophy. Especially clear was their description of deriving number from the null set: starting with nothing and the set it includes, one builds the entire number line from logical principles. These categories of sameness and difference, and the laws of thought attempting to limn their meaning (e.g., the Law of Identity, the Principle of Noncontradiction, the Principle of Sufficient Reason, etc.), can be seen as fundamental to the Western rationalist project of parsing reality quantitatively.
I expected the Nirenbergs to build upon that insight to further illuminate modern debates over positivism etc., but they never really did that work. Instead, they're quick to write off the whole enterprise as insufficiently aware of its own mental prejudices and then to use that to leap off into a sea of existentialism. All great and good, but their posing of the dilemma is a muddled thing that lacks the sharp horns of the best accounts (here I'm thinking of books by contemporary authors like Mary-Jane Rubenstein and Meghan O'Gieblyn). It gives the sense of depth because they're citing many of my favorite writers, but their take isn't especially additive to the tradition. I suppose I should lower my expectations and just be happy with the grab-bag of quotes, references, and anecdotes (for instance, Nabokov's precognitive dreams), but mostly it felt unfocused and self-indulgent to me. Also, if I can understand all the math you're discussing, you're not going very deep. I'm functionally innumerate but I was nonetheless left wanting more of what's intimated in the title here.
I found salutary the Nirenbergs' explication of the concern with sameness and difference that undergirds mathematics, and how the ghost of that concern haunts Western philosophy. Especially clear was their description of deriving number from the null set: starting with nothing and the set it includes, one builds the entire number line from logical principles. These categories of sameness and difference, and the laws of thought attempting to limn their meaning (e.g., the Law of Identity, the Principle of Noncontradiction, the Principle of Sufficient Reason, etc.), can be seen as fundamental to the Western rationalist project of parsing reality quantitatively.
I expected the Nirenbergs to build upon that insight to further illuminate modern debates over positivism etc., but they never really did that work. Instead, they're quick to write off the whole enterprise as insufficiently aware of its own mental prejudices and then to use that to leap off into a sea of existentialism. All great and good, but their posing of the dilemma is a muddled thing that lacks the sharp horns of the best accounts (here I'm thinking of books by contemporary authors like Mary-Jane Rubenstein and Meghan O'Gieblyn). It gives the sense of depth because they're citing many of my favorite writers, but their take isn't especially additive to the tradition. I suppose I should lower my expectations and just be happy with the grab-bag of quotes, references, and anecdotes (for instance, Nabokov's precognitive dreams), but mostly it felt unfocused and self-indulgent to me. Also, if I can understand all the math you're discussing, you're not going very deep. I'm functionally innumerate but I was nonetheless left wanting more of what's intimated in the title here.
August 25, 2024
I really like the premise of this book, focusing on "sameness" and "identity" laws in mathematics and their philosophical interpretations.
The authors go much further than this, putting these problems at the core of (mostly modern) history, which is an overview I disagree with, but which has provided much welcome food for thought.
The authors go much further than this, putting these problems at the core of (mostly modern) history, which is an overview I disagree with, but which has provided much welcome food for thought.
Want to Read
December 16, 2024Does this history and analysis consider pre-columbian mesoamerican number systems?
May 8, 2025
Fascinating book! Written in a very readable style.
Displaying 1 - 5 of 5 reviews