By Ernest Davis
Uncountable: A Philosophical History of Number and Humanity from Antiquity to the Present. By David Nirenberg and Ricardo L. Nirenberg. The University of Chicago Press, Chicago, IL, October 2021. 432 pages, $30.00.
Uncountable: A Philosophical History of Number and Humanity from Antiquity to the Present. By David Nirenberg and Ricardo L. Nirenberg. Courtesy of the University of Chicago Press.
In many cases, a thoughtful response to a question of the form “How many \(X\)s are there?” begins with “It depends on how you count.” The number of entities in a given category \(X\) is often not merely unknown, but indeterminate. The answer depends on what counts as an \(X\) and how the inquisitor individuates one entity from another; such decisions can be ill-defined and arbitrary. For example, consider the following question: “How many languages are currently spoken?” We must first decide on the definition of a legitimate language and the qualifications of “currently spoken,” then identify the relevant distinctions between languages. Are Klingon and Pig Latin legitimate languages? Are Moroccan Arabic and Iraqi Arabic different languages or different dialects? How about Swedish and Norwegian or French and Québécois? The problem becomes even more complicated if we look across time at continuously changing entities and ask questions such as “How many languages have been spoken in Europe since 1000 BCE?” These types of queries require determinations about whether medieval French and modern French are different languages.
Counting—as well as operations like addition that build on counting—is thus inherently dependent upon individuation, which is frequently problematic. Nonetheless, philosophers have often taken arithmetic statements such as "\(2+2=4\)" to be the ne plus ultra of indubitable, necessary truths. Uncountable: A Philosophical History of Number and Humanity from Antiquity to the Present—written by father-son team David (a medieval historian) and Ricardo L. Nirenberg (a mathematician and philosopher)—addresses this paradox and traces it through the history of philosophy and literature.
Uncountable has two stated goals. The first is “to convince you that it is not true of all things that two and two make four.” The second is to serve as “an exhortation to interrogate more self-consciously the consequences of extending laws of thought derived from one domain to others where the necessary conditions of ‘sameness’ may not apply.”
Chapter one opens with an account of the “crisis” that gripped the European intellectual world in the early 20th century. According to Oswald Spengler in his influential book The Decline of the West, a titanic struggle existed between two world views: one was organic and intuitive while the other was mechanical and mathematical. The authors of Uncountable recount how mathematicians like Bertrand Russell, L.E.J. Brouwer, and David Hilbert; physicists like Albert Einstein and Erwin Schrödinger; philosophers like John Dewey and Edmund Husserl; and poets like Paul Valéry addressed this dichotomy. The book then goes back in time; the next four chapters are a historical survey of philosophical and literary discussions about unity and number, sameness and difference, and the necessity of mathematical truths. Chapter two covers the pre-Socratic philosophers, chiefly Pythagoras, Parmenides, and Heraclitus. Chapter three includes Plato, Aristotle, and other classical philosophers, and chapter four examines philosophers from monotheistic religions — including Philo, Augustine of Hippo, the Epistles of the Brethren of Purity, Ibn Tufayl, and Simone Weil (Weil is described as “still fashionable;” the phrase is rather grating, particularly since she is the only woman who is discussed at length). Chapter five offers a whirlwind tour of several 17th- and 18th-century philosophers: René Descartes, Gottfried Wilhelm Leibniz, John Locke, David Hume, and Immanuel Kant.
The text summarizes the main contributions of each of these thinkers and provides a few quotes for every individual (which are generally very hard to understand). One by one, David and Ricardo Nirenberg then weigh the philosophers in the balances and find them wanting. They condemn most of them for exaggerating the scope and certainty of mathematics and a few for the opposite failing; none had seemingly found the proper happy medium.
In chapter six, the authors present two arguments to support the notion that \(2+2\) is not always \(4\). The first argument closely pertains to the history of the philosophy of sameness and difference, which I mentioned previously; if individuation is unreliable, then counting and addition are too. However, the Nirenbergs are much more interested in a second, much weaker argument; they appear to contend that counting involves physically gathering all things together. In some cases, doing so may be impossible — i.e., if the objects are too heavy or vulnerable to change as a result of the physical action of motion.
The authors pursue this second argument to the point of silliness. For instance, they write that philosopher David Lewis “creates the category [of ‘cats’] by taking the ‘fusion’ or ‘sum’ of all cats, consisting in putting together—into the same sack, as it were—all the cats there are in such [a] way that all the parts of cats—whiskers, tails, front halves, back halves, right sides, left sides, even each molecule in any cat—will be in this fusion or sack.’’ They then continue that “[I]n Reality, if those cats were thrown into the same sack, there would be not just fusion but violent confusion from which no cat would emerge unchanged.” Note that it is the Nirenbergs, not Lewis, who introduce the metaphor of the “sack,” which they then proceed to take literally.
More seriously, the authors present a third goal for the book at the end of chapter six. They coin the technical term “pathic” and its opposite “apathic,” which they define as follows: “We will call objects, items, things, categories, concepts, beings apathic if and only if whenever collected together or separated they remain the same.” The Nirenbergs then make two claims: (i) Being pathic or apathic is not an absolute feature of a thing but rather a contingent matter of perspective and context, and (ii) one can only properly apply mathematics to things if viewing them apathically.
Though I do not know for certain, I would guess that most of the aforementioned thinkers would have concurred that both of these claims are valid as general principles. However, some of them would certainly not have agreed that the claims are important or significant limitations on the assertion that \(2+2=4\). Plato would have probably viewed the pathic aspects of things as a limitation that is not shared by the ideal Forms.
The remainder of Uncountable examines a variety of intellectual issues through the lens of the contrast between pathic and apathic. Chapters seven and ten are discursions as to how these themes play out in 20th-century physics and literature; they mention W.H. Auden, Rainer Maria Rilke, Jorge Luis Borges, and Franz Kafka, among others. Chapter eight constitutes a critique of mathematized economics and social science, which the authors view as frequently violating the prior second principle. Chapter nine tackles the problem of time by contrasting the accounts of mathematics and physics with human experience.
Although the text discusses a large number of philosophers and writers, there are some glaring omissions. The most egregious is Ludwig Wittgenstein’s1 Remarks on the Foundations of Mathematics, which contains some observations that are quite similar to those of David and Ricardo Nirenberg:
This is how our children learn sums; for one makes them put down three beans and then another three beans and then count what is there. If the result at one time were \(5\), at another \(7\) (say because, as we should now say, one sometimes got added and one sometimes vanished of itself), then the first thing we said would be that beans were no good for teaching sums. But if the same thing happened with sticks, fingers, lines and most other things, that would be the end of all sums.
“But shouldn’t we then still have \(2+2=4?\)" — This sentence would have become unusable.
Another omission is George Orwell’s 1984, which is surely the most widely known literary discussion of whether \(2+2=4\):
“Freedom is the freedom to say that two plus two makes four.”
“Sometimes, Winston. Sometimes they are five. Sometimes they are three. Sometimes they are all of them at once. You must try harder. It is not easy to become sane.”
Though Orwell was not a philosopher, I would think that this dramatic scene—in which the denial of mathematical truth becomes a tool for tyranny—deserves a disquieting place among the many literary references in the opposite direction that the Nirenbergs adduce. Finally, I hope that I am not being too self-serving in pointing out that Philip Davis (my father) and Reuben Hersh discussed these issues at length in their jointly authored books: The Mathematical Experience and Descartes’ Dream.
I found the literary discussions in Uncountable often beautiful and even moving, the history of philosophy generally difficult but frequently eye-opening, and the authors’ explanations of their personal philosophy sometimes insightful but occasionally willfully obtuse. Ironically, the dichotomy between the pathic and apathic itself becomes a somewhat rigid framework into which the Nirenbergs cram a wide range of different issues. But the book is always thought-provoking, which is paramount in philosophy. Anyone who is interested in the difficult questions that surround abstract mathematics’ application to the real world and the accompanying risks and opportunities will find much to ponder in Uncountable.
1 Wittgenstein is mentioned in the book, but not his work on mathematics.
Ernest Davis is a professor of computer science at New York University’s Courant Institute of Mathematical Sciences.