## Metadata - Author: [[David Nirenberg and Ricardo L. Nirenberg]] - Full Title: Uncountable - Category: #books ## Highlights - We will call this comforting but unexamined extension of our habits of thought in search of illusory certainties “the expansive force of success.” ([Location 282](https://readwise.io/to_kindle?action=open&asin=B09418K8D3&location=282)) - It is perhaps a universal tendency to confuse, as an old quip has it, the customs of one’s tribe for the laws of the universe. ([Location 284](https://readwise.io/to_kindle?action=open&asin=B09418K8D3&location=284)) - Already in the early sixth century BCE, the Greek sage Pythagoras is said to have maintained that everything can be counted. The statement attributed to him, “all that is, insofar as it is, is number,” suggests not only that everything can be counted or measured but perhaps even that knowledge of numerical relations is the only true knowledge, numbers the only true being. ([Location 290](https://readwise.io/to_kindle?action=open&asin=B09418K8D3&location=290)) - Considerably closer to our own time and diction, the great logician and philosopher Alfred North Whitehead made a similar statement in An Introduction to Mathematics (1911): “Now, the first noticeable fact about arithmetic is that it applies to everything, to tastes and to sounds, to apples and to angels, to the ideas of the mind and to the bones of the body. The nature of the things is perfectly indifferent, of all things it is true that two and two make four.” ([Location 298](https://readwise.io/to_kindle?action=open&asin=B09418K8D3&location=298)) - The English word calculus comes from Latin for “pebble.” The ancients used pebbles to represent numbers as they counted, and the modern English word’s etymology serves to remind us that the powers of calculus derive from treating everything it touches as if it were a normal pebble: imperturbably always the same as itself, happily and unproblematically subject to the Identity Principle, remaining constant whether we collect them together or separate them. ([Location 324](https://readwise.io/to_kindle?action=open&asin=B09418K8D3&location=324)) - Calculus, like all of mathematics, depends on treating things as apathic. It divides a whole up into small parts—droplets, “infinitesimals”—applies its simplifying magic to the parts, and finally puts the parts together again with no change in them or in the whole. (Just as in geometry we can take a polygon, divide it up into triangles, then set them together again, and voilà, the original polygon.)25 We can apply this wondrous power to problems of enormous complexity, such as the motion of cooling fluid in a nuclear reactor’s core, of air over an airplane’s wing, or the enchanting vibrations of Stradivarius’s violins. ([Location 330](https://readwise.io/to_kindle?action=open&asin=B09418K8D3&location=330)) - Galileo famously ignored wind currents and viscosity in his model of free fall. If he had focused on turbulence (as James Clerk Maxwell once quipped), modern physics might not have gotten off the ground. ([Location 345](https://readwise.io/to_kindle?action=open&asin=B09418K8D3&location=345)) - when it comes to the study of the human, we need to become more aware of the losses our more lithic options require. ([Location 352](https://readwise.io/to_kindle?action=open&asin=B09418K8D3&location=352)) - One of the greatest lawyers of the sixteenth century, the “Sheikh of Islam” Ebü-s-Su‘ūd, chief jurisconsult of the Ottoman Empire, put it this way when asked about the standing of Sufi mystics in Muslim law: “Knowledge of Divine Truth is a limitless ocean. The sharī‘ah [law] is its shore. We [lawyers] are the people of the shore. The great Sufi masters are the divers in that limitless ocean. We do not argue with them.” ([Location 406](https://readwise.io/to_kindle?action=open&asin=B09418K8D3&location=406)) - whatever laws we choose for ourselves to live by, humility about their reach is a prerequisite for the preservation of humanity. ([Location 411](https://readwise.io/to_kindle?action=open&asin=B09418K8D3&location=411)) - Like Leibniz, Kant, and many other philosophers before him, Spengler distinguished between two ways of knowing the world, one organic, the other mechanical; one oriented toward image, the other toward law; one imaginative, the other scheming; one oriented toward the experience of time, the other toward mathematical number; one living, the other dead.9 Two ways of knowing, one more felicitous than the other. We trust you can tell on which side Spengler thought the danger lay. “Stiff forms are the negation of life, formulas and laws spread rigidity over the face of nature, numbers make dead.” ([Location 491](https://readwise.io/to_kindle?action=open&asin=B09418K8D3&location=491)) - For what makes Greece so important in our history of humanity is not its specific contributions to arithmetic or geometry or even its development of specific kinds of demonstrations, such as deductive proofs. What matters for our history is the ways in which the purposeful choices about sameness and difference that produced the necessary truths of mathematics in the Greek world came to be imagined in that world as necessary across many interrelated domains of thought, including those we today call physics, psychology, philosophy, and religion. ([Location 989](https://readwise.io/to_kindle?action=open&asin=B09418K8D3&location=989)) - “Equivalence relations are both the raw material and the machines in the factory of Greek proofs,” a historian of Greek mathematics has put it. ([Location 1027](https://readwise.io/to_kindle?action=open&asin=B09418K8D3&location=1027))