In the final weeks of World War I, Oswald Spengler published Der Untergang des Abendlandes, tamely translated as The Decline of the West. Its almost a thousand pages of turgid Teutonic prose swept over mangled Europe like a tidal wave, becoming the still-young century’s best-seller. (A second volume was published in 1922, to less rapturous attention.) It offered a diagnosis to a world convulsed in mass-produced death, an explanation of the “last spiritual crisis that will involve all Europe and America.” According to Spengler, the essence of modern civilization — its Faustian soul, he called it — was a type of mathematics that was created in the seventeenth century by Descartes, Galileo, Leibniz, and Pascal. That mathematics had proven powerful but also lethal, for “formulas and laws spread rigidity over the face of nature, numbers make dead.” Now the West and its mathematics, “having exhausted every inward possibility and fulfilled its destiny,” were dying together. Never mind that Spengler’s claim about the death of mathematics was incorrect. From Mussolini to Thomas Mann, everybody who was anybody claimed to have read the book. Plenty of people disagreed with the analysis. In 1920, on the brink of winning the Nobel Prize, Albert Einstein wrote to the mathematical physicist Max Born: “Sometimes in the evening one likes to entertain one of his propositions, and in the morning smiles about it.” He attributed Spengler’s “whole monomania” to his “school-child mathematics.” But the most acute critics recognized that Spengler represented a powerful stream of the Zeitgeist that saw in mathematics, as the writer Robert Musil put it, “the source of an evil intelligence that while making man the lord of the earth has also made him the slave of his machines.” (Ulrich, the protagonist of Musil’s great novel The Man Without Qualities, which was set in 1913 and published in 1930, is a mathematician, and his creator was himself a mathematically well-trained PhD.) Even the assassinations of that turbulent age were to be understood in mathematical terms. When Friedrich Adler murdered the prime minister of Austria in 1916, he invoked Einstein’s mathematizations of the universe, which Adler interpreted as legitimating a shift of frames of reference from nation to class. The lawyers who pleaded in his defense, on the other hand, argued that the assassin was not in his right mind, because he suffered from “an excess of the mathematical.” Fast forward a century. We, too, live in an age in which the nature of knowledge is intensely political, and in which the powers of number are rapidly expanding. 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. We stand on the threshold of new technologies — such as quantum computing — that promise to dwarf present powers of calculation. There is no realm of human life today exempted from quantification, a situation that one might think should constitute a crisis for our understanding of ourselves and our world. Yet very few people today would put the relationship of number and computation to other forms of knowledge anywhere near the top of the list of pressing questions confronting humanity, where we propose it belongs. We, the authors of this essay, one of whom is a mathematician, are certainly not hostile to mathematics, whose insights have extended usefully into many aspects of the world. Nor do we agree with Oswald Spengler, or with Edmund Husserl, Martin Heidegger, and numerous other modern philosophers who have sought the origins of “the radical life crisis of European humanity” (the phrase is Husserl’s) in some mistaken mathematical turn or other. The issue is not the legitimacy of mathematics, which is no issue at all. The issue is how we should think about both the powers and the limits of mathematics as we apply it to different realms of knowledge. We say powers and limits, because numbers have needs. The powers of mathematics depend on rules that do not apply to many things in the cosmos, from elementary particles to our own thoughts or mental states. The more we extend our mathematical reach toward those things, the more urgently we should all want to ask: what knowledge do we gain and what knowledge do we lose, and at what risk? 
That question should be one of the most urgent of our era. To answer it, we need to understand the peculiar needs of numbers, and the problems that arise when those needs are not met. Alexander Craigie, the narrator of “Blue Tigers,” one of Borges’ last short stories, learned that lesson the hard way. A Scottish logician living around 1900 in Lahore, the fictional Craigie was moved by dreams of blue tigers to scour the sub-continent in search of the implausibly colored felines. What he found instead, in the sandy channels of a mysterious region that was taboo to the neighboring villagers and of the same distinctive blue as the tigers in his dreams, were disks: “identical, circular, very smooth and a few centimeters in diameter.” He pocketed a handful and returned to his hut, where he removed some from his pocket. Opening his hand, he saw some thirty or forty disks, although he could have sworn that he had not taken more than ten from the channel. He could see that they had multiplied, so he put them in a pile and tried to count them one by one. “This simple operation proved impossible.” He would stare at any one of them, remove it with his thumb and index finger, and as soon as it was alone it was (they were?) many. “The obscene miracle” repeated itself over and over. The professor returned to Lahore. He carried out experiments, marking some with crosses, filing others, attempting to introduce some difference into their sameness by which he might distinguish them. He charted their increase and decrease, “trying
There's nothing Artificial about our Intelligence